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On the Minimum Average Distance Spanning Tree of the Hypercube
Abstract
Given an undirected and connected graph G, with a non-negative weight on each edge, the Minimum Average Distance (MAD) spanning tree problem is to find a spanning tree of G which minimizes the average distance between pairs of vertices. This network design problem is known to be NP-hard even when the edge-weights are equal. In this paper we make a step towards the proof of a conjecture stated by A.A. Dobrynin, R. Entringer and I. Gutman in 2001, and which says that the binomial tree B
n
is a MAD spanning tree of the hypercube H
n
. More precisely, we show that the binomial tree B
n
is a local optimum with respect to the 1-move heuristic which, starting from a spanning tree T of the hypercube H
n
, attempts to improve the average distance between pairs of vertices, by adding an edge e of H
n
-T and removing an edge e′ from the unique cycle created by e. We also present a greedy algorithm which produces good solutions for the MAD spanning tree problem on regular graphs such as the hypercube and the torus.
... In Sec. 2, the results of Wong and Dankelmann concerning the relations between MAD and DP spanning trees, together with the proofs are presented. In Sec. 3, we give a simple result which says that if G is an undirected graph with unit edge-lengths, then in MAD trees, any branch of size less than or equal to 3 √ n is distance preserving from its root. This result is improved in Sec. 4, where we present the main result of this paper. ...
... Proposition 3.1. In a MAD spanning tree of G, any branch B = (V , E ) of size less than 3 √ n, with root r, is distance preserving from r, with respect to the subgraph of G induced by V . ...
... Proof. Consider a MAD spanning tree T = (V , E) of G and a branch B = (V , E ) with root r and size less than 3 √ n. Assume that B is not distance preserving from r, with respect to the subgraph of G induced by V . ...
... But we believe that Hruska's conjecture is true,that is, stc(Q d ) = 2 d−1 .To show the upper bound, we use binomial trees. Binomial trees are introduced in the studies of the minimum average distance spanning tree of the hypercubes[9,22]. A d-level binomial tree B d is a spanning tree of Q d : B 1 is an edge Q 1 ; B d consists of two (d − 1)-level binomial trees and an edge between roots of the two trees (The root of B d is one of the roots of two B d−1 's.). ...
... SeeFig. 5for example, and see references[9,22] for formal definitions. From the construction of B d , it is easy to see that for any edge e ∈ B d , the smaller component C of B d − e induces a subcube Q δ for some δ < d. ...
Let G be a connected graph and T be a spanning tree of G. For e∈E(T), the congestion of e is the number of edges in G connecting two components of T−e. The edge congestion ofGinT is the maximum congestion over all edges in T. The spanning tree congestion ofG is the minimum congestion of G in its spanning trees. In this paper, we show the spanning tree congestion for the complete k-partite graphs and the two-dimensional tori. We also address lower bounds of spanning tree congestion for the multi-dimensional grids and the hypercubes.
... • Hypercube is a well known and popular interconnection network for multicomputers. The question whether the binomial tree is a MAD tree of the hypercube remains open (see [31]) even though Tchuente at el. [79] made a step towards this open problem. ...
There has been a great deal of interest in the computation of distances and shortest paths problem in graphs which is one of the central, and most studied, problems in (algorithmic) graph theory. In this paper, we survey the exact results of the static version of the all-pairs shortest paths problem and its variants namely, the Wiener index, the average distance, and the minimum average distance spanning tree (MAD tree in short) in graphs (focusing mainly on algorithmic results for such problems). Along the way we also mention some important open issues and further research directions in these areas.
It is required to transmit data through shorter path between sensor and base node for real time intrusion detection in wireless sensor networks (WSN) with a mobile base node. Because minimum Wiener index spanning tree (MWST) based routing approach guarantees lower average hop count than that of minimum spanning tree (MST) based routing method in WSN, it is known that MWST based routing is appropriate for real time intrusion detection. However, the minimum Wiener index spanning tree problem which aims to find a spanning tree which has the minimum Wiener index from a given weighted graph was proved to be a NP-hard. And owing to its high dependency on certain nodes, minimum Wiener index tree based routing method has a shorter network lifetime than that of minimum spanning tree based routing method. In this paper, we propose a multi-objective ant colony optimization algorithm to tackle these problems, so that it can be used to detect intrusion in real time in wireless sensor networks with a mobile base node. And we compare the results of our proposed method with MST based routing and MWST based routing in respect to average hop count, network energy consumption and network lifetime by simulation.
