Abstract

Abstract, Finding a minimum weighted complete matching on a set of vertices in which the distances satisfy the triangle inequality is of general interest and of particular importance when drawing graphs on a mechanical plotter. The "greedy " heuristic of repeatedly matching the two closest unmatched points can be implemented in worst-case time O(n log n), a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to n to find the minimum cost matching in a weighted graph. We show that, for an even number n of vertices whose distances satisfy the triangle inequality, the ratio of the cost of the matching produced by this greedy heuristic to the cost of the minimal lg matching is at most 3n 1, lg 0.58496, and there are examples that achieve this bound. We conclude that this greedy heuristic, although desirable because of its simplicity, would be a poor choice for this problem. Key words, graph algorithms, matching, greedy heuristic, analysis of algorithms Introduction, We begin with some motivation, the connection of which to our central topic will become clear later. The problem of drawing a graph G (V, E) on a mechanical plotter with prespecified vertex locations arises in numerous applications [7]. For example, in the

Citations