Showing papers in "SIAM Journal on Computing in 1982"
TL;DR: Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography.
Abstract: We define the set of planar boolean formulae, and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is NP-complete. Using these results, we are able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness for planar generalized geography.The NP-completeness of planar node cover and planar Hamiltonian circuit and line were first proved elsewhere [M. R. Garey and D. S. Johnson, The rectilinear Steiner tree is NP-complete, SIAM J. Appl. Math., 32 (1977), pp. 826–834] and [M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamilton circuit problem is NP-complete, SIAM J. Comp., 5 (1976), pp. 704–714].
796 citations
TL;DR: There is a distributed randomized algorithm that can route every packet to its destination without two packets passing down the same wire at any one time, and finishes within time $O(\log N)$ with overwhelming probability for all such routing requests.
Abstract: Consider $N = 2^n $ nodes connected by wires to make an n-dimensional binary cube. Suppose that initially the nodes contain one packet each addressed to distinct nodes of the cube. We show that the...
675 citations
TL;DR: A linear-time algorithm to recognize the class of vertex series-parallel (VSP) digraphs is presented and efficient methods to compute the transitive closure and transitive reduction of VSPDigraphs are obtained.
Abstract: We present a linear-time algorithm to recognize the class of vertex series-parallel (VSP) digraphs. Our method is based on the relationship between VSP digraphs and the class of edge series-paralle...
564 citations
TL;DR: The purpose of the present paper is to set up a categorical framework in which the known techniques for solving equations find a natural place, generalizing from least fixed-points of continuous functions over cpos to initial ones of continuous functors over $\omega $-categories.
Abstract: Recursive specifications of domains plays a crucial role in denotational semantics as developed by Scott and Strachey and their followers. The purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place. The idea is to follow the well-known analogy between partial orders and categories, generalizing from least fixed-points of continuous functions over cpos to initial ones of continuous functors over $\omega $-categories. To apply these general ideas we introduce Wand’s ${\bf O}$-categories where the morphism-sets have a partial order structure and which include almost all the categories occurring in semantics. The idea is to find solutions in a derived category of embeddings and we give order-theoretic conditions which are easy to verify and which imply the needed categorical ones. The main tool is a very general form of the limit-colimit coincidence remarked by Scott. In the concluding section we outline how compatibilit...
564 citations
TL;DR: This work gives necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes, and provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.
Abstract: A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-complete. This provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.
513 citations
TL;DR: A heuristic is proposed that delivers in O(n^3 ) steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.
Abstract: We propose a heuristic that delivers in $O(n^3 )$ steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.
503 citations
TL;DR: A consequence of these results is that $\omega $, the exponent for matrix multiplication, is a limit point, that is, it cannot be realized by any single algorithm.
Abstract: The main results of this paper have the following flavor: Given one algorithm for multiplying matrices, there exists another, better, algorithm. A consequence of these results is that $\omega $, th...
228 citations
TL;DR: It is shown that the problem of finding a minimum dominating set in a chordal graph is NP-complete, even when restricted to undirected path graphs, but exhibit a linear time greedy algorithm for the problem further restricted to directed path graphs.
Abstract: A set of vertices D is a dominating set for a graph if every vertex is either in D or adjacent to a vertex which is in D. We show that the problem of finding a minimum dominating set in a chordal graph is NP-complete, even when restricted to undirected path graphs, but exhibit a linear time greedy algorithm for the problem further restricted to directed path graphs. Streamlined to handle only trees, our algorithm becomes the algorithm of Cockayne, Goodman and Hedetniemi. An interesting parallel with graph isomorphism is pointed out.
179 citations
TL;DR: Coloring algorithms that run in time $O(\min (m(\log n)^2 ,n^2 \log n))$ are presented and can be used to find maximum cardinality matchings on regular bipartite graphs in the above time bound.
Abstract: A minimum edge coloring of a bipartite graph is a partition of the edges into $\Delta $ matchings, where $\Delta $ is the maximum degree in the graph. Coloring algorithms that run in time $O(\min (m(\log n)^2 ,n^2 \log n))$ are presented. The algorithms rely on an efficient procedure for the special case of $\Delta $ an exact power of two. The coloring algorithms can be used to find maximum cardinality matchings on regular bipartite graphs in the above time bound. An algorithm for coloring multigraphs with large multiplicities is also presented.
137 citations
TL;DR: In a general sequential model of computation, no restrictions are placed on the way in which the computation may proceed, except that parallel operations are not allowed, it is shown that in such an unrestricted environmentmega (N^2 /log N) = Omega (N + 1 / log N) is needed in order to sort N integers.
