A Criterion for Extractability of Mutual Information for a Triple of Strings

We say that the mutual information of a triple of binary strings a, b, c can be extracted if there exists a string d such that a, b, and c are independent given d, and d is simple conditional to each of the strings a, b, and c. It is proved that the mutual information between a, b, and c can be extracted if and only if the values of the conditional mutual informations I(a : b|c), I(a : c|b), and I(b : c|a) are negligible. The proof employs a non-Shannon-type information inequality (a generalization of the recently discovered Zhang–Yeung inequality).     

Worst-case optimal algorithm for XPath evaluation over XML streams

We consider the XPath evaluation problem: Evaluate an XPath query Q on a streaming XML document D; i.e., determine the set Q(D) of document elements selected by Q. We mainly consider Conjunctive XPath queries that involve only the child and descendant axes. Previously known in-memory algorithms for this problem use O(|D|) space and O(|Q||D|) time. Several previously known algorithms for the streaming version use Ω(dn) space and Ω(dn|D|) time in the worst case; d denotes the depth of D, and n denotes the number of location steps in Q. Their exponential space requirement could well exceed the O(|D|) space used by the in-memory algorithms. We present an efficient algorithm that uses O(d|Q|+nc) space and O((|Q|+dn)|D|) time in the worst case; c denotes the maximum number of elements of D that can be candidates for output, at any one instant. For some worst case Q and D, the memory space used by our algorithm matches our lower bound proved in a different paper; so, our algorithm uses optimal memory space in the worst case.     

Stochastic adaptive pole-zero assignment with convergence analysis

The adaptive control u_n is designed for the stochastic system A(z)y_n+1 = B(z)u_n+C(z)w_n+1 with unknown constant matrix coefficients in the polynomials A(z), B(z) and C(z) in the shift-back operator with the purposes that (1) the unknown matrices are strongly consistently estimated and (2) the poles and zeros are replaced in such a way that the system itself is transferred to A⁰(z)y_n+1 = B⁰(z)u_n⁰+_n+1 with given A⁰(z), B⁰(z) and u_n⁰ so that the pole-zero assignment error {_n+1} is minimized. The problem of adaptive pole-zero assignment combined with tracking is also considered in this paper. Conditions used are imposed only on A(z), B(z) and C(z).     

Independence and k-domination in graphs

A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbours in S. The k-domination number γ _k (G) is the minimum cardinality of a k-dominating set of G, and α(G) denotes the cardinality of a maximum independent set of G. Brook's well-known bound for the chromatic number χ and the inequality α(G)≥n(G)/χ(G) for a graph G imply that α(G)≥n(G)/Δ(G) when G is non-regular and α(G)≥n(G)/(Δ(G)+1) otherwise. In this paper, we present a new proof of this property and derive some bounds on γ _k (G). In particular, we show that, if G is connected with δ(G)≥k then γ _k (G)≤(Δ(G)?1)α(G) with the exception of G being a cycle of odd length or the complete graph of order k+1. Finally, we characterize the connected non-regular graphs G satisfying equality in these bounds and present a conjecture for the regular case.     

On the edge-tenacity of the middle graph of a graph

We consider the problem of efficiently breaking a graph into small components by removing edges. One measure of how easily this can be done is the edge-tenacity. Given a set of edges of G, the score of S is defined as sc(S)=[| S|+τ (G?S)]/[w(G?S)]. Formally, the edge-tenacity of a graph G is defined as T′(G)=min sc(S), where the minimum is taken over all edge-sets S of G. A subset S of E(G) is said to be a T′-set of G if T′(G)=sc(S). Note that if G is disconnected, the set S may be empty. For any graph G, τ(G?S) is the number of vertices in the largest component of G?S and w(G?S) is the number of components of G?S. The middle graph M(G) of a graph G is the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. In this paper, we give the edge-tenacity of the middle graph of specific families of graphs and its relationships with other parameters.     

