M-spline: Difference between revisions

Revision as of 23:52, 16 March 2013

This article is about the heuristic algorithm for the graph partitioning problem. For a heuristic for the traveling salesperson problem, see Lin–Kernighan heuristic.

Kernighan–Lin is a O(n³ ) heuristic algorithm for solving the graph partitioning problem. The algorithm has important applications in the layout of digital circuits and components in VLSI.^[1]^[2]

Description

Let $G (V, E)$ be a graph, and let $V$ be the set of nodes and $E$ the set of edges. The algorithm attempts to find a partition of $V$ into two disjoint subsets $A$ and $B$ of equal size, such that the sum $T$ of the weights of the edges between nodes in $A$ and $B$ is minimized. Let $I_{a}$ be the internal cost of a, that is, the sum of the costs of edges between a and other nodes in A, and let $E_{a}$ be the external cost of a, that is, the sum of the costs of edges between a and nodes in B. Furthermore, let

D_{a} = E_{a} - I_{a}

be the difference between the external and internal costs of a. If a and b are interchanged, then the reduction in cost is

T_{o l d} - T_{n e w} = D_{a} + D_{b} - 2 c_{a, b}

where $c_{a, b}$ is the cost of the possible edge between a and b.

The algorithm attempts to find an optimal series of interchange operations between elements of $A$ and $B$ which maximizes $T_{o l d} - T_{n e w}$ and then executes the operations, producing a partition of the graph to A and B.^[1]

Pseudocode

See ^[2]

 1  function Kernighan-Lin(G(V,E)):
 2      determine a balanced initial partition of the nodes into sets A and B
 3      A1 := A; B1 := B
 4      do
 5         compute D values for all a in A1 and b in B1
 6         for (n := 1 to |V|/2)
 7            find a[i] from A1 and b[j] from B1, such that g[n] = D[a[i]] + D[b[j]] - 2*c[a[i]][b[j]] is maximal
 8            move a[i] to B1 and b[j] to A1
 9            remove a[i] and b[j] from further consideration in this pass
 10           update D values for the elements of A1 = A1 \ a[i] and B1 = B1 \ b[j]
 11        end for
 12        find k which maximizes g_max, the sum of g[1],...,g[k]
 13        if (g_max > 0) then
 14           Exchange a[1],a[2],...,a[k] with b[1],b[2],...,b[k]
 15     until (g_max <= 0)
 16  return G(V,E)

References

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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[1]

[2]

@@ Line 1: / Line 1: @@
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+: ''This article is about the heuristic algorithm for the graph partitioning problem. For a heuristic for the traveling salesperson problem, see [[Lin–Kernighan heuristic]].''
+'''Kernighan–Lin''' is a O(n<sup>3</sup>&nbsp;) [[heuristic algorithm]] for solving the [[graph partitioning problem]]. The algorithm has important applications in the layout of digital circuits and components in [[VLSI]].<ref name="kl"/><ref name="ravikumar"/>
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+==Description==
+Let <math>G(V,E)</math> be a graph, and let <math>V</math> be the set of nodes and <math>E</math> the set of edges. The algorithm attempts to find a partition of <math>V</math> into two disjoint subsets <math>A</math> and <math>B</math> of equal size, such that the sum <math>T</math> of the weights of the edges between nodes in <math>A</math> and <math>B</math> is minimized. Let <math>I_{a}</math> be the ''internal cost'' of ''a'', that is, the sum of the costs of edges between ''a'' and other nodes in ''A'', and let <math>E_{a}</math> be the ''external cost'' of ''a'', that is, the sum of the costs of edges between ''a'' and nodes in ''B''. Furthermore, let
+:<math>D_{a} = E_{a} - I_{a}</math>
+be the difference between the external and internal costs of ''a''. If ''a'' and ''b'' are interchanged, then the reduction in cost is
+:<math>T_{old} - T_{new} = D_{a} + D_{b} - 2c_{a,b}</math>
+where <math>c_{a,b}</math> is the cost of the possible edge between ''a'' and ''b''.
+The algorithm attempts to find an optimal series of interchange operations between elements of <math>A</math> and <math>B</math> which maximizes <math>T_{old} - T_{new}</math> and then executes the operations, producing a partition of the graph to ''A'' and ''B''.<ref name="kl">{{cite journal|first1=B. W.|last1=Kernighan|authorlink1=Brian Kernighan|first2=Shen|last2=Lin|authorlink2=Shen Lin| year = 1970 | title = An efficient heuristic procedure for partitioning graphs | journal = Bell Systems Technical Journal|volume=49|pages=291–307}}</ref>
+==Pseudocode==
+See <ref name="ravikumar">{{cite book|last=Ravikumār|first=Si. Pi|coauthors=Ravikumar, C.P|title=Parallel methods for VLSI layout design|publisher=Greenwood Publishing Group|year=1995|pages=73|isbn=978-0-89391-828-6|oclc=2009-06-12|url=http://books.google.com/?id=VPXAxkTKxXIC}}</ref>
+<code>
+  '''function''' Kernighan-Lin(''G(V,E)''):
+      determine a balanced initial partition of the nodes into sets A and B
+      A1 := A; B1 := B
+      '''do'''
+         compute D values for all a in A1 and b in B1
+         '''for (n := 1 to |V|/2)'''
+            find a[i] from A1 and b[j] from B1, such that g[n] = D[a[i]] + D[b[j]] - 2*c[a[i]][b[j]] is maximal
+            move a[i] to B1 and b[j] to A1
+            remove a[i] and b[j] from further consideration in this pass
+           update D values for the elements of A1 = A1 \ a[i] and B1 = B1 \ b[j]
+        '''end for'''
+        find k which maximizes g_max, the sum of g[1],...,g[k]
+        '''if (g_max > 0)''' '''then'''
+           Exchange a[1],a[2],...,a[k] with b[1],b[2],...,b[k]
+     '''until (g_max <= 0)'''
+  '''return G(V,E)'''
+</code>
+==References==
+{{reflist}}
+{{DEFAULTSORT:Kernighan-Lin algorithm}}
+[[Category:Combinatorial optimization]]
+[[Category:Combinatorial algorithms]]
+[[Category:Heuristic algorithms]]

M-spline: Difference between revisions

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