Abstract: In a general sequential model of computation, no restrictions are placed on the way in which the computation may proceed, except that parallel operations are not allowed. We show that in such an unrestricted environment ${\text{TIME}} \cdot {\text{SPACE}} = \Omega (N^2 /\log N)$ in order to sort N integers, each in the range $[1,N^2 ]$.
136 citations
TL;DR: The ellipsoid method for linear programming is applied to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities.
Abstract: We show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any NP-complete combinatorial optimization problem unless NP = co-NP—a very unlikely event We also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities
TL;DR: A general theorem is proved which can be used to show that for a large number of matroid properties there is no good algorithm of a certain type for determining whether these properties hold for general matroids.
Abstract: A general theorem is proved which can be used to show that for a large number of matroid properties there is no good algorithm of a certain type for determining whether these properties hold for general matroids. Specifically, there exists no algorithm in which the matroid is represented by an independence test oracle (or an oracle polynomially related to an independence test oracle) and which solves the problem in question after a number of calls on the oracle which is bounded by a polynomial in the number of elements of the ground set of the matroid.
TL;DR: This work presents efficient $(O(\log ^2 n)$ parallel algorithms for two classical graph problems: planarity testing and finding triconnected components, using only a polynomial number of processors.
Abstract: We present efficient $(O(\log ^2 n))$ parallel algorithms for two classical graph problems: planarity testing and finding triconnected components. The algorithms use only a polynomial number of processors. Previous algorithms used $\Omega (n)$ operations, regardless of the number of available processors.
TL;DR: The number of essential multiplications required to multiply matrices of size N and N is bounded by CN^2 \log ^2 N, where N is the number of matrices in a N-dimensional model.
Abstract: The number of essential multiplications required to multiply matrices of size $N \times N$ and $N \times N^{0.172} $ is bounded by $CN^2 \log ^2 N$.
TL;DR: This paper shows that n(1 + \Theta (1/\sqrt M )) steps are both necessary and sufficient, if M memory cells are available to store values of the function, and explicitly considers the performance of the algorithm as a function of the amount of memory available and the relative cost of evaluating f and comparing sequence elements for equality.
Abstract: Given a function f over a finite domain D and an arbitrary starting point x, the sequence $f^0 (x),f^1 (x),f^2 (x), \cdots $ is ultimately periodic. Such sequences are typically the output of random number generators. The cycle problem is to determine the first repeated element $f^n (x)$ in the sequence. Previous algorithms for this problem have required $3n + O(1)$ operations. In this paper we show that $n(1 + \Theta (1/\sqrt M ))$ steps are both necessary and sufficient, if M memory cells are available to store values of the function. We explicitly consider the performance of the algorithm as a function of the amount of memory available and the relative cost of evaluating f and comparing sequence elements for equality.
TL;DR: It is shown that the problem of scheduling a two-processor n-job open shop nonpreemptively in order to minimize mean flow time is NP-complete even if input length is measured by the sum of the task lengths.
Abstract: It is shown that the problem of scheduling a two-processor n-job open shop nonpreemptively in order to minimize mean flow time is NP-complete even if input length is measured by the sum of the task lengths. The proof is similar in approach to that used by Garey, Johnson and Sethi to show NP-completeness of the two-processor flow shop mean flow problem. We assume previous results from their paper where possible and concentrate on those elements of the proof that are distinct from theirs.In addition, bounds are derived for the mean flow times of arbitrary and shortest processing time (SPT) first schedules for m-processor n-job systems in terms of the mean flow time of an optimal schedule.
TL;DR: Let t be a disjoint sum of tensors associated to matrix multiplication and the rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix multiplication, which leads to a trascendental equation defining a new exponent for Matrix multiplication.
Abstract: Let t be a disjoint sum of tensors associated to matrix multiplication. The rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix multiplication. This relation leads to a trascendental equation defining a new exponent for matrix multiplication.
TL;DR: This paper describes an algorithm to construct, for each expression in a given program text, a symbolic expression whose value is equal to the value of the text expression for all executions of the program.
Abstract: This paper describes an algorithm to construct, for each expression in a given program text, a symbolic expression whose value is equal to the value of the text expression for all executions of the program. We call such a mapping from text expressions to symbolic expressions a cover. Covers are useful in such program optimization techniques as constant propagation and code motion. The particular cover constructed by our methods is in general weaker than the covers obtainable by the methods of [Ki], [FKU], [RL], [R2] but our method has the advantage of being very efficient. It requires $O(m\alpha (m,n) + l)$ operations if extended bit vector operations have unit cost, where n is the number of vertices in the control flow graph of the program, m is the number of edges, l is the length of the program text, and $\alpha $ is related to a functional inverse of Ackermann’s function [T2]. Our method does not require that the program be well-structured nor that the flow graph be reducible.
