Abdulla is our Kennel Attendant at The Veterinary Surgery since June 2018 and he works hard to make sure that the boarding area is clean and comfortable during every pet's stay. When he's not looking after dogs in the boarding facility, he is busy with other task in/outside the clinic and making sure that surrounding of our clinic is very well maintained.
Al-Hussein became one of our veterinary technicians in June 2015 and has a special interest in surgery, health monitoring, and administering treatment. He earned his Bachelor of Science and Arts degree from Mindanao State University and his Doctor of Veterinary Medicine degree from Central Mindanao University. Over the years, he has had many noteworthy experiences in the field, including providing care for big cats like cheetahs and tigers. He loves being a veterinary technician and considers it a noble duty to take care of creatures that can't help themselves.
Al-Hussein is married and has a cat named Ottoman. In his spare time, he enjoys learning filmmaking and visual effects. He is also a Karate-do enthusiast.
Groomer and Veterinary Technician, graduated year 2015 from Nairobi Kenya; special interest in Animal Surgery, I enjoy and love working with Animals. My hobbies are swimming, playing piano.
Moid became one of our veterinary technicians in December 2017 and has a special interest in assisting in surgery, health monitoring, and administering treatment. He loves being a veterinary technician and considers it a noble duty to take care of creatures that can't help themselves.
In his spare time, he enjoys learning photography and editing. He is also a cricket enthusiast.
Asma has been part of our team since January 2014. Fueled by her love for pets, she earned her Bachelor of Veterinary Medicine in Egypt. She also earned her International Computer Driving License (ICDL). As one of our veterinary technicians, she now enjoys being part of a team that shares her love for pets. She especially loves working with dogs. In her spare time, she enjoys walking, drawing, and reading.
Huda is one of the veterinary technicians at The Veterinary Surgery. She has been part of our team since May 2014, after earning her Bachelor of Veterinary Medicine in Egypt. She has always loved pets, so pursuing a career as a veterinary technician was a natural choice for her. She also earned her International Computer Driving License (ICDL). When she's not at work, she loves reading, walking, cooking, and crafting.
Elbie joined our team in August 2016 as a veterinary technician. She attended Aklan State University in the Philippines, where she pursued her DVM. Elbie decided she wanted a career in the veterinary field because she had a desire to provide more than just emotional care for animals, she wanted to take it further and provide them with medical care and treatments when they're sick. In fact, her favorite part about working at our practice is sending patients home happy and healthy, after they came to her sick.
Elbie has three dogs (Pogi, Siomai, and Bo), and in her free time she loves to play with them, read, watch movies, and get a good night's sleep.
Edwin is one of the kennel attendants at The Veterinary Surgery and works hard to make sure that the boarding area is clean and comfortable during every pet's stay. Prior to joining our team, he earned an Associate in Industrial Technology (AIT) from Zamboanga City State Polytechnic College. When he's not looking after dogs in the boarding facility, he cares for her own pet—a cat named Totocha—and spends time with his family.
Vilma joined our team as a groomer in December 2006 and loves caring for the animals and making them look at their best. She also loves making her clients happy. She earned a degree in Agriculture from Zamboanga A.E. College and is married with three sons.
Siddhartha joined our team in February 2016, with several dog training and grooming awards under his belt. He earned a bachelor's degree from Computel. As one of our groomers, he loves working with pets and helping them to be better companions by training and grooming them. At home, he cares for his own dogs and cats.
Catherine worked as a front desk officer at The Veterinary Surgery. She has several years of experience in client relations and previously worked as Marketing Officer for private company, where she handled scheduling, accounting, payroll and finance. Catherine or Kate to her friends, have a rescued cat named Mark, such a loving soul, of course she enjoy the licking of her cat, She also love being outside and hanging out with family, friends, furry baby and colleagues.
Dorothy joined The Veterinary Surgery April of 2017, with a degree of Bachelor of Science in Commerce, Major in Management. With 20+ years of extensive experience, in various fields that includes works as HR Office Manager, Freight and Warehouse Specialist & Manufacturing Specialist. I am a dog lover by nature, but when I joined the practice, I come to love the cat as well.
Gwen worked as one of the Receptionist here in The Veterinary Surgery. She has several years of experience working in a private sector such as HR Assistant, Customer Service Representative, Team Leader and a Hotel Receptionist. She is a mother, with one child and a loving and very supportive daughter to her family. Just like anyone else, she has a Labrador in Philippines but unfortunately she passed away.
Sheila joined The Veterinary Surgery September of 2018; she has several years of experience from various fields that includes works as Office Assistant for a private association and Customer Support for Telecommunication Company. Sheila has adopted a 2 year old Chihuahua a very lovely dog and everyone in her family loves her so much. A family oriented person, always loves to gather every weekend and have family bonding.
Charmine joined The Veterinary Surgery in September 2014 after earning her Bachelor of Science in Business Administration degree in accounting management from the University of Makati. As one of our administrative assistants, she enjoys being able to help clients and meeting new people. She also loves assisting the veterinarians and her colleagues. Charmine is married with a daughter. She also has a dog. In her spare time, she enjoys cooking and cleaning.
Emily has been part of our receptionist team since October 2015. She studied to become a medical transcriptionist at MTC Academy of Davao in the Philippines. She chose to pursue a position in the veterinary field because of her interest in medicine and to become more familiar with the role of a doctor. She loves having knowledge of the evolution of the medical field and the exposure she receives to pharmacological and medical terminology. She also loves working with a great work force and with pets. Her own pets include two cats named Mau and Henry and three dogs named Tiny, George, and Tanya. In her spare time, she loves reading books, traveling, discovering music, and watching movies.
Mahmoud began working at The Veterinary Surgery in June 2005, after studying at Cairo University. As one of our pet relocation consultants, he helps pets travel safely and helps pet owners with the planning, including all the requirements. When he's not at work, he spends time with his family and cares for her African grey.
Mohamed has been with part of The Veterinary Surgery's team since December 2005. He received a degree in social service from the College of Social Service. As one of our pet relocation consultants, he loves being able to assist customers with their pet travel needs to make sure that every pet travels and arrives safely. He especially loves witnessing the happiness of clients when they receive their pets and being part of that process. Although cats are his favorite animal, he has a snake at home.
Hani has been part of The Veterinary Surgery's staff since December 2005. He earned his accounting degree from Cairo University. Although he deals with numbers all day, he also enjoys seeing the feline patients in particular; cats are his favorite animals. In his spare time, he enjoys watching television, playing games, and going out.
Inspired by her love for pets, Matilda joined our team in July 2012 after earning her Bachelor of Commerce degree from Indra Gandhi National Open High. She loves that every day is a new experience that allows her to learn something different. She also loves interacting with people from different parts of the world. She's married and has two dogs named Pepr and Mac. In her free time, she loves spending time with friends, her dogs, camping, and dancing. She also prides herself on her ability to make people smile.
Ibahadur has been part of The Veterinary Surgery's team since August of 2018. As one of our facility cleaners, he keeps the dog/cat recovery area clean for pets. He also walks the dogs, which he considers one of his favorite aspects of his position. When he's not at work, he enjoys doing small backyard gardening.
Bhim is one of the kennel attendants at The Veterinary Surgery since November of 2018 and ensuring keeping cages clean and sanitized especially in our Prep / Surgery area as his main focus; and reporting on the animal's condition to our Veterinarian and or Vet Tech. Assisting also in, nail trimming and grooming of animals as well.
Riyas is one of the kennel attendants at The Veterinary Surgery since September 2018 and ensuring keeping cages clean and sanitized; walking, feeding and picking up after the animals; and reporting on the animal's condition to our Veterinarian and or Vet Tech. Assisting also in bathing, nail trimming and grooming of animals as well.
Soduhang has been part of The Veterinary Surgery's team since July 2015. He attended Godawari College in Itahari, Nepal. As one of our facility cleaners, he keeps the recovery area clean for pets. He also walk the dogs, which he considers one of his favorite aspects of his position. When he's not at work, he enjoys reading books, writing, and playing his guitar.


