Showing papers by "Robert E. Tarjan published in 1981"

A data structure for dynamic trees

Abstract: We propose a data structure to maintain a collection of vertex-disjoint trees under a sequence of two kinds of operations: a link operation that combines two trees into one by adding an edge, and a cut operation that divides one tree into two by deleting an edge. Our data structure requires O(log n) time per operation when the time is amortized over a sequence of operations. Using our data structure, we obtain new fast algorithms for the following problems: (1) Computing deepest common ancestors. (2) Solving various network flow problems including finding maximum flows, blocking flows, and acyclic flows. (3) Computing certain kinds of constrained minimum spanning trees. (4) Implementing the network simplex algorithm for the transshipment problem. Our most significant application is (2); we obtain an O(mn log n)-time algorithm to find a maximum flow in a network of n vertices and m edges, beating by a factor of log n the fastest algorithm previously known for sparse graphs.

Fast Algorithms for Solving Path Problems

Abstract: Let G = (V,E) be a directed graph with a distinguished source vertex s. The single-source path expression problem is to find, for each vertex v, a regular expression P(s,v) which represents the set of all paths in G from s to v. A solution to this problem can be used to solve shortest path problems, solve sparse systems of linear equations, and carry out global flow analysis. We describe a method to compute path expressions by dividing G into components, computing path expressions on the components by Gaussian elimination, and combining the solutions. This method requires O(m $\alpha$(m,n)) time on a reducible flow graph, where n is the number of vertices in G, m is the number of edges in G, and $\alpha$ is a functional inverse of Ackermann''s function. The method makes use of an algorithm for evaluating functions defined on paths in trees. A simplified version of the algorithm, which runs in O(m log n) time on reducible flow graphs, is quite easy to implement and efficient in practice.

A Unified Approach to Path Problems

Abstract: We describe a general method for solving path problems on directed graphs. Such path problems include finding shortest paths, solving sparse systems of linear equations, and carrying out global flow analysis of computer programs. Our method consists of two steps. First, we construct a collection of regular expressions representing sets of paths in the graph. This can be done by using any standard algorithm, such as Gaussian or Gauss-Jordan elimination. Next, we apply a natural mapping from regular expressions into the given problem domain. We exhibit the mappings required to find shortest paths, solve sparse systems of linear equations, and carry out global flow analysis. Our results provide a general-purpose algorithm for solving any path problem, and show that the problem of constructing path expressions is in some sense the most general path problem.

Scheduling Unit–Time Tasks with Arbitrary Release Times and Deadlines

Abstract: The basic problem considered is that of scheduling n unit-time tasks, with arbitrary release times and deadlines, so as to minimize the maximum task completion time. Previous work has shown that this problem can be solved rather easily when all release times are integers. We are concerned with the general case in which noninteger release times are allowed, a generalization that considerably increases the difficulty of the problem even for only a single processor. Our results are for the one-processor case, where we provide an $O(n\log n)$ algorithm based on the concept of “forbidden regions”.

On a greedy heuristic for complete matching

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 $O(n^2 \log n)$, a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to $n^3 $ 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 matching is at most ${}_3^4 n^{\lg _2^3 } - 1$, $\lg _2^3 \approx 0.58496$, 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.