An efficient direct approach for computing shortest rectilinear paths among obstacles in a two-layer interconnection model

In this paper, we present a direct approach for routing a shortest rectilinear path between two points among a set of rectilinear obstacles in a two-layer interconnec-tion model. The previously best known direct approach for this problem takes O(n log 2 n) time and O(n log n) space. By using integer data structures and an implicit graph representation scheme, we reduce the time bound to O(n log 3=2 n) while maintaining the space bound to O(n log n). Comparing with the indirect approach for the problem, our algorithm is simpler to implement and faster when the input size is moderate. 1 Overview The problem of computing shortest rectilinear paths among a set of obstacles is an important problem in computational geometry and often arises in applied elds such as robot motion planning and VLSI wire routing. Considerable work has been done on solving this problem (see the survey 4] for results on this problem). In this paper, we consider one of its variations in VLSI: Routing a shortest rectilinear path between two points, s and d, in a widely adopted two-layer inter-connection model. This model consists of two layers: The horizontal routing layer H layer (allowing only horizontal routing) and the vertical routing layer V layer (allowing only vertical routing). On each layer, there are disjoint rectilinear non-penetrable obstacles. Each bend of a rectilinear path in this model implies a via between the two layers. Lee, Yang and Wong 5] recently proposed two interesting approaches for this problem. The rst approach (which we call the indirect approach) is to transform , in O(n log n) time and space, a problem instance of n obstacle edges in the two-layer model into one in a single layer in which both horizontal and vertical routings are allowed. Then by using any one-layer shortest path algorithm (e.g., 2, 6, 8]), a shortest rec-tilinear path in the two-layer model can be obtained in either O(n log 3=2 n) or O(n log n) time. However, as pointed out in 5], the transformation may incur heavy overhead and cause possible loss of information in the original problem instance (e.g., the number of extreme edges). Thus, the second approach (which we call the direct approach) was presented in 5], giving an easier-to-implement and faster algorithm for moderate input size. The direct approach rst constructs a shortest-path-preserving graph G 1 , and then uses Fredman and Tarjan's shortest path …