text:
A college degree and no job
The always insightful blog at Voices From El Salvador has a recent post about unemployment in El Salvador. The post describes some recent news articles where business complains that there are not enough qualified workers in El Salvador to fill their positions: [Marco Penado of Manpower El Salvador] says students are not graduating with the skills and experience that human resource managers are looking for, claiming that university students are more interested in the humanities rather than engineering or other technical skills. He also believes that too few Salvadorans speak English, and that in a globalized world defined by trade agreements, corporations that operate in El Salvador require employees that speak English. There are plenty of people looking for work, but there is a mismatch between their skills and the skills employers are looking for. Employers want universities to redesign their curriculum to line up more closely with the needs of corporations. But as the Voices pos
Environmental perils spur organization
As regular readers of this blog know, climate change and other environmental challenges regularly confront El Salvador. Those challenges have spawned a growing network of grass roots environmental organizations. A recent article titled Climate: Putting people over money from Al Jazeera English interviews the participants in this growing movement: A 2007 climate change study conducted by El Salvador's National Ministry of the Environment and Natural Resources focused on the Lower Lempa River and Bay of Jiquilisco areas of the central Pacific coast. The study found that this area can expect more of what it is already experiencing: increasing minimum and maximum temperatures, a shift in observed seasons, more frequent observations of extremely wet and extremely dry years, and intensified extreme event activity, including tropical storms and hurricanes. Against the backdrop of these dire predictions, the people are, however, forming a movement that is learning to protect and
An intro to Salvadoran food
Food blogger Sasha Martin has spent the past week writing about and cooking the food of El Salvador. Here's how she introduced the topic: Do you like colorful birds? What about ruins – ancient, gothic, and colonial? Step right this way. Meet El Salvador, a tiny country freckled with mighty volcanoes, thickly coated by lush tropics, and so much more. In this steamy dreamland, I discovered a theme: corn. You can read Sasha's week of posts on Salvadoran food, including recipes, at this link . Yum.
FMLN now has leadership of National Assembly
As of February 1, the presidency of the National Assembly is held for the first time by a member of the FMLN. Sigfrido Reyes, an FMLN legislator became the new head of the country's lawmaking body. Although the FMLN has held a plurality of seats in the National Assembly several times since the end of the civil war, it was the split of several right wing ARENA deputies to form the GANA party, which led to Reyes assuming his new position. The development and the reaction of the Salvadoran blogosphere are well described in a post by Omar Nieto on Global Voices . Some comment on the historic nature of another branch of government now being held by the FMLN. Others caution that the media and right wing are now poised to blame all problems of the country on the fact that the FMLN controls the presidency and the National Assembly. Sigfrido Reyes may have further political ambitions in Salvadoran presidential politics, but the National Assembly will need to deliver results for the
The scarce bean
Two news stories from the past week in the paper El Mundo highlight El Salvador's problem of food insecurity. The first was a story about the price of beans . In the past year, the price of a pound of red beans has increased 138% from $0.52 to $1.24 per pound. The price increase reflected the significant destruction of the bean crop by the damaging rains in 2010. The story noted that the price of corn also rose last year. The second was the report that the Salvadoran government will import approximately 9 million pounds of beans from China. The purchase will cost around $5 million, and will help alleviate the scarcity of beans for the Salvadoran markets. According to the story, Salvadoran officials say the Chinese bean is not the same as beans from Nicaragua, but they tested it, and it's a "buen frijol."
Obama to visit El Salvador March 22-23
Funes March 2010 visit to White House Barack Obama will visit El Salvador on March 22-23 as part of a three country tour of Latin America. The visit was announced during Obama's State of the Union address to Congress: This March, I will travel to Brazil, Chile, and El Salvador to forge new alliances across the Americas. Around the globe, we're standing with those who take responsibility -- helping farmers grow more food, supporting doctors who care for the sick, and combating the corruption that can rot a society and rob people of opportunity. In confirming the March dates for the US president's visit, Salvadoran head-of-state Mauricio Funes indicated that the fight against poverty would be a central theme of his discussions with Obama. According to the web site of the Salvadoran president : On the issue of poverty, President Funes hopes to advance the administration of the Bridge Initiative which seeks to use remittance flows as an asset to finance infrastructur
Growth of evangelical Christianity in El Salvador
In an article this week , The Economist magazine writes about the growth of evangelical Christianity at the expense of the Catholic church in El Salvador: More recently, the Catholic church's conservatism has shrunk its flock. Edgar López Bertrand , the founder of the Friends of Israel , says he could not become a Catholic priest because his parents were divorced. Now, the crowd outside his church includes teenage couples and not a few miniskirts. (Should relationship problems arise, the church offers a book called "Help! I'm married".) The gospel of prosperity, recklessly preached by some evangelical outfits, goes down well in poor countries: Costa Rica and Panama, twice as rich as their neighbours, remain strongly Catholic. Proximity to America has spurred the churches' growth. "Everything we know comes from the United States," says Edgar López Bertrand Jr, who runs Friends of Israel with his father. Media savvy is one useful import: his church broadcasts on television and radi
An audio visit to a Salvadoran coffee plantation
Chris Hallberg is from the Milwaukee area, and he recently narrated an audio visit to a coffee plantation west of San Salvador. You can listen here to the story which aired on public radio recently.
Five years under DR-CAFTA
The fifth anniversary of the effective date of the DR-CAFTA trade agreement is approaching in March. We can expect to see various writers trying to assess the impact of five years under this treaty and whether it was a good deal for El Salvador. I'll try to provide ways to think about DR-CAFTA's impact in posts over the next month. An article at alterinfos.org by Leonard Morin titled El Salvador - Free Trade's Dubious Blessings provides an assessment from the left, while providing space to acknowledge the viewpoints of supporters of DR-CAFTA. It's a good read to get you started on thinking about the impact of the trade pact. Here is an excerpt from one of the figures interviewed for the article: According to Gilberto García of the Center for Labor Studies and Support (Centro de Estudios y Apoyo Laboral, CEAL), "the Salvadoran and US authorities told us in 2003, 2004, and 2005 that the FTA [Free Trade Agreement] was going to be the solution, that the FTA was g
Salvadorans' view of the utility of democracy
A poll released by El Faro suggests that Salvadorans could abandon their democracy if they thought it would improve their daily lives: In a country with a history of military dictatorship and a democracy refounded just 19 years ago, almost half of people are willing to support the military overthrow of a democratic government if it helped solve their problems Only one seventh of Salvadorans value democracy as the system of government preferred to any other, while nearly half say they would be willing to support a military coup if the country continues without resolving its economic problems and public safety issues. These are two of the findings of a nationwide survey conducted for the purpose of finding out how much Salvadorans value their democracy in relation to the needs of the population. The information was gathered by the Analitika Research and Marketing Company between November 8 and 13 last year, through 1200 interviews with a margin of error of + /- 2.8 points. The r

redpajama_set_name: RedPajamaCommonCrawl


Education press releases - 3072 news release online.
Have You Secured Your Business With Email Protection Services Yet?
In an ever-changing world obsessed with Information Technology systems and processes, emails have become the lifeblood of every business organization.
Every student needs to put in a lot of efforts to earn a degree from the university. It is not only the gateway to one of the finest career opportunities but also provides the fame and position that you need to get as an individual.
MyAssignmenthelp.com is delighted to introduce its assignment help services in Gold Coast for students from Australia. We have the best assignment help experts of Gold Coast.
What Is The Placement Scenario Like At The Sri Balaji Society Institutes?
Since the dawn of the new millennium, India has witnessed an exponential rise in the number of multinational, transnational and global companies looking to control their operations from within the country.
International Students hailing from different parts of the world find Australia an attractive country to settle in with their future perspectives.
India has seen a radical shift in the perception regarding legal education in the last couple of decades. Because of its developing pertinence, law applicants can discover openings in courts as well as over various avenues across corporate houses, managerial administrations, among others.


text:
Q: How to hide keyboard from EditText after finish writing without lost the text Hell, I have a ListView with adapter when the app start the code put a EditText in my ListView after click in the EditText I can't write so I tried to solve this problem I did a focus and it's work fine and the app allow me to write but after done writing and I hide the keyboard the text will gone.
I wan't this editText to be normally like any other EditText when I clicked in I writed when I click done, the keyboard gone and my text stay there. How should I do that?
var List=ArrayList<Ticket>()
var adapter:ListAdabter?=null

override fun onCreate(savedInstanceState: Bundle?) {
    super.onCreate(savedInstanceState)
    setContentView(R.layout.activity_act2)

    List.add(Ticket("1"))
    adapter=ListAdabter(this,List)
    listview.adapter=adapter
}

fun View.showKeyboard() {
    this.requestFocus()
    val inputMethodManager = context.getSystemService(Context.INPUT_METHOD_SERVICE) as InputMethodManager
    inputMethodManager.showSoftInput(this, InputMethodManager.SHOW_IMPLICIT)
}

inner class ListAdabter :BaseAdapter{
    var ListLocal = ArrayList<Ticket>()
    var mCtx:Context?=null

    constructor(mCtx:Context,List:ArrayList<Ticket>):super(){
        this.ListLocal=List
        this.mCtx=mCtx
    }

    override fun getView(position: Int, convertView: View?, parent: ViewGroup?): View {
        val item = ListLocal[position]
        val layoutInf = LayoutInflater.from(mCtx)
        val view = layoutInf.inflate(R.layout.add_ticket,null)
        val edit = view.findViewById<EditText>(R.id.txtDes)
        edit.requestFocus()
        edit.showKeyboard()  //The Problem is Here
        return view
    }

    override fun getItem(position: Int): Any {
        return ListLocal[position]
    }

    override fun getItemId(position: Int): Long {
        return position.toLong()
    }

