# Fiduccia-Mattheyses algorithm

A classical approach to solve the Hypergraph bipartitioning problem is an iterative heuristic by Fiduccia and 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.
Example of FM

## 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