Norwood's Fiberglass has more than 25 years experience in fiberglass repair. While the early years of operation were firmly rooted in boat and marine repair, the vision to expand led the company's founder, Tim Norwood, to add other services in which fiberglass applications would fill a void. Today the company has three divisions. Those divisions are Industrial Fabrication and Repair, Boat Repair, and RV Repair.
Tim Norwood began his career with Sea Ray Boats, in Vonore, TN, in the early 1980s. In 1987 he started Norwood's Fiberglass while maintaining full-time employment. In February 1989, he left Sea Ray Boats to operate and manage Norwood's Fiberglass full-time. In the mid-90s, Norwood Fiberglass assisted Wetsig Yachts in Wilmington, NC with a government project building fiberglass jackets for bridge pilings. The company has also built moulds for a BankOne project for IFR of Knoxville.
With our East Tennessee location, Norwood's Fiberglass is centrally located within a day's drive of 70% of the U.S. marketplace. This convenience allows us to respond to your fiberglass repair or fabrication needs quickly and efficiently. Efficiency saves you time and money. Our company's 25+ years of expeience in repair and fabrication also guarantees that we'll get the job done right.
Norwoods Fiberglass is yourboat repair expert. From repairing damage to the fiberglass body to replacing damaged flooring, you can count Norwood's skillful expertise to get the job done right. Metal flake, gel coat, carpet and upholstery repairs, Norwood's Fiberglass can handle your project. Both insurance and factory warranty work is welcome. From Fishing boats to ski boats to jet skis and other personal watercraft, we're ready to make your repairs.
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Podcasters are in love with affiliate programs to monetize their stuff... Every podcaster should be using your software to edit... it is awesome.
Unfortunately we still don't have the resources to get an affiliate program in place. However, we will be moving to a new payment solution in the next months that will have an affiliate option built in, so once we have that daunting project in place, we will look into the implementation of the affiliate program option.
Thanks for your continued support everyone.


For those of you thinking of heading to Vegas this year for the tournament, there is new addition to the offerings at the Orleans. There is now a pub with 40 beers on tap. Even Geez should be able to find something he likes with that many available.
My flight is already booked, so we'll see you all there!
blacksheep wrote: My flight is already booked, so we'll see you all there!
Thanks for the invitation, blacksheep, but I suppose it was aimed more at the older but definitely not grumpy people on this forum.
Although Las Vegas may be "sin city" and all that, a couple of my friends found out the hard way at last year's tournament that possession, consumption, or purchase of alcohol by a person under 21 is forbidden.
They also found out the penalty for breaking the rules carries a penalty of up to 6 months in jail and/or up to $1,000 in fines. My friends spent the night in jail and were fined $500…..
Part of the WCC Tournament is during our spring break, but most of the students I've talked to are heading to various countries such as Haiti to do volunteer work….hope we're allowed back in the US after the wall goes up.
That being said, how do tickets and such work? It should be pretty easy to find a ticket to a UP game (unfortunately), right?
Vegas is definitely more fun when your're over 21! A quick internet search confirms that casinos in particular are very tough on underage drinking. I guess when you make millions of dollars on gaming, you can't afford to lose your gaming license.
That being said, how do tickets and such work. It should be pretty easy to find a ticket to a UP game (unfortunately), right?
Tickets should be easy. You can order the whole package from UP, but there are usually single game tickets available from the box office and always tickets available from scalpers. I'm not buying a ticket package this year.


