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Two speed Britain
Is the "north-south" divide intensifying and are local scale inequalities increasing?
Key Stage Four,
Key Stage Five,
Case study,
Changing places,
Urbanisation, migration and society
The global economic downturn and Britain's growing national debt have led the UK government to introduce a series of "austerity measures". They consist of dramatic cutbacks in public spending alongside higher taxes. How is the economic geography of Britain changing in response to these economic challenges?
This article examines evidence for an intensifying "north-south" divide and, at a local scale, emerging inequalities between neighbouring towns, both in the north and south. With unemployment now approaching 2.7 million, a new geography of uneven spaces is being created. What can the government do to help? The article concludes by analysing current growth pole policies, including the new generation of Enterprise Zones and the Regional Growth Fund.
In the Members' Area:
Britain's new economic challenge
The new "two-speed" economic geography of the UK
Meeting the challenge: possible solutions and futures
In common with people living in the USA, Japan and many Eurozone countries, most UK citizens are suffering increased economic hardship. Young people especially are finding it hard to get work; around one quarter of 18-24 year olds are now unemployed. What are the reasons for this current state of affairs?
Root causes of the financial difficulties currently experienced by many people and places include:
Economic recession During 2009, negative economic growth was recorded for two consecutive quarters in the UK, following a collapse of the banking sector in 2008. The root cause for this was the global credit crunch - a massive worldwide "bust" that followed boom years when banks engaged in high-risk lending (see Geography in the News article 'Credit crunch geography'). Economists now view the current period of reduced national output (beginning in 2008) as the longest economic downturn on record (Financial Times, 02 November 2011). Simply put, less new money is being created.
National debt In August 2011, the size of the UK's national debt (a long-term product of government spending exceeding tax revenues over many years, resulting in borrowing) reached £945 billion. This figure is equivalent to 61% of GDP, according to the Office for National Statistics (ONS). It has exploded in size (from around half that amount) since 2007, partly on account of enormous government rescue packages mounted for major British banks (Lloyds and Royal Bank of Scotland). To avoid the problem getting worse, the government has cut public spending.
To achieve its stated objective of curbing public expenditure, a series of "austerity measures" have been introduced by politicians. These include:
Tax rises including increased income tax (people earning over £150,000 now pay 50% on part of their annual earnings), VAT (now fixed at 20%), national insurance payments and alcohol duty.
Cuts and freezes in benefits including housing allowances, child benefit payments and maternity grants (Financial Times, 22 March 2011).
Public sector cutbacks including pay freezes, redundancies (with 400,000 jobs set to go over five years) and some pension payment changes.
Reduced arts and culture grants including a loss of funding for more than 200 arts bodies, following a 30% cut in the budget for Arts Council England.
As one might expect, the geographical impacts of the economic downturn (and the resulting austerity measures) are uneven, for several reasons:
People's wealth and incomes are not evenly distributed across the UK. Notably, there is a higher proportion of higher-income earners in the South-east, giving the region as a whole a greater ability to cope with rising prices and to also absorb tax rises.
Some towns have very high levels of workers in the public sector and have seen unemployment rise rapidly, along with the negative multiplier effect it brings (such as less spending in local shops). For instance, Sefton (a part of the Liverpool suburbs, including Crosby, Formby and Maghull) saw one of the largest rises in public sector employment of any area in the late 1990s. Half of all Sefton's growth came from expansion of government employment in the region, including police and health services administration.
In some places, arts projects, which have played a key role over the last 15 years in many regeneration and rebranding projects for peripheral regions (e.g. Sage Gateshead and the Salford Lowry), are now being scaled down or even scrapped. The virtuous effect of culture on urban regeneration was a key strategy of the Blair government; but it will be hard to fund flagship projects again on the scale seen in the 1990s and noughties (Financial Times, 30 March 2011).
The cutbacks have also taken place at a time when living costs are rising globally, notably fuel and food prices, due to a variety of factors (see 'Four corners of the food crisis'). Prime Minister David Cameron has acknowledged that most people will have to make sacrifices and that citizens may need to help one another as part of a "big society" initiative.
The new two-speed economic geography of the UK
During past periods of economic difficulty, such as the early 1980s, a strong "north-south divide" developed in the UK, measured in terms of regional performance across a range of economic and social indicators. Evidence for the same pattern - consisting of a more prosperous south and a struggling north - is again becoming highly visible. However, current regional trends are, on closer inspection, more complex than the simple idea of a north-south divide suggests. In both northern and southern regions, stories of "successful settlements" and "terminal towns" can be told.
The new north-south divide?
The current economic downturn has increased regional disparities. Although its impact has, so far, been less uneven than the recessions of 1980-83 and 1990-92, the UK has nonetheless become more unbalanced since 2005, according to the Office for National Statistics (Financial Times, 21 July 2011).
More than half of those people earning £150,000 - the threshold for the new 50p income tax rate - live in London and the South-east (Financial Times, 12 February 2011).
Local authorities with the greatest increases in unemployment - Hull, Wolverhampton, and Blaenau Gwent - all lie outside of southern regions (Financial Times, 21 July 2011).
Counties in the north have the highest percentage of boarded-up shops. Of 17 English counties with above-average rates, only two lie in southern England. The worst figure is for Lancashire in the North-west, at 18.7%.
Private sector growth looks less promising in northern and central England: for every extra private sector job generated there between 1998 and 2008, 10 were created in London and southern England, according to research by the Centre for Cities think-tank (Financial Times, 22 February 2011). In recent years, the north has been far more reliant in general on public sector job creation. However, this type of work is now becoming less available. The Work Foundation, another think-tank, warns that northern England could face a flight of young graduates to the South-east.
Source: University of Cambridge Local Examinations Syndicate, 1993
Uneven spaces
The geographical pattern is actually more complex than a simple "north-south divide". For instance, London's Brixton area has a very high number of shop vacancies and in neighbouring Camberwell there are 42 jobless people per job vacancy (this is seven times higher than the national average). Also, riots and looting took place recently in many poorer parts of London, including out-lying Hackney and Croydon. Parts of inner London share characteristics of decline with large parts of northern England and Wales.
Elsewhere in the country, strikingly divergent conditions can be found in settlements sited mere miles apart, such as Whitstable and Margate in Eastern England. The former has a high street vacancy rate of only 2%, well below the national average. A historic town, it attracts tourists and is home to many high-income retired people ("affluent greys") who provide one quarter of all spending. Some affluent Londoners have a second home in Whitstable. In contrast, Margate has a retail vacancy rate of almost 40%. It is in the bottom 1% of England's most deprived areas and has an unemployment rate three times higher than its neighbour, Whitstable.
This stark polarisation of fortunes amongst unequal neighbours is an emerging feature of the new economic geography of the UK. Possible factors that may help explain why close neighbours sometimes exhibit contrasting levels of resilience to the economic downturn include:
Reputation Some tourism "honeypots" such as Brighton, Harrogate and Stratford-upon-Avon remain popular with tourists, which helps to maintain local incomes.
Demography Selective in-migration of affluent greys or commuters has often focused on certain key settlements, such as Yorkshire's Hebden Bridge.
Public sector activity Settlements with a high proportion of public sector workers, such as Sunderland, Birkenhead and Swansea are suffering greater job losses than their neighbours (Financial Times, 24 January 2011).
Private sector activity The most buoyant cities, like Milton Keynes, Reading, Aberdeen and Bristol, continue to enjoy plenty of private-sector employment and have lower vulnerability to public sector job losses. In some cases, localised clusters of high-tech manufacturing have been the secret for success.
Accessibility Whether they are located in the north or south of the UK, highly accessible places, sited close to motorways, are continuing to receive inward investment from foreign companies (see our Geography in the News article on cars).
Size One recent study suggested that new manufacturing growth has been concentrated in medium-sized towns such as Aberdeen, Lancaster, Huddersfield and Telford; whilst larger, more traditional centres such as Birmingham and Sheffield have so far lagged behind (Financial Times, 24 January 2011).
The current state of the high street offers an important insight into any town's level of resilience. Some places have been hit badly by the partial closure or disappearance of retail chains such as Woolworths, Focus DIY, TJ Hughes, Habitat, HMV, Dixons and Oddbins. One quarter of all main high streets in the UK are in serious decline, with vacant premises left unfilled. In more prosperous towns, though, retail units do not stay empty for long; new bars or cafes move in.
Places with the worst-performing high streets were recently dubbed "terminal towns" by the retail analyst Colliers International (Financial Times, 17 June 2011). These settlements are struggling to find either public or private finance to spur future growth and are generally lacking in resilience. Colliers say 83 UK towns have reached a "tipping point" that has sent them into a downward spiral. With sales in shops falling at the fastest rate for 20 years, conditions are unlikely to improve soon.
Liverpool: "terminal town" or "supercity"?
Predictions for Liverpool are mixed. Some reports are extremely pessimistic. For instance, the inner city district of Bootle, with its struggling high street, was recently classed as a "terminal" place by the retail analyst Colliers International. Another district, Liverpool West Derby, has 33 applications per job vacancy. The city as a whole has been characterised as "vulnerable" by Centre for Cities, with 2.3% of the workforce potentially laid-off by public sector cutbacks.
A survey by HBSC Commercial Banking, however, predicts that Liverpool will emerge, phoenix-like, from the current economic downturn as a "supercity". The city is apparently poised to become a national and global hub for several clusters of important growth industries, including cultural and branding activities, as well as the renewable energy sector.
Wind farm plans Merseyside is on track to become a green energy (wind) hub. £15 billion will be invested in projects over the next ten years. Located on the North-west's coastline, with excellent land and quay facilities, Liverpool is an ideal location for offshore wind turbines. Construction and servicing of wind turbines will be sited Cammell Laird, which used to be a world-famous ship-building yard. The renewable energy industry should offer high job potential to the local community (Financial Times, 14 November 2010).
Enterprise zone and trade centre The new Liverpool Enterprise Zone covers an area where Peel Holdings, the property developer, has gained permission to build a new £6 billion multi-purpose development that includes a trade centre for Chinese companies. Liverpool City Council recently took a delegation to the Shanghai Expo; clearly, the mission was a success. This development helps make Liverpool an even more globally-connected place.
Liverpool One funding Liverpool One, the large flagship shopping complex in central Liverpool, managed to secure one of the year's largest loan packages during 2010, despite the fact that commercial loans have in general become harder to raise. Liverpool One has performed well since opening two years ago, in spite of the economic climate. The new loan comes from a group of banks including the Bank of Scotland, suggesting that the UK business community still views the Liverpool brand as a strong and successful asset.
Can development such as these provide a sustainable future for the city and its residents, including those in Bootle and Liverpool West Derby? Or will the city become an unequal space, with winners and losers?
To "bounce back" from the economic downturn, places need to find "pathways to resilience". The role of government can be to provide financial assistance to help with this. However, the UK's debt is now so great that the government can only afford to make a limited amount of public funds available, following its own austerity measures. What are the main regional aid priorities?
A new crop of "top-down" strategies have been targeted at a limited number of "hopeful" industries and business districts spread across the UK, especially those parts of northern and central England where public sector job losses are highest. Investment is being channelled in two ways: through the Regional Growth Fund (RGF) and a new generation of Enterprise Zones (EZs).
Regional Growth Fund
The Regional Growth Fund (RGF) is a £2.4 billion fund operating in three rounds over 3 years from 2011 to 2014. The objective is to stimulate private sector investment by providing support for specific projects and businesses that offer significant potential for long term economic growth (and the creation of additional sustainable private sector jobs). High-tech manufacturing firms in particular have been encouraged to apply for support. Massively over-subscribed, this scheme replaces the scrapped nine Regional Development Agencies (RDAs) and represents a policy shift towards greater "top-down" governance. Critics say around one-third less money is available under the new RGF than would have been the case if the RDAs had not been axed.
The first £1.3 billion was allocated in two rounds during 2011 (the 3rd round is expected to be announced in 2012):
