Showing papers in "SIAM Journal on Discrete Mathematics in 1988"
TL;DR: Efficient deterministic techniques for breaking symmetry in parallel are used to construct fast linear processor algorithms for several problems, including the problem of $(Delta + 1)$-coloring constant-degree graphs and 5-colors planar graphs.
Abstract: This paper describes efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. The primary technique allows us to 3-color a rooted tree in $O( \lg^* n )$ time on an EREW PRAM using a linear number of processors. These techniques are used to construct fast linear processor algorithms for several problems, including the problem of $( \Delta + 1)$-coloring constant-degree graphs and 5-coloring planar graphs. Lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs are also proved.
266 citations
TL;DR: This paper shows that the graph consisting of an n-cycle and a random matching has diameter about $\log _2 n$, which is very close to the best possible value.
Abstract: How small can the diameter be made by adding a matching to an n-cycle? In this paper this question is answered by showing that the graph consisting of an n-cycle and a random matching has diameter about $\log _2 n$, which is very close to the best possible value. It is also shown that by adding a random matching to graphs with certain expanding properties such as expanders or Ramanujan graphs, the resulting graphs have near optimum diameters.
193 citations
TL;DR: These techniques are used to characterize the graphs of infinitesimally rigid frameworks on other surfaces, such as the flat torus, the cylinder, cones, etc., using matroid unions of cycle and bicycle matroids of the graph.
Abstract: From the pattern of its rigidity matrix, we show that a k-frame on a graph (or multigraph) has the matroid structure of the union of k copies of the cycle matroid of the graph. This matrix pattern is applied to three central results about the rigidity of frameworks. An immediate corollary of this matroid union is a characterization of rigid bar and body frameworks in n-space (Tay’s Theorem). This is further specialized to characterize the independence and the rigidity of body and hinge structures in n-space (a new theorem). The two-frame, or union of two copies of the graphic matroid, is truncated to produce plane bar and joint frameworks giving a characterization of minimal infinitesimally rigid bar and joint frameworks in the plane (Laman’s Theorem). Finally, these techniques are used to characterize the graphs of infinitesimally rigid frameworks on other surfaces, such as the flat torus, the cylinder, cones, etc., using matroid unions of cycle and bicycle matroids of the graph.
145 citations
TL;DR: It is shown that First-Fit coloring requires at most $40\omega $ colors to color an interval graph with clique size $\omega$, and it follows that a polynomial time approximation algorithm for Dynamic Storage Allocation has a constant performance ratio of 80.
Abstract: It is shown that First-Fit coloring requires at most $40\omega $ colors to color an interval graph with clique size $\omega $. It follows that a polynomial time approximation algorithm for Dynamic ...
115 citations
TL;DR: This paper presents a new method for constructing wide-sense nonblocking generalized connectors with n inputs and outputs and size O, and with depth k and size $O ( n 1 + 1/k) and $O 1 - 1/ k.
Abstract: A new method for constructing wide-sense nonblocking networks is presented. Application of this method yields (among other things) wide-sense nonblocking generalized connectors with n inputs and outputs and size $O( n\log n )$, and with depth k and size $O ( n^{1 + 1/k} ( \log n )^{1 - 1/k} )$.
102 citations
TL;DR: This work gives a combinatorial characterization of a certain type of perfect threshold scheme, and investigates the maximum number of keys which a perfect (t, w)-threshold scheme can incorporate, as a function of t, w, and the total number of possible shadows, v.
Abstract: We investigate the combinatorial properties of threshold schemes. Informally, a (t, w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key, but no subset of fewer than t participants can determine the key. Our interest is in perfect threshold schemes: no subset of fewer than t participants can determine any partial information regarding the key. We give a combinatorial characterization of a certain type of perfect threshold scheme. We also investigate the maximum number of keys which a perfect (t, w)-threshold scheme can incorporate, as a function of t, w, and the total number of possible shadows, v. This maximum can be attained when there is a Steiner system S(t, w, v) which can be partitioned into Steiner systems S(t − 1. w, v). Using known constructions for such Steiner systems, we present two new classes of perfect threshold schemes, and discuss their implementation.
93 citations
TL;DR: This study focuses on (nonconvex) obnoxious center location problems on trees, and on the classical p-center problem on general networks, and in both cases better complexity bounds are derived.
