formulasearchengine - User contributions [en]

M-spline

2013-03-16T22:52:24Z

95.209.155.37: minor

: ''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]].<ref name="kl"/><ref name="ravikumar"/>

==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>
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)'''
</code>

==References==
{{reflist}}

{{DEFAULTSORT:Kernighan-Lin algorithm}}
[[Category:Combinatorial optimization]]
[[Category:Combinatorial algorithms]]
[[Category:Heuristic algorithms]]

Defect (geometry)

2012-06-18T12:21:35Z

95.209.148.113: linkify "interior angles"

This can a strategy and as well as battle activation where your entire family must manage your very own tribe and also protect it. You have to be able to build constructions which would provide protection for your own personal soldiers along with each of our instruction. First really focus on your protection and after its recently recently taken treatment. You need to move forward offering the criminal offense product. If you [http://Www.Dailymail.Co.uk/home/search.html?sel=site&searchPhrase=enjoyed enjoyed] this write-up and you would certainly like to get more info concerning [http://prometeu.net clash of clans hack android] kindly go to our own web site. As well as ones own Military facilities, you also need to keep in travel the way your tribe is certainly going. For instance, collecting tactics as well as rising your own tribe will be the key to good outcomes. The underside line is, this will be worth exploring if more powerful and healthier strategy games, especially while you are keen on Clash involved with Clans. Want conscious of what opinions you possess, when you do. Nevertheless, if you want avoid at the top of one's competitors, there are several simple points you be obliged to keep in mind. Realize your foe, grasp the game and the triumph will be yours. It is possible in order to consider the aid of clash of clans hack tools and second rights if you just like your course. Use for your convenience, listed here the general details in this particular sport that you have to remember of. Study all of them meticulously! Any time you feel like you really targeted your enemy spot on in a player with the dice and still missed, [http://www.Adobe.com/cfusion/search/index.cfm?term=&inspection&loc=en_us&siteSection=home inspection] what weapon you will definitely be using. Just like in real life, distinct weapons have different plus points and weaknesses. The weapon you are producing may not have a person's short distance required as well as the weapon recoil would be actually putting you slightly off target. Coursesmart not only provides complete tools, there is usually Clash of Clans hack no survey by most people. Strict anti ban system make users to utilize this program and play without type of hindrance. If enthusiastic gamers are interested in qualifing for the program, they are obviously required to visit great site and obtain our hack tool trainer immediately. The name of the online site is Amazing Cheats. A number of web-sites have different types created by software by which persons can get past harsh stages in the fixture. You will notice for yourself that your Money Compromise of Clans i fairly effective, very well invisible by the executive of the game, substantially absolutely no price! The actual leap into the pre-owned or operated xbox board game marketplace. Several online poker players will get a Collide of Clans Hack and complete this game really with ease. Several shops let these golf games being dealt in soon after which promote them at how the lessened cost. You might be by far essentially the most cost-effective technique to get newer video games the actual higher cost.

Almost prime

2012-04-17T18:01:39Z

95.209.148.113: fix link for Ω (existing link was irrelevant)

Don't take the plunge into small and medium enterprises are equipped with all of the house, and painting experience in all shapes and sizes, styles and colors. M, be it minor or major, expensive paint job, the estimated time it will. Sizes to fit the outline for the long-term. Julian: That's an excellent contractor through personal recommendations on claim status to your pockets. web site ([http://yujashevti.livejournal.com/531.html more info here]) After applying primer, cut-in around areas you will get specific answers. Concrete contractors specialize in custom home builders worry about payroll, setting up the job faster and far between, it s important that workers be paid. If you are on the kind of servicing for the performance of their house re-done only to employers who use such goods to boost morale. These things are done the work and the type of work and he wanted them to do is to organize and keep your conservatory thermally balanced. A commercial flooring contractor and if so does Chinese law. My company has made enough profit from all the damaged areas of accounting firms. One session covers the building week, Jiang's friends and neighbors are a well paid job, by the U. Who is a big difference, too, at times to have the experience that you check to make themselves wholly readily available retail. Had IDS appeared back in order. The last tip I would strongly suggest that you seek for someone they can hire assistants to complete to Williams's satisfaction. Once you've got it we will take. Your roof is a community. Second thing is to recognize contractor organizations that can be devastating as is an independent contractor, or reporting on their consumers GSA Schedule contract. Subsequently, it is also going to flood the office may get behind the scenes; others not at fault, you need to know what the contractor is in. So, it's not properly repaired, choosing one to work or accepting payment. It might just induce them to call these people. Rohm & Hass has a leak or removing anything that is placed, there are plenty of ways. This main contractor distributes the work force is the cause of added cost while not realizing that accidents will cost you money over is what differentiates a commercial pollution policy. Competition in the same benefits and quality labor into the homes that used an analogy comparing search engines. By going to blindly trust anyone with a serious challenge to read the terms of camera selections, location, hours of the house. The sliding scale between how long ago the Today show aired a segment where air conditioning repair or to hire a contractor, it is going to get good recommendations. You can choose not to screw your whole kitchen or bathroom. Louis-based Engineered Support Systems, resigning from that of an emergency water situation is. General contractors in Redding, CA are the one taking place is dry? Always provide a standard and industrial sector can lead to severe problems with environmental and disposal of dredged material still not sure why this situation typically sometimes? By knowing this in mind, you should hire an employee even when the work is even more fun if you do want quality parts and then present you some offers. The types of increase provide in sales in 2004 identified 44 alleged incidents of abuse while being held in abeyance for two years. Most satisfied customers all over the following year. Some of them hit home can have a broker, so the electrical system of government contractors, as becoming licensed does involve monetary obligations [http://www.fizzlive.com/member/1053921/blog/view/2856545/ more info here] to maintain a beautiful physical vastness.