Competition in the first round of funding, when £450 million was made available, was so high that just one in ten applications were successful. Nearly half of the 50 bids approved fell in the North-east and North-west. Successful recipients included the Haribo sweet factory near Wakefield, Bentley Motors in Crewe, Nissan Motors (North-east), Jaguar Land Rover (West Midlands) and the Yorkshire and Humber Carbon Trust. "The basic criteria we adopted was alleviating concentrated deprivation and the effect of public sector cuts. The East and South-east is clearly less of a priority than the North-east and that is reflected in how we judged the bids," said Vince Cable, business secretary (Financial Times, 12 April 2011). In contrast, London and the South-east received no money during the first round.
The second round of funding saw £950 million released across the entire UK. Of this, southern regions received just under £200 million (successful applicants included the Cornwall and Isles of Scilly local enterprise partnership and Portsmouth Naval Base Property Trust). In contrast, a further £463 million of funding was given to northern regions (winners included Newcastle University and Yorkshire's Sirius Minerals). In central England, grants were given to JCB (East Midlands) and Zytek Automotive (West Midlands).
Alongside this, the government has also put an additional £150 million into a "business angel" scheme. This is intended as a community micro-financing and bank-lending programme designed to support small companies. Some further details can be found in: HM Treasury financing business growth
The new Enterprise Zones (EZs) are a revival of the growth pole concept originally advocated by Peter Hall in the 1970s and first adopted by the Conservatives during the 1980s. Then, areas where the worst manufacturing and mining job losses had taken place were generally targeted as recipients for a 10-year period of tax breaks. In 2011, a select handful of business districts are once again being offered holidays on taxes and fast-track planning to encourage new investment from all-comers.
This time around, though, areas with strong medium-term growth prospects, rather than the most severely struggling locations, have been prioritised in order to give the new growth poles a high chance of success. This may be a sensible approach. The 1980s EZs were sometimes criticised for their lack of durability (when the 10 years came to an end, businesses often closed or moved on, notably so in Corby and Hartlepool, although Canary Wharf has a successful legacy). In contrast, the new recipient regions have already experienced some modest success prior to selection; with an additional boost from the government, it is hoped they will become highly dynamic and sustainable business hubs.
22 local areas were designated as Enterprise Zones during the first rounds of funding in 2011. More will almost certainly be announced as time passes. Of the first 22, 11 were hand-picked by central government, clearly illustrating the "top-down" nature of this aid strategy. Another 11 choices were made with some "bottom-up" input from local communities, following a competition launched among England's 38 local enterprise partnerships (LEPs).
The LEPs are partnership bodies composed of representatives from local authorities, universities, businesses and voluntary sector players. The bidding process was intended to make neighbouring LEPs work together to identify a single merit-worthy growth pole in their wider area, thereby getting beyond petty local rivalries within the region (Financial Times, 25 March 2011).
Several generalisations can be made about this latest phase of UK government assistance for economic development.
Overall, there is less money available than in the immediate past, reflecting the general climate of economic austerity in the UK since 2008.
Funding is being targeted at (i) areas where public sector job losses are high and (ii) businesses and business districts that already show signs of strength, rather than the worst-performing cases. The government believes buoyant areas are best placed for recovery and that some evidence of innate economic resilience should be displayed by the recipients of this sought-after funding.
Regional assistance and grants for businesses are very much centrally-controlled, with both the RGF and EZ "winners" chosen by the national government. Contemporary UK regional aid programmes can be characterised as "top-down" policies.
Regional aid is being used as a tool to leverage private sector investment. For every one pound released through the RGF, the government hopes to leverage more than five pounds of private sector investment, including inward investment from overseas TNCs, thereby creating many thousands of new jobs.
High-tech manufacturing, green technology and creative industries are strong candidates for government assistance.
Despite the measures that are being taken, national long-term economic recovery is still some way off. The signs of social unrest seen in UK cities during the summer of 2011 suggest that the RGF and EZs are far from being a panacea for all of the UK's current economic problems.
Re-branding and 'supercity' regeneration
Even if towns and businesses fail to gain support from the new EZ and RGF schemes, there are other routes to success that can be followed. These involve private investment, or support from charities, the National Lottery and agencies such as English Heritage.
A number of new re-branding schemes have recently got the go-ahead following pledges of support from a range of different players. These include:
Stoke-on-Trent Middleport Pottery is the last working Victorian pottery in Britain. It has been bought by a charity led by Prince Charles. The Prince's interest is in keeping the craft alive; so pottery will still be made on half the site with the remainder renovated to serve as a museum, restaurant and workshops for tradesmen. The scheme aims to double the number of employees on-site. Additional funding comes from the National Lottery, English Heritage, the European Union and private donations - as well as £1.7m from the RGF. Find out more at: Burleigh.co.uk
Corby Although Arts Council England has seen its funding slashed, it can still sponsor some projects directly. Corby Cube is a multi-use cultural centre in a town with one of the highest youth unemployment rates in the UK. The Arts Council will provide £140,000 in 2012, rising to £147,000 by 2015. This should be money well spent: according to one estimate, every £1 invested in culture generates another £2 as part of a multiplier effect based around the strengthening of local shops, restaurants or hotels. Find out more: The core at Corby Cube
Shoreditch (London) Many large companies including Google, Facebook, Intel and McKinsey have agreed to participate in the "Silicon Roundabout" scheme, with Facebook also agreeing to set up a permanent home in east London. The plan is to create a re-branded high-tech hub in east London that might one day become as successful as America's Silicon Valley. Find out more about Silicon Roundabout
North-east England With beautiful scenery and characterful urban locations, the region has always attracted film-makers. Now its screen presence should be boosted further by the £2.4m North East Creative Content Fund, a public-private investment partnership in the creative industries. Find out more at Northern Film and media
Finally, a recent survey by HBSC Commercial Banking optimistically predicts the economic map of the UK will, in the medium-term, be changed by the emergence of supercities. These will be hubs where clusters of private sector growth industries have specialised in technology sectors such as renewable energy, regenerative healthcare, plastic electronics and the space sector. Lynda Gratton, a professor at London Business School, told the Financial Times (02 June 2011): "We will see the rise of clusters of industries that the UK excels in. The UK already has a track record in sectors such as creativity, renewable energy and medicine, so it is incredibly well-positioned."
For instance, the report expects Glasgow to become an international force in renewable energy, one of seven supercities to capitalise on technologies or new ways of working.
Others are Bristol (new materials and advanced manufacturing), Newcastle (scientific research), Leeds (financial services), Liverpool (cultural and branding industries), London (creative industries) and Brighton ("capital of the UK's rebellious, alternative economy"). The report also sees regenerative healthcare driving half a dozen clusters including Edinburgh, York and Nottingham. It also predicts the emergence of clusters of photonics industries – based around the technology of transmitting information by light – in Strathclyde, West Midlands, the Cotswolds, the Thames Valley and Southampton.
"Growth fund favours manufacturing" Financial Times, 02 November 2011
"A timetable for austerity Britain" Financial Times, 22 March 2011
"Corby takes Cube route to urban regeneration" Financial Times, 31 March 2011
"Recession hits low-skilled and widens north-south divide" Financial Times, 21 July 2011
"Top 1% of earners set pay quarter of all income tax" Financial Times, 12 February 2011
"Regions reinvent to fill public sector gap" Financial Times, 22 February 2011
"Smaller towns foster manufacturing bounce" Financial Times, 24 January 2011
"Failure looms over 25% of high streets, says survey" Financial Times, 17 June 2011
"Merseyside tipped as hub for offshore wind industry" Financial Times, 14 November 2010
"North emerges as regional fund winner" Financial Times, 12 April 2011
HM Treasury financing business growth
"Flagship zones accentuate the positive" Financial Times, 25 March 2011
The core at Corby Cube
Silicon Roundabout
Northern Film and media
"Growth cities face severe test as cuts kick in" Financial Times, 24 January 2011
"Regions become rising stars in the film world" Financial Times, 16 February 2011
"Supercities to reshape business map" Financial Times, 02 June 2011
"Desire to preserve craftsmanship at heart of deal" Financial Times, 14 June 2011
"Planning policy offers key to survival" Financial Times, 05 July 2011
"Ten more Enterprise Zones to be confirmed within days" BBC News, 20 July 2011
"Enterprise zones announced to help boost growth" Guardian 17 August 2011
BIS Department for Business Innovation & Skills: Regional growth fund
Plymouth City Council: Regional Growth Fund Round 3 announced
Centre for Cities think-tank
Dr Simon Oakes is Chief Examiner for the International Baccalaureate diploma course in geography. He is also a senior examiner for A-level geography and GCSE citizenship
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text:
Rallying Asia
Today President Bush starts a week-long swing through Japan, the Philippines, Thailand, Singapore, Indonesia and Australia. He'll be giving thanks for help in Iraq and in the war on terror, and deploying carrots and sticks for more of the same. Bush will meet only with leaders and so he won't have to confront the fact that the people of most of these countries aren't at all happy with U.S. policy.
The trip, built around a drop-in at the three-day Asia-Pacific Economic Cooperation meeting in Bangkok, is designed, as National Security Advisor Condoleezza Rice put it in a pre-trip briefing, "to stress the crucial importance of our longstanding alliances in the region, to herald the transitions to democracy that are enhancing our partnerships, to express his gratitude to these nations for their cooperation and support in the global war on terror, and to work to strengthen the region's commitment to free and fair trade that benefits all."
In Tokyo on Friday Bush meets with Japanese Prime Minister Junichiro Koizumi, who is expected to offer $1.5 billion toward the reconstruction of Iraq; then he stops by the Phillipines on Saturday to meet with President Gloria Arroyo and to thank her for her country's contribution of soldiers, police and medical workers in Iraq.
After the APEC meeting, where he'll talk trade and security and have one-on-ones with the leaders of China, South Korea and, for some reason, Mexico, Bush heads to Singapore, and then to Bali, to mark the anniversary of the terrorist attack that killed over 200 people in a popular dance club. In Bali, Bush will meet with President Megawati Sukarnoputri, as well as leaders from the two largest moderate Muslim organizations in the country in an attempt to turn a new leaf with the world's most-populous Muslim nation.
But Bush has no plans to stop by the meetin of Organization of the Islamic Conference's meeting in Malaysia this week. Which may be a missed opportunity, given the recent study found that anger towards America has inreased massively in the Muslim world. It's unlikely that a brief meeting with select moderates in Indonesia will do much to reverse this.
The president will end his trip in Australia on the 22nd and 23rd, where he will meet with Prime Minister John Howard, a fairly solid Bush ally, and address the nation's parliament.
While the president is beating the war drum throughout the far East, each of the leaders he is visiting have their own agendas. With Japanese president Koizumi gearing up for November elections, which he is expected to win, Koizumi will have to be cautious about committing to sending troops to Iraq, as the Japanese public doesn't generally support the war. As the Asia Times reports, some analysts believe that Koizumi is hoping to trade his support in Iraq for an exchange rate intervention by the U.S. Federal Reserve, to help stabilize Japan's ailing currency. While Koizumi buddies up with Bush, Phillipine president Arroyo is expected to ask the U.S. president for military aid in fighting her domestic militant groups.
Although these foreign leaders might be warm to Bush's visit, the general public is less than welcoming. In Indonesia and the Phillipines, militant groups have issued threats against Bush — and needless to say the president's entourage will be heavily protected. This tour seems a great opportunity for the kind of public diplomacy recently called for to mend America's reputation in the world. But given that Bush's agenda seems to be to thank those who are "with us," and sound warnings to, or simply ignore, those who are "against us," the Asia jaunt holds little promise for increased understanding.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
In contemporary communication networks, the nodes perform only routing, \textit{i.e.}, they copy the data on incoming links to the outgoing links. In order to transmit messages generated simultaneously from multiple sources to multiple sinks the network may need to be used multiple times. This limits the throughput of the network and increases the time delay too. It is known that if intermediate nodes in a network are permitted to perform coding operations, \textit{i.e.}, encode data received on the incoming links and then transmit it on the outgoing links (each outgoing link can get differently encoded data), the throughput of the network increases. This is called network coding \cite{Yeung}. Thus, network coding subsumes routing.