Abstract: Several center location problems on general networks and even on tree networks are nonconvex. Known solution procedures are based on decomposing the problem into subproblems, finding each one of the local solutions independently, and then selecting among them the best solution. The purpose of this paper is to demonstrate that for some location models, it is possible to take advantage and use information that is common to certain subproblems in order to obtain better complexity bounds. The main tool implemented is a dynamic data structure that is used to maintain the objective (dynamically) as we move from one subproblem to the next.This study focuses on (nonconvex) obnoxious center location problems on trees, and on the classical p-center problem on general networks. In both cases better complexity bounds are derived.
83 citations
TL;DR: A polynomial bound on the length of the game in terms of the number of vertices of the graph provided the length is finite is proved.
Abstract: Bjorner, Lvasz, and Shor have introduced a chip firing game on graphs. This paper proves a polynomial bound on the length of the game in terms of the number of vertices of the graph provided the length is finite. The obtained bound is best possible within a constant factor.
79 citations
TL;DR: It is shown that for semicomplete digraphs T the T-colouring problem is NP-complete when T has more than one directed cycle, and polynomially decidable otherwise.
Abstract: The following problem, known as the H-colouring problem, is studied. An H-colouring of a directed graph D is a mapping $f:V( D ) \to V( H )$ such that $( f( x ),f( y ) )$ is an edge of H whenever $( x,y )$ is an edge of D. The H-colouring problem is the following. Instance: A directed graph D. Question: Does there exist an H-colouring of D? In this paper it is shown that for semicomplete digraphs T the T-colouring problem is NP-complete when T has more than one directed cycle, and polynomially decidable otherwise.
77 citations
TL;DR: A simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by m( T), a result first proved by Wilf.
Abstract: We give a simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by \[ m( T ) = \begin{cases} 2^{k - 1} + 1& {\text{if }} n = 2k, \\ 2^k & {\text{if }} n = 2k + 1, \end{cases}\] a result first proved by Wilf [SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 125–130]. We also characterize those trees achieving this maximum value. Finally we investigate some related problems.
73 citations
TL;DR: A calculation of the linearization coefficients of the Laguerre polynomials is proposed by means of analytic and combinatorial methods to extend to the case of an arbitrary $\alpha $ a combinatoric and analytic result due to Askey, Ismail, and Koornwinder and Even and Gillis.
Abstract: A calculation of the linearization coefficients of the (generalized) Laguerre polynomials $L_n^{( \alpha )} ( x )$ is proposed by means of analytic and combinatorial methods. This paper extends to the case of an arbitrary $\alpha $ a combinatoric and analytic result due to Askey, Ismail, and Koornwinder and Even and Gillis.
TL;DR: Graphs with fixed maximum degree in which broadcasting can be completed in near minimum time are described.
Abstract: Broadcasting is an information dissemination process in which a message is to be sent from a single originator to all members of a network by placing calls over the communication lines of the network. Several previous papers have investigated ways to construct sparse graphs (networks) in which this process can be completed in minimum time from any originator. The graphs produced by these methods contain high degree vertices. This paper describes graphs with fixed maximum degree in which broadcasting can be completed in near minimum time.
TL;DR: The only four cases in which polynomial algorithms can be expected to be expected (in the sense that all other cases are shown to be NP-hard) are identified are identified.
Abstract: A two-factor of G consists of disjoint cycles that cover $V( G )$. The authors consider the existence problem for two-factors in which the cycles are restricted to having lengths from a prescribed (possibly infinite) set of integers. Theorems are presented which derive the existence of such restricted two-factors in G from their existence in $G - u$ and $G - v $. The possibility of such theorems is then related to the complexity of the corresponding existence problem. In particular, the only four cases in which polynomial algorithms can be expected (in the sense that all other cases are shown to be NP-hard) are identified.
TL;DR: The problem of perfect hashing is generalized and some initial results are obtained and an improvement on earlier results for $( i, j )$-separating systems of partitions is provided.
Abstract: The problem of perfect hashing is generalized and some initial results are obtained. As a corollary, an improvement on earlier results for $( i, j )$-separating systems of partitions is provided.
TL;DR: It is shown thatGF ( q^n) has a self complementary normal basis overGF if and only if n is odd or $n \equiv 2(\bmod{\text{-}}4)$ and q is even.
Abstract: It is shown that $\mathrm{GF} ( q^n )$ has a self complementary normal basis over $\mathrm{GF} ( q )$ if and only if n is odd or $n \equiv 2(\bmod{\text{-}}4)$ and q is even. All existence proofs are constructive and can be readily employed to obtain such bases.