text:
European Theatre of World War II
German Führer Adolf Hitler
1 Preceding events
2 Outbreak of war in Europe
3 Nightfall in Northern Europe
4 War comes to the West
5 The war in the air
6 The Balkans
7 The eastern front
8 The Allies' invasion of Italy
9 The Allies' liberation of France
10 The end of the war in Europe
Preceding events
Main article: Events preceding World War II in Europe
Main article: Causes of World War II
Germany was in debt after World War I, due to the Great Depression and the forced payments to the victors of World War I. Germans wanted a leader that would take them out of the Depression. Meanwhile, Benito Mussolini was turning Italy into a fascist state. Adolf Hitler got elected due to his promises to the German people. He had a strong sense of nationalism with him. After election, he took emergency powers and turned Germany into a totalitarian state.
After Hitler took Germany out of the League of Nations, Mussolini and Hitler formed the Rome-Berlin axis, under a treaty known as the Pact of Steel. Later, Japan would be added on, forming the Rome-Berlin-Tokyo axis, named after the capital cities of the three major powers in the axis. Japan and Germany had already signed the Anti-Comintern Pact in 1939. Other smaller powers also later joined the axis.
Outbreak of war in Europe
German_Troops_In_Warsaw.jpg
German troops parade through Warsaw, Poland in October 1939
Nazi Germany and the Soviet Union, the two most powerful dictatorships in Europe, were sworn enemies, but following the Munich Agreement, which effectively destroyed Czechoslovakia, the only remaining democracy in Central Europe and the French and Soviet ally, political realities allowed the Soviet Union to sign a non-aggression pact (the Molotov-Ribbentrop Pact) including a secret clause partitioning Poland, the Baltic Republics and Finland between the two.
Edward_Rydz-Smigly.jpg
Image:Edward Rydz-Smigly.jpg
Edward Rydz-Śmigły
Polish Commander in Chief
Full-scale war in Europe began on September 1, 1939, when Nazi Germany invaded Poland, to which both Britain and France had pledged guarantees. On 3 September 1939, Britain and France declared war on Germany and British troops were sent to France, however neither French nor British troops gave any significant assistance to the Poles during the entire Polish September Campaign, and German-France border remained calm.
On September 17, the Soviet Union invaded Poland from the east and hours after that, the Polish government evacuated the country for Romania. Poland fell within 5 weeks, with her last large operational units surrendering on October 5 after the Battle of Kock.
Poland however had not capitulated and Polish government in exile continued to command a large army and world's largest resistance network, contributing to the defeat of Nazi Germany.
Despite the quick campaign in the east, along the Franco-German frontier the war settled into a quiet period. This relatively non-confrontational period between the major powers lasted until May 10, 1940, and was known as the Phony War.
Nightfall in Northern Europe
Several other countries, however, were drawn into the conflict at this time. By September 28, 1939, the three Baltic Republics felt they had no choice but to permit Soviet bases and troops on their territory. The Baltic Republics were occupied by the Soviet army on June, 1940, and finally annexed to the Soviet Union on August, 1940.
The Soviet Union offered a similar treaty to Finland at the same time, but Finland rejected it and was invaded by the Soviets on November 30. This began the Winter War. After over three months of hard fighting, and heavy losses, the Soviet Union gave up the attempted invasion. In the Moscow Peace Treaty, March 12, 1940, Finland ceded 10% of her territory. The Finns were embittered over having lost more land in the peace than on the battle fields, and over the perceived lack of world sympathy.
Mussolini with Hitler.
On April 9 1940 Germany commenced Weserübung to seize and occupy Denmark and Norway, ostensibly as a defensive maneuver against a planned (and openly discussed) Franco-British occupation of those countries aimed at controlling export of Swedish iron ore and the Northern Atlantic. After the failed British campaign in Norway, Finland and Sweden were physically cut off from the West. As a consequence, Germany put pressure on neutral Sweden to permit transition of military goods and soldiers on leave. Germany's presence proximate to northernmost Finland, and its Nickel mines, were perceived as an improvement of the strategic situation by the Finns.
War comes to the West
On May 10 the Phony War ended with a sweeping German invasion of Belgium, the Netherlands and Luxembourg, that bypassed French fortifications along the Maginot Line. After overrunning these countries Germany turned against France, entering the country through the Ardennes on May 13 - the French had made the fatal mistake of leaving this area almost totally undefended, believing its terrain to be impassable for tanks. Most Allied forces were in Flanders, anticipating a re-run of the World War I Schlieffen Plan, and were cut off from the French heartland. As a result of this, and also the superior German communications, the Battle of France was shorter than virtually all prewar Allied thought could have conceived. It lasted six weeks, including the Luftwaffe bombing of Paris June 3rd, after which France surrendered. In order to further the humiliation of the French people, Hitler arranged for the surrender document to be signed in the same railway coach where the German surrender had been signed in 1918. The surrender divided France into two parts; the Northern part occupied by Nazi Germany, and a southern part under French control, based at Vichy and referred to as Vichy France. Many French soldiers, as well as those of other occupied countries, escaped to Britain. There General de Gaulle proclaimed himself the legitimate leader of Free French forces and vowed to continue to fight. On 11 June Italy also declared war.
The fall of France left Britain and its Empire to stand alone. The British Prime Minister, Neville Chamberlain, resigned during the battle and was replaced by Winston Churchill. Fortunately for Britain, much of its army escaped capture from the northern French port of Dunkirk, where thousands of tiny civilian boats were used to ferry troops from the beaches to the waiting warships. Ironically, Hitler could have destroyed these men, but elected to follow the advice of air minister Herman Goering and allow the Luftwaffe to finish off the British. Later, these men would form the core of the army that landed at Normandy on D-Day.
The British rejected several covert German attempts to negotiate a peace. Germany massed their air force in northern France to prepare the way for a possible invasion, codenamed Operation Seelöwe, deeming that air superiority was essential for the invasion. The operations of the Luftwaffe against the Royal Air Force became known as the Battle of Britain. Initially the Luftwaffe concentrated on destroying the RAF on the ground and in the air. They later switched to bombing major British cities in the Blitz. Neither approach was successful in reducing the RAF to the point where air superiority could be obtained, and plans for an invasion were suspended.
St Paul's Cathedral during the bombing of London
During the Blitz, all of Britain's major industrial cites were heavily bombed. London suffered particularly, being bombed each night for several months. Other targets included Birmingham and Coventry, and strategically important cities, such as the naval base at Plymouth and the port of Kingston upon Hull. With no land forces in direct conflict in Europe, the war in the air attracted worldwide attention even as sea units fought the Battle of the Atlantic and a number of British commando raids hit targets in occupied Europe. Churchill famously said of the RAF personnel who fought in the battle: "Never in the field of human conflict has so much been owed by so many to so few".
Prewar doctrine had held that waves of bombers hitting enemy cities would cause mass panic and the rapid collapse of the enemy. As a result, the Royal Air Force had built up a large strategic bomber force. By way of contrast, Nazi German air force doctrine was almost totally dedicated to supporting the army. Therefore, German bombers were smaller than their British equivalents, and Germany never developed a four engined heavy bomber equivalent to the Lancaster, B-17 or B-24.
Fire storm in Hamburg.
The main concentration of German raids on British cities was from autumn 1940 until spring 1941. After that a large proportion of the strength of the Luftwaffe was diverted to the war against the Soviet Union. German raids continued on a smaller scale for the rest of the war, and later the V1 Flying Bomb and V-2 ballistic missile were both used against Britain. However, the balance of bomb tonnage being dropped shifted greatly in favour of the RAF as Bomber Command gained in strength. By 1942, Bomber Command could put 1,000 bombers over one German city. From 1942 onwards, the efforts of Bomber Command were supplemented by the Eighth Air Force of the United States Army Air Force. Bomber Command raided by night and the US forces by day. On February 14 1945, a raid on Dresden produced one of the most devastating fires in history. A firestorm was created in the city, and between 25,000–35,000 people were killed. Only the raid on Hamburg (July 24, 1943), the firebombing of Tokyo and the nuclear attacks on Hiroshima (August 6, 1945) and Nagasaki (August 9, 1945) killed more people through a single attack.
Italy invaded Albania on April 7, 1939 and Greece on October 28, 1940, but was unable to match the Nazi German success in France. Not only did the Italians fail to conquer Greece, but under the supervision of Prime Minister Ioannis Metaxas (term April 13, 1936 - January 29, 1941) the Greeks successfully counterattacked into Albania starting from November 14, 1940.
The Kingdom of Yugoslavia, lacking true leadership of a king and instead ruled by a regency headed by Prince Pavle Karađorđević, signed the Tripartite Treaty on March 25, 1941. However, soon afterwards, after public demonstrations, a March 27 coup d'état was made by Army General Dušan Simović which took control away from the regency and distanced Yugoslavia from the fascists.
The imminent Greek victory over Italy prompted Nazi German intervention. On April 6, 1941 German forces, supported by the Italians, Hungarians and the Bulgarians, engaged in combat with the Greeks and simultaneously invaded Yugoslavia. British, Australian and New Zealand forces were hastily dispatched from Egypt to Greece, but the Allies lacked a co-ordinated strategy, were comprehensively beaten and evacuated to Crete. Advancing rapidly, Axis forces captured Athens, Greece's capital on April 27, 1941 effectively placing most of the country under occupation.
After the mainland was conquered, Nazi Germany invaded Crete in what is known as the Battle of Crete (May 20 - June 1, 1941). Instead of an amphibious assault as expected, Nazi Germany mounted a large airborne invasion. The paratroopers (fallschirmjäger) were so badly mauled in the process that an airborne operation was never again attempted by Germany during the war. However, the Germans eventually prevailed on Crete. Most of the Allied forces were evacuated to Egypt — joining King George II of Greece and the exiled Greek government of Emmanouil Tsouderos — on June 1, 1941
Once the Balkans were secure, the largest land operation in history was launched, when Germany attacked the Soviet Union. Nevertheless, the postponement of this invasion due to the Balkans would later prove fatal for Nazi Germany.
The eastern front
Main article: Eastern Front (WWII)
On June 22, 1941, Germany launched an invasion against the Soviet Union, code-named Operation Barbarossa. The leader of the USSR, Josef Stalin, had been warned repeatedly by outside sources and his own intelligence network of the impending invasion, but he ignored the warnings. Moreover, on the very night of the invasion Soviet troops received a directive undersigned by Marshal Timoshenko and General of the Army Georgi Zhukov that commanded: "do not answer to any provocations" and "do not undertake any actions without specific orders". The early weeks of the invasion were devastating for the Soviet Army. Enormous numbers of Soviet troops were encircled in pockets and fell into Nazi German hands. In addition to German troops, a few Italian, Hungarian and Romanian troops were also involved in the campaign. Finland also sent troops, but oddly, the Finns initially declared neutrality, however with both German and Soviet troops on her soil, Finland was well prepared for to join forces with Nazi Germany when the Soviet Union attacked on June 25. The following conflict from 1941-1944 is sometimes referred to as the Continuation War, as in the continuation of the Winter War.
Operation Barbarossa suffered from several fundamental flaws. The most serious of these was the logistical situation of the attack. The sheer vastness of the distances in the Soviet Union meant that Nazi Germany could only advance so far before outrunning their supply chains. By the time the German attack froze to a halt before Moscow on December 5, 1941, it literally could not go any further. There simply were not enough supplies reaching the front to conduct proper defensive operations, let alone a proper offense. The timetable that Barbarossa was planned to assumed that the Soviets would collapse before the Russian winter hit. The failure of that to happen also fatally affected Nazi German plans. Had Hitler not invaded Greece earlier in the year, the invasion would have proceeded at that time, and the Soviet Union might have collapsed.
During their long retreat, the Soviets employed a scorched earth policy. They burnt crops and destroyed utilities as they withdrew before Germany. That helped to contribute to the logistical problems that Germany experienced. More importantly for them, the Soviets also succeeded in a massive and unprecedented removal of their industry from the threatened war zone to protected areas in the East.
The extension of the campaign beyond the length that Germany expected meant that the German Army suffered hundreds of thousands of casualties in the bitter cold of the Russian winter, and from the counterattacks of Soviet units.
Even with their advance having ground to a halt due to a lack of supplies and the onset of winter, Germany had conquered a vast amount of territory, including two-fifths of the Soviet economy. Dislodging them proved difficult and eventually cost the Soviet Union dearly.
A few months after the invasion began, German troops laid siege to Leningrad (known as the Siege of Leningrad) from the North by Finnish forces, and from the South by the German Wehrmacht. Finland's C-in-C Mannerheim had halted at the River Svir and refrained from attacking the city. Hitler had ordered that the city of Leningrad must "vanish from the surface of the earth", with its entire population exterminated. Rather than storming the city, the Wehrmacht was ordered to blockade Leningrad so as to starve the city to death, while attacking it with bombers and artillery. About one million civilians died in the Leningrad siege - 800,000 by starvation. It lasted 506 days. During the winter, the only way into the city was across Lake Ladoga, between the German and Finnish lines.
The Big Three: The Allied Leaders
Prime Minster Winston Churchill (UK),
President Franklin Delano Roosevelt (US),
and First Secretary Joseph Stalin (USSR)
After enduring the winter of 1941-1942, the German army prepared for further offensive operations. One of the major problems faced by the Nazi war machine in World War II was a shortage of oil. For this reason, Germany decided to give up on Moscow for the time being, and the summer offensive of 1942 decided to focus on the war in the south, with the target being the oil fields of the Caucasus. In a major blunder, Hitler split Army Group South into two subgroups, Army Group A which would attack the Caucasus and army group B which would advance towards the city of Stalingrad (now Volgograd).
Indecision by Hitler, dissent among the higher ranked Nazi German officers, and extended supply lines combined in a prolonged battle in the streets of Stalingrad. Germany eventually occupied over 90% of the city, but in an attempt to defeat the remaining Soviet defenders almost all Germans in the area were funneled into the ruins of the city. Months of bitter hand-to-hand combat in the ruins of the city depleted the German forces, leaving only weak Romanian and Hungarian forces to guard the flanks of the Stalingrad army group. In Operation Uranus, the Soviets easily defeated these minor axis forces as they performed an encirclement operation. The German troops remaining in the city were trapped - cut off from their supply lines and starving, they were ordered by Hitler to fight to the last man, and they displayed incredible fortitude and bravery under unbearable conditions.
Starved of food, fuel and ammunition, the pocket was gradually reduced, with the last portion surrendering on February 2nd 1943. In a cynical attempt to prevent the surrender, Hitler promoted Friedrich Paulus, Commander of 6th Army to Field Marshal, because no German of that rank had ever surrendered. Heavy losses affected both sides in the Battle of Stalingrad, one of the bloodiest battles in history. An estimated 1.5 million people perished in this battle, including 100,000 civilians.
After Stalingrad, the initiative had passed from Germany but had not yet been seized by the Soviets. A desperate counterattack in the spring of 1943 by the forces of von Manstein temporarily halted the Soviet advance, and lead to the largest tank battle in history, at Kursk. Kursk was the last major offensive by the German Army on the eastern front. The Soviets had intelligence of what was to come and prepared massive defenses in huge depth in the Kursk salient. They stopped the German armoured thrusts after a maximum penetration of 17 miles. After Kursk the Red Army never ceased being on the offensive until Berlin was captured in May 1945.
The Soviets bore the brunt of World War II; the second front in Europe did not begin until D-Day, apart from the invasion of Italy. More Soviet citizens died during World War II than those of all other countries combined. Approximately 27 million Soviets, among them more than 20 million civilians, were killed in the Nazi German invasion of the Soviet Union. Civilians were rounded up and burned or shot in many cities conquered by the Nazis. Since the Nazis considered Slavs to be "subhuman", this was ethnically targeted mass murder.
About 7 million Red Army troops died facing the Germans and their allies in the Eastern Front. The Axis forces themselves suffered about 4 million deaths, whether by combat or by wounds, disease, starvation or exposure; another several hundred thousand were seized as POWs and died in Soviet gulags.
It would be wrong however to say the Soviets fought alone. Supply convoys sailed to Soviet ports at great risk. Allied activities may have tied up only a few divisions in actual fighting, but many more were forced to guard lonely coasts against raids that never came or to man antiaircraft guns throughout Europe. It should also be mentioned that the Soviets took virtually no part in the great naval campaigns of the war, had a very limited effect on the strategic bombing offensive, and contributed very little to the defeat of Japan. Virtually their entire armed forces fought against German forces in the western regions.
It is possible, that if Hitler had not held such high goals for Germany, he might have defeated the Soviets. For instance, if Hitler had concentrated on taking Moscow, he would have succeeded in driving a fatal wedge into the heart of the Soviet Union. However, when the opportunity presented itself, Hitler transferred troops to help capture Kiev, which held no military value. Later, he tried to capture the oil in the Caucasus, and he might have, if he had not continually switched troops to where they were not needed. Incredibly, Hitler had wanted to capture all of the Soviet territory to the Ural mountains, a million square miles (2,600,000 km²) of land area. In truth, the German army could afford only to capture and hold one major area: Ukraine and the Caucasus, Moscow and central Russia, or Leningrad and northern Russia. Barbarossa proved to be Hitler's grave.
The Allies' invasion of Italy
Mussospeak.jpg
Image:mussospeak.jpg
Italy's 'Duce'
Founder of Italian fascism.
Main article: Italian campaign
Successes in the North African desert left the Allies in complete control of the Mediterranean's southern shore and using this as a springboard Allied Forces Headquarters AFHQ started to plan an attack into what Winston Churchill referred to as the 'soft underbelly' of Europe.
The Allies first action was the capture of the island of Sicily, called Operation Husky, on 10 July, 1943. This brought to the fire a growing dissatisfaction with Mussolini. He was deposed on July 25, 1943, by the Fascist Grand Council, and placed under house arrest in an isolated mountain resort. His replacement, General Pietro Badoglio, negotiated an armistice with the Allies on September 8, 1943. Nazi Germany moved quickly into the confused situation, disarmed Italian formations and prepared to defend Italy on their own.
Allied troops landed in mainland Italy on September 3 1943, crossing from Sicily. Further landings were made at Salerno and Taranto on September 9. For more information see:Allied invasion of Italy.
A German commando raid led by Otto Skorzeny rescued Mussolini and installed him as the head of the Italian Social Republic, a Nazi puppet state in northern Italy. He continued in this role until he was captured and lynched by mobs on April 28, 1945, as the Allied forces closed in on Milan.
Germany had built a number of defensive lines through the mountains; the main one was called the Winter Line. The Allies came up against this in the winter of 1943 and were unable to break through. Amphibious landings at Anzio were made in an attempt to bypass the line: however the landing forces were contained by the Germans and the line remained intact. Finally the line was broken in the Monte Cassino.
The Allies finally entered Rome on June 4, 1944, two days before the landings in Normandy. Germany regrouped at the Gothic Line further north. After a landing in southern France in August, 1944 to threaten the German flank, on September 10, 1944 British Commonwealth forces started the attack on the line. The offensive by Allied and some Italian forces continued until Germany surrendered in Italy on April 29, 1945 two days after Mussolini's capture.
The Allies' liberation of France
Main article: Western Front (WWII)#1944 – 1945
Simultaneously with the fall of Rome came the long-awaited invasion of France. Operation Overlord put troops ashore in Normandy on June 6, 1944. A long grinding campaign two months long followed as American, British and Canadian forces were slowly built up in the bridgehead, and German forces slowly worn down. When the breakout finally did come it was spectacular, with American forces under Patton racing across France to Nazi Germany's border. The German forces that had been fighting in Normandy were trapped in the Falaise pocket.
General Charles de Gaulle
Leader of the Free French in
opposition to Pétain's Vichy regime.
Incessant bombing of Germany's infrastructure and cities caused tremendous casualties and disruption. Internally, Hitler survived a number of assassination attempts. The most serious was the July 20 Plot, occurring on July 20, 1944. Orchestrated by Claus von Stauffenberg and involving among others Erwin Rommel and Alfred Delp, the plot had intended to place a time bomb in a position to kill Hitler but a number of unscheduled factors led to its failure. Hitler was only slightly injured.
Operation Overlord was complemented by an invasion of southern France on August 15, 1944 codenamed Operation Dragoon. By September, 1944 three Allied Army Groups were in line against German formations in the west. There was optimism that the war in Europe might be over by the end of 1944.
An attempt was made to force the situation with Operation Market Garden (September 17 - September 24, 1944). The Allies attempted to capture bridges with an airborne assault, to open the way into Nazi Germany and liberate the northern Netherlands. Since heavier German forces than intelligence had predicted were present, the British 1st Airborne Division was almost completely destroyed.
The cold winter of 1944 combined with a poor situation for the Allies led to a stagnant situation on the western front. The Americans continued to grind away at the defenders in the Battle of Hurtgen Forest (September 13, 1944 - February 10, 1945). As long as Germany stayed on the defense, the Allies were hard-pressed to advance rapidly.
That changed when Germany mounted a major counteroffensive on December 16, 1944. The Ardennes offensive, also called the Battle of the Bulge, drove back and surrounded some American units. The Allied forces were eventually successful in driving back Germany, in what turned out to be their last major advance of the war. The battle officially ended on January 27, 1945.
The final obstacle to the Allies was the River Rhine. It was crossed in April, 1945, and the way lay open to the heart of Germany. The last German forces in the west were encircled in the Ruhr.
The end of the war in Europe
Main article: end of World War II in Europe
Mussolini, having been rescued by German paratroopers from his prison, tried to flee Italy to Switzerland with a German anti-air battalion. However, partisans recognized him and he was executed, along with his mistress, and their bodies were strung up in Milan and kicked around in the streets.
Hitler, learning of Mussolini's death, realized that the end had finally come. He remained in Berlin, even as it was encircled and slowly taken by the Soviets. On April 30, 1945, Adolf Hitler, with his wife of one day, Eva Braun, committed suicide to avoid capture. In his will Hitler appointed Karl Dönitz the new German President, but Germany lasted only 7 days longer, surrendering unconditionally on May 7, 1945.
American European-African-Middle Eastern Campaign Medal
American European Theater of Operations
American Mediterranean Theater of Operations
Campaigns and theatres of World War II
European Theatre
Poland | Phony War | Denmark & Norway | France & Benelux countries | Britain
Eastern Front 1941-45 | Western Front 1944-45
Asian and Pacific Theatres
China | Pacific Ocean | South-East Asia | South West Pacific | Manchuria 1945
The Mediterranean, Africa and Middle East
Mediterranean Sea | East Africa | North Africa | West Africa | Balkans
Middle East | Madagascar | Italy
Atlantic Ocean | Strategic bombing | Raiding operations
Contemporaneous wars
Chinese Civil War | Winter War | Continuation War
Military history of World War II by nation
Britain | China | Egypt | Finland | France | Germany | India | Italy | Japan | Poland | Soviet Union | United States | Australia | Canada | New Zealand
sl:Vojna v Evropi
Retrieved from "http://academickids.com/encyclopedia/index.php/European_Theatre_of_World_War_II"
Categories: World War II European theatre