For example, consider the butterfly network \cite{Yeung} of Fig. \ref{b_fly} wherein each link can carry one bit per link use, source node $S$ generates bits $b_1$ and $b_2$, and both sink nodes $T_1$ and $T_2$ demand both source bits. With routing only, two uses of link $V_3-V_4$ are required while with network coding only one.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{b_fly.eps}
\caption{Butterfly Network (a) A network code and (b) Global encoding vectors}
\label{b_fly}
\vspace{-14pt}
\end{figure}

Above is an example of single-source multi-sink linear multicast network coding, wherein there is a single source ($S$), generating a finite number of messages, ($x_1,x_2$), and multiple sinks, each demanding all the source messages and the encoding operations at all nodes are linear. In general, there may be several source nodes, each generating different number of source messages, and several sink nodes, each demanding only a subset, and not all, of source messages. Decoding at sink nodes with such general demands is studied in this paper.

We represent a network by a finite directed acyclic graph $N=(V,E)$, where $V$ is the set of vertices or nodes and $E \subseteq V \times V$ is the set of directed links or edges between nodes. All links are assumed to be error-free. Let $[n]=\{1,2,\ldots,n\}$. The network has $J$ sources, $S_j,\,j\in [J]$, and $K$ sinks, $T_k,\,k\in [K]$. The source $S_j$ generates $\omega _j$ messages, $\forall j\in [J]$. Let $\omega=\sum _{j=1}^J \omega _j$ be the total number of source messages. The $\omega$-tuple of source messages is denoted by $x_{[\omega]}=(x_1,x_2,\ldots,x_\omega)$, where $x_i\in F,\,\forall i\in [\omega]$ and $F$ is a finite field. By $\mathbf{x}=(x_1,\ldots,x_{\omega})^T$ we denote the column vector of the source messages. The demand of the $k^{th}$ sink node is denoted by $D_k\subseteq [\omega]$. Given a set $I=\{i_1\ldots,i_l\}\subseteq [\omega]$, let $x_I=(x_{i_1},\ldots,x_{i_l})$, \emph{i.e.}, $x_{[\omega]}$ \emph{restricted} to $I$. Let $\{x_I\}=\{x_I\colon x_I\in F^I\}$, \emph{i.e.}, the set of all $I$-tuples over $F$. For a multi-variable binary-valued function $f(x_1,\ldots,x_\omega)$, the subset of $F^\omega$ whose elements are mapped to $1$ by $f(x_1,\ldots,x_\omega)$ is called its support and is denoted by  $\mathtt{supt}(f(x_{[\omega]}))$ and $\mathtt{supt}_{I}(f(x_{[\omega]}))$ denotes the $|I|$-tuples in the support restricted to $I$. A source message is denoted by edges without any originating node and terminating at a source node. Data on a link $e\in E$ is denoted by $y_e$.

A network code (NC) is a set of coding operations to be performed at each node such that the requisite source messages can be faithfully reproduced at the sink nodes. It can be specified using either local or global description \cite{Yeung}. The former specifies the data on a particular outgoing edge as a function of data on the incoming edges while the latter specifies the data on a particular outgoing edge as a function of source messages. Throughout the paper we use global description for our purposes.
\begin{definition}
Global Description of an NC \cite{Yeung}: An $\omega$-dimensional NC on an acyclic network over a field $F$ consist of $|E|$ global encoding maps $\tilde{f}_e : F^{\omega}\rightarrow F, \forall e\in E$ (i.e.,  $\tilde{f}_e ( \mathbf{x}) = y_e$).
\end{definition}
Let $e_i,i=1,\ldots,\omega$, be the incoming edges at the source, then $y_{e_i}=x_i$.

When the intermediate nodes perform only linear encoding operations then such an NC is said to be a linear network code (LNC).
\begin{definition}
Global Description of an LNC \cite{Yeung}: An $\omega$-dimensional LNC on an acyclic network over a field $F$ consist of $|E|$ $1\times\omega$ global encoding vectors $\mathbf{f}_e, ~ \forall e\in E$ such that $\mathbf{f}_e\cdot \mathbf{x}=y_e$.
\end{definition}
The global encoding vectors for the incoming edges at the source are standard basis vectors for the vector space $F^{\omega}$. The global encoding vectors of the LNC for butterfly network is given in Fig.~\ref{b_fly}(b).

Hereafter we assume that the network is feasible, \emph{i.e.}, demands of all sink nodes can be met using network coding and the global description of a network code (linear or non-linear) is given. If a sink node demands $\omega'$ ($\leq$ $\omega$) source messages, it will have at least $\omega'$ incoming edges. The decoding problem is to reproduce the desired source messages from the coded data received at the incoming edges. Thus, decoding amounts to solving for a specified set of $\omega'$ unknowns using a set of at least $\omega'$ simultaneous equations in $\omega$ unknowns. Hence, the global description of the NC is more useful for decoding.

While decoding of non-linear NC has not been studied, the common technique used for decoding a LNC is to perform Gaussian elimination \cite{Lun1,Lun2}, which requires $\mathcal{O}(n^3)$ operations, followed by backward substitution, which requires $\mathcal{O}(n^2)$ operations ($n$ is the number of variables) \cite{Num}. This is not recommendable when the number of equations and/or variables is very large. In such cases, iterative methods are used. Convergence and initial guess are some issues that arise while using iterative methods \cite{Strang}.

We propose to use the sum-product (SP) algorithm to perform iterative decoding at the sinks. A similar scheme for decoding multicast network codes using factor graph \cite{Frey} was studied in \cite{Salmond}. The authors considered the case of LNC. The problems associated with the proposed decoding scheme in \cite{Salmond} are:
\begin{itemize}
\item In order to construct the factor graph, full knowledge of network topology is assumed at the sinks which is impractical if the network topology changes. For a particular sink node (say $T$), the factor graph constructed will have $\omega +|E|$ variable nodes and $|E|+ |In(T)|$ factor nodes, where $In(T)$ is the set of incoming edges at node $T$.
\item Complete knowledge of local encoding matrix \cite{Yeung} of each node is assumed at the sinks which again is impractical since local encoding matrix for different nodes will have different dimensions and hence variable number of overhead bits will be required to communicate to downstream nodes which will incur huge overhead.
\end{itemize}
We also point out that the motivating examples, \textit{viz.}, Examples 1 and 4, given in \cite{Salmond} for which the proposed decoding method claims to exploit the network topology admits a simple routing solution and no network coding is required to achieve maximum throughput. Solving linear equations in boolean variables is also studied in \cite{Mez}.

The contributions and organization of the paper are as follows:
\begin{itemize}
\item In Section III-A we pose the problem of decoding of linear and non-linear NC as \emph{marginalize a product function problem} (MPF) and construct factor graph using the global description of network codes. For a particular sink node, the constructed graph will have fewer vertices than in \cite{Salmond} and hence the number of messages and operations performed will also be fewer. Unlike in \cite{Salmond}, our scheme requires only the knowledge of global encoding maps/vectors of incoming edges at a sink node and not the entire network structure and coding operation performed at each node.
\item In Sections III-B, we utilize \textit{traceback} instead of running multiple-vertex version of algorithm, thus, further reducing the number of operations. Some examples illustrating the proposed techniques are given in Section III-D.
\item We discuss utility and computational complexity of the proposed technique in Section III-C. For sink nodes which demand all the source messages, the notion of \emph{fast decodable network codes} is defined and a sufficient condition for the same is identified.
\end{itemize}
We present a brief overview of the SP algorithm in Section II and conclude the paper with a discussion on scope for further work in Section IV.

\section{The Sum-Product Algorithm and Factor Graphs}
In this section, we review the computational problem called the MPF problem and specify how SP algorithm can be used to efficiently solve such problems. An equivalent method to efficiently solve MPF problems is given in \cite{Aji} and is called the \emph{generalized distributive law} (GDL) or the \emph{junction tree algorithm}. The simplest example of SP algorithm offering computational advantage is the distributive law on real numbers, $a\cdot (b+c)=a\cdot b+a\cdot c$; the left hand side of the equation requires fewer operation than the right hand side. Generalization of addition and multiplication is what is exploited by the SP (or the junction tree) algorithm in different MPF problems. The mathematical structure in which these operations are defined is known as commutative semirings.

\begin{definition}
A commutative semiring \cite{Aji} is a set $R$, together with two binary operations ``$+$'' (\textit{addition}) and ``$\cdot$'' (\textit{multiplication}), which satisfy the following axioms:
\begin{enumerate}
\item The operation ``$+$'' satisfies closure, associative, and commutative properties; and there exists an element ``$0$'' (\textit{additive identity}) such that $r+0=r,\forall r\in R$.
\item The operation ``$\cdot$'' satisfies closure, associative, and commutative properties; and there exists an element ``$1$'' (\textit{multiplicative identity}) such that $r\cdot 1=r,\;\forall r\in R$.
\item The operation ``$\cdot$'' \textit{distributes} over ``$+$'', \textit{i.e.}, $r_1\cdot r_2 + r_1\cdot r_3 = r_1\cdot (r_2+r_3),\;\forall r_1,r_2,r_3\in R$
\end{enumerate}
\end{definition}

For different problems, we use different semirings with different notion of ``$+$ and $\cdot$''. Some examples are listed below.
\begin{enumerate}
\item Application of the SP algorithm to Fourier transform yields the FFT algorithm; the semiring is the set of complex numbers with the usual addition and multiplication \cite{Frey,Aji}.
\item ML decoding of binary linear codes is also an MPF problem and application of SP algorithm yields the Gallager-Tanner-Wiberg decoding algorithm over a Tanner graph; the semiring is the set of positive real numbers with ``$\min$'' as sum and ``$+$'' as product, called the min-sum semiring \cite{Frey,Aji}. The BCJR algorithm for decoding turbo codes and LDPC deocoding algorithm are some other applications of SP algorithm.
\item Application to the ML sequence estimation, for instance in decoding convolutional codes, yields the Viterbi algorithm \cite{Aji}; the semiring is again the min-sum semiring.
\item Recently, the GDL has been shown to reduce the ML decoding complexity of space-time block codes in \cite{LP}; the semiring applicable is the min-sum semiring of complex number. The authors introduced \textit{traceback} for GDL and used it to further lower the number of operations.
\end{enumerate}

Thus, both these algorithms subsume as special cases many well known algorithms.

