Fiduccia–Mattheyses algorithm explained

A classical approach to solve the Hypergraph bipartitioning problem is an iterative heuristic by Charles Fiduccia and Robert Mattheyses.^[1] This heuristic is commonly called the FM algorithm.

Introduction

FM algorithm is a linear time heuristic for improving network partitions.New features to K-L heuristic:

• Aims at reducing net-cut costs; the concept of cutsize is extended to hypergraphs.
• Only a single vertex is moved across the cut in a single move.
• Vertices are weighted.
• Can handle "unbalanced" partitions; a balance factor is introduced.
• A special data structure is used to select vertices to be moved across the cut to improve running time.
• Time complexity O(P), where P is the total # of terminals.

F–M heuristic: notation

Input: A hypergraph with a vertex (cell) set and a hyperedge (net) set

• n(i): # of cells in Net i; e.g., n(1) = 4
• s(i): size of Cell i
• p(i): # of pins of Cell i; e.g., p(1) = 4
• C: total # of cells; e.g., C = 13
• N: total # of nets; e.g., N = 4
• P: total # of pins; P = p(1) + … + p(C) = n(1) + … + n(N)
• Area ratio r, 0< r<1

Output: 2 partitions

• Cutsetsize is minimized
• |A|/(|A|+|B|) ≈ r

See also

• Graph partition
• Kernighan–Lin algorithm

Notes and References

1. Book: Fiduccia. Mattheyses. 19th Design Automation Conference . A Linear-Time Heuristic for Improving Network Partitions . 1982. 175–181 . 10.1109/DAC.1982.1585498 . 0-89791-020-6 . https://ieeexplore.ieee.org/document/1585498. 23 October 2013.