TL;DR: The SCAN policy, used to schedule read/write requests at a moving-arm disk device, when fast response over the entire disk area is at a premium is described and an analysis is presented which handles precisely the dependence structure between queues accumulated at different cylinders.
Abstract: This paper describes and analyzes the SCAN policy, used to schedule read/write requests at a moving-arm disk device, when fast response over the entire disk area is at a premium. An analysis is presented which handles precisely the dependence structure between queues accumulated at different cylinders. The arrival process of requests to each cylinder is assumed Poisson and homogeneous in time. A relatively efficient algorithm for evaluating numerically the mean waiting time at each cylinder is presented and its complexity analyzed. We discuss further extensions intended to capture additional details of realistic situations. These include distributed record lengths, skipping unreferenced cylinders and letting successive arrivals’ target cylinders be dependent variables.
TL;DR: The equivalence problem for deterministic two-way sequential transducers is shown to be decidable also for the general case and the result holds even when the devices are allowed to make some finite number of nondeterministic moves.
Abstract: The equivalence problem for deterministic two-way sequential transducers is a long time open problem which is known to be decidable for some restricted cases. Here, the problem is shown to be decidable also for the general case. In fact, the result holds even when the devices are allowed to make some finite number of nondeterministic moves.
TL;DR: Two sets of algorithms for searching and dynamic reorganization of linear lists are presented, with the well-known move-to-front and transposition algorithms at the extrema.
Abstract: Two sets of algorithms for searching and dynamic reorganization of linear lists are presented. Each set forms a spectrum, with the well-known move-to-front and transposition algorithms at the extrema. A linear ordering on the stationary expected search cost of the algorithms in each of the two spectra is established over a restricted set of probability distributions.
TL;DR: This paper presents a polynomial-time approximation algorithm for the maximum independent set problem on planar graphs, and finds an independent set that is necessarily larger in size than half a maximumIndependent set.
Abstract: In this paper we consider the maximum independent set problem in which one would like to find a maximum set of independent (i.e., pairwise nonadjacent) vertices in a given graph. The problem is NP-complete, and still remains so even if we restrict ourselves to the class of planar graphs. It has been conjectured that there exist no polynomial-time exact algorithms for any NP-complete problems. We present a polynomial-time approximation algorithm for the maximum independent set problem on planar graphs. For a given planar graph having any number n of vertices, our algorithm finds, in $O(n\log n)$ time, an independent set that is necessarily larger in size than half a maximum independent set. Thus the absolute worst case ratio of our algorithm is greater than $\tfrac{1}{2}$.
TL;DR: A property of Boolean functions of n variables is described and shown to imply lower bounds as large as $\Omega (n\log n)$ on the number of literals in any Boolean formula for any function with the property.
Abstract: A property of Boolean functions of n variables is described and shown to imply lower bounds as large as $\Omega (n\log n)$ on the number of literals in any Boolean formula for any function with the property. Formulas over the full basis of binary operations $( \wedge , \oplus ,{\text{ etc.}})$ are considered. The lower bounds apply to all but a vanishing fraction of symmetric functions, in particular, to all threshold functions with sufficiently large threshold and to the “congruent to zero modulo k” function for $k > 2$. In the case $k = 4$, the bound is optimal.
TL;DR: A number of characterizations are established of those families of graphs that are almost binary trees, in the sense that every graph in the family is embeddable in a binary tree within bounded cost.
Abstract: This paper studies embeddings of graphs in binary trees. The cost of such an embedding is the maximum distance in the binary tree between images of adjacent graph vertices. Several techniques for bounding the costs of such embeddings from above are derived; notable among these is an algorithm for embedding any outerplanar graph in a binary tree with a cost that is within a factor of 3 of optimal. A number of techniques for bounding the costs of such embeddings from below are developed; notable here are two techniques for inferring the presence of large separators in graphs. Finally, a number of characterizations are established of those families of graphs that are almost binary trees, in the sense that every graph in the family is embeddable in a binary tree within bounded cost.
TL;DR: It is shown that any implementation of Warshall’s transitive closure algorithm requires $\Omega (n)$ space, and that many familiar sorting algorithms exhibit similar behavior.
Abstract: Any Boolean straight-line program which computes the transitive closure of an $n \times n$ Boolean matrix by successive squaring requires time exceeding any polynomial in n if the space used is $o(n)$. This is the first demonstration of a “natural” algorithm which (1) has a polynomial time implementation and (2) has a small (e.g., $O(\log ^2 n)$) space implementation, but (3) has no implementation running in polynomial time and small space simultaneously. It is also shown that any implementation of Warshall’s transitive closure algorithm requires $\Omega (n)$ space, and that many familiar sorting algorithms exhibit similar behavior.
