A Fundamental Bi-partition Algorithm of Kernighan-Lin

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Presentation transcript:

A Fundamental Bi-partition Algorithm of Kernighan-Lin by Khaled Hadi 2/23/2019 7:07 PM Khaled Hadi

Outline Introduction Problem Definition Kernighan-Lin Algorithm (K-L Algorithm) K-L Algorithm : A Simple Example K-L Algorithm : A Weighted Example Time Complexity Drawbacks of the K-L Heuristic and Conclusion 2/23/2019 7:07 PM Khaled Hadi

Introduction 2/23/2019 7:07 PM Khaled Hadi

Levels of Partitioning The levels of partitioning: system, board, chip. Hierarchical partitioning: higher costs for higher levels. 2/23/2019 7:07 PM Khaled Hadi

Circuit Partitioning Objective: Partition a circuit into parts such that every component is within a prescribed range and the # of connections among the components is minimized. More constraints are possible for some applications. Cutset? Cut size? Size of a component? 2/23/2019 7:07 PM Khaled Hadi

Problem Definition: Partitioning k-way partitioning: Given a graph G(V, E), where each vertex v belongs toV has a size s(v) and each edge e belongs to E has a weight w(e), the problem is to divide the set V into k disjoint subsets V1, V2, …, Vk, such that an objective function is optimized, subject to certain constraints. Bounded size constraint: The size of the i-th subset is bounded by Min-cut cost between two subsets: Minimize , where p(u) is the partition # of node u. The 2-way, balanced partitioning problem is NP-complete, even in its simple form with identical vertex sizes and unit edge weights. 2/23/2019 7:07 PM Khaled Hadi

Kernighan-Lin Algorithm Kernighan and Lin, “An efficient heuristic procedure for partitioning graphs,” The Bell System Technical Journal, vol. 49, no. 2, Feb. 1970. An iterative, 2-way, balanced partitioning (bi-sectioning) heuristic. Till the cut size keeps decreasing Vertex pairs which give the largest decrease or smallest increase in cut size are exchanged. These vertices are then locked (and thus are prohibited from participating in any further exchanges). This process continues until all the vertices are locked. Find the set with the largest partial sum for swapping. Unlock all vertices. 2/23/2019 7:07 PM Khaled Hadi

Kernighan-Lin Algorithm: A Simple Example Each edge has a unit weight. Questions: How to compute cost reduction? What pairs to be swapped? Consider the change of internal & external connections. 2/23/2019 7:07 PM Khaled Hadi

Properties 2/23/2019 7:07 PM Khaled Hadi

Proof 2/23/2019 7:07 PM Khaled Hadi

Kernighan-Lin Algorithm: A Weighted Example Iteration 1: 2/23/2019 7:07 PM Khaled Hadi

A Weighted Example (cont’d) Iteration 1: 2/23/2019 7:07 PM Khaled Hadi

A Weighted Example (cont’d) 2/23/2019 7:07 PM Khaled Hadi

A Weighted Example (cont’d) 2/23/2019 7:07 PM Khaled Hadi

A Weighted Example (cont’d) 2/23/2019 7:07 PM Khaled Hadi

Kernighan-Lin Algorithm 2/23/2019 7:07 PM Khaled Hadi

Time Complexity 2/23/2019 7:07 PM Khaled Hadi

Extensions of K-L Algorithm 2/23/2019 7:07 PM Khaled Hadi

Drawbacks of the Kernighan-Lin Heuristic 2/23/2019 7:07 PM Khaled Hadi