\subsection{MPF Problems in Boolean Semiring}
A Boolean semiring is the set $\{0,1\}$ together with the usual Boolean operations $\vee$ (OR) and $\wedge$ (AND). We denote it by $R=(\{0,1\},\vee,\wedge)$. The elements $0$ and $1$ are the \emph{additive} and \emph{multiplicative identities} respectively. The MPF problem defined for this semiring is described below. Let $x_i,\,i\in [n]$ be $n$ variables taking values in finite alphabets $A_i,i\in [n]$. For $I=\{i_1,\ldots,i_k\}\subseteq [n]$, let $x_I=(x_{i_1},\ldots,x_{i_k})$ $A_I=A_{i_1}\times \ldots \times A_{i_k}$. Let $\mathcal{S}=\{S_1,S_2,\ldots,S_M\},\; S_j\subseteq [n]$, such that for each $j\in [M]$, there is a function $\alpha _j:A_{S_j}\rightarrow R$. The functions $\alpha _j$s are called the \emph{local kernels},  the set of variables in $x_{S_j}$ is called the \emph{local domain} associates with $\alpha _j$ and $A_{S_j}$ is the associated \emph{configuration space}. The \emph{global kernel}, $\beta: A_{[n]}\rightarrow R$ and its $i^{th}$ \emph{marginalization}, $\beta _i: A_{S_i}\rightarrow R$, are defined below.
\begin{align} \nonumber
\beta(x_1,x_2,\ldots,x_n)=\bigwedge _{j=1}^M \alpha _j(x_{S_j})
\end{align}

\begin{align}
\label{eqn_marg}
\beta_i(x_{i}) = \bigvee _{\{x_{[n]\backslash i}\}} \beta (x_1,x_2,\ldots,x_n)
\end{align}
\subsection{The SP Algorithm}
Brute force computation of marginalizations \eqref{eqn_marg} require $\mathcal{O}(A_{[n]})$ computations; the SP algorithm is an efficient way of computing these. It involves iteratively passing \emph{messages} along the edges of the \emph{factor graph}, $\mathcal{G}=(\mathcal{V\cup F},\mathcal{E})$, associated with the given MPF problem. The factor graph is a bipartite graph. Vertices in $\mathcal{V}$ are called variable nodes; one for each variable $x_i$, $\forall i\in [n]$ ($|\mathcal{V}|=n$).  The vertices in $\mathcal{F}$ are called the factor nodes; one for each local kernel $\alpha _j$, $\forall j\in [M]$ ($|\mathcal{F}|=M$). A variable node $x_i$ is connected to a factor node $\alpha _j$ iff $x_i$ is an argument of $\alpha _j$. For convenience we assume that for a variable node $x$ the local domain and local kernel are $x$ and $1$ respectively.

Let $N(x_i)$ denote the set of factor nodes adjacent to the variable node $x_i$, \emph{i.e.}, set of local kernels with $x$ as an argument and $N(\alpha _j)(=x_{S_j})$ denote the set of variable nodes adjacent to the factor node $\alpha _j$, \emph{i.e.}, the local domain of $\alpha _j$. The directed message passed from a variable node $x_i$ to an adjacent factor node $\alpha _j$ and vice versa are as follows:
\begin{align}
\label{eqn_msg1}
\mu _{x_i\rightarrow \alpha _j}(x_i)= \bigwedge _{\alpha '\in N(x_i)\backslash \alpha _j}\mu _{\alpha '\rightarrow x_i} (x_i)
\end{align}
\begin{align}
\mu _{\alpha _j \rightarrow x_i}= \bigvee _{\{x_{S_j\backslash i}\}} \alpha _j(x_{S_j}) \bigwedge _{x'\in \{x_{S_j\backslash i}\}}\mu _{x'\rightarrow \alpha _j}(x')
\end{align}

Depending on the requirement, we may need to evaluate marginal(s) at only one, a few or all variable nodes. The algorithm starts at the leaf nodes (nodes with degree one) with the leaves passing messages to the adjacent nodes. Once a vertex has received messages from all but one of its neighbor, it computes its own message and passes it to the neighbor from which it has not yet received the message. This message passing terminates when all the variable nodes at which marginals are required to be evaluated have received from all its neighbors. A node after receiving messages from all of its neighbors, computes its \emph{state} as the product of messages received from all the adjacent nodes. For a variable node, $x_i$, it is denoted by $\sigma_i(x_i)$ and is given as follows:
\begin{align}
\label{eqn_state_v}
\sigma_i(x_i) = \bigwedge _{\alpha ' \in N(x_i)}\mu_{\alpha ' \rightarrow x_i}(x_i),
\end{align}
Similarly, the state for a factor node is computed as follows:
\begin{align}
\label{eqn_state_f}
\sigma_{\alpha _j}(x_{S_j}) = \alpha _j(x_{S_j})\bigwedge _{x' \in \{x_{S_j}\}}\mu_{x' \rightarrow \alpha _j}(x'),
\end{align}

As stated in \cite{Frey, Aji}, after sufficient number of messages have been passed, the state of a variable node $x_i$ will be equal to $\beta_i(x_i)$.

To obtain the correct value of the required marginal(s), it is essential that the factor graph be free of cycles. If there are cycles these may not be the correct values. The cycles can be eliminated by \textit{stretching} variable nodes or \textit{clustering} variable or factor nodes (refer to \cite[Sec. VI]{Frey} for a detailed description). These methods are exemplified in Section III-D.

Both these graph transformations lead to enlargement of the local domain(s), and hence the configuration space of the node(s). In the new graph, the directed message passed from a vertex $v$ to $w$ is
\begin{equation}
\label{eqn_msg_mod}
\mu _{v\rightarrow w}(x_{S_v\cap S_w})=\bigvee _{\mathclap{\{x_{S_v\backslash S_w}\}}}\;\alpha _v(x_{S_v})\;\bigwedge _{\mathclap{v'\in N(v)\backslash w}}\;\mu _{v'\rightarrow v} (x_{S_{v'}\cap S_v}),
\end{equation}
where $N(v)$ is the set of neighboring vertices of $v$ and its state $\sigma_v(x_{S_v})$ is
\begin{align}
\label{eqn_state_mod}
\sigma_v(x_{S_v}) = \alpha _v(x_{S_v})\bigwedge _{v' \in N(v)}\mu_{v'\rightarrow v}(x_{S_{v'}\cap S_v})
\end{align}
These are the general forms of messages and states; \eqref{eqn_msg1}-\eqref{eqn_state_f} can be obtained from these.

Let $v^*$ be the node with the largest configuration space $A_{v^*}$ (choose any one if there are multiple such nodes). Then the number of operations required for computing messages and states in the SP algorithm will be $\mathcal{O}(A_{v^*})$. Thus, at the cost of possibly increased computational cost, the SP algorithm on the transformed graph yields the exact value of the marginals. In the sequel, we assume that the factor graph is acyclic.

\section{Decoding Network Codes using the Sum-Product Algorithm}
In this section, we show that decoding a NC is an MPF problem over a Boolean semiring. We provide a method to construct factor graph for decoding at a sink node using the SP algorithm.

Though the factor graph approach and the junction tree approach are equivalent formulations to solve MPF problems, we prefer the former because of the amount of preprocessing required to obtain junction tree as argued below:
\begin{enumerate}
\item The construction of a junction tree for an MPF problem requires: \textit{(a)} construction of a \textit{moral graph}, \textit{(b)} its \textit{minimum complexity triangulation} if it is not already triangulated, \textit{(c)} construction of the \textit{clique graph} of the triangulated moral graph, and \textit{(d)} finding a spanning tree which leads to minimum computational cost. To the nodes of this clique tree the local kernels and variables of the MPF problem are attached \cite{Aji} to obtain the junction tree (a kernel or a variable is attached to a node of clique tree iff its local domain is a subset of the local domain of the said clique tree node). Thus, the GDL always gives the exact solution of the MPF problems.
\item A factor graphs is easily described by the local kernels associated with the MPF problem; it is a bipartite graph involving set of variable and a set of local kernels of the MPF kernels as the two vertex sets. If it is acyclic, then the SP algorithm gives the exact solution, if not, it gives an approximate solution. The SP algorithm is known to perform well even if the factor graph has cycles, for example, in decoding of LDPC and turbo codes. Factor graphs with cycles can be transformed into acyclic ones to obtain exact solutions \cite[Sec.~VI]{Frey}.
\end{enumerate}

\subsection{NC Decoding as an MPF Problem}
Given an acyclic network $N=(V,E)$, the demands at each sink, $D_k,\,k\in [K]$ and a set of global encoding maps, $\{\tilde{f}_e:e\in E\}$, that satisfy all the sink demands, the objectives at a sink, say $k^{th}$, is to find the instance of $x_{D_k}$ that was generated by the source(s) using the data it receives on its incoming edges, \emph{i.e.},
\begin{align}
\label{eqn_mpf}
x_{D_k}^* &= \underset{D_k}{\mathtt{supt}} \bigvee _{x_{[\omega]}\in F^{\omega}} \underbrace{\bigwedge _{e \in In(T_k)} \delta \left( \tilde{f}_e (x_{[\omega]})\, , \, y_e \right)}_{\beta ^{(k)}(x_{[\omega]})}
\end{align}
Here $\beta ^{(k)}$ is the global kernel of the MPF problem at the $k^{th}$ sink and $\delta$ is a function that indicates whether its two input arguments are equal or not,\textit{i.e.} ,
\[
 \delta(a,b) =
  \begin{cases}
   0, & \text{if}\; a \neq b \\
   1, & \text{if}\; a = b
  \end{cases}
\]
For LNC, \eqref{eqn_mpf} becomes
\begin{align} \nonumber
x_{D_k}^*& = \underset{D_k}{\mathtt{supt}} \bigvee _{\mathbf{x}\in F^\omega}\; {\bigwedge _{e\in In(T_k)}\delta \left( \mathbf{f}_e \cdot \mathbf{x}\, , \, y_e \right)}
\end{align}

Thus, decoding a NC has the form of a special class of MPF problems over Boolean semiring wherein we are interested only in some coordinates (specified by $D_k$) of the $\omega$-tuples in the support set and not the value of the global kernel.

