The bondage numbers of extended de Bruijn and Kautz digraphs

In this paper, we consider the bondage number b(G) for a digraph G, which is defined as the minimum number of edges whose removal results in a new digraph with larger domination number. This parameter measures to some extent the robustness of an interconnection network with respect to link failures. By constructing a family of minimum dominating sets, we compute the bondage numbers of the extended de Bruijn digraph and the extended Kautz digraph. As special cases, we obtain for the de Bruijn digraph B(d, n) and the Kautz digraph K(d, n) that b(B(d, n)) = d if n is odd and d ⩽ b(B(d, n)) < 2d if n is even, and b(K(d, n)) = d + 1.     

The twin domination number in generalized de Bruijn digraphs

Let G=(V,A) be a digraph. A set T of vertices of G is a twin dominating set of G if for every vertex v∈V?T, there exist u,w∈T (possibly u=w) such that arcs (u,v),(v,w)∈A. The twin domination numberγ^∗(G) of G is the cardinality of a minimum twin dominating set of G. In this paper we investigate the twin domination number in generalized de Bruijn digraphs GB(n,d). For the digraphs GB(n,d), we first establish sharp bounds on the twin domination number. Secondly, we give the exact values of the twin domination number for several types of GB(n,d) by constructing minimum twin dominating sets in the digraphs. Finally, we present sharp upper bounds for some special generalized de Bruijn digraphs.     

Tighter Lower Bounds for Nearest Neighbor Search and Related Problems in the Cell Probe Model

We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n,d) cells, each containing poly(d, log n) bits. We get a lower bound of Ω(d/log n) on the search time, a significant improvement over the recent bound of Ω(log d) of Borodin et al. This should be contrasted with the upper bound of O(log log d) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.     

Acyclic and k-distance coloring of the grid

In this paper, we give a relatively simple though very efficient way to color the d-dimensional grid G(n₁,n₂,…,n_d) (with n_i vertices in each dimension 1?i?d), for two different types of vertex colorings: (1) acyclic coloring of graphs, in which we color the vertices such that (i) no two neighbors are assigned the same color and (ii) for any two colors i and j, the subgraph induced by the vertices colored i or j is acyclic; and (2) k-distance coloring of graphs, in which every vertex must be colored in such a way that two vertices lying at distance less than or equal to k must be assigned different colors. The minimum number of colors needed to acyclically color (respectively k-distance color) a graph G is called acyclic chromatic number of G (respectively k-distance chromatic number), and denoted a(G) (respectively χ_k(G)).The method we propose for coloring the d-dimensional grid in those two variants relies on the representation of the vertices of G_d(n₁,…,n_d) thanks to its coordinates in each dimension; this gives us upper bounds on a(G_d(n₁,…,n_d)) and χ_k(G_d(n₁,…,n_d)).We also give lower bounds on a(G_d(n₁,…,n_d)) and χ_k(G_d(n₁,…,n_d)). In particular, we give a lower bound on a(G) for any graph G; surprisingly, as far as we know this result was never mentioned before. Applied to the d-dimensional grid G_d(n₁,…,n_d), the lower and upper bounds for a(G_d(n₁,…,n_d)) match (and thus give an optimal result) when the lengths in each dimension are “sufficiently large” (more precisely, if ). If this is not the case, then these bounds differ by an additive constant at most equal to . Concerning χ_k(G_d(n₁,…,n_d)), we give exact results on its value for (1) k=2 and any d?1, and (2) d=2 and any k?1.In the case of acyclic coloring, we also apply our results to hypercubes of dimension d, H_d, which are a particular case of G_d(n₁,…,n_d) in which there are only 2 vertices in each dimension. In that case, the bounds we obtain differ by a multiplicative constant equal to 2.     

Finding Maximum Edge Bicliques in Convex Bipartite Graphs

A bipartite graph G=(A,B,E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v??A, vertices adjacent to v are consecutive in?B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G=(A,B,E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(nlog?³ nlog?log?n) time and O(n) space, where n=|A|. This improves the current O(n ²) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(n??(n)) and O(n) time, respectively, where n=min?(|A|,|B|) and ??(n) is the slowly growing inverse of the Ackermann function.     