TL;DR: Nonserial dynamic programming (DP), a simple elimination procedure, is shown to be optimal among all nonoverlapping comparison algorithms, including nondeterministic algorithms, and to give an exponential lower bound on the shortest admissible proof that a solution is optimal.
Abstract: We consider discrete optimization problems in which the only exploitable feature of the objective function is a limited form of decomposability. “Nonoverlapping comparison algorithms” are defined as a model of procedures which decompose the problem and apply Bellman’s principle of optimality. Nonserial dynamic programming (DP), a simple elimination procedure, is shown to be optimal among all nonoverlapping comparison algorithms, including nondeterministic algorithms. These results can give an exponential lower bound on the shortest admissible proof that a solution is optimal. Furthermore, if part of the search space is ruled out, a subset of the comparisons made by DP optimally searches the remainder. We suggest that the running time of DP is a useful measure of the “interaction complexity” of a problem, and that because of its simplicity DP is of practical as well as theoretical interest.
TL;DR: It is shown that each element of the Z-completion of P is a Z-join of elements of P iff Z is union complete, and some results on the internal structure of $P_f $ with regard to Z-joins are obtained.
Abstract: We show, for any subset system Z (as defined in Wright, Wagner, and Thatcher, T.C.S. 7 (1978), pp. 57–77) and any order preserving map $f:Q \to P$ of posets, the existence of a universal map $u_f :P \to P_f $ where $P_f $ is Z-complete and $u_f f$ is Z-continuous. This generalizes to arbitrary subset systems the result of Markowsky (T.C.S. 4 (1977), pp. 125–135) for chains, and the completions of Wright, Wagner, and Thatcher for union complete Z; our method, different from theirs, uses the time-honored direct construction of universal maps. Further, we obtain some results on the internal structure of $P_f $ with regard to Z-joins. Finally, we show that each element of the Z-completion of P is a Z-join of elements of P iff Z is union complete.
TL;DR: An algorithm for the determination of an Hamiltonian circuit in a 4-connected planar graph is presented, inspired by the proof of Tutte’s theorem which implies the existence of Hamiltonian circuits in 4- connected planar graphs.
Abstract: An algorithm for the determination of an Hamiltonian circuit in a 4-connected planar graph is presented. The timing for this algorithm depends on $n^3 $ (where n is the number of edges in the graph); the storage requirement also depends on $n^3 $. This paper completes the result of Garey, Johnson and Tarjan [SIAM J. Comput., 5 (1976), pp. 704–714] which claims that the problem is NP-complete for 3-connected planar graphs. This algorithm is inspired by the proof of Tutte’s theorem which implies the existence of Hamiltonian circuits in 4-connected planar graphs.
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TL;DR: An algorithm for testing primality is proposed (under the extended Riemann hypothesis) which is more efficient than ones proposed by Miller and Velu.
Abstract: Whether an odd number m is prime can be decided on the knowledge of the image of the function $a \mapsto a^{(m - 1)/2} (m)$. As a consequence, an algorithm for testing primality is proposed (under the extended Riemann hypothesis) which is more efficient than ones proposed by Miller [Pros. 7th ACM Symp. Theory of Computing, 1975, pp. 234–239] and Velu [SIGACT News, 10 (1978), pp. 58–59]. A probabilistic version is compared with the algorithm of Solovay and Strassen [SIAM J. Comput., 6 (1977), pp. 84–85; erratum, 7 (1978), p. 118].
TL;DR: This paper shows that, for any choice of a function $\sigma $ of n increasing to infinity with n more slowly than n, the algorithm can be adjusted so that, in probability, the time taken by the algorithm will be of order less than that of $n\sigma (n)$ as $n \to \infty $.
Abstract: This paper presents an algorithm for the traveling salesman problem in k-dimensional Euclidean space. For n points independently uniformly distributed in a set $\mathbb{E}$, we show that, for any choice of a function $\sigma $ of n increasing to infinity with n more slowly than n, we can adjust the algorithm so that, in probability, the time taken by the algorithm will be of order less than that of $n\sigma (n)$ as $n \to \infty $. The algorithm puts the n points in a cyclic order, and we also show that, with probability one, the length of the corresponding tour (that is, the sum of the n distances between adjacent points in the order given) will be asymptotic to the minimal tour length as $n \to \infty $. The latter is known (also with probability one) to be asymptotic to $\beta _k v(\mathbb{E})^p n^q $, where $\beta _k $ is a constant depending only on the dimension k, $v(\mathbb{E})$ is the volume of the set $\mathbb{E}$, $p = 1/k$, and $q = 1 - p$. Our result is stronger, and the algorithm is faster, ...