    override fun getCount(): Int {
       return ListLocal.size
    }

}

redpajama_set_name: RedPajamaStackExchange


text:
Understanding Europe
Articles by country:
Ceuta & Melilla
From file: A picture of migrants walking along the Roja river, near Ventimiglia, Italy northwest, towards the French border, June 26 2017 | Photo: Chiara Carenini /ANSA
migrant children
migrants in Italy
police intervention
Afghan girl hospitalized in shock after border pushback
Published on : 2021/03/30
By ANSA Published on : 2021/03/30
An Afghan girl suffering from shock was assisted by Turin's Regina Margherita children's hospital on Saturday. It is thought the girl was pushed back with other migrants on the border between Italy and France by French police last week.
A girl who fled Afghanistan began suffering a kind of post-traumatic shock, reliving her experience of war and terror in her home country after she and her family were among a group of Afghan migrants pushed back by French police last week in the Monginervo forest between France and Italy.
The shock was too intense for the ten-year-old girl, for whom seeing uniforms and weapons in the dark of the mountains reminded her of a bomb that had injured her in her home country when she was seven.
The medical staff at Turin's Regina Margherita children's hospital examined her and calmed her down prior to releasing her.
The prosecutors office for juvenile issues is assessing whether any crimes were committed.
"We are not surprized by mistreatment by the French police. We have received several reports," the prosecutor's office said, noting the seriousness of the situation on the border between Italy and France.
Mother's account
Migrant flows have changed in recent months and increasing numbers of families have been using the routes in the area, associations working in the mountains between Italy and France say.
A few days before the incident, the Afghan girl had been evicted from a squat in Oulx, in the upper Valle di Susa, where she had found shelter with the help of an international anarchist group while waiting to cross the border.
Between Thursday night and Friday, March 26 morning, on the paths of Monginevro with other Afghan families, she was pushed back by the French Gendarme.
"We were on the path when French police arrived. They surrounded us and yelled at us to stop. And I heard shots being fired...," the girl's mother told Red Cross workers who, after the pushback, took them to the Massi di Oulx shelter in Turin's upper Valle di Susa (Susa Valley).
The girl was stiff and could not talk or move and a neuropsychiatrist examined her and decided to hospitalize her as a precaution.
Reports were made to the Turin police station and initial inquiries and information exchanges were made. The French police deny that shots were fired to stop the migrants.
Girl hosted by a family
Between Saturday and Sunday, the Afghan girl spent the night with a family from the Valle di Susa along with her father, mother, and three younger siblings.
Their intention is to reach Germany, where some of their relatives live.
The 10-year-old girl spoke to investigators and told them what happened in the night between Thursday and Friday in the forests on the Italian border. The little girl did not mention at that time any shots being fired - as she had initially stated - but only armed men in uniform.
The girl sustained head injuries when she was seven from a bomb and it seems she was triggered by the scene at the border.
What exactly happened is not yet clear, but NGO Rainbow4Africa chief Paolo Narcisi, who has been working with refugees using the Val di Susa to get to France for years, said that "no human being should be subjected to treatment that harms their dignity."
A priest, Luigi Chiampo, known for rescuing migrants on the Alpine route added, "at the moment we are hosting many families. We are keeping a hold on the situation but the situation is very difficult."
'Illegal methods': The shadow zones employed by French border police on Italian soil
Migrants clash with authorities in Greece and Turkey
Fact check: Migrants along Bosnian-Croatian border face stand-off with Croatian authorities
Three investigated over 'illegal entry of migrants for $5,000'
Top stories / News
Connect with InfoMigrants on


text:
Q: Finding prime factors #include <iostream>
using namespace std;

void whosprime(long long x)
{
    bool imPrime = true;

    for(int i = 1; i <= x; i++)
    {
        for(int z = 2; z <= x; z++)
        {
            if((i != z) && (i%z == 0))
            {
                imPrime = false;
                break;
            }
        }

        if(imPrime && x%i == 0)
            cout << i << endl;

        imPrime = true;
    }
}

int main()
{
    long long r = 600851475143LL;
    whosprime(r);
}

I'm trying to find the prime factors of the number 600851475143 specified by Problem 3  on Project Euler (it asks for the highest prime factor, but I want to find all of them).  However, when I try to run this program I don't get any results.  Does it have to do with how long my program is taking for such a large number, or even with the number itself?
Also, what are some more efficient methods to solve this problem, and do you have any tips as to how can I steer towards these more elegant solutions as I'm working a problem out?
As always, thank you!

A: A C++ implementation using @user448810's pseudocode:
#include <iostream>
using namespace std;

void factors(long long n) {
    long long z = 2;
    while (z * z <= n) {
        if (n % z == 0) {
            cout << z << endl;
            n /= z;
        } else {
            z++;
        }
    }
    if (n > 1) {
        cout << n << endl;
    }
}

int main(int argc, char *argv[]) {
    long long r = atoll(argv[1]);
    factors(r);
}

// g++ factors.cpp -o factors ; factors 600851475143

Perl implementation with the same algorithm is below.
Runs ~10-15x slower (Perl 0.01 seconds for n=600851475143)
#!/usr/bin/perl
use warnings;
use strict;

sub factors {
    my $n = shift;
    my $z = 2;
    while ($z * $z <= $n) {
        if ( $n % $z ) {
            $z++;
        } else {
            print "$z\n";
            $n /= $z;
        }
    }
    if ( $n > 1 ) {
        print "$n\n"
    }
}

factors(shift);

# factors 600851475143

A: Your algorithm is wrong; you don't need i. Here's pseudocode for integer factorization by trial division:
define factors(n)

    z = 2

    while (z * z <= n)

        if (n % z == 0)
            output z
            n /= z

        else
            z++

    if n > 1
        output n

I'll leave it to you to translate to C++ with the appropriate integer datatypes.
Edit: Fixed comparison (thanks, Harold) and added discussion for Bob John:
The easiest way to understand this is by an example. Consider the factorization of n = 13195. Initially z = 2, but dividing 13195 by 2 leaves a remainder of 1, so the else clause sets z = 3 and we loop. Now n is not divisible by 3, or by 4, but when z = 5 the remainder when dividing 13195 by 5 is zero, so output 5 and divide 13195 by 5 so n = 2639 and z = 5 is unchanged. Now the new n = 2639 is not divisible by 5 or 6, but is divisible by 7, so output 7 and set n = 2639 / 7 = 377. Now we continue with z = 7, and that leaves a remainder, as does division by 8, and 9, and 10, and 11, and 12, but 377 / 13 = 29 with no remainder, so output 13 and set n = 29. At this point z = 13, and z * z = 169, which is larger than 29, so 29 is prime and is the final factor of 13195, so output 29. The complete factorization is 5 * 7 * 13 * 29 = 13195.
There are better algorithms for factoring integers using trial division, and even more powerful algorithms for factoring integers that use techniques other than trial division, but the algorithm shown above will get you started, and is sufficient for Project Euler #3. When you're ready for more, look here.

A: 600851475143 is outside of the range of an int
void whosprime(int x) //<-----fix heere ok?
{
    bool imPrime = true;

    for(int i = 1; i <= x; i++)
    {...
      ...

A: Edit: I'm wrong (see comments). I would have deleted, but the way in which I'm wrong has helped indicate what specifically in the program takes so long to produce output, so I'll leave it :-)
This program should immediately print 1 (I'm not going to enter a debate whether that's prime or not, it's just what your program does). So if you're seeing nothing then the problem isn't execution speed, there muse be some issue with the way you're running the program.

A: Try below code:
counter = sqrt(n)
i = 2;

while (i <= counter)
    if (n % i == 0)
        output i
    else
        i++

A: Here is my code that worked pretty well to find the largest prime factor of any number:
#include <iostream>
using namespace std;

// --> is_prime <--
// Determines if the integer accepted is prime or not
bool is_prime(int n){
    int i,count=0;
    if(n==1 || n==2)
      return true;
    if(n%2==0)
      return false;
    for(i=1;i<=n;i++){
    if(n%i==0)
        count++;
    }
    if(count==2)
      return true;
    else
      return false;
 }
 // --> nextPrime <--
 // Finds and returns the next prime number
 int nextPrime(int prime){
     bool a = false;
     while (a == false){
         prime++;
         if (is_prime(prime))
            a = true;
     }
  return prime;
 }
 // ----- M A I N ------
 int main(){

      int value = 13195;
      int prime = 2;
      bool done = false;

      while (done == false){
          if (value%prime == 0){
             value = value/prime;
             if (is_prime(value)){
                 done = true;
             }
          } else {
             prime = nextPrime(prime);
          }
      }
        cout << "Largest prime factor: " << value << endl;
 }

Keep in mind that if you want to find the largest prime factor of extremely large number, you have to use 'long' variable type instead of 'int' and tweak the algorithm to process faster.