Improved Algorithms for Even Factors and Square-Free Simple b-Matchings

Given a digraph G=(VG,AG), an even factor M?AG is a set formed by node-disjoint paths and even cycles. Even factors in digraphs were introduced by Geelen and Cunningham and generalize path matchings in undirected graphs. Finding an even factor of maximum cardinality in a general digraph is known to be NP-hard but for the class of odd-cycle symmetric digraphs the problem is polynomially solvable. So far the only combinatorial algorithm known for this task is due to Pap; its running time is O(n ⁴) (hereinafter n denotes the number of nodes in G and m denotes the number of arcs or edges). In this paper we introduce a novel sparse recovery technique and devise an O(n ³logn)-time algorithm for finding a maximum cardinality even factor in an odd-cycle symmetric digraph. Our technique also applies to other similar problems, e.g. finding a maximum cardinality square-free simple b-matching.     

On the k-tuple domination of generalized de Brujin and Kautz digraphs

A vertex u in a digraph G = (V, A) is said to dominate itself and vertices v such that (u, v) ∈ A. For a positive integer k, a k-tuple dominating set of G is a subset D of vertices such that every vertex in G is dominated by at least k vertices in D. The k-tuple domination number of G is the minimum cardinality of a k-tuple dominating set of G. This paper deals with the k-tuple domination problem on generalized de Bruijn and Kautz digraphs. We establish bounds on the k-tuple domination number for the generalized de Bruijn and Kautz digraphs and we obtain some conditions for the k-tuple domination number attaining the bounds.     

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.     

Implementing Shared Memory on Mesh-Connected Computers and on the Fat-Tree

We present deterministic upper and lower bounds on the slowdown required to simulate an (n, m)-PRAM on a variety of networks. The upper bounds are based on a novel scheme that exploits the splitting and combining of messages. This scheme can be implemented on an n-node d-dimensional mesh (for constant d) and on an n-leaf pruned butterfly and attains the smallest worst-case slowdown to date for such interconnections, namely, O(n^1/d(log(m/n))^1-1/d) for the d-dimensional mesh (with constant d) and O( ) for the pruned butterfly. In fact, the simulation on the pruned butterfly is the first PRAM simulation scheme on an area-universal network. Finally, we prove restricted and unrestricted lower bounds on the slowdown of any deterministic PRAM simulation on an arbitrary network, formulated in terms of the bandwidth properties of the interconnection as expressed by its decomposition tree.     

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R₁×R₂×?×Rk, where each Ri is a closed interval on the real line of the form [ai,ai+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph.It is known that for a graph G, . Recently it has been shown that for a graph G, cub(G)?4(Δ+1)lnn, where n and Δ are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G=(A∪B,E) with |A|=n₁, |B|=n₂, n₁?n₂, and Δ^′=min{ΔA,ΔB}, where ΔA=maxa∈Ad(a) and ΔB=maxb∈Bd(b), d(a) and d(b) being the degree of a and b in G, respectively, cub(G)?2(Δ^′+2)⌈lnn₂⌉. We also give an efficient randomized algorithm to construct the cube representation of G in 3(Δ^′+2)⌈lnn₂⌉ dimensions. The reader may note that in general Δ^′ can be much smaller than Δ.     

Restricted arc-connectivity of digraphs

Since interconnection networks are often modeled by graphs or digraphs, the edge-connectivity of a graph or arc-connectivity of a digraph are important measurements for fault tolerance of networks.The restricted edge-connectivity λ^′(G) of a graph G is the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G−S. A connected graph G is called λ^′-connected, if λ^′(G) exists.In 1988, Esfahanian and Hakimi [A.H. Esfahanian, S.L. Hakimi, On computing a conditional edge-connectivity of a graph, Inform. Process. Lett. 27 (1988), 195-199] have shown that each connected graph G of order n?4, except a star, is λ^′-connected and satisfies λ^′(G)?ξ(G), where ξ(G) is the minimum edge-degree of G.If D is a strongly connected digraph, then we call in this paper an arc set S a restricted arc-cut of D if D−S has a non-trivial strong component D₁ such that D−V(D₁) contains an arc. The restricted arc-connectivity λ^′(D) is the minimum cardinality over all restricted arc-cuts S.We observe that the recognition problem, whether λ^′(D) exists for a strongly connected digraph D is solvable in polynomial time. Furthermore, we present some analogous results to the above mentioned theorem of Esfahanian and Hakimi for digraphs, and we show that this theorem follows easily from one of our results.     

