A fast algorithm for finding interlocking sets

Abstract

Consider a set of intervals on the real line. Two such intervals [i, j] and [k, m] interlock if either i < k <j < m or k < i < m <j. An interlocking set is a set of intervals in which every two intervals are related in the symmetric-transitive closure of the interlock relation (Fig. 1). The problem considered here is to partition a set of intervals into maximal interlocking sets. An obvious approach to the problem of locating interlocking sets is to construct an interlock graph having one vertex for each interval and an edge between vertices x and y if and only if intervals x and y interlock. The maximal interlocking sets then correspond to the connected components of this graph. Constructing the interlock graph in a straightforward fashion requires time O(n2). In this paper we show how to find the interlocking sets in time ’ O(na(n)). Our method assumes that the endpoints of the intervals are presented in sorted order-if not, then the time is O(n log n). The determination of interlocking sets has an application in single-row routing [3,6]. There, a set of two-point nets lying in a single row is given. The goal is to find a routing of the nets that minimizes the maximum channel width (provided a routing exists). A solution to this routing prob-