Showing papers by "Robert E. Tarjan published in 1980"

Applications of a Planar Separator Theorem

Abstract: Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only $O(\sqrt n )$ vertices. This separator theorem, in combination with a divide-and-conquer strategy, leads to many new complexity results for planar graph problems. This paper describes some of these results.

Performance Bounds for Level-Oriented Two-Dimensional Packing Algorithms

Abstract: We analyze several “level-oriented” algorithms for packing rectangles into a unit-width, infinite-height bin so as to minimize the total height of the packing. For the three algorithms we discuss, we show that the ratio of the height obtained by the algorithm to the optimal height is asymptotically bounded, respectively, by 2, 1.7, and 1.5. The latter two improve substantially over the performance bounds for previously proposed algorithms. In addition, we give more refined bounds for special cases in which the widths of the given rectangles are restricted and in which only squares are to be packed.

Variations on the Common Subexpression Problem

Abstract: Let G be a directed graph such that for each vertex v in G, the successors of v are ordered Let C be any equivalence relation on the vertices of G. The congruence closure C* of C is the finest equivalence relation containing C and such that any two vertices having corresponding successors equivalent under C* are themselves equivalent under C* Efficient algorithms are described for computing congruence closures in the general case and in the following two special cases. 0) G under C* is acyclic, and (it) G is acychc and C identifies a single pair of vertices. The use of these algorithms to test expression eqmvalence (a problem central to program verification) and to test losslessness of joins in relational databases is described

The Pebbling Problem is Complete in Polynomial Space

Abstract: In this paper we study a pebbling problem that models the storage requirements of various kinds of computation. Sethi has shown this problem to be $NP$-hard and Lingas has shown a generalization to be P-space complete. We prove the original problem P-space complete by using a modification of Lingas’s proof. The pebbling problem is an example of a P-space complete problem not exhibiting any obvious quantifier alternation.

Two linear-time algorithms for five-coloring a planar graph

Abstract: A "sequential processing" algorithm using bicolor interchange that five-colors an n vertex planar graph in $O(n^2)$ time was given by Matula, Marble, and Isaacson [1972]. Lipton and Miller used a "batch processing" algorithm with bicolor interchange for the same problem and achieved an improved O(n log n) time bound [1978]. In this paper we use graph contraction arguments instead of bicolor interchange and improve both the sequential processing and batch processing methods to obtain five-coloring algorithms that operate in O(n) time.

Prime subprogram parsing of a program

Abstract: A parsing method based on the triconnected decomposition of a biconnected graph is presented. The parsing algorithm runs in linear time and handles a large class of flow graphs. The applications of this algorithm to flow analysis and to the automatic structuring of programs are discussed.

Biased 2-3 trees

Abstract: We describe a new data structure for maintaining collections of weighted items. The access time for an item of weight w in a collection of total weight W is proportional to log(W/w) in the worst case (which is optimal in a certain sense), and several other useful operations can be made to work just, as fast. The data structure is simpler than previous proposals, but the running time must be amortized over a sequence of operations to achieve the time bounds.

Linear expected-time algorithms for connectivity problems (Extended Abstract)

Abstract: Researchers in recent years have developed many graph algorithms that are fast in the worst case, but little work has been done on graph algorithms that are fast on the average. (Exceptions include the work of Angluin and Valiant [1], Karp [7], and Schnorr [9].) In this paper we analyze the expected running time of four algorithms for solving graph connectivity problems. Our goal is to exhibit algorithms whose expected time is within a constant factor of optimum and to shed light on the properties of random graphs. In Section 2 we develop and analyze a simple algorithm that finds the connected components of an undirected graph with n vertices in O(n) expected time. In Sections 3 and 4 we describe algorithms for finding the strong components of a directed graph and the blocks of an undirected graph in O(n) expected time. The time required for these three problems is Ω(m) in the worst case, where m is the number of edges in the graph, since all edges must be examined; but our results show that only O(n) edges must be examined on the average.*

Recent Developments in the Complexity of Combinatorial Algorithms

Abstract: : Several major advances in the area of combinatorial algorithms include improved algorithms for matrix multiplication and maximum network flow, a polynomial-time algorithm for linear programming, and steps toward a polynomial-time algorithm for graph isomorphism. This paper surveys these results and suggests directions for future research. Included is a discussion of recent work by the author and his students on dynamic dictionaries, network flow problems, and related questions.