Since the solution $x_{D_k}^*$ is unique, individual coordinates $j\in D_k$ can be separately computed, \emph{i.e.},
\begin{equation}
\label{eqn_marg1}
\begin{aligned}
x_j^* = \underset{j}{\mathtt{supt}} \bigvee _{x_j\in F}\;\beta _j^{(k)}(x_j) \\
\beta _j^{(k)}(x_j) = \bigvee _{\{x_{[\omega]\backslash j}\}}\beta ^{(k)}(x_{[\omega])},
\end{aligned}
\end{equation}
where $\beta _j^{(k)}(x_j)$ is the $j^{th}$ marginalization of the global kernel $\beta ^{(k)}$.

The factor graph for decoding at sink $T_k,k\in [K]$ is constructed as follows:
\begin{enumerate}
\item Install $\omega$ \textit{variable nodes}, one for each source message. These vertices are labeled by their corresponding source messages, $x_i$.
\item Install $|In(T_k)|$ \textit{factor nodes} and label them $\tilde{f}_e, e\in In(T_k)$. The associated local domain of each such vertex is the set, $S\subseteq x_{[\omega]}$, of source messages that participate in that encoding map and the local kernel is $\delta(\tilde{f}_e(x_{[\omega]})\, , \, y_e)$. These vertices are labeled by their corresponding local kernels, $\tilde{f}_e$.
\item A variable node is connected to a factor node iff the source message corresponding to that variable node participates in the encoding map corresponding to the said factor node.
\end{enumerate}
We use thicker lines for factor nodes to differentiate them from variable nodes. The factor graph so constructed will be a bipartite graph. General form of a factor graph and the same for the two sink nodes of the butterfly network are given in Fig.~\ref{junc_grph} (cf. \cite[Fig. 3]{Salmond}).
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{junc_grph.eps}
\caption{(a) General form of a factor graph (b) Factor graphs for $T_1$ and (c) $T_2$ of the butterfly network. Local kernels are given adjacent to the function vertices.}
\label{junc_grph}
\vspace{-7pt}
\end{figure}

Messages and states are computed using \eqref{eqn_msg_mod} and \eqref{eqn_state_mod} respectively. As stated before, once a node (say $v$), has received message from all the adjacent nodes, its state, $\sigma _v(x_{S_v})$, can be computed. Let $x_{D_k}\cap x_{S_v}= x_{B}$. The value of the subset $B$ of the requisite source messages at the $k^{th}$ sink node is
\begin{equation}
x_B^*=\underset{B}{\mathtt{supt}} \bigvee _{\{x_{S_v}\}} \alpha _v(x_{S_v})\bigwedge _{v'\in N(v)} \mu _{v'\rightarrow v} (x_{S_{v'}\cap S_v})
\end{equation}

As specified in Section II, the SP algorithm yields the correct value of the source messages if the factor graph is a tree. If not then the cycles in the factor graph will have to be removed.

\subsection{Traceback}
Since we are interested in the value of the source messages and not the value of the marginalizations of the global kernel, we can use \emph{traceback} \cite{LP} to further lower the number of computations.

Assume that the single-vertex SP algorithm is run with vertex $v$ as the root (all messages are directed towards $v$) and the value $x_{S_v}^*\subseteq x_{D_k}$ of some source messages in the demand set of the $k^{th}$ sink has been ascertained. Now, partition the local domain of a neighboring node $w$, as $x_{S_w}=x_A \cup x_B$, where $A=S_w\backslash S_v$ and $B=S_w\cap S_v$. Since $x_{S_v}^*$ is known, the value $x_B^*$ that causes $\sigma_w(x_A,x_B)$ to take value $1$ is also known. The value of the source messages $x^*_A$ can then be obtained using \eqref{eqn_state_mod} as follows:
\begin{equation}
\begin{aligned}
\label{eqn_trcbk}
x^*_A & =\underset{A}{\mathtt{supt}} \bigvee _{\{x_A\}} \sigma_w(x_A,x_B^*)\\
& = \underset{A}{\mathtt{supt}} \bigvee _{\{x_A\}} \mu_{v\rightarrow w}(x_B^*)\; \lambda_w(x_A,x_B^*)\\
 & = \underset{A}{\mathtt{supt}} \bigvee_{\{x_A\}} \lambda_w(x_A,x_B^*),
\end{aligned}
\end{equation}
where
\[\lambda_w(x_A,x_B)=\alpha (x_{S_w})\bigwedge_{v'\in N(w)\backslash v} \mu_{v'\rightarrow w}(x_{S_{v'}\cap S_w})\]
is the partial state computed at $w$ while passing the message $\mu_{w \rightarrow v}(x_B)$ to the root $v$. Thus, $\mu_{v\rightarrow w}(x_B^*)$ does not need to be computed or passed, leading to saving of operations.

Here we have exploited the fact that we require only $\mathtt{supt}_A \bigvee _{x_A} \sigma_w$ and not the value of $\sigma_w$ which would have required passing of the message $\mu_{v\rightarrow w}(x_B^*)$ from $v$ to $w$ too. Hence, the traceback step reduces the computational complexity.

The traceback step can be used repeatedly until values of all the source messages in $x_{D_k}$ are obtained. This can be done by using \eqref{eqn_trcbk} on other neighbors of $v$ and then neighbors of neighbors of $v$ and so on. This can lead to considerable reduction in number of operations and is exemplified in Section III-D.

\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{trcbk.eps}
\caption{(a) Combination network (b) Factor graph with cycles for $T_{495}$ and (c) one obtained by clustering equation nodes $\tilde{f}_6-\tilde{f}_{12}$ $(\tilde{f}=\bigwedge_{i=6}^{12}\tilde{f}_i)$}
\label{trcbk}
\end{figure*}

\subsection{Computational Complexity}
We suggest using SP algorithm for decoding a network code only when the code is either non-linear or it is linear but the number of messages is very large. For linear network codes with manageable value of $\omega$, Gaussian elimination with backward substitution is advisable. If using SP algorithm for decoding network codes (when warranted) leads to computational complexity strictly better than brute-force decoding complexity, then the code is called \emph{fast decodable network code}.

We now discuss the computational complexity of SP algorithm based decoding of network codes. As in \cite{Frey}, by complexity we mean the number of semiring operations required to obtain the desired source message. For convenience, we assume that all the source messages take value from a $q$-ary alphabet; results for variable alphabet size can be obtained similarly.

As stated above, in order to recover the requisite source messages at a sink we need only run single-vertex SP algorithm followed by traceback steps. For a given sink node, if the factor graph constructed using the method given in Section III-A is cycle-free and the network code is such that the local domains of all factor nodes have cardinality at most $l\, (<\omega)$, then the number of operations required for decoding using the SP algorithm is $\mathcal{O}(q^l)$. If the sink demands all the source messages, then the brute-force decoding would require $\mathcal{O}(q^{\omega}) (> \mathcal{O}(q^l))$ operations. Thus, an acyclic factor graph with at most $l\, (<\omega)$ variables per equation is a sufficient condition for fast decodability of the network code at a sink which demands all the source messages.

If the graph is not cycle-free then we remove the cycles using the methods specified above and let $m\leqslant \omega$ be the size of maximum cardinality local domain in the transformed cycle-free factor graph. The number of computations required now will be $\mathcal{O}(q^m)\leqslant \mathcal{O}(q^{\omega})$ and the code is fast decodable iff $m<\omega$.

\subsection{Illustrations}
We now present some examples illustrating use of the SP algorithm to decode NC.

\begin{example}
Consider the butterfly network of Fig.\ref{b_fly}. Here $q=\omega =2$. The factor graphs for two sink nodes are given in are given in Fig.~\ref{junc_grph}(b) and (c). The messages passed and state computations for decoding at $T_1$ are as follows:
\begin{align} \nonumber
\mu_{x_2\rightarrow \mathbf{f}_{V_4-T_1}}(x_2)&= 1 \\ \nonumber
\mu_{\mathbf{f}_{V_1-T_1}\rightarrow x_1}(x_1)&= \delta(x_1,y_{V_1-T_1}) \\   \nonumber
\mu_{\mathbf{f}_{V_4-T_1}\rightarrow x_1}(x_1)&= \bigvee _{x_2}\delta(x_1+x_2,y_{V_4-T_1})  \\
\nonumber
\mu_{x_1\rightarrow \mathbf{f}_{V_4-T_1}}(x_1)&= \mu_{\mathbf{f}_{V_1-T_1},x_1}(x_1)  \\ \nonumber
\sigma_{x_1}(x_1)&= \mu_{\mathbf{f}_{V_1-T_1},x_1}(x_1) \; \mu_{\mathbf{f}_{V_4-T_1}\rightarrow x_1}(x_1)
\end{align}
\begin{align}
 \nonumber
\mu_{\mathbf{f}_{V_4-T_1}\rightarrow x_2}(x_2)&= \bigvee _{x_1}\delta(x_1+x_2,y_{V_4-T_1})\; \mu_{\mathbf{f}_{V_1-T_1}\rightarrow x_1}(x_1)  \\ \nonumber
\sigma_{x_2}(x_2)&= \mu_{\mathbf{f}_{V_4-T_1},x_2}(x_2)
 \end{align}
It is easy to verify that $\sigma_{x_i}(x_i)=\beta_i(x_i), i=1,2$. Hence, $x_i^*=\mathtt{supt}_{x_i} \bigvee _{x_i}\sigma_{x_i}(x_i), i=1,2$.
Similar computations apply for $T_2$ also. This network code is not fast decodable since the number of computations required is $\mathcal{O}(q^{\omega})$. \hfill $\square$
\end{example}

Now we present a multicast network which admits no linear solution over $\mathbb{F}_2$, but a non-linear solution exists.

\begin{example}
Consider the multicast combination network given in Fig.\ref{trcbk}(a). All messages take value from $F_2$. Each sink is connected to a distinct size $8$ subset of intermediate node. Multicast, in $F_2$, is feasible over such a network iff a $(12,32,5)$ binary error correcting code exists \cite{Riis}. The Nadler's code is one such systematic code with the requisite parameters. Note that since there exist no $(12,32,5)$ binary linear code, the above multicast network, admits no linear solution over $F_2$.
\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{fano.eps}
\caption{The network $\mathcal{N}_3$ of \cite{Insuff}.}
\label{fano}
\end{figure*}

Apart from the systematic part of the Nadler's code, the $7$ redundant bits are encoded using non-linear functions \cite{Wet};these are:
\begin{align}
\nonumber \tilde{f}_i &= x_i, \;\;\; i\in [5] \\
\nonumber \tilde{f}_6 &= x_1+x_2+x_3+(x_1+x_5)(x_3+x_4)
\\
\nonumber \tilde{f}_7 &= x_1+x_2+x_4+(x_1+x_3)(x_4+x_5)
\\
\nonumber \tilde{f}_8 &= x_1+x_2+x_5+(x_1+x_4)(x_3+x_5)
\\
\nonumber \tilde{f}_9 &= x_2+x_3+x_4+x_1x_4+x_4x_5+x_5x_1 \\
\nonumber \tilde{f}_{10} &= x_2+x_3+x_5+x_1x_3+x_3x_5+x_5x_1
\\
\nonumber \tilde{f}_{11} &= x_2+x_4+x_5+x_1x_3+x_3x_5+x_5x_1 \\
\nonumber \tilde{f}_{12} &= x_1+x_2+x_3+x_4+x_5+ x_3x_4+ x_4x_5+ x_5x_3
\end{align}
These functions are the global encoding maps of the $12$ source to intermediate node links. The intermediate nodes simply route the data on incoming edges to the connected sink nodes, hence, the global encoding maps of the incoming and outgoing edges of an intermediate node is same.

