The bisection problem for graphs of degree 4 (configuring transputer systems)

Abstract

It is well known that for each k≥3 there exists such a constant c _k and such an infinite sequence {G _n} ⁿ⁼⁸_∞ of k-degree graphs (each G _n has exactly n vertices) that the bisection width of G _n is at least c _k ·n. It this paper some upper bounds on ck's are found. Let σk(n) be the maximum of bisection widths of all k-bounded graphs of n vertices. We prove that

$$\sigma _k \left( n \right) \leqslant \frac{{\left( {k - 2} \right)}}{4} \cdot n + o\left( n \right)$$

for all k=2^r, r≥2. This result is improved for k=4 by constructing two algorithms A and B, where for a given 4-degree-bounded graph G _n of n vertices

• (i) A constructs a bisection of G _n involving at most n/2+4 edges for even n≤76 (i.e., σ₄(n)≤n/2+4 for even n≤76)

• (ii) B constructs a bisection of G _n involving at most n/2+2 edges for n≥256 (i.e. σ₄(n)≤n/2+2 for n≥256).

The algorithms A and B run in O(n ²) time on graphs of n vertices, and they are used to optimize hardware by building large transputer systems.

extended abstract

On the leave of Comenius University, Bratislava

The work of this author has been supported by the grant Mo 285/4-1 from the German Research Association (DFG)