Edward M. Reingold

Graph drawing by force-directed placement

We present a modification of the spring‐embedder model of Eades [Congressus Numerantium, 42, 149–160, (1984)] for drawing undirected graphs with straight edges. Our heuristic strives for uniform edge lengths, and we develop it in analogy to forces in natural systems, for a simple, elegant, conceptually‐intuitive, and efficient algorithm.

Tidier Drawings of Trees

Various algorithms have been proposed for producing tidy drawings of trees–drawings that are aesthetically pleasing and use minimum drawing space. We show that these algorithms contain some difficulties that lead to aesthetically unpleasing, wider than necessary drawings. We then present a new algorithm with comparable time and storage requirements that produces tidier drawings. Generalizations to forests and m-ary trees are discussed, as are some problems in discretization when alphanumeric output devices are used.

Backtrack programming techniques

The purpose of this paper is twofold. First, a brief exposition of the general backtrack technique and its history is given. Second, it is shown how the use of macros can considerably shorten the computation time in many cases. In particular, this technique has allowed the solution of two previously open combinatorial problems, the computation of new terms in a well-known series, and the substantial reduction in computation time for the solution to another combinatorial problem.

Binary Search Trees of Bounded Balance

A new class of binary search trees, called trees of bounded balance, is introduced. These trees are easy to maintain in their form despite insertions and deletions of nodes, and the search time is only moderately longer than in completely balanced trees. Trees of bounded balance differ from other classes of binary search trees in that they contain a parameter which can be varied so the compromise between short search time and infrequent restructuring can be chosen arbitrarily.

Efficient generation of the binary reflected gray code and its applications

Algorithms are presented to generate the n-bit binary reflected Gray code and codewords of fixed weight in that code. Both algorithms are efficient in that the time required to generate the next element from the current one is constant. Applications to the generation of the combinations of n things taken k at a time, the compositions of integers, and the permutations of a multiset are discussed.

The complexity of drawing trees nicely

SummaryWe investigate the complexity of producing aesthetically pleasing drawings of binary trees, drawings that are as narrow as possible. The notion of what is aesthetically pleasing is embodied in several constraints on the placement of nodes, relative to other nodes. Among the results we give are: (1) There is no obvious “principle of optimality” that can be applied, since globally narrow, aesthetic placements of trees may require wider than necessary subtrees. (2) A previously suggested heuristic can produce drawings on n-node trees that are Θ(n) times as wide as necessary. (3) The problem can be reduced in polynomial time to linear programming; hence, if the coordinates assigned to the nodes are continuous variables, then the problem can be solved in polynomial time. (4) If the placement is restricted to the integral lattice then the problem is NP-hard, as is its approximation to within a factor of about 4 per cent.

The complexity of pursuit on a graph

Abstract A robber and k cops choose starting vertices in a graph and move in alternation from vertex to vertex along the edges of the graph; capture occurs if a cop ever shares a vertex with the robber. If k is not fixed and if either the graph is directed or initial positions are given, we show that the problem is EXPTIME-complete. Similar techniques lead to the PSPACE- and EXPTIME-completeness of some other combinatorial games that were previously only known to be NP- and PSPACE-hard.

On a Greedy Heuristic for Complete Matching

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

Determining the majority

O(n^2 \log n)

Index assignment for multichannel communication under failure

, a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to