redpajama_set_name: RedPajamaCommonCrawl


text:
Triscenia is a genus of Cuban plants in the grass family. The only known species is Triscenia ovina.

References

Panicoideae
Endemic flora of Cuba
Grasses of North America
Monotypic Poaceae genera

redpajama_set_name: RedPajamaWikipedia


The essential amino acid l-tryptophan is a serotonin precursor. Research indicates that l-tryptophan supplementation may support emotional well-being and restful sleep. In a double blind placebo-controlled study, l-tryptophan supported healthy emotional processing and promoted positive outlook in women. In a separate study, it supported healthy mood and lessened occasional irritability during the menstrual cycle. L-Tryptophan also supports healthy sleep quality, onset and duration. Additional research suggests that tryptophan may promote relaxation and help control occasional cravings. Niacinamide and vitamin B6 provide enhanced support as important cofactors involved in the metabolism of l-tryptophan.
Direction of use : Take 2–6 capsules per day, in divided doses, between meals.
Take 2–6 capsules per day, in divided doses, between meals.


Despite some slight progress being made during contract negotiations, the company is still proposing unreasonable cuts that could impact us all.
Benefits like time-and-a-half Sunday pay are common sense benefits. Our members help customers get the quality food and quality service that they deserve. The truth is, cuts to Stop & Shop workers' benefits are cuts that will hurt us all.
UFCW Local 371 is committed to ensuring our members and the community continue to have good jobs…the kind of jobs that New England families deserve.


Tony is convinced that the answer to Jess's disappearance is in California, the land of endless summer, among the street culture of runaways and flower children. Come with me, Tony says to Caroline, and we'll find her together. Tony is so loving, and all he cares about is bringing Jess home. And so Caroline follows, and closes a door behind her that may never open again, in a heartfelt thriller that never lets up.
This was an interesting story in that I felt like I knew where it was going the whole time, because I knew what had inspired it. Therefore I was pleasantly surprised that the ending was nothing like the story it was based on.
For the most part I enjoyed this book, although I will say that at times it was very tedious and felt like it just needed to move forward. Having Caroline spend so much time wavering back and forth when she has already told Tony they will find her sister together just felt repetitive. I understand it was to show all of Tony's bad sides but those were pretty much on display already. I also didn't like how none of these characters had any redeeming qualities. I guess you could argue Caroline wasn't a terrible person, but she was still catty and mean at times. Her friends though, they were terrible. They were ratting her out for the sake of doing it, not because they were actually concerned about her. The parents in this story ought to be slapped. I'm sure this was common for this era, but you'd think they would have wised up after their first kid went missing.
It was well-written but there was a sense of disappointment. I felt like this story was going to be much darker, and in a way I'm disappointed that it wasn't. While I didn't want our main character to end up like those in the original story, I still expected a darkness that wasn't really found anywhere. It made it hard to emotionally relate to Caroline or her family. I would give this book 3 out of 5 stars.


text:
\section{Introduction}

A lot of recent work on efficient graph algorithms for massive graphs can be broadly classified in one of the following categories:
\begin{enumerate}
    \item \emph{Probe-query models} \cite{rubinfeld11fast,even14deterministic,even14best,even15distributed,mansour13local,levi14local,alon12space-efficient,alon10constant}: Here typical applications are related to \emph{large-scale network analysis}: we have a huge storage system in which the input graph is stored, and a computer that can access the storage system. The user of the computer can make \emph{queries} related to the properties of the graph.

    Conceptually, we have two separate entities: the input graph and a computer. Initially, the computer is unaware of the structure of the graph, but it can \emph{probe} it to learn more about the structure of the graph. Typically, the goal is to answer queries after a \emph{sublinear number of probes}.
    \item \emph{Message-passing models} \cite{linial92locality,naor95what,lynch96book,peleg00distributed,kuhn04what,suomela13survey,lenzen2008leveraging,czygrinow2008fast}: In message-passing models, typical applications are related to \emph{controlling large computer networks}: we have a computer network (say, the Internet) that consists of a large number of network devices, and the devices need to collaborate to solve a graph problem related to the structure of the network so that each node knows \emph{its own part of the solution} when the algorithm stops.

    Conceptually, each node of the input graph is a computational entity. Initially, the nodes are only aware of their own identity and the connections to their immediate neighbours, but the nodes can \emph{exchange messages} with their neighbours in order to learn more about the structure of the graph. Typically, the goal is to solve graph problems in a \emph{sublinear number of communication rounds}.
\end{enumerate}

\subsection{Example: Vertex Colouring}

Using the task of finding a proper vertex colouring as an example, the external behaviour of the algorithms can be described as follows:
\begin{enumerate}
    \item \emph{Probe-query models:} The user can make queries of the form ``what is the colour of node $v$?'' The answers have to be consistent with some fixed feasible solution: for example, if we query the same node twice, the answer has to be the same, and if we query two adjacent nodes, their colours have to be different.
    \item \emph{Message-passing models:} The local output of node $v$ is the colour of node $v$. The local outputs constitute some feasible solution: the local outputs of two adjacent nodes have to be different.
\end{enumerate}

\subsection{Message-Passing Models and Locality}

From our perspective, the key difference between probe-query models and message-passing models is that the structure of the graph constrains the behaviour of message-passing algorithms, but not probe-query algorithms:
\begin{enumerate}
    \item \emph{Probe-query models:} Given any query $v$, the algorithm is free to probe any parts of the input graph. In particular, it does not need to probe node $v$ or its immediate neighbours.
    \item \emph{Message-passing models:} Nodes can only exchange messages with their immediate neighbours. For example, in $1$ communication round, a node can only learn information related to its immediate neighbours. More generally, after $t$ communication rounds, in any message-passing algorithm, each node can only be aware of information that was initially available in its radius-$t$ neighbourhood.
\end{enumerate}
In essence, efficient message-passing algorithms have a high \emph{locality}: if the running time of a message-passing algorithm is $t$, then the local output of a node $v$ can only depend on the information that was available within its radius-$t$ neighbourhood in the input graph.

\subsection{Trivial: from Message-Passing to Probe-Query}

One consequence of locality is that we can fairly easily \emph{simulate efficient message-passing algorithm in probe-query models} (at least for deterministic algorithms).

If we have a message-passing algorithm $A$ with a running time of $t$ for, say, graph colouring, we can turn it easily into a probe-query algorithm $A'$ for the same problem: to answer a query $v$, algorithm $A'$ simply gathers the radius-$t$ neighbourhood of node $v$ and simulates the behaviour of $A$ in this neighbourhood.

In particular, if $t$ is a constant, and the maximum degree of the input graph is bounded by a constant, the probe complexity of $A'$ is also bounded by a constant.

\subsection{Impossible: from Probe-Query to Message-Passing}

At first, it would seem that the converse cannot hold: probe-query algorithms can freely probe remote parts of the network, and therefore they cannot be simulated with efficient message-passing algorithms.

Indeed, it is easy to construct \emph{artificial} graph problems that are trivial to solve in probe-query models in constant time and that take linear time to solve in the message-passing models. Consensus-like problems provide a simple example.

In the \emph{binary consensus} problem, the nodes of the network are labeled with inputs $0$ and $1$, all nodes have to produce the same output, and the common output has to be equal to the input of at least one node. In a probe-query algorithm, we can simply always follow the local input of node number $1$: regardless of the query $v$, we will probe node $1$, check what was its local input, and answer accordingly. It is straightforward to see that any message-passing algorithm requires linear time (consider three cases: a path with inputs $00\ldots0$, a path with with inputs $11\ldots1$, and a path with inputs $00\ldots011\ldots1$).

