Showing papers on "Dominating set published in 2003"

A Polynomial-Time Approximation Scheme for the Minimum-Connected Dominating Set in Ad Hoc Wireless Networks

Abstract: A connected dominating set in a graph is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected. A minimum-connected dominating set is such a vertex subset with minimum cardinality. An application in ad hoc wireless networks requires the study of the minimum-connected dominating set in unit-disk graphs. In this paper, we design a (1 + 1/s)-approximation for the minimum-connected dominating set in unit-disk graphs, running in time nO((s log s)2). © 2003 Wiley Periodicals, Inc.

Topology management in ad hoc networks

Abstract: The efficiency of a communication network depends not only on its control protocols, but also on its topology. We propose a distributed topology management algorithm that constructs and maintains a backbone topology based on a minimal dominating set (MDS) of the network. According to this algorithm, each node determines the membership in the MDS for itself and its one-hop neighbors based on two-hop neighbor information that is disseminated among neighboring nodes. The algorithm then ensures that the members of the MDS are connected into a connected dominating set (CDS), which can be used to form the backbone infrastructure of the communication network for such purposes as routing. The correctness of the algorithm is proven, and the efficiency is compared with other topology management heuristics using simulations. Our algorithm shows better behavior and higher stability in ad hoc networks than prior algorithms.

Approximating the Domatic Number

Abstract: A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, $\delta$ the minimum degree, and $\Delta$ the maximum degree. We show that every graph has a domatic partition with $(1 - o(1))(\delta + 1)/\ln n$ dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a $(1 + o(1))\ln n$-approximation algorithm for domatic number, since the domatic number is always at most $\delta + 1$. We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every $\epsilon > 0$, a $(1 - \epsilon)\ln n$-approximation implies that $NP \subseteq DTIME(n^{O(\log\log n)})$. This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with $(1 - o(1))(\delta + 1)/\ln \Delta$ dominating sets, where the "o(1)" term goes to zero as $\Delta$ increases. This can be turned into an efficient algorithm that produces a domatic partition of $\Omega(\delta/\ln \Delta)$ sets.

Constant-time distributed dominating set approximation

Abstract: Finding a small dominating set is one of the most fundamental problems of traditional graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary parameter k and maximum degree Δ, our algorithm computes a dominating set of expected size O(kΔ2/k log Δ|DSOPT|) in O(k2) rounds where each node has to send O(k2Δ) messages of size O(logΔ). This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.

Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons

Abstract: Motivated by routing issues in ad hoc networks, we present polylogarithmic-time distributed algorithms for two problems. Given a network, we first show how to compute connected and weakly connected dominating sets whose size is at most O(logΔ) times optimal, Δ being the maximum degree of the input network. This is best-possible if NP ⊈ DTIME[nO(log log n)] and if the processors are limited to polynomial-time computation. We then show how to construct dominating sets which satisfy the above properties, as well as the "low stretch" property that any two adjacent nodes in the network have their dominators at a distance of at most O(log n) in the network. (Given a dominating set S, a dominator of a vertex u is any v ∊ S such that the distance between u and v is at most one.) We also show our time bounds to be essentially optimal.

Distributed dominant pruning in ad hoc networks

Abstract: Efficient routing among mobile hosts is an important function in ad hoc networks. Routing based on a connected dominating set is a promising approach, where the search space for a route is reduced to the hosts in the set. A set is dominating if all the hosts are either in the set or neighbors of hosts in the set. The efficiency of dominating-set-based routing mainly depends on the overhead introduced in the formation of the dominating set and the size of the dominating set. In this paper, we first review a distributed formation of a connected dominating set called marking process and dominating-set-based routing. Then a generalization of two existing rules (called rules 1 and 2). We prove that the vertex set derived by applying rule k is still a connected dominating set. When implemented with local neighborhood information. Rule k is more effective in reducing the dominating set derived from the marking process than the combination of rules 1 and 2, and has the same communication complexity and less computation complexity. Simulation results confirm that rule k outperforms rules 1 and 2, especially in relatively dense networks with unidirectional links.