A: short and clear vesion:
    int main()
    {
        int MAX = 13195;

        for (int i = 2; i <= MAX; i++)
        {
             while (MAX % i == 0)
             {
                  MAX /= i;
                  cout <<  i << ", " << flush;   // display only prime factors
             }
        return 0;
    }

A: This is one of the easiest and simple-to-understand solutions of your question.
It might not be efficient like other solutions provided above but yes for those who are the beginner like me.
int main() {

int num = 0;
cout <<"Enter number\n";
cin >> num;
int fac = 2;
while (num > 1) {
    if (num % fac == 0) {
        cout << fac<<endl;
        num=num / fac;
    }
    else fac++;
}
return 0;

}

A: # include <stdio.h>
# include <math.h>
void primeFactors(int n)
{
    while (n%2 == 0)
    {
        printf("%d ", 2);
        n = n/2;
    }
    for (int i = 3; i <= sqrt(n); i = i+2)
    {
        while (n%i == 0)
        {
            printf("%d ", i);
            n = n/i;
        }
    }
   if (n > 2)
        printf ("%d ", n);
}
int main()
{
    int n = 315;
    primeFactors(n);
    return 0;
}

A: Simple way :
#include<bits/stdc++.h>
using namespace std;
typedef long long int ll;

ll largeFactor(ll n)
{
        ll ma=0;
        for(ll i=2; i*i<=n; i++)
        {
            while(n%i == 0)
            {
                n=n/i;
                ma=i;
            }
        }
        ma = max(ma, n);
        return ma;
}

int main()
{
    ll n;
    cin>>n;
    cout<<largeFactor(n)<<endl;
    return 0;
}

Implementation using prime sieve ideone.

A: Since 600851475143 is out of scope for int as well as single long type wont work here hence here to solve we have to define our own type here with the help of typedef.
Now the range of ll is some what around 9,223,372,036,854,775,807.
typedef long long int LL

A: Try this code. Absolutely it's the best and the most efficient:
long long number;
bool isRepetitive;

for (int i = 2; i <= number; i++) {
    isRepetitive = false;
    while (number % i == 0) {
        if(!isRepetitive){
            cout << i << endl;
            isRepetitive = true;
        }
        number /= i;
    }
}

Enjoy! ☻


text:
\section{Background and Related Work}
\label{sec:bg}

Before describing our experimental study, we briefly introduce
recommendation systems and collaborative filtering techniques.

\subsection{Recommendation Systems}

Broadly speaking, any software system which actively suggests an
item to purchase, to subscribe, or to invest can be regarded as a
recommender system. In this broad sense, an advertisement also
can be seen as a recommendation. We mainly consider, however, a
narrower definition of  ``personalized''
recommendation system that base recommendations on user specific
information.

There are two main approaches to personalized recommendation systems:
content-based filtering and collaborative filtering. The former makes
explicit use of domain knowledge concerning users or items. The domain knowledge may correspond to user information such
as age, gender, occupation, or location \cite{Debnath2008}, or to item
information such as genre, producer, or length in the case of movie
recommendation.

The latter category of collaborative filtering (CF) does not use user
or item information with the exception of a partially observed rating
matrix. The rating matrix holds ratings of items (columns) by users (rows) and is typically binary, for example like vs. do
not like, or ordinal, for example, one to five stars in Netflix movie
recommendation. The rating matrix may also be gathered implicitly
based on user activity, for example a web search followed by click through may be interpreted as a positive value judgement for the chosen
hyperlink \cite{Hu2008}. In general, the rating matrix is extremely
sparse, since it is unlikely that each user experienced and provided
ratings for all items.

Hybrid collaborative and content-based filtering strategies combine
the two approaches above, using both the rating matrix, and user and
item information.
\cite{Pavlov2002,Ziegler2004,Pennock2000,Melville2001,Kim2004,Basu1998,Pennock2000,Su2007}.
Such systems typically obtain improved prediction accuracy over
content-based filtering systems and over collaborative filtering
systems.

In this paper we focus on a comparative study of collaborative
filtering algorithms. There are several reason for not including
content-based filtering. First, a serious comparison of
collaborative filtering systems is a challenging task in itself.
Second, experimental results of content-based filtering are
intimately tied to the domain and are not likely to transfer from
one domain to another. Collaborative filtering methods, on the other
hand, use only the rating matrix which is similar in nature across
different domains.

\subsection{Collaborative Filtering}

Collaborative filtering systems are usually categorized into two subgroups:
memory-based and model-based methods.

Memory-based methods simply memorize the rating matrix and issue
recommendations based on the relationship between the queried user
and item and the rest of the rating matrix. Model-based methods fit
a parameterized model to the given rating matrix and then issue
recommendations based on the fitted model.

The most popular memory-based CF methods are neighborhood-based
methods, which predict ratings by referring to users whose ratings
are similar to the queried user, or to items that are similar to the
queried item. This is motivated by the assumption that if two users
have similar ratings on some items they will have similar ratings on
the remaining items. Or alternatively if two items have similar
ratings by a portion of the users, the two items will have similar
ratings by the remaining users.

Specifically, user-based CF methods \cite{Breese1998} identify users
that are similar to the queried user, and estimate the desired
rating to be the average ratings of these similar users. Similarly,
item-based CF~\cite{Sarwar2001} identify items that are similar to
the queried item and estimate the desired rating to be the average
of the ratings of these similar items.  Neighborhood methods vary
considerably in how they compute the weighted average of ratings.
Specific examples of similarity measures that influence the
averaging weights are include Pearson correlation, Vector cosine,
and Mean-Squared-Difference (MSD). Neighborhood-based methods can be
extended with default votes, inverse user frequency, and case
amplification~\cite{Breese1998}. A recent neighborhood-based method
\cite{Sun2011} constructs a kernel density estimator for incomplete
partial rankings and predicts the ratings that minimize the
posterior loss.

Model-based methods, on the other hand, fit a parametric model to the
 training data that can later be used to predict unseen ratings and
 issue recommendations. Model-based methods include
cluster-based
CF~\cite{Ungar1998,Chee2001,Connor2001,Sarwar2002,Xue2005}, Bayesian
classifiers~\cite{Miyahara2000,Miyahara2002}, and regression-based
methods \cite{Vucetic2005}. The slope-one method \cite{Lemire2005}
fits a linear model to the rating matrix, achieving fast computation
and reasonable accuracy.

A recent class of successful CF models are based on low-rank matrix
factorization. The regularized SVD method \cite{Billsus1998}
factorizes the rating matrix into a product of two low rank matrices
(user-profile and item-profile) that are used to estimate the
missing entries.  An alternative method is Non-negative Matrix
Factorization (NMF)~\cite{Lee1999} that differs in that it constrain
the low rank matrices forming the factorization to have non-negative
entries. Recent variations are Probabilistic Matrix Factorization
(PMF)~\cite{Salakhutdinov2008a}, Bayesian
PMF~\cite{Salakhutdinov2008b}, Non-linear PMF~\cite{Lawrence2009},
Maximum Margin Matrix Factorization
(MMMF)~\cite{Srebro2005,Rennie2005,DeCoste2006}, and Nonlinear
Principal Component Analysis (NPCA)~\cite{Yu2009}.

\subsection{Evaluation Measures}

The most common CF evaluation measure for prediction accuracy are
the mean absolute error (MAE) and root of the mean square error
(RMSE):
\begin{align*}
MAE &= \frac{1}{n} \sum_{u, i} |p_{u, i} - r_{u, i}|\\
RMSE &= \sqrt{\frac{1}{n} \sum_{u, i} (p_{u, i} - r_{u, i})^2}
\end{align*}
where $p_{u, i}$ and $r_{u, i}$ are the predicted and observed rating
for user $u$ and item $i$, respectively. The sum above ranges over a
labeled set that is set aside for evaluation purposes (test
set). Other evaluation measures are precision, recall, and F1 measures
~\cite{Breese1998}.

Gunawardana and Shani~\cite{Gunawardana2009} argue that different
evaluation metrics lead different conclusion concerning the relative
performance of the CF algorithms. However, most CF research papers
motivate their algorithm by examining a single evaluation measure.
In this paper we consider the performance of different CF algorithms
as a function of the problem parameters, measured using several
different evaluation criteria.

\subsection{Related Work}

Several well-written surveys on recommendation systems are
available. Adomavicius and Tuzhilin~\cite{Adomavicius2005}
categorized CF algorithms available as of 2006 into content-based,
collaborative, and hybrid and summarized possible extensions. Su and
Khoshgoftaar~\cite{Su2009} concentrated more on CF methods,
including memory-based, model-based, and hybrid methods. This survey
contains most state-of-the-art algorithms available as of 2009,
including Netflix prize competitors. A recent textbook on
recommender systems introduces traditional techniques and explores
additional issues like privacy concerns~\cite{Jannach2011}.

There are a couple of experimental studies available. The first
study by Breese et al.~\cite{Breese1998} compared two popular
memory-based methods (Pearson correlation and vector similarity) and
two classical model-based methods (clustering and Bayesian network)
on three different dataset. A more recent experimental comparison of
CF algorithms~\cite{Huang2007} compares user-based CF, item-based
CF, SVD, and several other model-based methods, focusing on
e-commerce applications. It considers precision, recall, F1-measure
and rank score as evaluation measures, with comments about the
computational complexity issue. This however ignores some standard
evaluation measures such as MAE or RMSE.

\subsection{Netflix Prize and Dataset}

The Netflix competition \footnote{http://www.netflixprize.com} was
held between October 2006 and July 2009, when BellKor's Pragmatic
Chaos team won the million-dollar-prize. The goal of this
competition was improving the prediction accuracy (in terms of RMSE)
of the Netflix movie recommendation system by 10\%. The winning team
used a hybrid method that used temporal dynamics to account for
dates in which ratings were reported ~\cite{Koren2008,Bell2010}. The
second-placed place, The Ensemble~\cite{Sill2009}, achieved
comparable performance to the winning team, by linearly combining a
large number of models.