Bounded dilation maps of hypercubes into Cayley graphs on the symmetric group

LetG andH be graphs with |V(G)≤ |V(H)|. Iff:V(G) →V(H) is a one-to-one map, we letdilation(f) be the maximum of dist _H (f x),f(y)) over all edgesxy inG where dist _H denotes distance inH. The construction of maps fromG toH of small dilation is motivated by the problem of designing small slowdown simulations onH of algorithms that were originally designed for the networkG. LetS(n), thestar network of dimension n, be the graph whose vertices are the elements of the symmetric group of degreen, two verticesx andy being adjacent ifx o (1,i) =y for somei. That is,xy is an edge ifx andy are related by a transposition involving some fixed symbol (which we take to be 1). Also letP(n), thepancake network of dimension n, be the graph whose vertices are the elements of the symmetric group of degreen, two verticesx andy being adjacent if one can be obtained from the other by reversing some prefix. That is,xy is an edge ifx andy are related byx o (1,i(2,i-1) ⋯ ([i/2], [i/2]) =y. The star network (introduced in [AHK]) has nice symmetry properties, and its degree and diameter are sublogarithmic as functions of the number of vertices, making it compare favorably with the hypercube network. These advantages ofS(n) motivate the study of how well it can simulate other parallel computation networks, in particular, the hypercube. The concern of this paper is to construct low dilation maps of hypercube networks into star or pancake networks. Typically in such problems, there is a tradeoff between keeping the dilationsmall and simulating alarge hypercube. Our main result shows that at the cost ofO (1) dilation asn→ ∞, one can embed a hypercube of near optimum dimension into the star or pancake networksS(n) orP(n). More precisely, lettingQ (d) be the hypercube of dimensiond, our main theorem is stated below. For simplicity, we state it only in the special case when the star network dimension is a power of 2. A more general result (applying to star networks of arbitrary dimension) is obtained by a simple interpolation. This author's research was done during the Spring Semester 1991, as a visiting professor in the Department of Mathematics and Statistics at Miami University.     

A class of hybrid collocation methods for third-order ordinary differential equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x _n+j , i = 0(1)k and at an off grid point x = x _n+u , where k is the step number of the method and u is an arbitrary rational number in (x _n , x _n+k ). A predictor of order 2k ? 1 is also proposed to cater for y _n+k in the main method. Taylor series expansion is employed for the calculation of y _n+1, y _n+2, y _n+u and their higher derivatives. Evaluation of the resulting method at x = x _n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.     

The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

A Hamiltonian cycle C=? u ₁, u ₂, …, u _n(G), u ₁ ? with n(G)=number of vertices of G, is a cycle C(u ₁; G), where u ₁ is the beginning and ending vertex and u _i is the ith vertex in C and u _i ≠u _j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C ₁, C ₂, …, C _k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B _n )=n?1 if n≥4, where B _n is the n-dimensional bubble-sort graph.     

A Simple Optimal Parallel Algorithm for a Core of a Tree

A core of a tree T = (V, E) is a path in T which minimizes ∑_vVd(v, P), where d(v, P), the distance from a vertex v to path P, is defined as min_uPd(v, u). We present an optimal parallel algorithm to find a core of T in O(log n) time using O(n/log n) processors on an EREW PRAM machine, where n is the number of vertices of tree T.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

New Plain-Exponential Time Classes for Graph Homomorphism

A homomorphism from a graph G to a graph H (in this paper, both simple, undirected graphs) is a mapping f:V(G)→V(H) such that if uv∈E(G) then f(u)f(v)∈E(H). The problem Hom (G,H) of deciding whether there is a homomorphism is NP-complete, and in fact the fastest known algorithm for the general case has a running time of O ^*(n(H) ^cn(G)) (the notation O ^*(⋅) signifies that polynomial factors have been ignored) for a constant 0<c<1. In this paper, we consider restrictions on the graphs G and H such that the problem can be solved in plain-exponential time, i.e. in time O ^*(c ^n(G)+n(H)) for some constant c.     

<Emphasis Type="Italic">f</Emphasis>-Sensitivity Distance Oracles and Routing Schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V,E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s,t,F), where s and t are vertices and F is a set of forbidden edges such that |F|≤f, returns an estimate of the distance between s and t in G(V,E∖F). For an integer parameter k≥1, the size of the data structure is O(fkn ^1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k−2)(f+1), and the query time is O(|F|⋅log ² n⋅log log n⋅log log d), where d is the distance between s and t in G(V,E∖F).     

On Bounded Leg Shortest Paths Problems

Let V be a set of points in a d-dimensional l _p -metric space. Let s,t∈V and let L be a real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set has length of at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s–t Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and a real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper we obtain the following results:     

Centralities of a network and its line graph: an analytical comparison by means of their irregularity

Given a network G, it is known that the Bonacich centrality of the bipartite graph B(G) associated with G can be obtained in terms of the centralities of the line graph L(G) associated with G and the centrality of the network G+gr (whose adjacency matrix is obtained by adding to the adjacency matrix A(G) the diagonal matrix D=b_ij, where b_ii is the degree of node i in G) and conversely. In this contribution, we use the centrality of G to estimate the centrality of G+gr and show that the error committed is bounded by some measure of the irregularity of G. This estimate gives an analytical comparison of the eigenvector centrality of G with the centrality of L(G) in terms of some irregularity measure of G.     