Resolving domination in graphs

Abstract: For an ordered set $W =\lbrace w_1, w_2, \cdots , w_k\rbrace $ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r(v|W) = (d(v, w_1),d(v, w_2) ,\cdots , d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $\dim G$. A set $S$ of vertices in $G$ is a dominating set for $G$ if every vertex of $G$ that is not in $S$ is adjacent to some vertex of $S$. The minimum cardinality of a dominating set is the domination number $\gamma (G)$. A set of vertices of a graph $G$ that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called the resolving domination number $\gamma _r(G)$. In this paper, we investigate the relationship among these three parameters.

Global Defensive Alliances in Graphs

Abstract: A defensive alliance in a graph G =( V,E) is a set of vertices S V satisfying the condition that for every vertex v 2 S, the number of neighbors v has in S plus one (counting v) is at least as large as the number of neighbors it has in V S. Because of such an alliance, the vertices in S, agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in V S. A defensive alliance S is called global if it eects every vertex in V S, that is, every vertex in V S is adjacent to at least one member of the alliance S. Note that a global defensive alliance is a dominating set. We study global defensive alliances in graphs.

Power-aware broadcasting and activity scheduling in ad hoc wireless networks using connected dominating sets

Abstract: In ad hoc mobile wireless networks, owing to host mobility, broadcasting is expected to be more frequently used to find a route to a particular host, to page a host, and to alarm all hosts. A straightforward broadcasting by flooding is usually very costly and will result in substantial redundancy and more energy consumption. Power consumption is an important issue since most mobile hosts operate on battery. Broadcasting based on a connected dominating set is a promising approach, where only nodes in the dominating set need to relay the broadcast packet. A set is dominating if all the nodes in the system are either in the set or are neighbors of nodes in the set. Wu and Li proposed a simple and efficient distributed algorithm for calculating connected dominating set in ad hoc wireless networks, where connections of nodes are determined by their geographical distances. In general, nodes in the connected dominating set consume more energy to handle various bypass traffic than nodes outside the set. To prolong the life span of each node and, hence, the network by balancing the energy consumption in the system, nodes should be alternately chosen to form a connected dominating set. Activity scheduling deals with the way of rotating the role of each node among a set of given operation modes (e.g. dominating nodes versus dominated nodes). In this paper, we propose to apply the notion of power-aware connected dominating set to broadcasting and activity scheduling. The effectiveness of the proposed method in prolonging the life span of the network is confirmed through simulation. Copyright © 2003 John Wiley & Sons, Ltd.

A zonal algorithm for clustering an hoc networks

Abstract: A Mobile Ad Hoc Network (MANET) is an infrastructureless wireless network that can support highly dynamic mobile units. The multi-hop feature of a MANET suggests the use of clustering to simplify routing. Graph domination can be used in defining clusters in MANETs. A variant of dominating set which is more suitable for clustering MANETs is the weakly-connected dominating set. A cluster is defined to be the set of vertices dominated by a particular vertex in the dominating set. As it is NP-complete to determine whether a given graph has a weakly-connected dominating set of a particular size, we present a zonal distributed algorithm for finding small weakly-connected dominating sets. In this new approach, we divide the graph into regions, construct a weakly-connected dominating set for each region, and make adjustments along the borders of the regions to produce a weakly-connected dominating set of the entire graph. We present experimental evidence that this zonal algorithm has similar performance to and provides better cluster connectivity than previous algorithms.

Self-stabilizing algorithms for minimal dominating sets and maximal independent sets

Abstract: In the self-stabilizing algorithmic paradigm for distributed computation each node has only a local view of the system, yet in a finite amount of time, the system converges to a global state satisfying some desired property. In this paper we present polynomial time self-stabilizing algorithms for finding a dominating bipartition, a maximal independent set, and a minimal dominating set in any graph.

Constant-time distributed dominating set approximation

Abstract: Finding a small dominating set is one of the most fundamental problems of traditional graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary parameter $k$, our algorithm computes a dominating set of expected size $\bigO{k\Delta^{2/k}\log\Delta|\dsopt|}$ in $\bigO{k^2}$ rounds where each node has to send $\bigO{k^2\Delta}$ messages of size $\bigO{\log\Delta}$. This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.