Consider decoding at the $495^{th}$ sink whose global encoding maps of incoming edges are $\tilde{f}_i,\;i=5,\ldots,12$. The factor graph constructed using the method stipulated above will have cycles (Fig.~\ref{trcbk}(c)). To eliminate the cycles, we cluster $\tilde{f}_i,\;i=6,\ldots,12$ function vertices into a single one with the local kernel as the product of $7$ original ones (Fig.~\ref{trcbk}(c)). The computational complexity of SP decoding is same as that of brute-force decoding ($\mathcal{O}(q^5)$). This is true for all the sink nodes and hence this code is not fast decodable for any of the sinks. \hfill $\square$
\end{example}

In the next example we present a network with general demands at sinks and employ the SP algorithm for decoding a vector non-linear network code for it. We also demonstrate usefulness of traceback for saving computations of some messages in the factor graph.

\begin{example}
Consider the network given in Fig.~\ref{fano}. The sinks (nodes $37-46$) have general demands which are specified by variables below them. In \cite{Insuff}, the authors showed that this network admits no linear solution over any field and gave a vector non-linear solution. The source message $x_i,i\in[5]$ are 2-bit binary words ($q=4,\,\omega = 5$), $+$ denotes addition in ring $\mathbf{Z}_4$, $\oplus$ denotes the bitwise XOR and the function $t(\cdot)$ reverses the order of the 2-bit input.

The factor graphs for nodes $37$, $40$ and $43$, denoted by $\mathcal{G}_{37}, \mathcal{G}_{40}$ and $\mathcal{G}_{43}$ respectively, are given in Fig.~\ref{fano_junc}; the same for other nodes have similar structure and can be constructed using method given in Section III-A. Note that $\mathcal{G}_{40}$ has a cycle of length $4$ and $\mathcal{G}_{43}$ has two cycles of length $6$ each. The cycles are removed as follows:
\begin{itemize}
\item The cycle in $\mathcal{G}_{40}$ is removed by clustering the variable vertices $x_1$ and $x_2$; the local domain and kernel of the new variable vertex are $(x_1,x_2)$ (union of the local domains of the clustered vertices) and  $1$ (product of the local kernels of the clustered vertices) (Fig.~\ref{fano_junc}).
\item The cycles $C_1$ and $C_2$ in $\mathcal{G}_{43}$ are removed by deleting the dotted edges and \textit{stretching} variable vertex $x_3$ around the respective cycles; this involves adding the stretched variable to all the local domain in the cycle and leaving the local kernels unchanged (Fig.~\ref{fano_junc}).
\end{itemize}

\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{fano_junc2.eps}
\caption{The factor graphs for sinks with labels $37$, $40$ and $43$ in network of Fig.~\ref{fano}.}
\label{fano_junc}
\end{figure*}
We infer from $\mathcal{G}_{43}$ that the number of computations required to reproduce all the source messages at $V_{43}$ is only $\mathcal{O}(q^3)$ instead of $\mathcal{O}(q^5)$ (as brute-force decoding would have required) and hence this code is fast decodable for $V_{43}$. The decoding process at $V_{43}$, using single-vertex SP algorithm with node ``f'' in $\mathcal{G}_{43}$ as root followed by traceback to compute $x_1,x_2,x_4,x_5$, is demonstrated below:
\begin{align}
\nonumber & \mu _{k\rightarrow j}=\mu _{j\rightarrow i}(x_3,x_5)=\delta (t(x_3)+x_5,y_{35-43})
\\
\nonumber & \mu _{a\rightarrow b}=\mu _{b\rightarrow c}(x_2,x_3)=\delta (x_2+x_3,y_{33-43})
\\
\nonumber & \mu _{i\rightarrow h}=\mu _{h\rightarrow g}(x_3,x_4)= \bigvee _{x_5} \delta (x_4+x_5,y_{36-43}) \mu _{j\rightarrow i}(x_3,x_5)
\\
\nonumber & \mu _{c\rightarrow d}=\mu _{d\rightarrow e}(x_1,x_3)= \bigvee _{x_2} \delta (x_1+x_2,y_{31-43}) \mu _{b\rightarrow c}(x_1,x_2)
\\
\nonumber & \mu _{e\rightarrow f}(x_3)=\bigvee _{x_1}\delta (x_1+x_3,y_{32-43})\mu _{d\rightarrow e}(x_1,x_3)
\\
\nonumber & \mu _{g\rightarrow f}(x_3)=\bigvee _{x_4}\delta (t(x_3)+x_4,y_{34-43})\mu _{h\rightarrow g}(x_3,x_4)
\end{align}
At ``f'', decoding of $x_3$ is performed as follows:
\begin{align}
\nonumber x_3^*&=\underset{x_3}{\mathtt{supt}}\bigvee _{x_3}\sigma _f(x_3) =\underset{x_3}{\mathtt{supt}}\,\sigma _f(x_3) \\\nonumber &= \underset{x_3}{\mathtt{supt}}\,\mu _{e\rightarrow f}(x_3)\,\mu _{g\rightarrow f}(x_3)
\end{align}
We can now use traceback \eqref{eqn_trcbk} to compute $x_1$ and $x_2$ without having to compute $\mu _{f\rightarrow e},\mu _{e\rightarrow d}$ or $\mu _{d\rightarrow c}$ as follows:
\begin{align}\nonumber
x_1^*&=\underset{x_1}{\mathtt{supt}}\bigvee _{x_1}\sigma _e(x_1,x_3^*) = \underset{x_1}{\mathtt{supt}}\,\lambda _e(x_1,x_3^*)\\ \nonumber & =\underset{x_1}{\mathtt{supt}}\, \delta (x_1+x_3^*,y_{32-43})\mu _{d\rightarrow e}(x_1,x_3^*)
\\
\nonumber
x_2^*&=\underset{x_2}{\mathtt{supt}}\bigvee _{x_2}\sigma _c(x_1^*,x_2,x_3^*) = \underset{x_2}{\mathtt{supt}}\,\lambda _c(x_1^*,x_2,x_3^*)\\ \nonumber & =\underset{x_2}{\mathtt{supt}}\, \delta (x_1^*+x_2,y_{31-43})\mu _{b\rightarrow c}(x_2,x_3^*)
\end{align}
Note that $\mu _{d\rightarrow e}$ and $\mu _{b\rightarrow c}$ were already computed. Without traceback, $x_1$ and $x_2$ are decoded at variable nodes ``d'' and ``b'' respectively as follows:
\begin{align*}
\mu _{f\rightarrow e}(x_3)&=\mu _{g\rightarrow f}(x_3)\\
\mu _{e\rightarrow d}(x_1,x_3)&=\mu _{f\rightarrow e}(x_3) \delta (x_1+x_3,y_{32-43})\\
x_1^*&=\underset{x_1}{\mathtt{supt}}\bigvee _{x_3}\,\mu _{e\rightarrow d}(x_1,x_3) \mu _{c\rightarrow d}(x_1,x_3)\\
\mu _{d\rightarrow c}(x_1,x_3)&=\mu _{e\rightarrow d}(x_1,x_3)\\
\mu _{c\rightarrow b}(x_2,x_3)&= \bigvee _{x_1}\,\mu _{d\rightarrow c}(x_1,x_3)\delta (x_1+x_2,y_{31-43})\\
x_2^*&=\underset{x_2}{\mathtt{supt}}\bigvee _{x_3}\,\mu _{c\rightarrow b}(x_2,x_3) \mu _{a\rightarrow b}(x_2,x_3)
\end{align*}
Similarly $x_4$ and $x_5$ can be obtained. The number of semiring operations required to compute all the messages passed and states computed is given in Table~I.

\begin{table}[ht]
\normalsize
  \centering

  \begin{tabular}{>{\centering}m{0.2in} >{\centering}m{1in} >{\centering}m{0.6in} >{\centering\arraybackslash}m{0.6in}}
    \toprule
  & \textbf{Messages/States} & \textbf{No. of $\bigwedge$} & \textbf{No. of $\bigvee$} \\
    \midrule
C1 & $\mu _{k\rightarrow j},\mu _{j\rightarrow i}$ & $0$ & $0$ \\
C2 & $\mu _{a\rightarrow b},\mu _{b\rightarrow c}$ & $0$ & $0$ \\
C3 & $\mu _{i\rightarrow h},\mu _{c\rightarrow d}$ & $q^3$ & $q^2(q-1)$ \\
C4 & $\mu _{h\rightarrow g},\mu _{d\rightarrow e}$ & $0$ & $0$ \\
C5 & $\mu _{e\rightarrow f},\mu _{g\rightarrow f}$ & $q^2$ & $q(q-1)$ \\ \hline
C6 & $x_3^*$ & $q$ & $0$ \\
C7 & $x_1^*,x_4^*$ & $q$ & $0$ \\
C8 & $x_2^*,x_5^*$ & $q$ & $0$ \\ \hline
C9 & $\mu _{f\rightarrow e},\mu _{f\rightarrow g}$ & $0$ & $0$ \\
C10& $\mu _{e\rightarrow d},\mu _{g\rightarrow h}$ & $q^2$ & $0$ \\
C11& $x_1^*$ & $q^2$ & $q(q-1)$ \\
C12& $\mu _{d\rightarrow c},\mu _{h\rightarrow i}$ & $0$ & $0$ \\
C13& $\mu _{c\rightarrow b},\mu _{i\rightarrow j}$ & $0$ & $0$ \\
C14& $x_2^*$ & $q^2$ & $q(q-1)$ \\
    \bottomrule
    \end{tabular}
    \vspace{5pt}
\caption{}
\vspace{-20pt}
  \end{table}

The number of computations required with traceback are 2 (C1+C2+$\ldots$+C5)+C6+C7+C8 which is $2(q^3+q^2)+5q=180$ products ($\wedge$) and $2(q^2(q-1)+q(q-1))=120$ sums ($\vee$).Without traceback, the number of operations required are 2(C1$\ldots$+C5)+C6+2(C9+C10)+C11+2(C12+C13)+C14 which is $2q^3+6q^2=224$ products  and $2q^2(q-1)+4q(q-1)=144$ sums. Thus, running single-vertex SP algorithm followed by traceback step affords computational advantage over multiple-vertex version.