Two New Perspectives on Multi-Stage Group Testing

The group testing problem asks to find d?n defective elements out of n elements, by testing subsets (pools) for the presence of defectives. In the strict model of group testing, the goal is to identify all defectives if at most d defectives exist, and otherwise to report that more than d defectives are present. If tests are time-consuming, they should be performed in a small constant number s of stages of parallel tests. It is known that a test number O(dlogn), which is optimal up to a constant factor, can be achieved already in s=2 stages. Here we study two aspects of group testing that have not found major attention before. (1) Asymptotic bounds on the test number do not yet lead to optimal strategies for specific n,d,s. Especially for small n we show that randomized strategies significantly save tests on average, compared to worst-case deterministic results. Moreover, the only type of randomness needed is a random permutation of the elements. We solve the problem of constructing optimal randomized strategies for strict group testing completely for the case when d=1 and s≤2. A byproduct of our analysis is that optimal deterministic strategies for strict group testing for d=1 need at most 2 stages. (2) Usually, an element may participate in several pools within a stage. However, when the elements are indivisible objects, every element can belong to at most one pool at the same time. For group testing with disjoint simultaneous pools we show that Θ(sd(n/d)^1/s ) tests are sufficient and necessary. While the strategy is simple, the challenge is to derive tight lower bounds for different s and different ranges of d versus n.     

On the Expansion and Diameter of Bluetooth-Like Topologies

The routing capabilities of an interconnection network are strictly related to its bandwidth and latency characteristics, which are in turn quantifiable through the graph-theoretic concepts of expansion and diameter. This paper studies expansion and diameter of a family of subgraphs of the random geometric graph, which closely model the topology induced by the device discovery phase of Bluetooth-based ad hoc networks. The main feature modeled by any such graph, denoted as BT(r(n),c(n)), is the small number c(n) of links that each of the n devices (vertices) may establish with those located within its communication range r(n). First, tight bounds are proved on the expansion of BT(r(n),c(n)) for the whole set of functions r(n) and c(n) for which connectivity has been established in previous works. Then, by leveraging on the expansion result, nearly-tight upper and lower bounds on the diameter of BT(r(n),c(n)) are derived. In particular, we show asymptotically tight bounds on the diameter when the communication range is near the minimum needed for connectivity, the typical scenario considered in practical applications.     

On the K best integer network flows

We address the problem of finding the K best integer solutions of a linear integer network flow problem. We design an O(f(n,m,L,U)+KmS(n,m,L)) time and O(K+m) memory space algorithm to determine the K best integer solutions, in a directed network with n nodes, m arcs, maximum absolute value cost L, and an upper bound U on arc capacities and node supplies. f(n,m,L,U) is the best time needed to solve the minimum cost flow problem in a directed network and S(n,m,L) is the best time to solve the single-source shortest path problem in a network with non-negative lengths. The introduced algorithm efficiently determines a “proper minimal cycle” by taking advantage of the relationship between the best solutions. This way, we improve the theoretical as well as practical memory space bounds of the well-known method due to Hamacher. Our computational experiments confirm this result.     

Hamiltonian connectivity of the WK-recursive network with faulty nodes

A graph G is said to be Hamiltonian-connected if there is a Hamiltonian path between every two distinct nodes of G. Let F denote the set of faulty nodes of G. Then, G is |F|-node Hamiltonian-connected if G-F is Hamiltonian-connected. We use K(d,t) to denote a WK-recursive network of level t, each of whose basic modules is a d-node complete graph. Compared with other networks, it is rather difficult to construct a Hamiltonian path between two arbitrary nodes in a faulty WK-recursive network. In this paper, we show that K(d,t) is (d-4)-node Hamiltonian-connected. Since the connectivity of K(d,t) is d-1, the result is optimal in the worst case.     