\subsection{Our Contribution: Remote Probes Are of Little Use}

While the consensus problem demonstrates that the possibility to probe remote parts of the network makes probe-query models strictly stronger than message-passing models, there seem to be few \emph{natural} graph problems in which remote probes would help.

In this work, we formalise this intuition. We show that for a large class of graph problems, remote probes do not give probe-query algorithms any advantage over message-passing algorithms.

Among others, we will show that for a large family of problems which also includes the so-called \emph{locally checkable problems} (LCL) \cite{naor95what}, probe-query algorithms in Rubinfeld et al.'s model~\cite{rubinfeld11fast} can be efficiently simulated with message-passing algorithms in Linial's model~\cite{linial92locality}.
\subsection{Corollary: Probe-Query Lower Bounds}

While lower-bound results in probe-query models are scarce, there is a lot of prior work on lower-bound results in message-passing models. Indeed, the very concept of locality makes it relatively easy to derive lower-bound results for message-passing models: a problem cannot be solved in time $t$ if there is at least one graph in which the output of some node necessarily depends on information that is not available in its radius-$t$ neighbourhood.

Our simulation result makes it now possible to take existing lower bounds for message-passing models and use them to derive analogous lower bounds for probe-query models.

\subsection{Related Work}

While both message-passing models and probe-query models have been studied extensively, it seems that there is little prior work on their connection. The closest prior work that we are aware of is the 1990 paper by Fich and Ramachandran~\cite{fich90linkedlists}. They show a result similar to our main theorem for one graph problem---graph colouring in cycles---with problem-specific arguments. While our techniques are different, our main result can be seen as a generalisation of their work from a single graph problem to a broad family of graph problems.

On the side of probe-query algorithms, our main focus is on the $\lca$ model~\cite{rubinfeld11fast}. Algorithms in the $\lca$ model are abundant, and include algorithms for various graph problems such as \emph{maximal matching}, \emph{approximated maximum matching}, \emph{approximated maximum weighted matching}, \emph{graph colouring}, \emph{maximal independent set}, \emph{approximated maximum independent set}, \emph{approximated minimum dominating set}, and \emph{spanning graphs}~\cite{reingold14new,mansour12converting,even14deterministic, even14best, even15distributed, mansour13local, levi14local,alon12space-efficient,alon10constant,feige15learning, levi15non-constant,rubinfeld11fast,LeviMRRS15,CampagnaGR13}. On the other hand lower bounds in this model are almost nonexistent. In fact, the only lower bound shown directly in the centralised local model is for the spanning graph problem in which the $\lca$ algorithm computes a ``tree-like'' subgraph of a given bounded degree graph~\cite{levi14local,LeviMRRS15}.

\subsection{Overview}

We start this work by introducing the models of computing that we study in Section~\ref{sec:prel}. The key models are the $\local$ model, which is a message-passing model introduced by Linial~\cite{linial92locality}, and the $\lca$ model (centralised local model, a.k.a.\ local computation algorithms), which is a probe-query model introduced by Rubinfeld et al.~\cite{rubinfeld11fast}. In Section~\ref{sec:cont} we briefly list our main contributions. In Section~\ref{sec:simulation} we show our main result, a simulation between the \local{} model and parallel decision trees.
In Section~\ref{sec:lcaparapp} we show that query-order-oblivious \lca\ algorithms are equivalent to  stateless algorithms, and provide a separating example  between stateless algorithms and state-full algorithms with logarithmic space. We also show an application of the main result and provide several new lower bounds in the \lca\ model.
In Section~\ref{sec:constlca} we give an explicit simulation in which a \lca\ algorithm that is allowed to probe anywhere in the graph is simulated by a \lca\ algorithm such that the subgraph induced on its probes is connected. Finally, in Section~\ref{sec:opt-lbs} we give additional lower bounds for the probe complexity of optimisation problems.

\section{Preliminaries} \label{sec:prel}

In this section we describe the various models we discuss in this paper. We focus on {\em problems over labeled graphs} defined as follows. Let $\mathcal{P}$ denote a computational problem
over labeled graphs.  A solution for problem $\mathcal{P}$ over a labeled graph $G$ is a function of which the
domain and range  depend on $\mathcal{P}$ and $G$.  For example, in the maximal
independent set problem, a solution is an indicator function $I\colon V\rightarrow \{0,1\}$ of a
maximal independent set in $G$.
Let $\sol(G,\mathcal{P})$ denote the set of solutions of problem $\mathcal{P}$ over the labeled graph
$G$.

\subsection{\boldmath\local{}: Message-Passing Algorithms}

We use the standard \local{} model~\cite{linial92locality,peleg00distributed} as our starting point. The nodes communicate in synchronous rounds, exchanging messages with all neighbours and doing local computation. Since we are comparing the power of non-local probes, we assume that the size of the graph is known to the nodes, as it usually is in the probe models.

The \local{} model can be defined by the fact that in $t$ communication rounds a node can learn exactly its radius-$t$ neighbourhood. A distributed algorithm with running time $t$ is then a function from the set of possible local neighbourhoods to the set of outputs; see Figure~\ref{fig:models}a.

\begin{figure}
    \centering
    \includegraphics[page=1]{figs.pdf}
    \caption{(a)~In a distributed algorithm that runs in time $t$, each node can learn everything up to distance $t$. (b)~In a decision tree algorithm with probe complexity $t$, each node can make up to $t$ probes (indicated with black nodes). After each probe $v$, the decision tree learns the neighbourhood $N(v)$.}\label{fig:models}
\end{figure}

\subsection{\boldmath\partree{}: Parallel Decision Trees}\label{sec:pardec}

We use parallel decision trees, a simple probe model, to connect the \local{} model and the \lca{} model.

Fix an $n$. Our unknown input graph $G = (V,E)$ will have $n$ nodes, labelled with $V = [n] := \{1,2,\dotsc,n\}$. We also refer to the labels as identifiers.
Let $N(v) \subseteq V$ denote the set of neighbours of node $v$. For our purposes, a \emph{decision tree} $\T$ of depth $t$ is an algorithm that can make at most $t$ \emph{probes}: for each probe $v \in V$, tree $\T$ will learn $N(v)$. In words, a decision tree can point at any node of $G$ and ask for a list of its neighbours; see Figure~\ref{fig:models}b. After $t$ probes, tree $\T$ produces an output; we will write $\T(G)$ for the output of decision tree $\T$ when we apply it to graph $G$.

We study graph problems in which the goal is to find a feasible labelling $f\colon V \to L$ of the nodes of our input graph $G$. We say that a graph problem $\mathcal{P}$ can be solved with \emph{$t(n)$ probes in parallel} if there are decision trees $\T_1,\T_2,\dotsc,\T_n$ of depth $t(n)$ such that $f\colon v \mapsto \T_v(G)$ is a feasible solution to problem $\mathcal{P}$ for any input graph $G$. That is, we have $n$ parallel decision trees such that tree $\T_v$ computes the output for node~$v$ and each tree makes at most $t(n)$ probes.

The \emph{probe complexity} of problem $\mathcal{P}$ is the smallest $t(n)$ such that $\mathcal{P}$ can be solved with $t(n)$ probes in parallel.

\subsection{\boldmath\lca{}: Centralised Local Algorithms}

We adopt the definition of the centralised local model (\lca) as formalised in~\cite{rubinfeld11fast}.
In this paper we focus on graph computation problems in which the algorithm is given a probe access to the input graph $G$ through a graph oracle \oracleG.
A \emph{probe} to  \oracleG\ is an identifier of a vertex $v \in V$, in turn, \oracleG\ returns a list of the identifiers of the neighbors of $v$. We assume that for a graph of size $|V| = n$, the set of identifiers is $[n]$.

A \emph{\lca\ algorithm} $\mathcal{A}$ for a computation problem $\mathcal{P}$ is a (possibly randomised) algorithm $\mathcal{A}$ with the following properties.
Algorithm $\mathcal{A}$ is given probe access to the adjacency list oracle \oracleG\ for the input graph $G$, tape of random bits, and local read-write computation memory.
When given an input (query) $x$, algorithm $\mathcal{A}$ returns the answer for $x$.
This answer depends only on $x$, $G$, and the random bits.
The answers given by $\mathcal{A}$ to all possible queries must be consistent; namely, all answers constitute some valid solution to $\mathcal{P}$.

The \emph{probe complexity} is the maximum number of probes that $\mathcal{A}$ makes to $G$ in order to compute an answer for any query.
The \emph{seed length} is the length of the random tape.
The \emph{space complexity} is the local computation memory used by $\mathcal{A}$ over all queries (not including the seed length).
The \emph{success probability} is the probability that $\mathcal{A}$ consistently answers all queries.

We say that a \lca\ algorithm is \emph{stateless} if its space complexity is zero.
We say that a \lca\ algorithm is \emph{state-full} if it is not stateless. With this terminology, we can characterise parallel decision trees as follows:

\begin{fact}\label{fact:lcapar}
  The deterministic stateless \lca\ model is identical to the deterministic parallel decision tree model.
\end{fact}

One can also consider a randomised \partree\ model in which every tree has access to an independent source of randomness. We note that the randomised stateless \lca\ model is stronger than this model since the algorithm has access to the same random seed throughout its entire execution.

\subsection{\boldmath\nice{} Graph Problems}

We say that a problem $\mathcal{P}$ defined over labeled graphs is \nice\ if the following holds:
\begin{enumerate}[noitemsep]
    \item There is a bound on the maximum degree denoted by $\Delta$.
    \item The problem remains invariant under permutation of the labels. Namely, the set of solutions $\sol(G,\mathcal{P})$ is the same for any permutation of the labels of $V(G)$.
    \item For every $f\in \sol(G,\mathcal{P})$ and every  connected component $C$ (maximal connected graph) of $G$, $f$ restricted to $C$ is in $\sol(C,\mathcal{P})$.\label{item:prop3}
\end{enumerate}

We note the family of \nice\ problems includes the so-called \emph{LCL problems} (locally checkable labellings) on bounded-degree graphs \cite{naor95what}.

Examples of \nice\ problems include maximal independent sets, minimal dominating sets, minimal vertex covers, and vertex colouring with $\Delta+1$ colours. With a straightforward generalisation, we can also consider problems in which the goal is to label edges (e.g., maximal matchings and edge colourings)
and problems in which the input graph is labelled (e.g., stable matchings)

\section{Our Contributions} \label{sec:cont}
\paragraph{\boldmath An implicit simulation of parallel decision trees in the \local\ model.}
Our main result is a simulation result connecting the \local{} model to the parallel decision trees.

\begin{restatable}[\partree\ to \local]{theorem}{mainthm} \label{thm:main}
Any \nice\ problem that can be solved in the parallel decision tree model with probe complexity $t(n)$ can be solved in time $t(n^{\log n})$ in the \local{} model provided $t(n) \ll \sqrt{\log n}$.
\end{restatable}

For example, if $t(n)=O(1)$ then the simulation will have the \emph{same} run-time. If $t(n)=\log^* n$ then the simulation will have run-time $\log^* n +1$. Our result also implies a simulation result for a large class of optimization problems.