Although the Netflix competition has finished, the dataset used in
that competition is still used as a standard dataset for  evaluating CF methods. It has 480,046 users and 17,770
items with 95,947,878 ratings. This represents a sparsiy level of
1.12\% (total number of entries divided by observed entries). Older
and smaller standard datasets include MovieLens (6,040 users, 3,500
items with 1,000,000 ratings), EachMovie (72,916 users, 1,628 items
with 2,811,983 ratings), and BookCrossing (278,858 users, 271,379
items with 1,149,780 ratings).

\section{Discussion}
\label{sec:conclusion}

In addition to the conclusions stated in Section~\ref{sec:exp}, we have identified seven groups of CF methods, where CF methods in the same group share certain experimental properties:

\begin{itemize}
  \setlength{\itemsep}{0cm}
  \setlength{\parskip}{0cm}

  \item Baselines: Constant, User Average, Item Average
  \item Memory-based methods: User-based, Item-based (with and
  without default values)
  \item Matrix-Factorization I: Regularized SVD, PMF, BPMF, NLPMF
  \item Matrix-Factorization II: NMF
  \item Others (Individually): Slope-one, NPCA, Rank-based CF.
\end{itemize}

Table~\ref{tab:summary} displays for each of these groups the dependency, accuracy, computational cost, and pros and cons.

\begin{sidewaystable*}
\centering \footnotesize
\begin{tabular}{c|c|c|c|c|c|c|c|c} \hline \hline
\multicolumn{2}{c|}{Category} & Baselines & Memory-based &
\multicolumn{2}{|c|}{Matrix-factorization} &
\multicolumn{3}{|c}{Others} \tabularnewline \hline

\multicolumn{2}{c|}{Algorithms} & Constant, & User-based, & Reg-SVD, & NMF & Slope-one & NPCA & Rank-based \\
\multicolumn{2}{c|}{} & User Average, & Item-based, & PMF, BPMF, & & & \\
\multicolumn{2}{c|}{} & Item Average & w/default & NLPMF & & &
\tabularnewline \hline \hline

Dependency & Size & \textbf{Very low} & Low & High & Low & High &
\textbf{Very low} & Fair \tabularnewline \cline{2-9}

& Density & \textbf{Very low} & High & Very high & Low & Very high &
High & Fair \tabularnewline \hline

Accuracy & Dense & Very poor & Good & \textbf{Very good} & Good &
Good & Poor & Fair \tabularnewline \cline{2-9}

& Sparse & Poor & Very poor & Fair & \textbf{Very good} & Poor &
Poor & Fair \tabularnewline \hline

\multicolumn{2}{c|}{Asymmetric accuracy} & Poor & Fair &
\textbf{Very good} & Good & Good & \textbf{Very good} & \textbf{Very
good} \tabularnewline \hline

\multicolumn{2}{c|}{HLU/NDCG} & Very poor & Fair & \textbf{Very
good} & Fair & Good & Fair & Fair \tabularnewline \hline

\multicolumn{2}{c|}{Kendall's Tau/Spearman} & Very poor & Fair &
Good & Fair & Fair & Fair & Fair \tabularnewline \hline

Computation & Train & \textbf{No} & \textbf{No} & Slow & Fast &
\textbf{Very fast} & Slow & No \tabularnewline \cline{2-9}

& Test & \textbf{Very fast} & Very slow & Fair & Fair & Fast & Slow
& Very slow \tabularnewline \hline

\multicolumn{2}{c|}{Memory consumption} & \textbf{Low} & High & High
& High & \textbf{Low} & Very high & High \tabularnewline \hline

\multicolumn{2}{c|}{Adjustable parameters} & \textbf{No} & Few &
Many & Many & \textbf{No} & Few & Few \tabularnewline \hline

\multicolumn{2}{c|}{Overall Merits}
 & Computation
 & Do not need
 & Perform best
 & Perform best
 & Perform well
 & Perform well
 & Perform well
\tabularnewline

\multicolumn{2}{c|}{}
 & takes
 & to train.
 & with high-
 & with sparse
 & in spite of
 & when using
 & when using
\tabularnewline

\multicolumn{2}{c|}{}
 & little time.
 &
 & density data.
 & data. Train
 & short time.
 & asymmetric
 & asymmetric
\tabularnewline

\multicolumn{2}{c|}{}
 &
 &
 &
 & is fast.
 &
 & measures.
 & measures.
\tabularnewline \hline

\multicolumn{2}{c|}{Overall Demerits}
 & Accuracy is
 & Testing takes
 & Many parameters
 & Many
 & Perform poorly
 & Uses extremely
 & Computation
\tabularnewline

\multicolumn{2}{c|}{}
 & very poor.
 & very long time.
 & should be
 & parameters
 & without
 & large memory
 & takes too
\tabularnewline

\multicolumn{2}{c|}{}
 &
 & Uses lots
 & adjusted.
 & should be
 & large/dense
 & during
 & long time.
\tabularnewline

\multicolumn{2}{c|}{}
 &
 & of memory.
 & Computation
 & adjusted.
 & dataset.
 & training.
 &
\tabularnewline

\multicolumn{2}{c|}{}
 &
 &
 & takes long.
 &
 &
 &
 &
\tabularnewline \hline \hline

\end{tabular}
\caption{Summary of Pros and Cons for Recommendation Algorithms}
\label{tab:summary}
\end{sidewaystable*}

We repeat below some of the major conclusions. See
Section~\ref{sec:exp} for more details and additional conclusions.

\begin{itemize}
  \setlength{\itemsep}{0cm}
  \setlength{\parskip}{0cm}

\item Matrix-Factorization-based methods generally have the highest accuracy. Specifically,
regularized SVD, PMF and its variations perform best as far as MAE
and RMSE, except in very sparse situations, where NMF performs the
best. Matrix-factorization methods perform well also in terms of the
asymmetric cost and rank-based evaluation measures. NPCA and
rank-based CF work well in these cases as well. The Slope-one method
performs well and is computationally efficient. Memory-based
methods, however, do not have special merit other than simplicity.

\item All algorithms vary in their accuracy, based on the user count, item count, and density. The strength and nature of the dependency, however, varies from algorithm to algorithm and bivariate relationships change when different values are assigned to the third variable. In general cases, high dependence on the user count and item count is correlated with high dependency on density, which appeared to be the more influential
factor.

\item There is trade-off between better accuracy and other
factors such as low variance in accuracy, computational efficiency,
memory consumption, and a smaller number of adjustable parameters.
That is, the more accurate algorithms tend to depend highly on
dataset size and density, to have higher variance in accuracy, to be
less computationally efficient, and to have more adjustable
parameters. A careful examination of the experimental results can
help resolve this tradeoff in a manner that is specific to the
situation at hand. For example, when computational efficiency is
less important, Matrix-Factorization methods are the most
appropriate, and when computational efficiency is important,
slope-one could be a better choice.
\end{itemize}

This experimental study, accompanied by an open source software that allows reproducing our experiments, sheds light on how CF algorithms compare to each other, and on their dependency on the problem parameters. The conclusions described above should help practitioners, implementing recommendation systems, and researchers examining novel state-of-the-art CF methods.

\section{Introduction}
\label{sec:intro}

Collaborative filtering is a rapidly advancing research area. Classic
methods include neighborhood methods such as memory based or user
based collaborative filtering. More recent methods often revolve
around matrix factorization including singular value decomposition and
non-negative matrix factorization. New methods are continually
proposed, motivated by experiments or theory. However, despite the
considerable research momentum there is no consensus or clarity on which method works best.

One difficulty that is undoubtedly central to the lack of clarity is
that the performance of different methods differ substantially based
on the problem parameters. Specifically, factors such as number of
users, number of items, and sparsity level (ratio of observed to
total ratings) affect different collaborative filtering methods in
different ways. Some methods perform better in sparse setting while
others perform better in dense settings, and so on.

Existing experimental studies in collaborative filtering either do
not compare recent state-of-the-art methods, or do not investigate
variations with respect to the above mentioned problem parameters.
In this paper we do so, and concentrate on comparing both classic
and recent state-of-the-art methods. In our experiments we control
for the number of users, the number of items, sparsity level and
consider multiple evaluation measure, and computational cost.

Based on our comparative study we conclude the following.

\begin{enumerate}
\item Generally speaking, Matrix-Factorization-based methods perform
best in terms of prediction accuracy. Nevertheless, in some special
cases, several other methods have a distinct advantage over
Matrix-Factorization methods.

\item The prediction accuracy of the different algorithms depends on the number of users, the number of items, and density, where the nature and degree of dependence differs from algorithm to algorithm.

\item There exists a complex relationship between
prediction accuracy, its variance, computation time, and memory
consumption that is crucial for choosing the most appropriate recommendation system algorithm.
\end{enumerate}

The following sections describe in detail the design and implementation of the experimental study and the experimental results themselves.

