Abstract
The compaction step of integrated circuit design motivates associating several kinds of graphs with a collection of non-overlapping rectangles in the plane. These graphs are intended to capture various visibility relations amongst the rectangles in the collection. The contribution of this paper is to propose time- and cost-optimal algorithms to construct two such graphs, namely, the dominance graph (DG, for short) and the visibility graph (VG, for short). Specifically, we show that with a collection of n non-overlapping rectangles as input, both these structures can be constructed in θ(log n) time using n processors in the CREW model.
Highlights
Two important design methodologies central to the fabrication of integrated circuits are symbolic layout and compaction
We show that any parallel algorithm solving these problems must take we note that Schlag et al [12] have established f(log n) time in the CREW even if an infinite number of a similar lower bound for the problem of constructing the processors and memory cells are used
The dominance graph corresponding to a collection of n iso-oriented, non-overlapping, rectangles in the plane can be constructed in O(log n) time using O(n) processors in the CREW
Summary
Two important design methodologies central to the fabrication of integrated circuits are symbolic layout and compaction. The dominance graph of a collection of non-overlapping rectangles is "a directed graph whose vertices are precisely the rectangles in the collection; two vertices u and v are linked by a directed 2 LOWER BOUNDS edge (u, v) when the rectangle corresponding to u is directly above the rectangle corresponding to v It is easy The purpose of this section is to provide lower bounds to see that the notion of direct aboveness captures the that establish both the time- and cost-optimality of our intuitive idea that for compaction purposes we only need algorithms in the CREW model of computation. The problem of constructing the visibility graph of a collection of n non-overlapping rectangles in the plane has a lower bound of 2(n log n) in the algebraic decision-tree model. The problem of constructing the dominance graph of a set of n non-overlapping rectangles in the plane has a lower bound of f(log n) on the CREW, regardless of the number of processors and memory cells used. The problem of constructing the visibility graph of a set of n iso-oriented, non-overlapping rect-
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