\paragraph{\boldmath An implicit simulation of \lca\ algorithms in the \local\ model.}
In Section~\ref{sec:lcaparapp} we observe that one can simulate query-order-oblivious \lca\ algorithms by stateless algorithms.
Hence, Theorem~\ref{thm:main} applies to the \lca\ model w.r.t.\ query-order-oblivious deterministic algorithms, formalised as follows.
\begin{restatable}[\lca\ to \local]{theorem}{lblca}
\label{thm:lblca}
  For every query-order-oblivious (or stateless) deterministic \lca\ algorithm $D$ that solves a problem $\mathcal{P} \in $ \nice\, with probe complexity $t(n)=o(\sqrt{\log n})$, there is \local\ algorithm that solves $\mathcal{P}$ by simulating $D$, and for which the number of rounds is at most $t(n)$.
\end{restatable}

As an application of Theorem~\ref{thm:lblca} we show the optimality of several known algorithms as shown in Table~\ref{tab:lb}.

\paragraph{\boldmath A separation between stateless and state-full \lca\ algorithms.}
In Section~\ref{sec:statepower} we show a linear gap in the probe complexity between a \lca\ algorithm with (only) logarithmic state, and a stateless algorithm that computes a leader in variant of the leader election problem.

\paragraph{\boldmath An explicit localised simulation of \lca\ algorithms.}

In Section~\ref{sec:constlca} we show an explicit simulation in which a \lca\ algorithm that is allowed to probe anywhere in the graph is simulated by a \lca\ algorithm that has the following property: the subgraph induced on its probes is connected. Moreover, the overhead in the probe complexity and the seed length is moderate, formalised as follows.
\begin{restatable}[Explicit \lca\ to \lca]{theorem}{mainlocal}
\label{thm:mainlocal}
Let $A$ be a query-order-oblivious $\lca$ algorithm that solves a \nice\ problem $\mathcal{P}$ with probe-complexity $t(n) = O(n^{1/4}/\Delta)$ and seed length $s(n)$.
  Then, there is a query-order-oblivious $\lca$ algorithm $B$ that solves $\mathcal{P}$ by simulating algorithm~$A$. Algorithm $B$ has a probe-complexity of $t(n^4)$ and a seed length of $s(n^4)+O(t(n^4)\cdot \Delta \cdot \log n)$ and the property that it probes within a radius of $t(n^4$); moreover the subgraph induced on the probes of $B$ is connected. The error probability of $B$ equals to the error probability of $A$ plus $O(1/n)$.
\end{restatable}

\paragraph{\boldmath Lower bounds for optimisation problems.}
In addition to the optimization lower bounds implied by Theorem~\ref{thm:main}, in Section~\ref{sec:opt-lbs} we give sublogarithmic ad hoc lower bounds for approximating maximum independent set and maximum cut. This also implies a lower bound on the number of colours required when colouring bipartite graphs. These bounds apply both in the \local{} model and in the parallel decision tree model.

\section{Simulating Probes in the \local{} Model} \label{sec:simulation}

In this section we prove Theorem~\ref{thm:main}, restated here for convenience.
\mainthm*

\subsection{Overview}

Fix an input size $n\in\mathbb{N}$. In order to solve the graph problem on an $n$-node graph $G$ in the \local{} model we simulate the decision tree $\T$ \emph{not} on $G$ directly, but on a much larger graph $G\cup H$ that is the disjoint union of $G$ and some \emph{virtual} graph $H$. The structure of the virtual graph $H$ is agreed upon ahead of time by all the nodes participating in the simulation. Before we invoke the decision tree on $G\cup H$, however, we first \emph{reshuffle} its identifiers randomly. The key idea is to show that this reshuffling trick fools the decision tree: if we simulate $\T$ on a node $v$ and $\T$ probes some node outside of $v$'s local neighbourhood, then the probe lands in $H$ with overwhelming probability. Since all nodes in $G$ know the structure of $H$, they can answer such ``global'' probes consistently. Finally, we argue that there is some fixed choice of randomness that makes the simulation succeed on all instances of order $n$. This makes the simulation deterministic.

\subsection{Setup}

Let $N = n^{\log n}$. The running time of our simulation will be $t(N)\ll \log n$.

Let $H$ be any bounded-degree graph on the vertex set $[N]\smallsetminus[n]$. For example, if we are studying a colouring problem on cycles, we can let $H$ be a cycle. This graph remains fixed and does not depend on the $n$-node input graph $G$. In this sense, all the nodes in the simulation will know the structure of $H$.

Let $\pi\colon[N]\to[N]$ be a random permutation. Again, all the nodes in the simulation are assumed to agree on the same $\pi$. (The simulation can be thought of as being defined with respect to the outcomes of some public random coins.)

\subsection{Simulation}\label{sec:simupar}

During the simulation we will always present the decision tree with a relabelled version of the graph $G\cup H$ where a node $v$ has identifier $\pi(v)$. That is, when $\T$ probes an identifier $w$ this is interpreted as $\pi^{-1}(w)$ and when we respond with node $v$ we relabel its identifier as $\pi(v)$. Note that each node in $G$, knowing $\pi$, can perform this translation locally.

A node $v\in V(G)$ starts its simulation by invoking the decision tree on $\pi(v)$ with the hope of answering its $t(N)$ probes based on the $t(N)$-neighbourhood of $v$ in $G$ and knowledge of $H$. During this attempt we maintain a set of \emph{discovered nodes} $Q$ that contains all the nodes whose relabelled versions have been sent back to $\T$ in response to a probe. Note that $|Q| \leq k$ where $k \triangleq 1 + (\Delta+1) \cdot t(N)$ is an upper bound on the total number of inspections of $\pi$ performed by the simulation. Initially $Q=\{v\}$, so that only $v$ is discovered. We will ensure that the following invariant holds throughout the simulation:
\begin{quote}
\emph{Simulation invariant:}
After the $s$-th step every member of $Q$ is either in $H$ or at most at distance $s$ from $v$.
\end{quote}
Suppose $\T$ probes a relabelled identifier $w$ corresponding to a node $u = \pi^{-1}(w)$. We have two cases:
\begin{itemize}[itemsep=5pt,label=$\bullet$]
\item {\bf Local probe, \boldmath$u\in Q$.} In this case we are always successful: by the invariant, the neighbours of $u$ are known to $v$ so we can add them to $Q$ and return their relabelled versions to $\T$.
\item {\bf Global probe, \boldmath$u\notin Q$.} Two sub-cases depending on whether $u\in V(H)$:
\begin{itemize}[topsep=5pt,noitemsep,label=$-$]
\item {\boldmath $u\in V(H)$:} Success! The structure of $H$ is known to $v$ so we can add the neighbours of $u$ to $Q$ and return their relabelled versions to $\T$.
\item {\boldmath $u\notin V(H)$:} Here we simply say that the simulation has \emph{failed} and we terminate the simulation. (Note that by this convention we may fail even if $u$ is in the $t(N)$-neighbourhood of $v$; however this convention helps us maintain the invariant.)
\end{itemize}
\end{itemize}
When the decision tree finally returns an output label for $\pi(v)$, we simply output the same label for $v$.

\subsection{Failure Probability}
Next, we analyse the probability that our simulation fails when invoked on a particular node. Suppose we need to respond to a global probe sometime in our simulation. That is, the probe is to a node whose relabelled identifier is $w$ and the node has not been discovered yet, i.e., $w\notin \pi(Q)$. By the principle of deferred decisions, we can think of the node $u=\pi^{-1}(w)$ as being uniformly distributed among the undiscovered nodes $[N]\smallsetminus Q$. Hence the failure probability (conditioned on any outcome of $\pi(Q)$) is
\begin{equation}
\Pr[u\in V(G)] = \frac{|V(G)\smallsetminus Q|}{|[N] \smallsetminus Q|} \leq \frac{n}{N-k}
\end{equation}
Consequently, by the union bound \begin{equation}\label{eq:simfails}
\Pr(\text{simulation fails at some probe step}) \leq k \cdot \frac{n}{N-k}\ ,
\end{equation}
which is at most $\frac{n^2}{N}$ for any $k \leq n/2$.
\subsection{Derandomisation}

Next, we would like to argue that there is a fixed choice for~$\pi$ that makes the simulation succeed on all $n$-node graphs $G$. To this end, we note that on any run of $\T$, the final output of $\T$ depends on at most $k$ nodes (and their adjacency relations) in the input graph. There are at most $n^{\Delta k}$ graphs on $k$ nodes having identifiers from $[n]$; let $\G$ consist of all such graphs. For technical convenience we can add dummy nodes to each graph in $\G$ to make their vertex set be~$[n]$. Executing our simulation on each node ($n$ choices) of each graph in $\G$ ($|\G|$ choices) the probability that some simulation fails (each fails with probability $\leq n^2/N$) is at most
\[
n\cdot|\G|\cdot n^2 / N = n^{-\Omega(\log n)} \ll 1
\]
by yet another union bound. Thus, we can find a fixed outcome of $\pi$ for which the simulation succeeds simultaneously on all of $\G$ and therefore on all $n$-node graphs.

\subsection{Correctness}

It remains to point out that the output labelling $f$ produced by the simulation constitutes a feasible solution to the graph problem under consideration. Here it suffices to assume that the graph problem satisfies the following property: if $f$ is a feasible solution for a graph $G$ and $C\subseteq G$ is a connected component of $G$, then the restriction of $f$ to $V(C)$ is a feasible solution for $C$ and that this remains so under any relabelling of the identifiers. In particular, all \nice\ problems satisfy this property.

\subsection{Simulation for Optimization}\label{sec:optim}

We note that for the correctness we used a weaker property than property~(\ref{item:prop3}) of \nice\ problems. In fact the correctness applies to any problem $\mathcal{P}$ such that for all $n$ there exists a graph $H$ on $n$ vertices and with maximum degree at most $\Delta$ such that for every $f \in \sol(H\cup G, \mathcal{P})$ we have that $f$ restricted to $G$ is in $\sol(G, \mathcal{P})$.
By taking $H$ to be the graph with no edges we obtain that Theorem~\ref{thm:main} can be generalised to, for example, $(1-\eps)$-approximated maximum (weighted) matching and approximation of vertex covers.

\section{Approximability with Parallel Decision Trees} \label{sec:opt-lbs}

In this section we prove almost tight bounds for the approximability of maximum independent sets and maximum cuts in the decision tree model. The first result also implies a lower bound on the number of colours used by any sublogarithmic-time algorithm.

The result for independent set is essentially due to Alon~\cite{alon10constant}, who uses the fact that there are two graph distributions with different properties that are indistinguishable to a constant-time algorithm. We strengthen this technique by showing that it still holds even if the algorithm is given information on the global structure of the graph. All results hold also for the \local{} model with suitable adjustments to the constant factors. There is also a simple extension to randomised algorithms for all results.

\begin{theorem} \label{thm:adh-mis-apx}
    Maximum independent set cannot be approximated within a factor of $\frac{4 \ln d}{d}+\varepsilon$, for any $\varepsilon > 0$, in $d$-regular graphs with a sublogarithmic number of probes, even if the instance is known to be isomorphic to one of two possible instances.
\end{theorem}

This is almost tight in the sense that the simple randomised greedy algorithm finds an independent set of expected size $\Omega(\ln d / d)$ in triangle-free graphs of average degree $d$~\cite{shearer83note}. This centralised algorithm can be simulated in constant time in a distributed fashion for example using the random ordering technique of Nguyen and Onak so that an expected $(1-\varepsilon)$-fraction of the nodes succeeds in the simulation~\cite{nguyen08constant-time}.