\hfill $\square$
\end{example}

\section{Discussion}
In this paper, we proposed a SP algorithm based decoder for decoding NC. Subsequently, a method for constructing the factor graph for a given sink node using the global encoding maps (or vectors in case of LNC) of the incoming edges and demands of the sink was provided. The graph so constructed had fewer nodes and led to fewer message being passed lowering the number of operations as compared to the scheme of \cite{Salmond}. Next we discussed how cycles in factor graph affect the solution of the MPF problem and illustrated with examples how to circumvent them. We introduced and discussed the advantages of traceback over multiple-vertex SP algorithm. Next, for the sinks demanding all the source messages, we introduced the concept of fast decodable network codes and provided a sufficient condition for a network code to be fast decodable.

redpajama_set_name: RedPajamaArXiv


text:
Monday Brief: Crypto goes to Washington
Trump's "not a fan" and Congress raked Calibra's David Marcus over the coals, but this is the start, not the end, of digital currency.
Photo: Caleb Perez on Unsplash
Last week, David Marcus, head of Calibra and Facebook's point man for its proposed Libra currency, underwent two days of grilling by Congress.
Facebook's ambitious crypto-currency proposal has unleashed all manner of skepticism. There are a lot of genuine questions and concerns over a powerful corporation with 2.7 billion users issuing money. Here's a short clip of Marcus getting raked over the coals by a Congressman, Sean Duffy (R-Wisconsin), who makes a theatrical statement against the proposal.
Duffy, brandishing a U.S. twenty-dollar note, extolled the virtues of anyone being able to use it, and wanted to know whether Facebook might be able to determine who can use its money, just as it can expel certain individuals from using its site.
Marcus came off weaselly in that exchange. But Marcus had to remain non-committal, because Duffy (whether he knew it or not) , exposed one of the biggest challenges: the tension between the need for privacy, and the demands any financial service must meet to know its clients.
If Marcus said yes, Facebook will have the right to say who can and can't use Libra, he'd reveal excessive corporate power. But if Marcus had said no, then he would be opening the company to charges of negligence. (The more interesting answer would have been to note that some massive percentage of U.S. dollar bills are dusted with traces of cocaine, but that probably wouldn't have endeared him to the House Committee on Financial Services.)
President Trump and Fed Reserve governor Jay Powell also made statements against crypto-currencies. Trump tweeted that he's "not a fan" of bitcoin and suggested Facebook should get a bank license. (Calibra has applied for a money-operator license.)
The attacks have also come from the political left. Here's Nobel economist Joseph Stiglitz denouncing Libra as a Facebook plot to let bad actors launder money. But his biggest argument turns on the lack of competition in payments, the oligopoly in payments technology for high fees – yet he seems oblivious to the point that Libra could be a real competitor. He also takes an odd tangent on its stablecoin blueprint; I don't think Stiglitz is on firm ground here.
Stiglitz's arguments seem flawed but he makes the central point that a currency requires trust, and he wonders if Facebook is worthy of our trust.
The clash in Washington appears to have caused Facebook and Calibra some pause. The tone has shifted, from Libra as a decentralized, bitcoin-like system to one in which the Libra Association would have to assume a more direct governance role.
And CNBC's Jim Cramer says that although he likes the idea of Libra, he thinks Facebook made a political mistake and should buy Square instead, if it wants to be in payments.
But I don't quite agree. Cramer's looking at this from the point of a shareholder in tech companies, and he's just interested in short-term stock movements. Facebook's probably the most vulnerable Big Tech company to regulation and even the possibility of a forced breakup, and Libra may not win it new friends in Washington. But the vision behind Libra is far bigger than just getting a chunk of credit-card fees.
Money is a cultural and political concept, as much as a technological or financial one. It's important to raise questions about Libra, its governance, the balance of privacy and law enforcement, and its potential impact on currencies, financial markets, and banking services.
For the past few years, as regulators around the world have grasped with crypto-currencies, we have seen offshore centers such as Hong Kong, Gibraltar and Switzerland take the lead. The only large, onshore financial market that has taken a friendly approach is Japan, but regulators there have lost some of their ardor.
The U.S. SEC and CFTC have taken two stances that have chilled the industry: essentially declaring all tokens securities, and then bringing the hammer down on players accused of breaching securities rules. It's one reason why a lot of the innovation has taken place elsewhere.
But now Facebook has forced a discussion about digital money at the highest levels. The outcome is likely to be a deal to support Libra in ways that regulators and Congress believe doesn't threaten the dollar and threads the needle of privacy, universality, fungibility, and KYC rules.
If so, this will put digital, programmable money at the heart of the international financial system. But it will also prove a shock to the vast majority of blockchain and crypto companies, because the rules of the game will have changed, and they'll be living in Facebook's world.
Related Topics:BitcoinblockchainCalibracrypto-currencyDavid MarcusDonald TrumpFacebookfeaturedJim CramerJoseph StiglitzLibraregulationUSA
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Leap Day Legs | Nourish. Nature. Love.
Nourish your body. Live through nature. Love your life.
A workout dedicated to Leap Day! (and your legs)!
Do 29 reps of each exercise and then repeat the circuit as many times as you can in 29 minutes! Focusing on plyometrics-jumping, leaping movements, and getting your heart rate up. Only take as much rest as you absolutely need, not want, in between each exercise.
welcome. this is the place where i share healthy tips-for the mind, body and spirit. whether it be fitness tips, nutrition tips, encouraging quotes, my own personal healthy happenings, what have you. enjoy.
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HomeEntertainmentAFRIMA: Burna Boy wins 2022 artiste of the year award
AFRIMA: Burna Boy wins 2022 artiste of the year award
Nigerian singer and music producer, Damini Ogulu, popularly known as "Burna Boy", has won the All-Africa Music Awards (AFRIMA) artiste of the year 2022.
Burna boy, who was absent during the award presentation in Dakar, Senegal, also won the award for the album of the year 2022.
The News Agency of Nigeria (NAN) reports that the 8th AFRIMA, tagged the 'Teranga' edition, had lots of Nigerian artistes win several categories of the award.
David Adeleke, known as "Davido", won the best artiste duo or group in African Electro; the best male artiste in African inspirational music and artiste with the best African collaborations.
While Adekunle Gold got the best artiste duo or group in African pop, Wizkid emerged winner of the best male artiste in West Africa.
Also, Ismael Lo from Senegal won the AFRIMA legend award, Brazil's Anitta got the best global act award while the best soundtrack in movies, series and documentaries went to Tems.
Azaya from Guinea emerged the best artiste, duo or group in African traditional music and Cavemen featuring Made Kuti from Nigeria got the award as the best African duo, group or band.
For the best male artiste in Northern Africa, ALA from Tunisia emerged winner while Gutu Abera from Ethiopia won the best artiste duo or group in African Jazz.
Morocco musician, Faouzia, bagged the best female artiste in Northern Africa and Best African act in diaspora (female category).
Also, Fally Ipupa won the best male artiste in central Africa while Teddy Afro from Ethiopia became the best artiste duo and group in African raggae, ragga and dancehall music.
NAN reports that African artistes including PSquare, Fave, Black O, Black Sheriff and Diere, among others entertained their fans with energising performances.
Adekunle Gold
AFRIMA
AFRIMA: Ugandan DJ Lito wins best African DJ award


Commissioned by Show me design & art gallery and in collaboration with Sala da Frente this exhibition is composed by a series of pieces specially designed for this event.
This collection was developed during a four days retreat which took place at Casa do Adro in Barcelos.
Melt Project started in 2006 as a reflection about plastic consumption in contemporary society. This object aim to redefine the life of plastic disposals and show how it can be transformed into alternative everyday objects.


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Home > Releases > Unemployment in States and Local Areas (all other areas) > Civilian Labor Force in Jefferson County, KY
Civilian Labor Force in Jefferson County, KY (KYJELFN)
Nov 2022: 401,893 | Persons | Monthly | Updated: Dec 29, 2022
Nov 2022: 401,893 (+ more)
Nov 2022: 401,893
Oct 2022: 402,075
Sep 2022: 403,957
Aug 2022: 405,574
Jul 2022: 403,338
Persons,
Civilian Labor Force in Jefferson County, KY 2005-06-08 2022-12-29
Unemployment in States and Local Areas (all other areas) 2005-06-08 2022-12-29
Persons 2016-03-18 2022-12-29
These data come from the Current Population Survey (CPS), also known as the household survey.
Civilian Labor Force includes all persons in the civilian noninstitutional population ages 16 and older classified as either employed or unemployed.
Employed persons are all persons who, during the reference week (the week including the 12th day of the month), (a) did any work as paid employees, worked in their own business or profession or on their own farm, or worked 15 hours or more as unpaid workers in an enterprise operated by a member of their family, or (b) were not working but who had jobs from which they were temporarily absent because of vacation, illness, bad weather, childcare problems, maternity or paternity leave, labor-management dispute, job training, or other family or personal reasons, whether or not they were paid for the time off or were seeking other jobs. Each employed person is counted only once, even if he or she holds more than one job.
Unemployed persons are all persons who had no employment during the reference week, were available for work, except for temporary illness, and had made specific efforts to find employment some time during the 4 week-period ending with the reference week. Persons who were waiting to be recalled to a job from which they had been laid off need not have been looking for work to be classified as unemployed.
For more details, see the release's https://www.bls.gov/lau/laufaq.htm>frequently asked questions.
Unemployment in States and Local Areas (all other areas)
Counties by State, Monthly, Not Seasonally Adjusted: Kentucky
Unemployment Rate in Jefferson County, KY
Civilian Labor Force in Jefferson County, KY
Annual, Not Seasonally Adjusted
Jefferson County, KY Counties Kentucky States U.S. Regional Data
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Jefferson County, KY Louisville Kentucky Civilian Labor Force Labor County or County Equivalent Bureau of Labor Statistics Monthly Not Seasonally Adjusted Public Domain: Citation Requested United States of America


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GENERAL LEE'S COLLEGE
A bland history of Washington and Lee University written by a faculty member who proceeds chronologically without much anecdotal relief. Crenshaw describes the Virginia institution's founding in 1749 as Augusta Academy, its frequent changes of name and president, and its consistently moderate Southern character. He stresses the importance of two events: the changing of the college's name to Washington Academy in 1798 (following a bequest from George Washington) and the arrival of Robert E. Lee as president in 1865. The former provided funds to enlarge college programs, the latter provided post-Civil War prestige. More interesting, however, is the college's mid-18th century stance: ""mildly pro-slavery,"" its suppression of a former president's history for forty years because of his anti-slavery views, and the one brief era of ""progressive"" education (1830-1834) when all classes were voluntary. There is special mention of the law school and of the gradual broadening of the curriculum from the standard classical education to the now standard liberal arts cum science program.
Pub Date: May 8th, 1969


