Constant-Time Algorithms for Dominating Problem on Circular-Arc Graphs

Journal of Taiwan Normal University: Mathematics, Science & Technology

1999, 44(1), 31-42

Constant-Time Algorithms for Dominating

Problem on Circular- Arc Graphs

Shun -Shii Lin Ching- Fung Lee Department of lnformation and Computer Edu回tiOD

National Taiwan Normal Univen.ity

Abstract

The objective of this paper is to solve the dominating problem on circular-arc graphs in 0(1) time. This problem has not been solved in 0(1) time before, even on the ideal PRAM model. In this paper,

we take advantage of 伽 characteristics of the P ARBS (pro翩。r arrays 叫th reconfigurable bus sys-tems), which can ∞nnect the inner buses in 0(1) time. We use 0(n 2) processors in the study. By com-bining the characteristics of PARBS and improving the methods of [14][1呵， we are able to derive ∞n­

stant-time algorithrns for this problem.

Keywords: circular-arc graphs, dominating problem, PARBS (processor arrays with reconfigurable bus systems).

I ntroduction

A graph is an ordered pair G=(V, E), where V is a finite set of n =

I

I

and Eç;;;; {抖， y) 抖， yE V, x 爭 y} is a set of m =

I

I

S2, ... , Sn.l} be a family of sets with each Si (0 ~三

1 ~三 n-1) being a set. A graph G is an intersection

graph of S if there is a one-to-one correspondence between V and S such that the vertices in V are adjacent if and only if their corresponding sets have a nonemp句 intersection [5]. There are many applications for circular-arc graphs, such as genetics

[呵， course scheduling [旬， the channel assignment problem in computer-aided design [的， and so on. These applications rise some interesting problems on circular-arc graphs. There are many related re-searches as can be found in [2] [4] [7] [11] [12] [13]. The set S is then called the intersection model of G

[1] [3] [7] [8]. When S is a set of circular-arcs on a circle, G is called a circular-arc graph [5]. A

circular-arc graph is called a proper circular-arc graph if there is no circular-arc containing the other ar臼 or

contained by the other arcs, in the given set of cir-cular-arcs. For instance, Fig. 1 gives a set of proper

circular-ar臼. If a circular-arc graph is not proper, it is called a general circular-arc graph. For instance,

Fig. 2 gives a set of general circular-arcs. Given a set A of circular-arcs, LARGE(A) denotes the set

of ar臼 which are not contained by any other ar臼

[1月. For instance, in Fig. 2, LARGE(A) = {1, 2, 3,

4, 8, 10, 11}. Note that arc 5 doesn't belong to LARGE(A), because arc 5 is contained by arc 4.

The processor am句's with reconfigurable bω 矽:5"­

tems model (abbreviated to PARBS) consists of a VLSI array of pro臼ssors connected to a reconfig-urable bus system which can be used to dynamical-Iy obtain various interconnection patterns between the processors. Each processor of P ARBS has four inner ports and outer ports. The four inner ports