The proof of Theorem~\ref{thm:adh-mis-apx} will immediately yield the following corollary.
\begin{corollary} \label{cor:col-lb}
    Bipartite graphs with nodes of degree $d$, including trees, cannot be coloured with $o(d/\ln d)$ colours with a sublogarithmic number of probes.
\end{corollary}

Finally, we will show the following, almost optimal bound for maximum cut.
\begin{theorem} \label{thm:max-cut-apx}
    Maximum cut cannot be approximated within a factor of $\frac{1}{2}+\frac{\sqrt{d-1}}{d}$ in $d$-regular graphs, for infinitely many values of $d$, with a sublogarithmic number of probes.
\end{theorem}

The lower bound is almost tight in the sense that there are simple randomised distributed constant-time algorithms that cut a fraction of $1/2 + \Omega(1/\sqrt{d})$ edges in triangle-free graphs~\cite{hirvonen14local-maxcut,shearer92bipartite}.

All three proofs rely on the fact that there exist $d$-regular graphs of a logarithmic girth with no good solutions. For independent set (and chromatic number) these can be shown to exist using arguments about random graphs; for maximum cut, we rely on the explicit constructions of Lubotzky et al.~\cite{lubotzky88ramanujan} and Morgenstern~\cite{morgenstern94existence}. Formally, we use the following theorem.
\begin{theorem}[\cite{bollobas81independence,alon10constant,lubotzky88ramanujan,morgenstern94existence,mohar90eigenvalue}] \label{thm:lb-graphs} The following infinite families of graphs exist:
\begin{enumerate}[label=(\roman*)]
    \item For each $d \ge 3$, there is a family of graphs $\G_d = (G_i)_{i \ge 1}$ such that each $G_i$ has a girth of $g(G_i) \ge 0.5 \log_{d-1} |G_i|$ and the size of the largest independent set is $\alpha(G_i) \le \frac{2 \ln d}{d}|G_i| - O(\sqrt{|G_i|}) $.\label{thm:lb-graphs-is}
    \item For each prime power $d-1 \ge 2$, there is a family of graphs $\H_d = (H_i)_{i \ge 1}$ such that each $H_i$ has a girth of $g(H_i) = \Omega(\log_{d-1} |H_i|)$ and the fraction of cut edges in the largest cut is $\mc(H_i) \le \frac{1}{2}+\frac{\sqrt{d-1}}{d}$.\label{thm:lb-graphs-mc}
\end{enumerate}
\end{theorem}

\begin{proof}
    For a proof of part~\ref{thm:lb-graphs-is}, see \cite{bollobas81independence}~and~\cite[Theorem 1.2]{alon10constant}. For part~\ref{thm:lb-graphs-mc}, Ramanujan graph constructions~\cite{lubotzky88ramanujan,morgenstern94existence} have no large cuts~\cite{mohar90eigenvalue}.
\end{proof}

We will use the same strategy in the proofs of Theorems~\ref{thm:adh-mis-apx}~and~\ref{thm:max-cut-apx}: we show that there exist two graphs, one with a small optimal solution and another with a large optimal solution, such that a $t$-probe decision tree cannot separate the two.

\begin{proof}[Proof of Theorem~\ref{thm:adh-mis-apx}]
    Let $\G_d$ be a family of graphs as in part~\ref{thm:lb-graphs-is} of Theorem~\ref{thm:lb-graphs}. For an arbitrary $G \in \G_d$ with $|G| = n$, we have the following construction. Let $A_G$ be the graph consisting of two disjoint copies of $G$, that is, for each node $v \in G$ there are two copies $v_1, v_2 \in V(A_G)$ and for each edge $\{u,v\} \in E(G)$ two copies $\{u_1, v_1\}, \{u_2, v_2\} \in E(A_G)$. Similarly, let $B_G$ be the \emph{bipartite double cover} of $G$, that is, a graph on $2n$ nodes such that for each node $v \in V(G)$ there are two copies $v_1, v_2 \in V(B_G)$ and for each edge $\{u,v\} \in E(G)$ there are two edges $\{u_1, v_2 \}, \{u_2, v_1 \} \in E(H)$. We identify $V = V(A_G) = V(B_G)$ in the natural way. Note that clearly $\alpha(B_G) = n$.

    Let $\id \colon V \to [2n]$ be an identifier assignment on $A_G$ and $B_G$. We let the algorithm know $G$, $A_G$, $B_G$ and $\id$. The input is then promised to be either $A_G$ or $B_G$ with the following random perturbation $p$: for each pair of nodes $v_1$ and $v_2$ independently, we swap the identifiers of the nodes with probability $1/2$. Denote the graph distributions produced this way by $p(A_G)$ and $p(B_G)$.

    Denote by $Q(v,t)$ the sequence of $t(n)$ probes done by $\T_v$, where $t(n) = o(\log n)$. This sequence can be seen as a subset of edges revealed to the algorithm. Now it is easy to see for an arbitrary node $v$ that, since the graph induced by nodes in $Q(v,t)$ does not contain any cycles (that is, $t < g(G)/2$), we have $\Pr[Q(v,t) | p(A_G)] = \Pr[Q(v,t) | p(B_G)]$; consider a rooted version of each subtree in $Q(v,t)$ and note that both $p(A_G)$ and $p(B_G)$ have the same probability to produce that subtree.

    Finally, given that the distribution of the outputs of $\T_v$ is the same in $p(A_G)$ and $p(B_G)$ for each node $v$, and that any $t(n) = o(\log n)$ time algorithm $A$ must have $\sum_{v \in V(A_G)} \E[\A(v)] = \sum_{v \in V(B_G)} \E[\A(v)] \le 2\alpha(G)$, there must exist an identifier assignment on $B_G$ such that the size of the independent set produced by $\A$ is at most the expectation, that is $2\alpha(G)$.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:col-lb}]
    Clearly $\chi(G) \ge n/(\alpha(G)n)$. Therefore there must be an identifier assignment such that $A_G(u) \ge \chi(G)$ for some node $u$ in any graph $G$. Now let $T = \F \cup S$ be the disjoint union of the probe view of $A$ at $u$ when run in $G$ and some arbitrary tree $S$ such that $|T| = |G|$, with some arbitrary identifier assignment on $S$. Clearly the probe view of $A$ at $u$ is the same in $G$ and in $T$, and therefore $A_T(u) = A_G(u) \ge \chi(G)$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:max-cut-apx}]
    The proof proceeds almost exactly as in the proof of Theorem~\ref{thm:adh-mis-apx}, except that we use the family of graphs from part~\ref{thm:lb-graphs-mc} of Theorem~\ref{thm:lb-graphs}. The key property is again that a bipartite double cover of a graph admits a cut in which all edges are cut edges.
\end{proof}

\section{Centralised Local Model and Parallel Decision Trees}\label{sec:lcaparapp}

In this section we observe that one can simulate query-order-oblivious \lca\ algorithms by stateless algorithms.
Recall that a deterministic stateless \lca\ algorithm is a \partree\ algorithm (Fact~\ref{fact:lcapar}). Hence, the simulation result (Theorem~\ref{thm:main}) applies to the \lca\ model w.r.t.\ query-order-oblivious deterministic algorithms.
We then summarise the obtained lower bounds and show the optimality of several known algorithms.
We conclude with a separation between stateless and state-full \lca\ algorithms: we show a linear gap in the probe complexity between a \lca\ with (only) logarithmic state and a stateless algorithm that computes a leader in variant of the leader election problem.

\subsection{\boldmath Query-Order-Oblivious vs. Stateless \lca\ Algorithms}

We say that a \lca\ algorithm is \emph{query-order-oblivious} if the (global) solution that the algorithm computes does not depend on the input sequence of queries.
Even et al.~\cite{even14best} state that a stateless \lca\ algorithm is also query-order-oblivious.
In the following lemma we prove that the converse is also true, that is, for every query-order-oblivious \lca\ algorithm there is a stateless algorithm that can simulate it.

\begin{lemma}\label{lemma:qoo}
  For every query-order-oblivious \lca\ algorithm $C$ there is a stateless \lca\ algorithm $S$ that simulates $C$. Moreover, the probe complexities of $C$ and $S$ are equal.
\end{lemma}
\begin{proof}
  The stateless algorithm $S$ simply invokes $C$ with its initial state and the same random seed for every input query.
\end{proof}

\subsection{\boldmath \lca\ vs. \local}\label{sec:lcaapp}
Parnas and Ron~\cite{parnas07approximating} observed that given a deterministic \local\ algorithm $D$ that performs $r$ rounds of communication, there is a \lca\ algorithm $C$ that simulates $D$ with probe complexity which is $O(\Delta^r)$.
Even et al.~\cite{even14best} observed that if a deterministic \lca\ algorithm $C$ probes in an $r$-neighborhood of each queried vertex, then there is a deterministic \local\ algorithm $D$ that simulates $C$ in $r$ communication rounds.
Theorem~\ref{thm:main} implies that for some \lca\ algorithms there is a \local\ (implicit) simulation such that the number of communication rounds is asymptotically equal to the probe complexity, even though the \lca\ algorithm probes outside of the $r$-neighborhood of a queried vertex. This argument allows carrying lower bounds to the \lca\ model from the \local\ model. In Section~\ref{sec:constlca} we show an explicit simulation in which a \lca\ algorithm that is allowed to probe anywhere in the graph is simulated by a \lca\ algorithm that probes along connected components only.

Theorem~\ref{thm:main} and Fact~\ref{fact:lcapar} along with Lemma~\ref{lemma:qoo} imply the following theorem.
\lblca*

\iffalse
Theorem~\ref{thm:lblca}, and Lemma~\ref{lemma:qoo} implies the following corollary.
\begin{corollary}
    For every query-order-oblivious deterministic \lca\ algorithm $C$ with probe complexity $P=o(\sqrt{\log n})$, there is a \local\ algorithm $D$ that simulates $A$. Moreover, the number of rounds of $D$ is at most $P$.
\end{corollary}
\fi

In Section~\ref{sec:optim} we observed that Theorem~\ref{thm:main} also applies to several optimization problems.
In Table~\ref{tab:lb} we summarise (1)~the known \lca\ algorithms, their probe complexities as well as the obtained approximation ratios, and (2)~corresponding \local\ lower bounds. By Theorem~\ref{thm:lblca} all stated lower bounds apply to deterministic, query-order-oblivious \lca\ algorithms.