Maximal independent set, weakly-connected dominating set, and induced spanners in wireless ad hoc networks

Abstract: A maximal independent set (MIS) S for a graph G is an independent set and no proper superset of S is also independent. A set S is dominating if each node in the graph is either in S or adjacent to one of the nodes in S. The subgraph weakly induced by S is the graph G′ such that each edge in G′ has at least one end point in S. A set S is a weakly-connected dominating set (WCDS) of G if S is dominating and G′ is connected. G′ is a sparse spanner if it has linear edges. The nodes of WCDS have been proposed in the literature as clusterheads for clustered wireless ad hoc networks. In this paper, we present two distributed algorithms for constructing a WCDS for wireless ad hoc networks in linear time. The first algorithm has an approximation ratio of 5, and requires O(n log n) messages, while the second algorithm has a larger approximation ratio, and requires only O(n) messages. Both of these algorithms are used to obtain sparse spanners. The spanner obtained by the second algorithm has a topological dilation of 3, and a geometric dilation of 6. Both of these algorithms are based on the construction of a MIS. The first algorithm requires the construction of a spanning tree. The second algorithm is fully localized, and does not depend on the spanning tree, which makes the maintenance of the WCDS simpler, and guarantees the maintenance of the same approximation ratio.

Paired-domination of Trees

Abstract: Let G= (V, E) be a graph without isolated vertices. A set S⊂V is a paired-dominating set if it dominates V and the subgraph induced by S,≤S\ge, contains a perfect matching. The paired-domination number γp(G) is defined to be the minimum cardinality of a paired-dominating set S in G. In this paper, we present a linear-time algorithm computing the paired-domination number for trees and characterize trees with equal domination and paired-domination numbers.

New metrics for dominating set based energy efficient activity scheduling in ad hoc networks

Abstract: In a multi-hop wireless network, each node is able to send a message to all of its neighbors that are located within its transmission radius. In a flooding task, a source sends the same message to all the network. Routing problem deals with finding a route between a source and a destination. In the activity-scheduling problem, each node decides between active or passive state. We present a scheme whose goal is to prolong network life while preserving connectivity. Each node is either active or has an active neighbor node. Routing and broadcasting are restricted to active nodes that create such dominating set. Activity status is periodically updated during a short transition period. The main contribution of this article is to propose new metrics for previously studied source-independent localized dominating sets, based on combinations of node degrees and remaining energy levels, for deciding activity status.

A Synchronous Self-stabilizing Minimal Domination Protocol in an Arbitrary Network Graph

Abstract: In this paper we propose a new self-stabilizing distributed algorithm for minimal domination protocol in an arbitrary network graph using the synchronous model; the proposed protocol is general in the sense that it can stabilize with every possible minimal dominating set of the graph.

Bipartite Domination and Simultaneous Matroid Covers

Abstract: Damaschke, Muller, and Kratsch [Inform. Process. Lett., 36 (1990), pp. 231--236] gave a polynomial-time algorithm to solve the minimum dominating set problem in convex bipartite graphs $B=(X \cup Y,E)$, that is, where the nodes in Y can be ordered so that each node of X is adjacent to a contiguous sequence of nodes. Gamble et al. [Graphs Combin., 11 (1995), pp. 121--129] gave an extension of their algorithm to weighted dominating sets. We formulate the dominating set problem as that of finding a minimum weight subset of elements of a graphic matroid, which covers each fundamental circuit and fundamental cut with respect to some spanning tree T. When T is a directed path, this simultaneous covering problem coincides with the dominating set problem for the previously studied class of convex bipartite graphs. We describe a polynomial-time algorithm for the more general problem of simultaneous covering in the case when T is an arborescence. We also give NP-completeness results for fairly specialized classes of the simultaneous cover problem. These are based on connections between the domination and induced matching problems.

Multifacility ordered median problems on networks: a further analysis

Abstract: In this paper, we address the ordered p-median problem, which includes as special cases most of the classical multifacility location problems discussed in the literature. Finite dominating sets (FDS) are known for particular instances of this problem: p-median, p-center, and p-centdian. We find an FDS for the ordered p-median problem. This set allows us to gain a better insight into the general FDS structure of network location problems. This FDS is later used to present the first polynomial time algorithm for p-facility ordered median problems on tree networks. This result is combined with some approximation algorithms to give an O(log M log log M) approximate solution of these problems on general networks, where M is the number of vertices. © 2002 Wiley Periodicals, Inc.