TL;DR: A fast procedure for determining sets of nodes to interchange in order to improve an initial partition by interchanging nodes between the subsets of the initial partition.
Abstract: There is a class of graph partitioning algorithms which improve an initial partition by interchanging nodes between the subsets of the initial partition. These algorithms tend to require long running times because usually many trials must be made before the right combination of nodes to interchange is found. In this paper we describe a fast procedure for determining sets of nodes to interchange in order to improve an initial partition.
TL;DR: This paper gives time algorithms for finding a minimum weighted total dominating set and aminimum weighted maximal irredundant set in a weighted interval graph, i.e., one that represents n intersecting intervals on the real line, each having a (possibly negative) real weight.
Abstract: In an undirected graph, a subset X of the nodes is a total dominating set if each node in the graph is a neighbor of some node in X. In contrast, X is an irredundant set if the closed neighborhood of each node in X is not contained in the union of closed neighborhoods of the other nodes in X. This paper gives $O( n\log n )$ and $O( n^4 )$ time algorithms for finding, respectively, a minimum weighted total dominating set and a minimum weighted maximal irredundant set in a weighted interval graph, i.e., one that represents n intersecting intervals on the real line, each having a (possibly negative) real weight.
TL;DR: It will be proved that for a bipartite graph G and an arbitrary probability distribution P on its vertex set the entropies of G and $\bar G$ add up to the entropy of P.
Abstract: The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This concept is at the core of a new bounding technique for graph covering problems and has furnished the best known bounds for the problem of perfect hashing.The basis of the technique is the sub-additivity of graph entropy with respect to the union of graphs. The tightness of the bounds depends on whether or not we have equality rather than just sub-additivity. As a first step in this analysis, we are investigating whether for a given graph G the entropies of G and $\bar G$ add up to the entropy of the complete graph on the same vertex set, i.e., the entropy of the underlying probability distribution.We shall prove that for a bipartite graph G and an arbitrary probability distribution P on its vertex set the entropies of G and $\bar G$ add up to the entropy of P. Related problems will be discussed.The results have interesting connections with the Ford–Fulkerson theory of network...
TL;DR: The results are surprising in that they improve significantly on the solution usually used in practice, namely: alternation of east and west orientations and north and south orientations.
Abstract: The problem of finding strongly connected orientations (one-way street assignments) for graphs which arise from city streets is studied Specifically, the grid graphs consisting of $n_1 + 1$ east-west avenues and $n_2 + 1$ north-south streets, for $n_1 ,n_2 $ sufficiently large, are studied In general, it is difficult to find strongly connected orientations of graphs which are optimal according to any of a variety of criteria However, for the grid graphs in question, optimal strongly connected orientations according to several important criteria are described The results are surprising in that they improve significantly on the solution usually used in practice, namely: alternation of east and west orientations and north and south orientations
TL;DR: This paper describes a hierarchy of such classes of functions that are shown to strictly contain each other, in increasing order of generality, the threshold, the saddle-free, the pseudomodular, the completely unimodal, the unimmodal, and the unimin functions.
Abstract: When an injective pseudo-Boolean function $f:B^n \to \mathbb{R}$ is minimized, where $B^n = \{ 0,1 \}^n$ is the set of vertices of the unit-hypercube, it is natural to consider so-called greedy vertex-following algorithms. These algorithms construct a sequence of neighbouring (Hamming distance 1) vertices with decreasing f-value. The question arises as to when such algorithms will find the global optimum given any starting point. This paper describes a hierarchy of such classes of functions that are shown to strictly contain each other. These classes are, in increasing order of generality, the threshold, the saddle-free, the pseudomodular, the completely unimodal, the unimodal, and the unimin (respectively, unimax) functions. Some considerations as to the complexity of the above-mentioned class of algorithms are also made.
TL;DR: It is shown (among other things) that for any rational $p/q\geqq 1$, there is a 3-uniform hypergraph H with $\tau* ( H ) = p/q$.
Abstract: The fractional covering number$\tau^*$ of a hypergraph $H = ( V, E )$ is defined to be the minimum possible value of $\sum_{x \in V} t( x )$ where t ranges over all functions $t : V \to \mathbb{R}$ which satisfy $\sum_{x \in e} t ( x ) \geqq 1$ for all edges $e \in E$. In the case of ordinary graphs G, it is known that $2\tau^* ( G )$ is always an integer. By contrast, it is shown (among other things) that for any rational $p/q\geqq 1$, there is a 3-uniform hypergraph H with $\tau^* ( H ) = p/q$.