\begin{table}[htb]
\centering
\begin{tabular}{lllll}
\toprule
Problem
 & \multicolumn{2}{l}{\lca\ upper bounds}
 & \multicolumn{2}{l}{\local\ lower bounds} \\
 &  (deterministic, 0-space) \\
 & \# probes && \# rounds \\
\midrule
\mis     & $O(\log^* n)$ & \cite{even14best} &  $\Omega(\log^* n)$ & \cite{linial92locality}\\
\mm     & $O(\log^* n)$ & \cite{even14best} &  $\Omega(\log^* n)$ & \cite{linial92locality}\\
\dcolor  & $O(\log^* n)$ & \cite{even14best} &  $\Omega(\log^* n)$ & \cite{linial92locality}\\
$(1-\eps)$-\mcm & $\poly(\log^* n)$ & \cite{even14best} & $\Omega(\log^* n)$ & \cite{lenzen2008leveraging,czygrinow2008fast}\\
$(1-\eps)$-\mwm & $\poly(\min\{\Gamma,n/\eps\}\cdot\log^*n)$ & \cite{even14best} & $\Omega(\log^* n)$ & \cite{lenzen2008leveraging,czygrinow2008fast}\\
\bottomrule
\end{tabular}
\caption{\mis\ denotes maximal independent set, \mm\ denotes maximal matching, \dcolor\ denotes $\Delta+1$ vertex colouring,  $(1-\eps)$-\mcm\ denotes $(1-\eps)$-approximated maximum cardinality matching, and
  $(1-\eps)$-\mwm\ denotes $(1-\eps)$-approximated maximum weighted matching.
All the upper bounds presented in this table are of algorithms which are deterministic and stateless.
All the upper bounds are presented under
the assumption that $\Delta = O(1)$ and $\eps = O(1)$.
For weighted graphs, the ratio between the maximum to
minimum edge weight is denoted by $\Gamma$.
}\label{tab:lb}
\end{table}

\subsection{\boldmath The Power of State-full \lca\ Algorithms}\label{sec:statepower}
In this section we prove that the ratio between the probe complexity of a stateless \lca\ algorithm and a state-full algorithm is $\Omega(n)$  in a variant of the leader-election problem, defined as follows. This linear gap occurs even if the state-full algorithm uses only logarithmic number of bits to encode its state.

\begin{definition}[Two-path leader-election]
The input is a graph $G=(V,E)$, where $|V| \geq 6$.
In this graph there are two vertex-disjoint paths $p_i=(v^i_1,v^i_2,v^i_3)$ for $i \in \{1,2\}$. The rest of the vertices are of degree zero.
The ID set is $\{1,\ldots,n\}$ where $n$ is the number of vertices.\footnote{Note that the Two-path leader-election problem requires $\Omega(n)$ probes in the stateless \lca\ model even when the ID domain is exactly $\{1, \ldots, n\}$.}

The local output is $1$ in exactly one vertex in $\{v^1_2, v^2_2\}$. All the the other vertices output $0$.
\end{definition}

Note that a state-full \lca\ algorithm requires at most $O(\log n)$ space. The state-full algorithm proceeds as follows. The algorithm output ``no'' if the queried vertex has degree $0$ or~$1$. Otherwise, if the queried vertex is the first ``middle'' vertex, then it outputs $1$ and writes its ID to the state.
The next queried vertices will output $0$, even if their degree is $2$.

The linear gap between stateless \lca\ algorithms and state-full algorithms is formalised in the following lemma.
\begin{theorem}
  Every deterministic stateless \lca\ algorithm requires $\Omega(n)$ probes to compute a leader in the Two-path leader-election problem.
\end{theorem}
\begin{proof}
Let $\mathcal{A}$ be a deterministic algorithm for the Two-path leader-election problem.
Let the triplet $\ell^i = (\ell_1^i, \ell_2^i, \ell_3^i)$ denote the labeling of $p_i$. Namely, $\ell_j^i$ is the identifier of $v_j^i$.
Let $m$ denote the size of the domain of $\ell^i$, namely the number possible labelings of $p_i$, i.e., $m= n (n-1) (n-2) = \Theta(n^3)$.
Observe that the answers to probes on $G$ depend only on the labels of $p_1$ and $p_2$ and the same is true for the output of $\mathcal{A}$ (every vertex that is not in $p_1$ or $p_2$ is an isolated vertex).

The proof has two parts. In the first part we prove that there is a labeling of $p_1$ and a constant $c$ such that $\mathcal{A}$ outputs $1$ on $v_2^1$ for at least $cN$ of the possible labelings of $p_2$ and $0$ on $v_2^1$ for at least $cN$ of the possible labelings of $p_2$.
In the second part we assume in contradiction that the algorithm requires less than $\Omega(n)$ probes and show that there exists a legal labeling of $p_1$ and $p_2$ such that $\mathcal{A}$ errs.

We now give the first part of the proof.
Let $T_1, \ldots, T_N$ denote all possible labelings of $p_1$.
Consider the $N$ by $N$ matrix $M$ that encodes the decision of the deterministic algorithm, that is, $M(i,j)$ equals to the algorithm's output for the query $v_2^1$ when $p_1$ is labeled by $T_i$ and $p_2$ is labeled by $T_j$.
Note that some pairs of labelings are illegal (their intersection should be empty). Let $N' = \Theta(n^3) = \gamma n^3$ denote the number of legal labels in each row (or column) of $M$.
Let $S$ be the set of rows in which the number of $1$'s is in $[\frac 13 N', \frac 23 N']$. If $S \neq \emptyset$ then we are done.
Otherwise, let $L_0$ ($L_1$) be the set of rows in which the number of $0$'s ($1$'s) is greater than the number of $1$'s ($0$'s). Before we continue with the proof, we prove the following property of $L_0$ and $L_1$.

\begin{lemma}
If $S = \emptyset$ then $L_i \geq N/4$ for every $i\in \{0, 1\}$.
\end{lemma}
\begin{proof}
From the fact that the algorithm always outputs $1$ on exactly one of the elements in $(v_2^1, v_2^2)$ we derive that in $M$ the number of $1$'s equals to the number of $0$'s.
Now, assume for the sake of contradiction that $L_i < N/4$ for some $i \in \{0,1\}$.
Since every row in $L_i$ has at most $N'$ $i$'s we obtain that the total number of $i$'s in $M$ is at most $|L_i| \cdot N' + (N- |L_i|) \cdot (N'/3) < (N/2) N'$, in contradiction to the fact that the number of $i$'s is exactly $(N/2) N'$.
\end{proof}

Let $i\in \{0,1\}$. Consider $M$ restricted to $L_i$.
Since in each row in $L_i$ the number of $i$'s is at least $\frac 23 N'$ we obtain that the total number of $i$'s in this restriction is at least $\frac 23 N' |L_i|$.\
Let $X$ denote average of the number of $i$'s in each column restricted to $L_i$.
Thus we obtain that $X \geq \frac 23 N' |L_i|\frac{1}{N}$.
Let $c \triangleq \frac{2|L_i|}{N'}(\frac23 \frac{N'}{N} -\frac 12)$, and let $a$ be the fraction of columns for which the number of $i$'s in each column restricted to $L_i$ is greater than $cN'$.
Then $X \leq |L_i|\cdot a + cN' (1-a)$.
Assume towards contradiction that $a \leq 1/2$.
Hence, $X \leq |L_i|/2 + cN'/2 < 2/3 N' |L_i|\frac{1}{N}$, a contradiction.
Hence, there exists a constant $c$ such that for strictly more than $1/2$ of the columns it holds that at least $cN'$ elements are $1$ (0).
We derive that there exists a columns such that both the number of $1$'s and the number of $0$'s is at least $cN'$ as desired.

Consider the query $v^1_2$. Observe that in each probe for which the answer is ``isolated vertex'', the algorithm eliminates at most $3\cdot n^2$ possible labeling of $p_2$.
Thus, after at most $c\gamma n^3/ (3n^2) = \Omega(n)$ probes, at least one ``zero'' labeling and one ``one'' labeling remains, which concludes the proof of the lemma.
\end{proof}

\section{\boldmath Localizing Stateless \lca\ Algorithms} \label{sec:constlca}
In this section we give a constructive, polynomial blow-up, randomised \lca\ simulation.

\mainlocal*

\subsection{Novelty}
We observe that in the \lca\ model an explicit simulation is possible, since, unlike the \local\ and \partree\ models, a \lca\ algorithm uses the same random seed for all queries.
This random seed is a costly resource and we try to minimize its length.
Known randomised implementations of greedy algorithms in the \lca\ domain require explicit random ordering constructions~\cite{alon12space-efficient,reingold14new} over the vertices or edges~\cite{alon12space-efficient,mansour12converting,mansour13local,reingold14new}.
In our implementation of the simulation, on the other hand, we use a permutation over the labels, which is a stronger requirement than a random ordering.
This requirement comes from the fact that in the simulation each vertex has a unique identifier and that the set of identifiers is assumed to be known.
In what follows we propose an implementation which builds on techniques by~\cite{v009a015,knr09}.

For many graph problems the simulation presented in this section proves that one can design a \lca\ algorithm that only performs ``close'' probes. On one hand, this brings closer the two models of  \lca\ and \local. On the other hand, the single source of randomness that a \lca\ algorithm possesses is an advantage for the \lca\ algorithm, so they are still far apart.

\begin{comment}
  \begin{remark}
  Note that the $\Delta$ factor in the seed ``overhead'' is a consequence of our probe model in which the answer to a probe $v$ is its (up to) $\Delta$ neighbors. Let us  consider the alternative probe model in which a probe is a pair $(v,i)$ where $v \in V$ and $i \in [\Delta]$, and the answer to such probe is the $i$th neighbor of $v$ and ``null'' if there is no such neighbors. In case we consider the latter probe model, then the aforementioned $\Delta$ factor is omitted from the seed overhead and may  reappear in the probe-complexity.
\end{remark}
\end{comment}

\subsection{Notations and Definitions}
Let $S_n$ denote the set of all permutations on $[n]$.
\begin{definition}[Statistical Distance]
Let $D_1$, $D_2$ be distributions over a finite set $\Omega$. The statistical distance between $D_1$ and $D_2$ is
$$
\|D_1 - D_2 \|= \frac 12 \sum_{\omega\in \Omega} |D_1(\omega) - D_2(\omega)| .
$$
Alternatively,
$$
\|D_1 - D_2 \|= \max_{A\subseteq \Omega} \left|\sum_{\omega\in A} D_1(\omega) - \sum_{\omega \in A} D_2(\omega)\right| .
$$
\end{definition}
We say that $D_1$ and $D_2$ are $\epsilon$-close if $\|D_1 - D_2\| \leq \epsilon$.

\begin{definition}
  Let $n,k \in \mathbb{N}$, and let $\mathcal{F} \subseteq S_n$ be a multiset of permutations. Let $\eps \geq 0$. The multiset $\mathcal{F}$ is $k$-wise $\eps$-dependent if for every $k$-tuple of distinct elements $(x_1,\ldots,x_k) \in [n]^k$, the distribution $(f(x_1),
  f(x_2),\ldots,f(x_k))$, when $f \simur \mathcal{F}$ is $\eps$-close to the uniform distribution over all $k$-tuples of distinct elements of $[n]$.
\end{definition}
As a special case, a multiset of permutations is $k$-wise independent if it is $k$-wise $0$-dependent.