Weakly-connected dominating sets and sparse spanners in wireless ad hoc networks

Abstract: A set S is dominating if each node in the graph G = (V, E) is either in S or adjacent to at least one of the nodes in S. The subgraph weakly induced by S is the graph G' = (V, E') such that each edge in E' has at least one end point in S. The set S is a weakly-connected dominating set (WCDS) of G if S is dominating and G' is connected G' is a sparse spanner if it has linear edges. In this paper, we present two distributed algorithms for finding a WCDS in O(n) time. The first algorithm has an approximation ratio of 5, and requires O(n log n) messages. The second algorithm has a larger approximation ratio, but it requires only O(n) messages. The graph G' generated by the second algorithm forms a sparse spanner with a topological dilation of 3, and a geometric dilation of 6.

Dominating-Set-Based Searching in Peer-to-Peer Networks

Abstract: The peer-to-peer network for sharing information and data through direct exchange has emerged rapidly in recent years. The searching problem is a basic issue that addresses the question “Where is X”. Breadth-first search, the basic searching mechanism used in Gnutella networks [4], floods the networks to maximize the return results. Depth-first search used in Freenet [3] retrieves popular files faster than other files but on average the return results are not maximized. Other searching algorithms used in peer-to-peer networks, such as iterative deepening [9], local indices [9], routing indices [2] and NEVRLATE [1] provide different improved searching mechanisms. In this paper, we propose a dominating-set-based peer-to-peer searching algorithm to maximize the return of searching results while keeping a low cost for both searching and creating/maintaining the connected-dominating-set (CDS) of the peer-to-peer network. This approach is based on random walk. However, the searching space is restricted to dominating nodes. Simulation has been done and results are compared with the one using regular random walk.

Perfect connected-dominant graphs

Abstract: If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number ∞c(G) of G. A graph G is called a perfect connected-dominant graph if ∞(H) = ∞c(H) for each connected induced subgraph H of G. We prove that a graph is a perfect connected-dominant graph if and only if it contains no induced path P5 and induced cycle C5.

Upper total domination in claw-free graphs

Abstract: A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S (other than itself). The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We establish bounds on Γt(G) for claw-free graphs G in terms of the number n of vertices and the minimum degree δ of G. We show that $\Gamma _t(G) \le 2(n+1)/3$ if $\delta \in \{ 1,2\}, \Gamma _t(G) \le 4n/(\delta + 3)$ if $\delta \in \{ 3,4\}$, and $\Gamma _t(G) \le n/2$ if δ ≥ 5. The extremal graphs are characterized. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 148–158, 2003

Dominating set algorithms for sparse placement of full and limited wavelengthconverters in WDM optical networks

Abstract: Optical wavelength converters are expensive, and their technology is still evolving Deploying full conversion capability in all nodes of a large optical network would be prohibitively costly We present a new algorithm for the sparse placement of full wavelength converters based on the concept of a k-minimum dominating set (k-MDS) of graphs The k-MDS algorithm is used to select the best set of nodes that will be equipped with full-conversion capability To allow placement of full wavelength conversion at any arbitrary number of nodes, we introduce a HYBRID algorithm and compare its performance with the simulation-based k-BLK approach We also extend the k-MDS algorithm to the case of limited conversion capability by using a scalable and cost-effective node-sharing switch design Compared with full search algorithms previously proposed in the literature our algorithm has better blocking performance, has better time complexity, and avoids the local minimum problem The performance benefit of our algorithms is demonstrated by simulation

A self-stabilizing distributed algorithm for minimal total domination in an arbitrary system graph

Abstract: In a graph G = (V, E), a set S /spl sube/ V is said to be total dominating if every v /spl isin/ V is adjacent to some member of S. When the graph represents a communication network, a total dominating set corresponds to a collection of servers having a certain desirable backup property, namely, that every server is adjacent to some other server. Self-stabilization, introduced by Dijkstra (1974, 1986), is the most inclusive approach to fault tolerance in distributed systems. We propose a new self-stabilizing distributed algorithm for finding a minimal total dominating set in an arbitrary graph. We also show how the basic ideas behind the proposed protocol can be generalized to solve other related problems.