TL;DR: For Euclidean location problems with uniformly distributed customers, the results imply that hexagonal partitioning of the region is asymptotically optimal and that the uniform distribution is the worst possible.
Abstract: A zero-sum game with a maximizer that selects a point x in given polygon R in the plane and a minimizer that selects K points $c_1 ,c_2 , \cdots ,c_K $ in the plane is considered; the payoff is the Euclidean distance from x to the closest of the points $c_j $, or any monotonically nondecreasing function of this quantity. Lower and upper bounds on the value of the game are derived by considering, respectively, the maximizer’s strategy of selecting a uniformly distributed random point in R and the minimizer’s strategy of selecting K members of a (uniformly) randomly positioned grid of centers that induce a covering of R by K congruent regular hexagons. The analysis shows that these strategies are asymptotically optimal (for $K \to \infty $). For Euclidean location problems with uniformly distributed customers, the results imply that hexagonal partitioning of the region is asymptotically optimal and that the uniform distribution is asymptotically the worst possible.
TL;DR: This paper separates deterministic algorithms from randomized ones, as there are randomized algorithms whose expected number of comparisons is O ( n^{1 + 1/k} ) and improves the previous known bounds by a factor of $( \log n )^{2/( 2^k - 1 )} .
Abstract: The worst case number of comparisons needed for sorting or selecting in rounds is considered. The following results are obtained. (a) For every fixed $k\geqq 2$, $\Omega ( n^{1 + 1/k} ( \log n )^{1/k} )$ comparisons are required to sort n elements in k rounds. ($O ( n^{1 + 1 / k} \log n )$ are known to be sufficient.) This improves the previous known bounds by a factor of $( \log n )^{1/k} $, which separates deterministic algorithms from randomized ones, as there are randomized algorithms whose expected number of comparisons is $O ( n^{1 + 1/k} )$. (b) For every fixed $k\geqq 2$, $\Omega ( n^{1 + 1/( 2^k - 1 )} ( \log n )^{2/( 2^k - 1 )} )$ comparisons are required to select the median from n elements in k rounds. ($O ( n^{1 + 1/ ( 2^k - 1 ) } ( \log n )^{2 - 2/ ( 2^k - 1 ) } )$ are known to be sufficient.) This improves the previous known bounds by a factor of $( \log n )^{2/( 2^k - 1 )} $ and separates the problem of finding the median from that of finding the minimum, as $O( n^{1 + 1/( 2^k - 1 ) } )$ c...
TL;DR: The main result shown here is that the frequency of pairs of binary trees a given distance apart is described by a limiting Poisson distribution, with e^{ -1 / 8} \approx 88$ percent of all pairs maximally distant.
Abstract: The symmetric difference metric has been useful in comparing phylogenetic trees derived from DNA sequence data. The main result shown here is that the frequency of pairs of binary trees a given distance apart is described by a limiting Poisson distribution, with $e^{ -1 / 8} \approx 88$ percent of all pairs maximally distant. Asymptotic bounds on the distribution are derived, and the asymptotic mean and variance of the normalized metric on the class of all phylogenetic trees is also calculated. The results rely on simple combinatorial constructions and analytic properties of appropriate generating functions.
TL;DR: A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows.
Abstract: A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets $X_i ,Y_i $ of $X,i = 1, \cdots ,n$, satisfying $X_i \cap Y_i = \O $, find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by $X\backslash Y_i $ has a connected component containing $X_i $.Two other problems related to this one are the following ones. (2) Given a set X and subsets $X_1 ,X_2 , \cdots ,X_n $ such that $X = \cup _{i = 1}^n X_i $, find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph $G_i $ induced by $X_i $ in G is connected. (3) Given a set X and subsets $X_1 ,X_2 , \cdots ,X_n $ such that $X = \cup _{i = 1}^n X_i $, find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of $\{ 1, \cdots ,n \}$, the subgraph induced by $ \cap _{i \in I} X_i $ is co...
TL;DR: A new negative Steinitz-type theorem is established; the face lattices of simplicial k-polytopes with $k + 4$ vertices cannot be characterized locally, and the affine Gale diagrams corresponding to a given simplicial complex are characterized.