\subsection{Techniques}
We shall use the following results from previous work in the proof of Theorem~\ref{thm:mainlocal}.
The following theorem is an immediate corollary of a theorem due to Alon and Lovett~\cite[Theorem~1.1]{v009a015}.
\begin{theorem}\label{thm:alon}
Let $\mu$ be a distribution taking values in $S_n$ which is $k$-wise $\epsilon$-dependent. Then there exists a distribution $\mu'$ over permutations which is $k$-wise independent, and such that the
statistical distance between $\mu$ and $\mu'$
is at most $O(\epsilon n^{4k})$.
\end{theorem}

Additionally, we build on the following construction in our \lca\ simulation \simu. This construction enables accesses to a  uniform random permutation from a $k$-wise $\epsilon$-dependent family of permutations by using a seed of length $O(k\cdot \log n  + \log(1/\epsilon))$.
\begin{theorem}[{\cite[Theorem~5.9]{knr09}}]\label{thm:kaplan}
There exists $\mathcal{F} \subseteq S_n$, such that $\mathcal{F}$ is $k$-wise $\epsilon$-dependent. $\mathcal{F}$ has description length $O(k\cdot \log n  + \log(1/\epsilon))$, and time complexity $\poly(\log n, k, \log (1/\epsilon))$.
\end{theorem}

\subsection{\boldmath Simulating Queries in the \lca\ Model}
Recall that the simulation in Section~\ref{sec:simupar} simulates a \partree\ algorithm via a distributed \local\ algorithm.
In order to prove Theorem~\ref{thm:mainlocal} we consider a similar simulation with the difference that now both the simulation and the simulated algorithm are in \lca.
However, the simulation has the property that it is limited to query \oracleG\ only on $Q$ (as defined in Section~\ref{sec:simupar}).
Since a \lca\ is equipped with a random seed we can use a randomised simulation and consequently the blow-up in the probe-complexity will be significantly smaller.
In particular, in this section we consider $N = O(n^4)$, that is, the size of the augmented graph $G \cup H$ is polynomial in the size of the input graph $G$.
We are interested in keeping the seed length small and so we show next that the additional random seed that is required for the simulation is small.

\begin{comment}
The listing of the simulation is in \moti{Algorithm}~\ref{alg:simu}.
\begin{algorithm}
  \caption{Simulation of \lca\ algorithm $A$: \simu}
  \label{alg:simu}
  \begin{algorithmic}[1]
  \Require Access to \oracleG\,
  access to a permutation $\pi:[N] \rightarrow [N]$ and its inverse, and a random seed $r \in \{0,1\}^{s(N)}$ of $A$.
  \State On query $v \in V(G)$:
  \While {$A$ performs probes}
  \State Let $\{p_i\}_i$ denote the sequence of probes generated by $A(r,\pi(q))$.
  \For {$i$}
    \State If $\pi^{-1}(p_i) \in V(G)$ then $\hat {p_i} \gets \oracleGm (p_i)$,
    \State Otherwise,  $\hat {p_i} \gets \emptyset$.
  \EndFor
   \State Answer $A$ with $\pi(\{p_i\}_i)$.
  \EndWhile

  \State Output:  $g$.
  \end{algorithmic}
\end{algorithm}
\end{comment}

\subsection{Proof of Theorem~\ref{thm:mainlocal}}
Let $A$ be as in Theorem~\ref{thm:mainlocal} and let \simuA\ denote its simulation on $G\cup H$ as described  above.
Since the size of $G\cup H$ is $N$ the size of the random seed required by $A$ is $s(N)$.
For a fixed a fixed random seed $r\in \{0,1\}^{s(N)}$,
a fixed query $q\in V(G)$ and a fixed permutation $\pi\in S_N$ let $\simu(G, r, \pi, q)$ be the indicator variable for the event that the simulation succeeds in simulating $A$ with random seed $r$ on input $q$ where $\pi$ is the permutation which is used to relabel the vertices in $G\cup H$.

Let $\mathcal{F}$ be a family of $k$-wise independent permutations over $[N]$ where  $k \triangleq 1 + (\Delta+1) \cdot t(N)$.

\begin{lemma}\label{lemma:unitokwise}
For every $q \in V(G)$ and $r\in \{0,1\}^{s(N)}$,
$$\Pr_{\pi\simur \mathcal{F}}(\simu(G, r, \pi, q) = 0) \leq \frac{kn}{N-k} $$
\end{lemma}
\begin{proof}
Fix $r\in \{0,1\}^{s(N)}$, $\pi\in S_N$ and $q\in V(G)$.
We will prove that
$$
\Pr_{\pi\simur \mathcal{F}}(\simu(G, r, \pi, q) = 0) = \Pr_{\pi\simur S_N}(\simu(G, r, \pi, q) = 0),
$$
and then the lemma will follow from Equation~\ref{eq:simfails}.
Recall that the simulation relabels the vertices by accessing both to $\pi$ and $\pi^{-1}$.
Now consider the sequence of inspections the simulation makes to $\pi$ and $\pi^{-1}$ by the order they occurred.\footnote{The construction stated in Theorem~\ref{thm:kaplan} computes $\pi(v)$ for some $v$ in time $\poly(\log n, k, \log (1/\epsilon))$. This construction does not describe a time efficient way to access $\pi^{-1}$. As time complexity is not the focus of this paper, we implement the inverse access in a straightforward manner.}
The first inspection to $\pi$ is $\pi(q)$.
Then, according to the decisions of $A$ and the answers from \oracleG, the simulation continues to inspect both $\pi$ and $\pi^{-1}$ on at most $k-1$ locations.

Let $(v, u, b) \in N \times N \times \{-1, 1\}$ represent a single inspection and answer as follows:
If $b = 1$ then the interpretation is that the simulation inspects $\pi$ at index $v$ and the answer is $u$ and if $b = -1$ then the interpretation is that the simulation inspects $\pi^{-1}$ at $v$ and the answer is $u$.

For a fixed $G$, $r$, $\pi$ and $q$ the sequence of (possibly adaptive) inspections and answers is fixed. Let it be denoted by $\sigma = (v_1, u_1, b_1), \ldots,(v_{\ell}, u_{\ell}, b_{\ell})$ where $\ell \leq k$.
We say that a permutation $\pi$ agrees with the sequence $\sigma$ iff $\pi^{b_i}(v_i) = u_i$ for every $i\in [\ell]$.
Clearly, we can replace $\pi$ with any permutation $\pi'$ which agrees with $\sigma$ and the outcome (failure or success) remains unchanged, that is $\simu(G, r, \pi, q) = \simu(G, r, \pi', q)$.
We say that $\sigma$ is {\em positive} if $b_i = 1$ for every $i\in [\ell]$.
We say that a pair of sequences $\sigma$ and $\sigma'$ are {\em equal} if any permutation that agrees with $\sigma$ also agrees with $\sigma'$ and vice versa.
Observe that from any sequence $\sigma$ we can obtain an equal sequence, $\sigma'$, which is positive by performing the following replacements: If there exists $j$ such that $b_j = -1$ (this means that $\pi^{-1}(v_j) = u_j$ which is equivalent to $v_j = \pi(u_j)$) then replace $(v_j, u_j, -1)$ with $(u_j, v_j, 1)$.
Therefore we obtain that $S_N$ can be partitioned into equivalence classes where in each class $C$:
(1)~all the permutations agree with some positive sequence $Q$ of length at most $k$, and (2)~$\simu(G, r, \pi, q)$ is fixed when $\pi$ is taken from $C$.
Since for every positive sequence $\sigma$ we have
$$
\Pr_{\pi\simur \mathcal{F}}[\pi\text{ agrees with } \sigma] = \Pr_{\pi\simur S_N}[\pi\text{ agrees with } \sigma]\ ,$$
we obtain the desired result.
\end{proof}

\begin{corollary}\label{coro:n3}
For a fixed graph $G$ and a fixed random seed $r \in \{0,1\}^{s(N)}$,
the probability that the simulation works when $\pi \simur \mathcal{F}$ is at least $1- O(1/n)$.
\end{corollary}

\begin{proof}
Recall that the simulation succeeds if and only if $\simu(G, r, \pi, q) = 1$ for every $q \in V(G)$.
Since $t(N) = O(n/\Delta)$, from Lemma~\ref{lemma:unitokwise} and the union bound over the $n$ possible queries we obtain that the simulation succeeds with probability at least $1 - \frac{kn^2}{N-k} = 1 - O(1/n)$.
\end{proof}

Let $\mathcal{F}'$ be a $k$-wise $\epsilon$-dependent family of permutations.
\begin{corollary}\label{coro:simerr}
For a fixed graph $G$ and a fixed random seed $r \in \{0,1\}^{s(N)}$,
the probability that the simulation works when $\pi \simur \mathcal{F}'$ is at least $1- O(1/n) - O(\epsilon N^{4k})$.
\end{corollary}
\begin{proof}
For a fixed graph $G$ and random seed $r$ let $S\subseteq S_N$ denote the subset of permutations $\pi$ for which the simulation works.
By Corollary~\ref{coro:n3}, $\Pr(\pi \in S) \geq 1- O(1/n)$ when $\pi \simur \mathcal{F}$.
By Theorem~\ref{thm:alon} the statistical distance between the uniform distribution over $\mathcal{F}$ and the uniform distribution over $\mathcal{F}'$ is at most $O(\epsilon N^{4k})$.
Therefore $\Pr(\pi \in S) \geq 1- O(1/N)-O(\epsilon N^{4k})$ when $\pi \simur \mathcal{F}'$ as desired.
\end{proof}

We are now ready to prove the main result of this section.
\begin{proof}[Proof of Theorem~\ref{thm:mainlocal}]
In the simulation we shall use the construction from Theorem~\ref{thm:kaplan} to obtain a random access to a permutation $\pi$ over $[N]$ where $N = n^4$, using a random seed of length $O(k \cdot \log N + \log \gamma)$ where $\gamma = O(n N^{4k})$ and access time $\poly(\log N, k, \log \gamma)$.
By Corollary~\ref{coro:simerr} the simulation works with probability at least $1 - O(1/n)$ as desired.
\end{proof}

\iffalse
-------------------------------------------------------
\moti{Do it all}
- Derandomzie (this implies also for the partree model)
- Find seed that captures both pi and inverse pi
- the poly of nkr is ``really'' poly?
- edge probes

- Approximation coro for ``empty'' aux in the LCA SIM.
- Extend LCL
- Do we need Alon? (what does he requires? almost eps or delta dep.?)
- Do we need to go through uniform dist.?

Future research:
- Extend lca sim to: Statfull, (weak) randomised and to see if this new LB applies to some random work?

We are interested in a constructive, low blow-up simulation.
As said in an earlier email (see this thread) one can consider the randomised simulation as a solution.
The downside of this solution, as observed by Juho, is that it requires from the Distributed algorithm to have shared randomness. This observation eliminates this path to a constructive simulation of a Centrzlied Local Alg by a Distributed one. Having said that, the randomised simulation still gives a constructive simulation in the Centralised model world, that is: for every \lca\ algorithm that probes where every it wants, there is a \lca\ that probes only along connected components.

We think that this new statement is interesting, since it somewhat diminishes the "power" of centralised local model to added value of having shared randomness (at least w.r.t. the dependency of the probe complexity to $n$).
\moti{discussing what we can localise.}

\fi

redpajama_set_name: RedPajamaArXiv


text:
Judging by what the kids are wearing on Newbury Street these days, we'd say they're all about the old-school. Take them way back this summer with the Plimoth Plantation Summer Programs.
Alicia Keys is 27-years-old, and has won 11 Grammys. Try not to think too much about this as she sings her award-winning neo-soul-infused pop at the Garden tonight.
David McCullough started the John Adams lovefest, HBO kept it going, and now the Boston Landmarks Orchestra has embraced the Quincy native and Founding Father. The group debuts a new work titled John Adams: The Voice Heard 'Round the World and honors artist Clara Wainwright at its Fundraising Gala.
If yesterday's heat didn't warm you up, check out Cuban jazz trumpeter Arturo Sandoval as he plays with the Boston Pops as part of JazzFest.


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