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22
Power assignment for kconnectivity in wireless ad hoc networks
 J. Combinatorial Optimization
, 2005
"... seeks a power assignment to the nodes in a given wireless ad hoc network such that the produced network topology is kconnected and the total power is the lowest, In this paper, we present several approximation algorithms for this problem. Specifically, we propose a Ykapproximatiun algorithm for an ..."
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Cited by 12 (0 self)
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seeks a power assignment to the nodes in a given wireless ad hoc network such that the produced network topology is kconnected and the total power is the lowest, In this paper, we present several approximation algorithms for this problem. Specifically, we propose a Ykapproximatiun algorithm for any k 2 3, a (kt E H (k))approximation algorithm for k (2k 1) 5 TI where n is the network size, a (k+ 2 [(k i 1) /21)approximatiun algorithm for 2 5 k 5 7, a &approximation algorithm for k = 3, and a 9approximation algorithm fur k = 4. index Terms I;connectivity, power assignment, wireless ad hoc sensor networks I.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Directed st Numberings, Rubber Bands, and Testing Digraph kVertex Connectivity
"... Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f( ..."
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Cited by 10 (2 self)
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Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91102. Using this characterization, a directed graph can be tested for kvertex connectivity by a Monte Carlo algorithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in expected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of st numberings, we give a combinatorial construction of a directed st nulmberiug for any 2vertex connected directed graph.
Finding Four Independent Trees
"... Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(V  2) algorithm for finding three independent spanning trees in a 3 ..."
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Cited by 4 (0 self)
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Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2connected graph. Cheriyan and Maheshwari gave an O(V  2) algorithm for finding three independent spanning trees in a 3connected graph. In this paper we present an O(V  3) algorithm for finding four independent spanning trees in a 4connected graph. We make use of chain decompositions of 4connected graphs. ∗ Partially supported by NSF VIGRE grant † Supported by CNPq (Proc: 200611/003) – Brazil ‡ Partially supported by NSF grant DMS0245530 and NSA grant MDA9040310052
Optimal Independent Spanning Trees on Hypercubes
, 2004
"... Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertexdisjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is ..."
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Cited by 3 (1 self)
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Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertexdisjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is optimal if the average path length of the trees is the minimum. Any kdimensional hypercube has k independent spanning trees rooted at an arbitrary vertex. In this paper, an O(kn) time algorithm is proposed to construct k optimal independent spanning trees on a kdimensional hypercube, where n = 2 k is the number of vertices in a hypercube.
Directional Routing via Generals stNumberings
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
"... We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on ..."
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Cited by 3 (1 self)
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We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in di#erent directions from the same source are nodedisjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3connected graphs have 3directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to kconnected graphs. We also showthat a directed acyclic graph (dag) that is kconnected to a set of sinks has a kdirectional embedding in (k  1)space with the sink set as the extreme nodes.
A Network Management Architecture for Robust Packet Routing in Optical Access Networks
 IEEE Journal on Selected Areas in Communications
, 2002
"... We describe an architecture for optical local area network (LAN) or metropolitan area network (MAN) access. The architecture allows for bandwidth sharing within a wavelength and is robust to both link and node failures. The architecture can be utilized with an arbitrary, linkredundant mesh netwo ..."
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Cited by 2 (2 self)
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We describe an architecture for optical local area network (LAN) or metropolitan area network (MAN) access. The architecture allows for bandwidth sharing within a wavelength and is robust to both link and node failures. The architecture can be utilized with an arbitrary, linkredundant mesh network (noderedundancy is necessary only to handle all node failures), and assumes neither the use of a star topology nor the ability to embed such a topology within the physical mesh. Reservation of bandwidth is performed in a centralized fashion at a (replicated) head end node, simplifying the implementation of complex sharing policies relative to implementation on a distributed set of routers. Unlike a router, however, the head end does not take any action on individual packets and, in particular, does not buffer packets. The architecture thus avoids the difficulties of processing packets in the optical domain while allowing for packetized shared access of wavelengths. In this paper, we describe the route construction scheme and prove its ability to recover from single link and single node failures, outline a flexible medium access protocol and discuss the implications for implementing specific policies, and propose a simple implementation of the recovery protocol in terms of state machines for perlink devices.
Independent Tree Spanners  FaultTolerant Spanning Trees with Distance Guarantees
 In Proceedings 24rd International Workshop on GraphTheoretic Concepts in Computer Science, WG'98
, 1998
"... . For any fixed rational parameter t 1, a (tree) tspanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General tspanners and their variants have multiple applications in the field of communi ..."
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Cited by 2 (2 self)
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. For any fixed rational parameter t 1, a (tree) tspanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General tspanners and their variants have multiple applications in the field of communication networks, distributed systems, and network design. In this paper, we combine the two concepts of simple structured, sparse tspanners and faulttolerance by examining independent tree tspanners. Given a root vertex r, this is a pair of tree tspanners, such that the two paths from any vertex to r are edge disjoint or internally vertex dijoint, respectively. It is shown that a pair of independent tree tspanners can be found in linear time for t ! 3, whereas the problem for arbitrary t 4 is NPcomplete. As a less restrictive concept, we also treat tree trootspanners, where the distance constraint is relaxed. Here, we show that the problem of finding an independent pair of su...
Independent Spanning Trees With Small Depths in Iterated Line Digraphs
, 1997
"... . We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn ..."
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. We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths. Keywords. independent spanning trees, line digraphs, vertexconnectivity, de Bruijn digraphs, Kautz digraphs, interconnection networks, broadcasting. 1 Introduction Unless stated otherwise each digraph of this paper is nite and may have loops but not multiarcs. Let G be a digraph. Then V (G) and A(G) denote the vertex set and the arc set of G, respectively. Let (u; v) 2 A(G). Then we say that u is adjacent to v, and v is adjacent from u. Also, it is said that (u; v) is incident to v and incident from u. Let (v; w) 2 A(G). Then we say that (u; v) is adjacent to (v; w), and (v; w) is adjacent from (u; v). Let ...
Independent Spanning Trees with Small Stretch Factors
, 1996
"... A pair of spanning trees rooted at a vertex r are independent if for every vertex v the pair of unique tree paths from v to the root r are disjoint. This paper presents the first analysis of the path lengths involved in independent spanning trees in 2edgeconnected and 2vertexconnected graphs. We ..."
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A pair of spanning trees rooted at a vertex r are independent if for every vertex v the pair of unique tree paths from v to the root r are disjoint. This paper presents the first analysis of the path lengths involved in independent spanning trees in 2edgeconnected and 2vertexconnected graphs. We present upper and lower bounds on the stretch factors of pairs of independent spanning trees, where the stretch factor of a spanning tree is defined to be the maximum ratio between the length of paths in the tree to the root to the length of the shortest path in the graph to the root. We prove that if G is a 2edgeconnected graph with the property that every edge lies on a cycle of size at most h than we can construct in linear time a pair of edgeindependent spanning trees whose stretch factors are bounded by O(h). In fact, we prove a more general result, namely that the stretch factor of both independent trees can be bounded by a minimax length of ears with respect to a certain class of ...