It's been a busy few days. Let me start by showing you my quilting progress! I Pieced 20 more of my Windmill blocks while Richard was doing some adjusting on my longarm. These 20 bring my total to 320...only 208 more to go!!!! LOL!
I did get this customer quilt finished. This is the one that I posted in the last post. It was hand pieced out of t-shirts and polyester. Soooo stretchy to quilt.
After he did some adjusting, I loaded this little quilt of mine. Not really pieced just a piece of airplane fabric with a border sewn around it. A nice sized baby quilt. This makes me 5 of my own for the year. If you remember I have a goal to quilt 15 of my own this year. So I'm off to a pretty good start.
The tension was almost perfect but still a hair off. You can't really tell it on this quilt because I redid any issues that I had on it.
I also loaded another one of mine on the longarm but did get a pic of it. It's not finished so I will get a pic of it then. It will make number 6 quilted for the year!
Now just a few various pics... in this laundry basket I have a few of the quilts that are coming up to either be pieced or quilted. You can see a bunch of t-shirts in there. Those will be used for a t-shirt quilt that I will be making for a childhood friend of mine. She contacted me a few weeks ago asking if I could do it. The blue and white is a hand pieced bow tie quilt that belongs to a customer. It needs to be quilted and the binding done. Underneath that is my Courthouse Steps that also needs to be quilted. I have the backing and batting ready and the binding is already cut. I'm very anxious to get it quilted.
Then over on my cutting table I have a few more projects. First you see my Jacob's Ladder that needs the borders sewn on then quilted. The Churn Dash needs the binding done.
It is a cold weekend so I made a big pot of chili for us to eat. We got a little snow Thursday night so Hunter didn't have school on Friday. A long weekend for him. Richard is also off this weekend but Andrew has to work all weekend.
In other news...my momma went back to work. It's a long story that I will not completely get into. She wasn't able to have her surgery if you remember because of her blood pressure and blood sugar being too high. So that was put off until it was under control. She had to go back due to some insurance reasons. I talked to her today and she was having some pain and had ice on the shoulder. Not good at all. So we will just have to wait to see how this all goes for her. She is planning on retiring later this year as well.
Printed Notebooks – Are They Right for your Business?


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Home ⁄ Articles ⁄ "PRIME" MINISTER.. By Desmond Kelly 'the Star of eLanka'
"PRIME" MINISTER.. By Desmond Kelly 'the Star of eLanka'
Nov 18, 2017 Posted by eLanka admin In Articles, DESMOND KELLY	Tagged "PRIME" MINISTER.. By Desmond Kelly, "PRIME" MINISTER.. By Desmond Kelly 'the Star of eLanka', Desmond Kelly	Comments 0
Something a little bit different, especially for those Lankan/Aussies & ex-Patriots around the World, who are still "Internetless" for any reason whatsoever, I take my chance to pay a "sincere tribute" to the Man who, I believe, was the "Prime" Minister of Ceylon, Mr.D.S. (Don Stephen) Senanayake, the very first, & to my knowledge, the very best "Father of Ceylon", as he was known.
Personally, I do not "think, too much" about Politicians., in total, nowadays. I do not think about "POLITICS" at all, if I can help it, either in Sri Lanka OR Australia, but, unfortunately, everything stems from the "Political Situation" in one's Country of residence.
Strangely, however, the first two individual Prime Ministers I remember who were undoubtedly the very best in the two Countries I have been involved with, Ceylon & Australia. In Ceylon, the best was D.S.SENANAYAKE & , in Australia, R.G.MENZIES. Their "Parties" do not matter in the slightest, their "Personalities", mattering the most. This "story" features the "Father of Ceylon" the Right Honourable Don Stephen Senanayake.
Firstly, the word "caste" is thoroughly distasteful to me, but this "term" was very important to most Sri Lankans. Just the one example, people of "high-caste"
were not supposed to marry people of "low-caste".
Also, "Surnames" in S.L. seems to indicate "caste" and the name "Senanayake" was right up there in the top echelons, so to speak. Especially amoungst the Sinhalese, there were many "high-caste" Names. To be honest, I really don't know any "low-caste" ones.
Anyway, Don Stephen was born in Ceylon on the 20th of October 1883, in the little town of Mirigama, which, in turn, boasted a littler Village with the strange name of "Botale"(English translation :- "Miri-Town of the bottle"). His father was Mudaliyar Don Spater Senanayake & his mother was Dona Catherina Elizabeth Perera Gunasekera Senanaayake, simply called Dona. His siblings were two brothers, Fredrick Richard & Don Charles & one sister, Maria Frances.
He completed his education at the renowned, private Anglican-boys' school, St.Thomas' College, Mutwal, moving to St.Thomas' Mount Lavinia (in other words, he was always a proud "Thomian" right from the start..
Later, he joined the Surveyor General's Department as a clerk, after which he joined his father's rubber plantation, where he worked as a partner with his dad.
I am not certain if it was an "arranged" marriage, but his partner in life was formerly Miss Molly Dunuwila.
They had one son, Dudley Senanayake. Because there isn't too much that I can write about Dudley's mum, all I can say is that very much later, there was a song written for her called " Good golly, Miss Molly" and, about Dudley, all I know is that, like his father, he was a kind, honourable guy who followed in his dad's footsteps to become the next P.M., but, unlike his father, he wasn't a strong leader & was "replaced" very quickly, but did stage a "come-back" to the job, a few years after..
In December, 1947, Britain decided to give Ceylon it's independence & after being instrumental in achieving this independence without any political mayhem at all, D.S.Senanayake became the very first Prime Minister of the Country after the 4th of February 1948. He "served" the Island, becoming famous around the World, as "The Father Of Ceylon", in my opinion, he not only "looked the part", he WAS a leader in every sense of the word. Those were the days of British Ceylon, English was the main Language, and "The Right Honourable" P.M.D.S.Senanayake looked more like a "Singlishman" as I would call him, especially as he rode his horse on Galle-face Green regularly, very unfortunately, this pleasent habit cost him is life on the 22nd of March, 1952, when the horse stumbled, the P.M. was thrown off & died as a result, leaving his wife, family & the entire Nation grieving the death of a superb man, an intelligent one and, as far as I am concerned, a "PRIME" MINISTER in anyone's book.
Desmond Kelly.
Star.of eLanka.
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Sri Lanka Design Festival (SLDF) - 2020 15 - 17 January 2021
Australia Day Celebrations at the BBA
358 haughto Road, Clayton 3169
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Ore was wonderful to work with, and even though I was VERY scantily clad I found myself actually relaxing and having fun in the moment. She gave directions on where to put everything from my feet to my hands (which was great, because I normally never know what to do with my hands — I'm awkward AF). Not only did I feel empowered when I saw the photos, but the experience itself during the shoot made me feel like a beautiful woman, both inside and out. Ore's cheers from behind the camera are good for the ego (trust me). There was one sneak peek at the pictures during our session that had me squealing "OMG IS THAT ME?!?" and the proud smile on her face was priceless. You can tell she really loves doing this.
Thank you so much, Ore!! These photos are all beautiful. I'm so thankful and [my girlfriend] is super happy with them, too. You rock. Keep it up, because you have a gift.
Working with Ore is always such a breeze. She has the perfect mixture of professionalism and fun working behind the camera. She knows how to capture those candid moments by knowing when to give you words of encouragement and when to just keep taking pictures because whatever you're thinking while you're in the moment is perfect and she knows not to mess it up. She has an eye for angles and how to make you look absolutely amazing you'll wonder if it's even you in that picture, I'm not kidding! Every session I've had always goes by so quickly, we talk, we pose, we talk some more. Before I know it she has what she needs and we are done. It's happened at the end of a shoot I'm about to hug her thank you and then realized I didn't have anything on but my shoes! Her personality and vibe make for a very nonjudgmental environment where you can freely express how you feel and she captures it perfectly.


Professionalism in dealing Cassioli foods and beverages in total safety.
Today, for food companies that deal in a highly competitive and modern, it is essential to keep costs and timing of the supply chain but, above all, it is essential to ensure the safety and health of consumers treating foods with the right mode suitable to preserve its freshness by preventing the risk of contamination and deterioration.
The automation systems of storage and picking is the added value that ensures the entire chain of the food industry To achieve high performance security and profitability of the plant is the goal to be reached to allow, in the long run, an overall improvement of the efficiency of the company is reflected in significant competitive advantages over competitors.
CASSIOLI, among dozens of systems installed worldwide, has developed customized solutions for its clients, depending on the food to transport and store, installing plants partially or fully automated.


As with previous years, the Special Events category is particularly variable as it is dependent on whether performances that cannot be classified into other categories take place.
The Special Events category experienced a substantial increase in revenue by 44.1% from $20.3m in 2015 to $29.3m in 2016. Attendance in the Special Events category declined substantially by 66.8%. The average ticket price increased by 304.4% from $42.26 to $170.90. The increase in revenue and average ticket price were primarily driven by the Royal Edinburgh Military Tattoo performances held in Melbourne. This was the first time that the Royal Edinburgh Military Tattoo performed in Victoria, and prior to this had only performed outside of the UK on three occasions.
Victoria was the only state to experience an increase in revenue of 340.2% and an increase in attendance of 34.7%.


text:
Q: Uso de Break no Python Eu fiz esse código que vai passando de fase, porém, quando se erra eu usei o Break para finalizar o jogo... mas ele apresenta a mensagem de Game Over e mesmo assim continua para a próxima fase... alguém pode me ajudar?
Segue o código:
#menu
hotel = [['*', '*', '_', 'G'], ['R','_', '*', '*']]

print('Seja bem vindo ao hotel dos animais!')
nome = input('Qual o seu nome, para que possamos iniciar? ')

print('{}, vamos iniciar nossa jogatina!...:'.format(nome))
sleep(5)
regras= [['Regras: O rato não pode ficar ao lado do gato.'],['O cão não pode ficar ao lado do osso.'], ['O gato não pode ficar ao lado do cão.'], ['O queijo não pode ficar ao lado do gato.']]
for linha in regras:
  print(linha)
sleep(5)

#fase1
print('Na fase 1 você deve alocar o Rato e o Gato na seguinte matriz que representa os quartos')

sleep(5)

for linha in hotel:
  print(linha)

sleep(5)

posRato = int(input("Qual a posição que o Rato deve ficar? "))
posGato = int(input('Qual a posição que o gato deve ficar? '))
while True:
  if posRato == 5 and posGato == 4:
    print('Parabéns {}, você segue para a fase 2!'.format(nome))
  else:
    print('Ops {}, infelizmente você perdeu!'.format(nome))
    break

A: Parece que o trecho que cita na dúvida não está descrito no código, mas o break irá parar o loop do while, se o restante do código estiver com a mesma indentação não será executado, mas caso esteja fora do while irá ser executado mesmo com o break, pois ele está parando somente o loop.

A: Você não colocou o break na condição correta que você quer que ele pare o programa, por exemplo se você colocar o break depois do print do if, ele vai fazer com que o programa pare sem executar o else.

redpajama_set_name: RedPajamaStackExchange


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