Hamiltonian connectivity and globally 3∗-connectivity of dual-cube extensive networks

In 2000, Li et al. introduced dual-cube networks, denoted by DC_n for n?1, using the hypercube family Q_n and showed the vertex symmetry and some fault-tolerant hamiltonian properties of DC_n. In this article, we introduce a new family of interconnection networks called dual-cube extensive networks, denoted by DCEN(G). Given any arbitrary graph G, DCEN(G) is generated from G using the similar structure of DC_n. We show that if G is a nonbipartite and hamiltonian connected graph, then DCEN(G) is hamiltonian connected. In addition, if G has the property that for any two distinct vertices u,v of G, there exist three disjoint paths between u and v such that these three paths span the graph G, then DCEN(G) preserves the same property. Furthermore, we prove that the similar results hold when G is a bipartite graph.     

Computing functions on asynchronous anonymous networks

In an “anonymous” network the processors have no identity numbers. We investigate the problem of computing a given functionf on an asynchronous anonymous network in the sense that each processor computesf(I) for any inputI = (I(v ₁),...,I(v _n )), whereI(v _i) is the input to processorv _i ,i = 1, 2,...,n. We address the following three questions: (1) What functions are computable on a given network? (2) Is there a “universal” algorithm which, given any networkG and any functionf computable onG as inputs, computesf onG? (3) How can one find lower bounds on the message complexity of computing various functions on anonymous networks? We give a necessary and sufficient condition for a function to be computable on an asynchronous anonymous network, and present a universal algorithm for computingf(I) on any networkG, which acceptsG andf computable onG, as well as {I(v _i )}, as inputs. The universal algorithm requiresO(mn) messages in the worst case, wheren andm are the numbers of processors and links in the network, respectively. We also propose a method for deriving a lower bound on the number of messages necessary to solve the above problem on asynchronous anonymous networks.     

Weighted Mean of a Pair of Graphs

G and G′, with d(G, G′) being the edit distance of G and G′, the weighted mean of G and G′ is a graph G″ that has edit distances d(G, G″) and d(G″, G′) to G and G′, respectively, such that d(G, G″) + d(G″, G′) = d(G,G′). We'll show formal properties of the weighted mean, describe a procedure for its computation, and give examples. Received April 9, 2001     

Group path covering and distance two labeling of graphs

For a positive integer d, an L(d,1)-labeling f of a graph G is an assignment of integers to the vertices of G such that |f(u)−f(v)|?d if uv∈E(G), and |f(u)−f(v)|?1 if u and u are at distance two. The span of an L(d,1)-labeling f of a graph is the absolute difference between the maximum and minimum integers used by f. The L(d,1)-labeling number of G, denoted by λd,1(G), is the minimum span over all L(d,1)-labelings of G. An L^′(d,1)-labeling of a graph G is an L(d,1)-labeling of G which assigns different labels to different vertices. Denote by the L^′(d,1)-labeling number of G. Georges et al. [Discrete Math. 135 (1994) 103-111] established relationship between the L(2,1)-labeling number of a graph G and the path covering number of Gc, the complement of G. In this paper we first generalize the concept of the path covering of a graph to the t-group path covering. Then we establish the relationship between the L^′(d,1)-labeling number of a graph G and the (d−1)-group path covering number of Gc. Using this result, we prove that and for bipartite graphs G can be computed in polynomial time.     

The (strong) rainbow connection numbers of Cayley graphs on Abelian groups

A path in an edge-colored graph G, whose adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. The strong rainbow connection number src(G) of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices u and v of G are connected by a rainbow path of length d(u,v). In this paper, we give upper and lower bounds of the (strong) rainbow connection numbers of Cayley graphs on Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.