Shortest Path Geometric Rounding

Abstract

Exact implementations of algorithms of computational geometry are subject to exponential growth in running time and space. In particular, coordinate bit-complexity can grow exponentially when algorithms are cascaded : the output of one algorithm becomes the input to the next. Cascading is a significant problem in practice. We propose a geometric rounding technique: shortest path rounding . Shortest path rounding trades accuracy for space and time and eliminates the exponential cost introduced by cascading. It can be applied to all algorithms which operate on planar polygonal regions, for example, set operations, transformations, convex hull, triangulation, and Minkowski sum. Unlike other geometric rounding techniques, shortest path rounding can round vertices to arbitrary lattices, even in polar coordinates, as long as the rounding cells are connected. (Other rounding techniques can only round to the integer grid.) On the integer grid, shortest path rounding introduces less combinatorial change and geometric error than the other rounding methods. Three algorithms are given for shortest path rounding, one of which we have used in industrial application software since 1992. In combination with recent advances in exact floating point evaluation of numerical primitives, shortest path geometric rounding yields a practical solution to numerical issues in computational geometry. Geometric algorithms can be implemented exactly on floating point input coordinates; the exact output coordinates can be rounded to accurate floating point approximations; and the cost of each arithmetic operation is only a little more than if it were implemented as a single hardware floating point operation.