Heuristics for planar minimum-weight perfect metchings

Several linear-time approximation algorithms for the minimum-weight perfect matching in a plane are proposed, and their worst- and average-case behaviors are analyzed theoretically as well as experimentally. A linear-time approximation algorithm, named the “spiral-rack algorithm (with preprocess and with tour),” is recommended for practical purposes. This algorithm is successfully applied to the drawing of road maps such as that of the Tokyo city area. I. INTRODUCTION Consider n (an even number) points in a plane. The problem of finding the minimumweight perfect matching, i.e., determining how to match the n points in pairs so as to minimize the sum of the distances between the matched points, as well as Euler’s problem of unicursal traversing on a graph, is of fundamental importance for optimizing the sequence of drawing lines by a mechanical plotter ([2-5, 81; details are discussed in Sec. V). The algorithm which exactly solves this problem in 0(n3) time [6] seems to be too complicated from the practical point of view. Even approximation algorithms of O(n2) or O(n log n) [lo] would not be satisfactory or need some improvement for the application to real-world problems of a size, say, n greater than lo4. In contrast with the matching problem, an Eulerian path can be found in linear time in the’number of edges. In this paper, linear-time* approximation algorithms are proposed for the matching problem in a unit square; their worst-case performances are analyzed theoretically; their average-case performances are investigated both theoretically and experimentally for the case where n points are uniformly distributed on the unit square; and an application to the drawing of a road map is shown. The quality of an approximate solution is measured by the absolute cost of the matching, i.e., the sum of the distances *We adopt the RAM model of computation which executes an arithmetic operation such as addition, multiplication, or integer division (hence, the “floor” operation) in a unit time [ 11.