\section{Experimental Study}
\label{sec:exp}

We describe below some details concerning our experimental study, and
then follow with a description of our major findings.

\subsection{Experimental Design}
\label{subsec:expdesign}

To conduct our experiments and to facilitate their reproducability we
implemented the PREA\footnote{http://prea.gatech.edu} toolkit, which
implements the 15 algorithms listed in Table~\ref{tab:algorithm}. The
toolkit is available for public usage and will be updated with
additional state-of-the-art algorithms proposed by the research
community.

There are three elementary baselines in Table~\ref{tab:algorithm}: a
constant function (identical prediction for all users and all
items), user average (constant prediction for each user-based on
their average ratings), and item average (constant prediction for
each item-based on their average ratings). The memory-based methods
listed in Table~\ref{tab:algorithm} are classical methods that
perform well and are often used in commercial settings. The methods
listed under the matrix factorization and others categories are more
recent state-of-the-art methods proposed in the research literature.

\begin{table}[!h]
\centering \footnotesize
\begin{tabular}{c|l} \hline
Category & Algorithms \tabularnewline \hline \hline

& Constant \\
Baseline & User Average \\
& Item Average \tabularnewline \hline

& User-based \cite{Su2009} \\
Memory & User-based w/ Default \cite{Breese1998} \\
-based& Item-based \cite{Sarwar2001} \\
& Item-based w/ Default \tabularnewline \hline

& Regularized SVD \cite{Paterek2007} \\
Matrix & NMF \cite{Lee2001} \\
Factorization & PMF \cite{Salakhutdinov2008a} \\
-based & Bayesian PMF \cite{Salakhutdinov2008b} \\
& Non-linear PMF \cite{Lawrence2009} \tabularnewline \hline

& Slope-One \cite{Lemire2005} \\
Others & NPCA \cite{Yu2009} \\
& Rank-based CF \cite{Sun2011} \tabularnewline \hline

\end{tabular}
\caption{List of Recommendation Algorithms used in Experiments}
\label{tab:algorithm}
\end{table}

In our experiments we used the Netflix dataset, a standard benchmark
in the CF literature that is larger and more recent than alternative
benchmarks. To facilitate measuring the dependency between
prediction accuracy and dataset size and density, we sorted the
rating matrix so that its rows and columns are listed in order of
descending density level. We then realized specific sparsity pattern
by selecting the top $k$ rows and $l$ columns and subsampling to
achieve the required sparsity.

Figure~\ref{fig:netflix_density} shows the density level of the
sorted rating matrix. For instance, the top right corner of the
sorted rating matrix containing the top 5,000 users and top 2,000
items has 52.6\% of density. In other words, there 47.4\% of the
ratings are missing. The density of the entire dataset is around
1\%. We subsample a prescribed level of density which will be used
for training as well as 20\% more for the purpose of testing. We
cross validate each experiment 10 times with different train-test
splits. The experiments were conducted on a dual Intel Xeon X5650
processor (6 Core, 12 Threads, 2.66GHz) with 96GB of main memory.

\begin{figure*}
\centering{}
\includegraphics[width=17.5cm]{netflix_density}
\caption{Rating density (cumulative in the left panel and
non-cumulative in the right panel) in Netflix rating matrix, sorted by
descending density of rows and columns. See text for more detail.}
\label{fig:netflix_density}
\end{figure*}

\subsection{Dependency on Data Size and Density}
\label{subsec:dependency}

We start by investigating the dependency of prediction accuracy on
the dataset size (number of users and number of items) and on the
rating density. Of particular interest, is the variability in that
dependency across different CF algorithms. This variability holds
the key to determining which CF algorithms should be used in a
specific situation. We start below by considering the univariate
dependency of prediction accuracy on each of these three quantities:
number of users, number of items, and density level. We then
conclude with an investigation of the multivariate dependency
between the prediction accuracy and these three variables.

\subsubsection{Dependency on User Count} \label{sec:userCount}

Figure~\ref{fig:var} (top row) graphs the dependency of mean
absolute error (MAE) on the number of users with each of the three
panels focusing on CF methods in a specific category (memory-based
and baselines, model-based, and other). The item count and density
were fixed at 2,000 and 3\%, respectively. The RMSE evaluation
measure shows very similar trend.

We omitted the simplest baseline of constant prediction rule, since its
performance is much worse than the others. The default voting variants
of the user-based and item-based CF methods did not produce noticable
changes (we graphed the variant which worked the best).

\begin{figure*}
    \centering{}
    \includegraphics[width=17.5cm]{var_user_mae}
    \includegraphics[width=17.5cm]{var_item_mae}
    \includegraphics[width=17.5cm]{var_density_mae}
    \caption{Prediction loss as a function of user count (top) item
      count (middle), density (bottom).}
    \label{fig:var}
\end{figure*}

To compare the way in which the different algorithms depend on the
user count quantity, we fitted a linear regression model to the curves
in Figure~\ref{fig:var} (top row).  The slope $m$ and intercept $b$
regression coefficients appear in Table~\ref{tab:dependency_reg}. The
intercept $b$ indicates the algorithm's expected MAE loss when the
number of users approach 0. The slope $m$ indicates the rate of decay
of the MAE.

Looking at Figure~\ref{fig:var} and Table~\ref{tab:dependency_reg}, we
make the following observations.

\begin{enumerate}
\item Matrix factorization methods in general show better performance
  when the number of users gets sufficiently large ($>3,000$).
\item Overall, the best performing algorithm is regularized SVD.
\item When the number of users is sufficiently small there is very
  little difference between matrix factorization methods and the
  simpler neighborhood-based methods.
\item Item average, item-based, regularized SVD, PMF, BPMF, and NLPMF
  tend to be the most sensitive to variation in user count.
\item Constant baseline, user average, user-based, NMF, NPCA, and
  rank-based are relatively insensitive to the number of users.
\item There is stark difference in sensitivity between the two popular
neighborhood-based methods: user-based CF and item-based CF.
\item User-based CF is extremely effective for low user count but has an
  almost constant dependency on the user count. Item-based CF
  performs considerably worse at first, but outperforms all other
  memory-based methods for larger user count.
\end{enumerate}

\begin{table*}[!htb]
\centering \footnotesize
\begin{tabular}{l|r|r|r|r|r|r} \hline
Algorithm & \multicolumn{2}{|c|}{User Count} & \multicolumn{2}{|c|}{Item Count} & \multicolumn{2}{|c}{Density} \\

\cline{2-7}

& $10^{-6}m$ & $b$ & $10^{-6}m$ & $b$ & $m$ & $b$\tabularnewline
\hline \hline

Constant                    &  -1.7636 & 0.9187 & -13.8097 & 0.9501 & -0.0691 & 0.9085\\
User Average                &  -2.5782 & 0.8048 &  -3.5818 & 0.8006 & -0.6010 & 0.8094\\
Item Average                &  -6.6879 & 0.8593 &  +1.1927 & 0.8155 & -0.2337 & 0.8241\\
User-based                  &  -3.9206 & 0.7798 & -18.1648 & 0.8067 & -4.7816 & 0.9269\\
User-based (Default values) &  -3.6376 & 0.7760 & -19.3139 & 0.8081 & -4.7081 & 0.9228\\
Item-based                  & -10.0739 & 0.8244 &  -3.2230 & 0.7656 & -4.7104 & 0.9255\\
Item-based (Default values) & -10.4424 & 0.8271 &  -3.6473 & 0.7670 & -4.9147 & 0.9332\\
Slope-one                   &  -5.6624 & 0.7586 &  -7.5467 & 0.7443 & -5.1465 & 0.9112\\
Regularized SVD             &  -7.6176 & 0.7526 & -14.1455 & 0.7407 & -2.2964 & 0.7814\\
NMF                         &  -4.4170 & 0.7594 &  -5.7830 & 0.7481 & -0.9792 & 0.7652\\
PMF                         &  -6.9000 & 0.7531 & -14.7345 & 0.7529 & -6.3705 & 0.9364\\
Bayesian PMF                & -11.0558 & 0.7895 & -23.7406 & 0.7824 & -4.9316 & 0.8905\\
Non-linear PMF              &  -8.8012 & 0.7664 & -14.7588 & 0.7532 & -2.8411 & 0.8135\\
NPCA                        &  -4.1497 & 0.7898 &  -7.2994 & 0.7910 & -3.5036 & 0.8850\\
Rank-based CF               &  -3.8024 & 0.7627 &  -7.3261 & 0.7715 & -2.4686 & 0.8246

\tabularnewline \hline

\end{tabular}
\caption{Regression coefficients $y = mx + b$ for the curves in
  Figure~\ref{fig:var}. The variable $x$ represents user count, item
  count, or density, and the variable $y$ represents the MAE
  prediction loss.}
\label{tab:dependency_reg}
\end{table*}

\subsubsection{Dependency on Item Count} \label{sec:itemCount}

In analogy with Section~\ref{sec:userCount} we investigate here the
dependency of the prediction loss on the number of items, fixing the
user count at and density at 5,000 and  3\%, respectively.  Figure~\ref{fig:var} (middle row) shows the MAE as a function of the
number of items for three different categories of CF
algorithms. Table~\ref{tab:dependency_reg} shows the regression
coefficients (see description in Section~\ref{sec:userCount}).

