Showing papers by "Robert E. Tarjan published in 1975"

Efficiency of a Good But Not Linear Set Union Algorithm

Abstract: TWO types of instructmns for mampulating a family of disjoint sets which partitmn a umverse of n elements are considered FIND(x) computes the name of the (unique) set containing element x UNION(A, B, C) combines sets A and B into a new set named C A known algorithm for implementing sequences of these mstructmns is examined It is shown that, if t(m, n) as the maximum time reqmred by a sequence of m > n FINDs and n -- 1 intermixed UNIONs, then kima(m, n) _~ t(m, n) < k:ma(m, n) for some positive constants ki and k2, where a(m, n) is related to a functional inverse of Ackermann's functmn and as very slow-growing

Bounds on Backtrack Algorithms for Listing Cycles, Paths, and Spanning Trees

Abstract: Backtrack algorithms for listing certain kinds of subgraphs of a graph are described and analyzed. Included are algorithms for listing all spanning trees, all cycles, all simple cycles, or all of c...

Algorithmic aspects of vertex elimination on directed graphs.

Abstract: We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse systems of linear eauations. We give efficient algorithms to: (1) calculate the fill-in produced by any elimination ordering; (2) find a perfect elimination ordering if one exists; and (3) find a minimal elimination ordering. We also show that problems (1) and (2) are at least as time-consuming as testing whether a directed graph is transitive, and that the problem of finding a minimum ordering is NP-complete.

Graph theory and Gaussian elimination.

Abstract: This paper surveys graph-theoretic ideas which apply to the problem of solving a sparse system of linear equations by Gaussian elimination Included are a discussion of bandwidth, profile, and general sparse elimination schemes, and of two graph-theoretic partitioning methods Algorithms based on these ideas are presented

Solving path problems on directed graphs.

Abstract: This paper considers path problems on directed graphs which are solvable by a method similar to Gaussian elimination. The paper gives an axiom system for such problems which is a weakening of Salomaa''s axioms for a regular algebra. The paper presents a general solution method which requires O($n^3$) time for dense graphs with n vertices and considerably less time for sparse graphs. The paper also presents a decomposition method which solves a path problem by breaking it into subproblems, solving each subproblem by elimination, and combining the solutions. This method is a generalization of the "reducibility" notion of data flow analysis, and is a kind of single-element "tearing". Efficiently implemented, the method requires O(m $\alpha$(m,n)) time plus time to solve the subproblems, for problem graphs with n vertices and m edges. Here $\alpha$(m,n) is a very slowly growing function which is a functional inverse of Ackermann''s function. The paper considers variants of the axiom system for which the solution methods still work, and presents several applications including solving simultaneous linear equations and analyzing control flow in computer programs.