Kernighan–Lin algorithm

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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(n2 log(n)) 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[edit]

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_{old} - T_{new} = D_{a} + D_{b} - 2c_{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_{old} - T_{new} and then executes the operations, producing a partition of the graph to A and B.[1]

Pseudocode[edit]

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[edit]

  1. ^ a b Kernighan, B. W.; Lin, Shen (1970). "An efficient heuristic procedure for partitioning graphs". Bell System Technical Journal 49: 291–307. doi:10.1002/j.1538-7305.1970.tb01770.x. 
  2. ^ a b Ravikumār, Si. Pi; Ravikumar, C.P (1995). Parallel methods for VLSI layout design. Greenwood Publishing Group. p. 73. ISBN 978-0-89391-828-6. OCLC 2009-06-12.