Looking at Figure~\ref{fig:var} and Table~\ref{tab:dependency_reg}, we
make the following observations that are largely in agreement with the
observations in Section~\ref{sec:userCount}.

\begin{enumerate}
\item Matrix factorization methods in general show better performance
  when the number of items gets sufficiently large ($>1,000$).
\item Overall, the best performing algorithm is regularized SVD.
\item When the number of users is sufficiently small there is very
  little difference between matrix factorization methods and the
  simpler neighborhood-based methods.
\item User-based, regularized SVD, PMF, BPMF, and NLPMF tend to be the most sensitive to variation in item count.
\item Item average, item-based, and NMF tend to be less
  sensitive to variation in item count.
\item There is stark difference in sensitivity between the two popular
neighborhood-based methods: user-based CF and item-based CF.
\item Item-based CF is extremely effective for low item count but has an
  almost constant dependency on the item count. User-based CF
  performs considerably worse at first, but outperforms all other
  memory-based methods significantly for larger user count.
\item Combined with the observations in Section~\ref{sec:userCount},
  we conclude that slope-one, NMF, and NPCA are insensitive to
  variations in both user and item count.  PMF and BPMF are relatively
  sensitive to variations in both user and item count.
\end{enumerate}

\subsubsection{Dependency on Density} \label{sec:density}

Figure~\ref{fig:var} (bottom row) graphs the dependency of the MAE
loss on the rating density, fixing user and item count at 5,000 and
2,000. As in Section~\ref{sec:userCount},
Table~\ref{tab:dependency_reg} displays regression coefficients
corresponding to performance at near 0 density and sensitivity of
the MAE function to the density level.

Looking Figure~\ref{fig:var} (bottom row) and
Table~\ref{tab:dependency_reg} we make the following observations.

\begin{enumerate}
\item The simple baselines (user average and item average) work
  remarkably well for low density.
\item The best performing algorithms seem to be regularized SVD.
\item User-based CF and item-based CF show a remarkable similar
  dependency on density level. This is in stark contrast to their
  different dependencies on the user and item count.
\item As the density level increases the differences in prediction
  accuracy of the different algorithm shrink.
\item User-based, item-based, slope-one, PMF, and BPMF are largely
dependent on density.
\item The three baselines and NMF are relatively independent from density.
\item The performance of slope-one and PMF degrade significantly at
  low densities, performing worse than the weakest
  baselines. Nevertheless both algorithms feature outstanding
  performance at high densities.
\end{enumerate}

\subsubsection{Mutlivariate Dependencies between Prediction Loss, User Count, Item Count, and Density}

The univariate dependencies examined previously show important
trends but are limited since they examine variability of one
quantity while fixing the other quantities to arbitrary values. We
now turn to examine the dependency between prediction loss and the
following variables: user count, item count, and density. We do so
by graphing the MAE as a function of user count, item count, and
density (Figure~\ref{fig:mae1}--\ref{fig:mae2}) and by fitting
multivariate regression models to the dependency of MAE on user
count, item count, and density
(Table~\ref{tab:dependency_combined}).

Figure~\ref{fig:mae1}--\ref{fig:mae2} shows the equal height
contours of the MAE as a function of user count ($x$ axis) and item
count ($y$ axis) for multiple density levels (horizontal panels) and
for multiple CF algorithms (vertical panels). Note that all contour
plots share the same $x$ and $y$ axes scales and so are directly
comparable. Intervals between different contour lines represent a
difference of 0.01 in MAE and so more contour lines represent higher
dependency on the $x$ and $y$ axis. Analogous RMSE graphs show
similar trend to these MAE graphs.

\begin{figure}
    \centering{}
    \includegraphics[width=17cm]{mae1}
    \caption{MAE Contours for simple method (Lower values mean better performance.)}
    \label{fig:mae1}
\end{figure}

\begin{figure}
    \centering{}
    \includegraphics[width=17cm]{mae2}
    \caption{MAE Contours for advanced method (Lower values mean better performance.)}
    \label{fig:mae2}
\end{figure}

Table~\ref{tab:dependency_combined} displays the regression
coefficients $m_u, m_i, m_d$ corresponding to the linear model
\begin{equation}
y = m_u x_u + m_i x_i + m_d x_d + b \label{eq:regression_combined}
\end{equation}
where $x_u$, $x_i$, and $x_d$ indicate user count, item count, and
density, $b$ is the constant term, and $y$ is the MAE.

Based on Figure~\ref{fig:mae1}--\ref{fig:mae2} and
Table~\ref{tab:dependency_combined} we make the following
observations.

\begin{enumerate}
\item The univariate relationships discovered in the previous section for fixed values of the remaining two variables do not necessarily hold in general. For example, the conclusion that PMF is relatively sensitive to user count and item count at 1\% sparsity level and 3\% sparsity level is not longer valid for 5\% density levels. It is thus important to conduct a multivariate, rather than univariate, analysis of the dependency of the prediction loss on the problem parameters.
\item The shape of the MAE contour curves vary from algorithm to algorithm. For example, the contour curves of constant, user average, user-based, and user-based with default values are horizontal, implying that these algorithms depend largely on the number of items, regardless of user count. On the other hand, item average, item-based, and item-based with default values show vertical contour lines, showing a high dependency on the user count.
\item Higher dependency on user count and item count is correlated with high dependency on density.
\item The last column of Table~\ref{tab:dependency_combined} summarizes dependency trends based on the absolute value of the regression coefficients and their rank. Generally speaking, baselines are relatively insensitive, memory-based methods are dependent on one variable (opposite to their names), and matrix factorization methods are highly dependent on both dataset size and density.
\end{enumerate}

\begin{table*}[!htb]
\centering \scriptsize
\begin{tabular}{l|r|r|r|r|l} \hline

Algorithm & $m_u$ & $m_i$ & $m_d$ & $b$ & Summary\tabularnewline
\hline \hline

Constant                & (15) -0.0115 &  (8) -0.0577 & (15) -0.0002 & 0.9600 & Weekly dependent on all variables.\\
User Average            & (13) -0.0185 & (14) -0.0164 & (13) -0.0188 & 0.8265 & Weekly dependent on all variables.\\
Item Average            &  (8) -0.0488 & (15) +0.0003 & (14) -0.0078 & 0.8530 & Weekly dependent on all variables.\\
User-based              & (11) -0.0282 &  (3) -0.0704 &  (1) -0.2310 & 0.9953 & Weekly dependent on user count.\\
User-based (w/Default)  & (12) -0.0260 &  (7) -0.0598 &  (5) -0.2185 & 0.9745 & Weekly dependent on user count.\\
Item-based              &  (3) -0.0630 & (12) -0.0172 &  (4) -0.2201 & 0.9688 & Weekly dependent on item count.\\
Item-based (w/Default)  &  (2) -0.0632 & (13) -0.0167 &  (2) -0.2286 & 0.9751 & Weekly dependent on item count.\\
Slope-one               &  (1) -0.0746 &  (4) -0.0702 &  (8) -0.1421 & 0.9291 & Strongly dependent on all variables.\\
Regularized SVD         &  (7) -0.0513 &  (9) -0.0507 & (11) -0.0910 & 0.8371 & Strongly dependent on dataset size.\\
NMF                     &  (9) -0.0317 & (10) -0.0283 & (12) -0.0341 & 0.7971 & Weekly dependent on all variables.\\
PMF                     &  (5) -0.0620 &  (2) -0.1113 &  (3) -0.2269 & 0.9980 & Strongly dependent on all variables.\\
Bayesian PMF (BPMF)     &  (4) -0.0628 &  (1) -0.1126 &  (6) -0.1999 & 0.9817 & Strongly dependent on all variables.\\
Non-linear PMF (NLPMF)  &  (6) -0.0611 &  (6) -0.0599 &  (9) -0.1165 & 0.8786 & Strongly dependent on dataset size.\\
NPCA                    & (14) -0.0184 & (11) -0.0213 &  (7) -0.1577 & 0.9103 & Weekly dependent on dataset size.\\
Rank-based CF           & (10) -0.0295 &  (5) -0.0687 & (10) -0.1065 & 0.8302 & Strongly dependent on item count.

\tabularnewline \hline

\end{tabular}
\caption{Regression coefficients for the model $y = m_u z_u + m_i z_i + m_d z_d + b$ \eqref{eq:regression_combined}) where $y$
is MAE, $z_u$, $z_i$, and $z_d$ are inputs from
$x_u$, $x_i$, and $x_d$, normalized to achieve similar scales:  $z_u = x_u / 10,000$, $z_i = x_i / 3,000$, and $z_d = x_d /
0.05$. The rank on each variable is indicated in parenthesis with rank
1 showing highest dependency.)} \label{tab:dependency_combined}
\end{table*}

\subsection{Accuracy Comparison}
\label{subsec:accuracy}

Figure~\ref{fig:best_unlimited} shows the best performing algorithm
(in terms of MAE) as a function of user count, item count,
and density. We make the following conclusions.

\begin{enumerate}
\item The identity of the best performing algorithm varies is non-linearly dependent on user count, item count, and density.
\item NMF is dominant low density cases while BPMF works well for high density cases (especially for high item and user count).
\item Regularized SVD and PMF perform well for density levels 2\%-4\%.