Abstract: Affine Gale diagrams are of one dimension lower than the well-known Gale transforms, and thus k-polytopes with $k + 4$ vertices can be represented by planar point configurations. The underlying algebraic reduction is due to Bokowski [6], while similar geometric arguments were used before by Perles [12]. In this paper we consider affine Gale diagrams as a special case of oriented matroid duality, and we apply this technique to several convex geometrical problems. As the main result we establish a new negative Steinitz-type theorem in the spirit of [25]; the face lattices of simplicial k-polytopes with $k + 4$ vertices cannot be characterized locally. We answer two questions posed in [11] concerning Kleinschmidt’s 4-polytope Q with a facet of nonarbitrary shape [15], and we describe another such 4-polytope P with minimal number of facets. We characterize the affine Gale diagrams corresponding to a given simplicial complex, and we discuss as an example Mobius’ torus with 7 vertices. Finally, we prove a parti...
TL;DR: It is shown that the problem of realizing a metric by a graph or network with minimum total edge-length is, depending on the version, NP-hard or NP-complete.
Abstract: It is shown that the problem of realizing a metric by a graph or network with minimum total edge-length is, depending on the version, NP-hard or NP-complete.In particular, Discrete Metric Realization (DMR) is NP-complete “in the strong sense,” where DMR is defined as follows:INSTANCE. An n-by-n integer-entry distance matrix $D = ( d_{i,j} )$ and a positive integer k;QUESTION. Is there a graph $G = \langle V,E \rangle $ with distinguished vertices $v _1 ,v _2 , \cdots ,v _n $ that realize D, i.e., the number of edges in a shortest path between $v _i $ and $v _j $ is exactly $d_{i,j} $ for each $1\leqq i\leqq j\leqq n$, and such that the total number of edges of G is at most k?
TL;DR: A lower bound of $\Omega ( ( \log n )^{1 /2} )$ steps is proved, improving the previous lower bound and using Ramsey-theoretic and combinatorial arguments.
Abstract: We consider the problem of element distinctness. Here n synchronized processors, each given an integer input, must decide whether these integers are pairwise distinct, while communicating via an infinitely large shared memory.If simultaneous write access to a memory cell is forbidden, then a lower bound of $\Omega ( \log n )$ on the number of steps easily follows (from S. Cook, C. Dwork, and R. Reischuk, SIAM J. Comput., 15 (1986), pp. 87–97.) When several (different) values can be written simultaneously to any cell, then there is an simple algorithm requiring $O ( 1 )$ steps.We consider the intermediate model, in which simultaneous writes to a single cell are allowed only if all values written are equal. We prove a lower bound of $\Omega ( ( \log n )^{1 /2} )$ steps, improving the previous lower bound of $\Omega ( \log \log \log n )$ steps (F. E. Fich, F. Meyer auf der Heide, and A. Wigderson, Adv. in Comput., 4 (1987), pp. 1–15).The proof uses Ramsey-theoretic and combinatorial arguments. The result imp...
TL;DR: This paper considers a variety of combinatorial and number-theoretic questions that arise from considerations of uniqueness in the theory of measurement, and three types of assumptions that allow different inequality and equality comparisons between certain subsets of differences are considered.
Abstract: This paper considers a variety of combinatorial and number-theoretic questions that arise from considerations of uniqueness in the theory of measurement. Given $n + 1$ objects linearly ordered from worst (smallest) to best (largest), let $d_i $ stand for the difference between objects i and $i + 1$. Three types of assumptions that allow different inequality and equality comparisons between certain subsets of differences are considered. The three types of assumptions arise from problems concerning the uniqueness of finite algebraic difference measurement and finite subjective probability measurement. If the number of equality comparisons is sufficient to imply that the $d_i $ in $d = ( d_1 ,d_2 , \cdots ,d_n )$ are unique up to multiplication by a positive constant, then d is said to be unique. Results on the number of unique d’s and relationships among the components of such d’s are obtained for each of the three types.
TL;DR: An $O( n^3 \log n )$ algorithm is presented that, for a given tournament on n vertices, produces a ranking with fit at least $\frac{1}{2} + c_1 n^{3/ 2} $, where $c_1 = \frac {1}{8}\pi ^{ - 1 / 2} £.
Abstract: An $O( n^3 \log n )$ algorithm is presented that, for a given tournament on n vertices, produces a ranking with fit at least $\frac{1}{2} \begin{pmatrix} n \\ 2 \end{pmatrix} + c_1 n^{3/ 2} $, where $c_1 = \frac{1}{8}\pi ^{ - 1 / 2} $.