On Space Efficient Two Dimensional Range Minimum Data Structures

The two dimensional range minimum query problem is to preprocess a static m by n matrix (two dimensional array) A of size N=m⋅n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every algorithm enabled to access A during the query and using a data structure of size O(N/c) bits requires Ω(c) query time, for any c where 1≤c≤N. This lower bound holds for arrays of any dimension. In particular, for the one dimensional version of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits which can be preprocessed in O(N) time to support O(clog ² c) query time. For c=O(1), this is the first O(1) query time algorithm using a data structure of optimal size O(N) bits. For the case where queries can not probe A, we give a data structure of size O(N⋅min {m,log n}) bits with O(1) query time, assuming m≤n. This leaves a gap to the space lower bound of Ω(Nlog m) bits for this version of the problem.     

On Word Equations in One Variable

For a word equation E of length n in one variable x occurring # _x times in E a resolution algorithm of O(n+# _x log n) time complexity is presented here. This is the best result known and for the equations that feature #_x < \fracnlogn\#_{x}<\frac{n}{\log n} it yields time complexity of O(n) which is optimal. Additionally it is proven here that the set of solutions of any one-variable word equation is either of the form F or of the form F∪(uv)⁺ u where F is a set of O(log n) words and u, v are some words such that uv is a primitive word.     

Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology

This paper discusses how to count and generate strings that are ``distinct' in two senses: p -distinct and b -distinct. Two strings x on alphabet A and x' on alphabet A' are said to be p -distinct iff they represent distinct ``patterns'; that is, iff there exists no one—one mapping from A to A' that transforms x into x' . Thus aab and baa are p -distinct while aab and ddc are p -equivalent. On the other hand, x and x' are said to be b -distinct iff they give rise to distinct border (failure function) arrays: thus aab with border array 010 is b -distinct from aba with border array 001 . The number of p -distinct (resp. b -distinct) strings of length n formed using exactly k different letters is the [k,n] entry in an infinite p' (resp. b' ) array. Column sums p[n] and b[n] in these arrays give the number of distinct strings of length n . We present algorithms to compute, in constant time per string, all p -distinct (resp. b -distinct) strings of length n formed using exactly k letters, and we also show how to compute all elements p'[k,n] and b'[k,n] . These ideas and results have application to the efficient generation of appropriate test data sets for many string algorithms. Received December 21, 1995; revised April 28, 1997.     

Solving shortest paths efficiently on nearly acyclic directed graphs

Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min operations on trigger vertices, thereby making the computation of shortest paths through non-trigger vertices easier. A previously presented algorithm computed the 1-dominator set in O(mn) worst-case time, which allowed it to be integrated as part of an O(mn+nrlogr) time all-pairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1-dominator set in just O(m) time. This can be integrated as part of the O(m+rlogr) time spent solving single-source, improving on the value of r obtained by the earlier tree-decomposition single-source algorithm. In addition, a new bidirectional form of 1-dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bidirectional 1-dominator set can similarly be computed in O(m) time and included as part of the O(m+rlogr) time spent computing single-source. This paper also presents a new all-pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous all-pairs time complexity from O(mn+nr²) to O(mn+r³).     

Algorithms for (0, 1, d)-graphs with d constrains

Let G be a graph with vertex set V(G). Let n, k, d be non-negative integers such that n+2k+d≤|V(G)|?2 and |V(G)|?n?d are even. A matching which saturates exactly |V(G)|?d vertices is called a defect-d matching of G. If when deleting any n vertices the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect-d matching, then G is said to be an (n, k, d)-graph. We present an algorithm to determine (0, 1, d)-graphs with d constraints. Moreover, we solve the problem of augmenting a bipartite graph G=(B, W) to be a (0, 1, d)-graph by adding fewest edges, where d=∥B|?|W∥. The latter problem is applicable to the job assignment problem, where the number of jobs does not equal the number of persons.     

The maximum gain of increasing the number of preemptions in multiprocessor scheduling

We consider the optimal makespan C(P, m, i) of an arbitrary set P of independent jobs scheduled with i preemptions on a multiprocessor with m identical processors. We compare the ratio for such makespans for i and j preemptions, respectively, where i < j. This ratio depends on P, but we are interested in the P that maximizes this ratio, i.e. we calculate a formula for the worst case ratio G(m, i, j) defined as G(m,i,j)=max\fracC(P,m,i)C(P,m,j),{G(m,i,j)=\max \frac{C(P,m,i)}{C(P,m,j)},} where the maximum is taken over all sets P of independent jobs.