\end{enumerate}

Analogous RMSE graphs show similar trends with regularized SVD outperforming other algorithms in most regions.

\begin{figure*}
    \centering{}
    \includegraphics[width=17.5cm]{best_unlimited}
    \caption{Best-performing algorithms in MAE for given user count, item count, and density.}
    \label{fig:best_unlimited}
\end{figure*}

\subsection{Asymmetric and Rank-based Metrics}
\label{subsec:metrics}

We consider here the effect of replacing the MAE or RMSE with other
loss functions, specifically with asymmetric loss and with
rank-based loss.

Asymmetric loss is motivated with the fact that recommending an
undesirable item is worse than avoiding recommending a desirable
item. In other words, the loss function $L(a,b)$, measuring the
effect of predicting rating $b$ when the true rating is $a$ is an
asymmetric function. Specifically, we consider the loss function
$L(a,b)$ defined by the following matrix (rows and columns express
number of stars on a 1-5 scale)
\begin{equation}
L(\cdot,\cdot)=\begin{pmatrix}
0 & 0 & 0 & 7.5 & 10 \\
0 & 0 & 0 & 4 & 6 \\
0 & 0 & 0 & 1.5 & 3 \\
3 & 2 & 1 & 0 & 0 \\
4 & 3 & 2 & 0 & 0
\end{pmatrix}.
\label{eq:asym}
\end{equation}
This loss function represents two beliefs: 1) Difference among items
to be recommended is not important. Assuming that we issue
recommendations with rating 4 or 5, no loss is given between the
two. In the same way, we do not penalize error among items which
will not be recommended. 2) We give severer penalty for recommending
bad items than for missing potentially preferable items. For the
latter case, the loss is the exact difference between the prediction
and ground truth. For the former case, however, we give higher
penalty. For example, penalty is 10 for predicting worst item with
true score 1 as score 5, higher than 4 for the opposite way of
prediction. In many practical cases involving recommendation
systems, asymmetric loss functions provide a more realistic loss
function than symmetric loss functions such as MAE or RMSE.

Rank-based loss function are based on evaluating a ranked list of
recommended items, presented to a user. The evaluation of the list
gives higher importance to good recommendations at the top of the
list, than at the bottom of the list. One specific formula, called
half life utility (HLU)~\cite{Breese1998,Huang2007} assumes an
exponential decay in the list position. Formally, the utility
function associated with a user $u$
\begin{equation}
R_u = \sum_{i=1}^{N} \frac{max(r_{u,i} - d,
0)}{2^{(i-1)/(\alpha-1)}} \label{eq:rank}
\end{equation}
where $N$ is the number of recommended items (length of the list),
$r_{u,i}$ is the rating of user $u$ for item $i$ in the list, and
$d$ and $\alpha$ are constants, set to $d = 3$, and $\alpha = 5$ (we
assume $N=10$). The final utility function is $R_u$ divided by the
maximum possible utility for the user, average over all test users
\cite{Breese1998}. Alternative rank-based evaluations are based on
NDCG~\cite{Jarvelin2002}, and Kendall's Tau, and Spearman's Rank
Correlation Coefficient~\cite{Marden1996}.

\subsubsection{Asymmetric Loss}

Figure~\ref{fig:asym1}--\ref{fig:asym2} shows equal level contour
plots of the asymmetric loss function \eqref{eq:asym}, as a function
of user count, item count, and density level. We make the following
observations.

\begin{enumerate}
\item The shape and density pattern of the contour lines differ from the shape of the contour lines in the case of the MAE.
\item In general, regularized SVD outperforms all other algorithms.
Other matrix factorization methods (PMF, BPMF, and NLPMF) perform
relatively well for dense data. With sparse data, NMF performs well.
\end{enumerate}

\begin{figure}
    \centering{}
    \includegraphics[width=17cm]{asym1}
    \caption{Asymmetric Loss Contours for simple methods (Lower values mean better performance.)}
    \label{fig:asym1}
\end{figure}

\begin{figure}
    \centering{}
    \includegraphics[width=17cm]{asym2}
    \caption{Asymmetric Loss Contours for simple methods (Lower values mean better performance.)}
    \label{fig:asym2}
\end{figure}

\subsubsection{Rank-based Evaluation Measures}

Figure~\ref{fig:hlu1}--\ref{fig:hlu2} show equal level contours of
the HLU function \eqref{eq:rank}. Figure~\ref{fig:best_hlu} shows
the best performing algorithm for different user count, item count,
and density. We make the following observations.

\begin{enumerate}
\item The contour lines are generally horizontal, indicating that performance under HLU depend largely on the number of items and is less affected by the number of users.
\item The HLU score is highly sensitive to the dataset density.
\item  Regularized SVD outperforms other methods (see Figure~\ref{fig:best_hlu}) in most settings. The simple baseline item average is best for small and sparse datasets. A similar comment can be made regarding NPCA. NMF and slope-one perform well for sparse data, though they lag somewhat the previous mentioned algorithms.
\end{enumerate}

Other rank-based loss functions based on NDCG, Kendall's Tau, and
Spearman show similar trends.

\begin{figure}
    \centering{}
    \includegraphics[width=17cm]{hlu1}
    \caption{Half-Life Utility Contours for simple method (Higher values mean better performance.)}
    \label{fig:hlu1}
\end{figure}

\begin{figure}
    \centering{}
    \includegraphics[width=17cm]{hlu2}
    \caption{Half-Life Utility Contours for advanced method (Higher values mean better performance.)}
    \label{fig:hlu2}
\end{figure}

\begin{figure*}
    \centering{}
    \includegraphics[width=17.5cm]{best_hlu}
    \caption{Best-performing algorithms in HLU for given user count, item count, and density.}
    \label{fig:best_hlu}
\end{figure*}

\subsection{Computational Considerations}
\label{subsec:computation}

\begin{figure*}
    \centering{}
    \includegraphics[width=17.5cm]{time_bw}
    \caption{Computation time (Train time + Test time) for each algorithm.
    Legend on the right indicates relation between color scheme and computation time.
    Time constraints (5 minutes and 1 minutes) used in this article
    are marked as well. User count increases from left to right, and item count increases
    from bottom to top in each cell. (Same way to Figure~\ref{fig:mae1}--\ref{fig:mae2}.)}
    \label{fig:time}
\end{figure*}

As Figure~\ref{fig:time} shows, the computation time varies significantly between different algorithms. It is therefore important to consider computational issues when deciding on the appropriate CF algorithm. We consider three distinct scenarios, listed below.\\

\begin{itemize}
\setlength{\itemsep}{0cm}
\setlength{\parskip}{0cm}

\item \textbf{Unlimited Time Resources:} We assume in this case that we can afford arbitrarily long computation time. This scenario is realistic in some cases involving static training set, making offline computation feasible.

\item \textbf{Constrained Time Resource:} We assume in this cases
some mild constraints on the computation time. This scenario is
realistic in cases  where the training set is periodically updated,
necessitating periodic re-training with updated data. We assume here
that the training phase should tale within an hour or so. Since
practical datasets like Netflix full set are much bigger than the
subsampled one in our experiments, we use much shorter time limit: 5
minutes and 1 minute.

\item \textbf{Real-time Applications:} We assume in this case that severe constraints on the computation time. This scenario is realistic in cases where the training set changes continuously. We assume here that the training phase should not exceed several seconds.\\
\end{itemize}

Figure~\ref{fig:best_unlimited} and~\ref{fig:time} show the best
performing CF algorithm (in terms of MAE) in several different time
constraint cases, as a function of the user count, item count and
density.

\begin{figure*}
    \centering{}
    \includegraphics[width=17.5cm]{best_constrained}
    \caption{Best-performing algorithms with varied constraints.}
    \label{fig:best_constrained}
\end{figure*}

We make the following observations.

\begin{enumerate}

\item When there are no computation constraints, the conclusions from the previous sections apply. Specifically, NMF performs best for sparse dataset, BPMF performs best for dense dataset, and regularized
SVD and PMF perform the best otherwise (PMF works well with smaller user count while Regularized SVD works well smaller item counts).

\item When the time constraint is 5 minutes, Regularized SVD, NLPMF,
NPCA, and Rank-based CF (the ones colored darkly in
Figure~\ref{fig:time}) are not considered. In this setting, NMF
works best for sparse data, BPMF works best for dense and large data
and PMF works best otherwise.

\item When the time constraint is 1 minutes, PMF and BPMF are additionally excluded from
consideration. Slope-one works best in most cases, except for the
sparsest data where NMF works best.

\item In cases requiring real-time computation, the user average is the best algorithm, except for a small region where item average is preferred.
\end{enumerate}

redpajama_set_name: RedPajamaArXiv


text:
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