The d-dimensional k-ary hypercube tree Tk(d) is a generalization of the hypercube tree, also known in the literature as the spanning binomial tree. We present some of its structural properties and investigate in detail its average distance. For instance, it is shown that the binary hypercube tree has the anomaly of having two nodes in its centre as opposed to having one in hypercube trees of arity k>2. However, in all dimensions, the centre, centroid and median coincide. We show that its total distance is , which is minimum. Consequently, for d≥2, its average distance is μ(Tk(d))=(2d(k−1)k )/(k −1)−2/k, whose limiting value is 2. This answers a generalization of a conjecture of Dobrynin et al. [Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), pp. 211–249] for the binary hypercube by the affirmative.
This paper actually did not give a citation to my work, although clearly ought to have done.) 5. H. Furstenberg, Y. Katznelson, B. Weiss, Ergodic theory and con- gurations in sets of positive density, in: Mathematics of Ramsey Theory, J. Nesetril, V. Rodl, eds., Algorithms and Combinatorics 5, Springer Verlag, 1990, 184-198
In this paper, we present exact and approximate algorithms for the problem of optimal network design, which was first studied by Ridley [15], Stairs [17] and Scott [16], among others. Given a network, the problem consists in selecting a subset of links that minimizes the sum of shortest routes between all nodes subject to a budget constraint. Congestion effects are not considered. First, we refine a branch-and-bound algorithm proposed by Hoang [7] and we compare the resuling algorithm to an algorithm developed by Boyce et al. [1]. Our conclusion is that these exact algorithms perform adequately on medium size problems; however, for larger problems, approximate methods based on heuristic approaches are necessary. We then introduce two new heuristic approaches and test their computational efficiency as well as that of two other already known heuristics. The new heuristics presented produce excellent results. Finally, we consider more general versions of the problem.
Combinatorial Chemistry is a powerful new technology in drug design and molecular recognition. It is a wetlaboratory methodology aimed at "massively parallel" screening of chemical compounds for the discovery of compounds that have a certain biological activity. The power of the method comes from the interaction between experimental design and computational modeling. Principles of "rational" drug design are used in the construction of combinatorial libraries to speed up the discovery of lead compounds with the desired biological activity. This paper presents algorithms, software development and computational complexity analysis for problems arising in the design of combinatorial libraries for drug discovery. We provide exact polynomial time algorithms and intractability results for several Inverse Problems - formulated as (chemical) graph reconstruction problems - related to the design of combinatorial libraries. These are the first rigorous algorithmic results in the literature. We also present results provided by our combinatorial chemistry software package OCOTILLO for combinatorial peptide design using real data libraries. The package provides exact solutions for general inverse problems based on shortest-path topological indices. Our results are superior both in accuracy and computing time to the best software reports published in the literature. For 5-peptoid design, the computation is rigorously reduced to an exahustive search of about 2% of the search space; the exact solutions are found in a few minutes.
Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NP-hard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomial-time approximation scheme valid for both versions of the problem. In particular, we show how to build a spanning tree of an n-vertex weighted graph with routing cost within (1+ε) from the minimum in time O(n O(1 ε) ). Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a generalization of the minimum routing cost tree problem where the routing costs of different pairs are weighted by different requirement amounts. We observe that a randomized O(log 2 n)-approximation for this problem follows directly from a recent result of Bartal, where n is the number of nodes in a metric graph. This also yields the same approximation for the generalized sum-of-pairs alignment problem in computational biology.
The Optimal Network problem (as defined by Scott [16]) consists of selecting a subset of arcs that minimizes the sum of the shortest paths between all nodes subject to a budget constraint. This paper considers the worst-case behavior of heuristics for this prob'em. Let n be the number of nodes in the network and e be a constant between 0 and 1. For a general class of Optimal Network Problems, we show that the question of finding a solution which is always less than n times the optimal solution is NP-complete. This indicates that all polynomial-time heuristics for the problem most probably have poor worst-case performance. An upper bound for worst-case heuristic performance of 2n times the optimal solution is also derived. For a restricted version of the Optimal Network problem we describe a procedure whose maximum percentage of error is bounded by a constant. This research was supported, in part, by the U. S. Department of Transportation under Contract DOT-TSC-1058, Transportation Advanced Research Program (TARP).
Gossip-based protocols for group communication have attractive scalability and reliability properties. The probabilistic gossip schemes studied so far typically assume that each group member has full knowledge of the global membership and chooses gossip targets uniformly at random. The requirement of global knowledge impairs their applicability to very large-scale groups. In this paper, we present SCAMP (Scalable Membership protocol), a novel peer-to-peer membership protocol which operates in a fully decentralized manner and provides each member with a partial view of the group membership. Our protocol is self-organizing in the sense that the size of partial views naturally converges to the value required to support a gossip algorithm reliably. This value is a function of the group size, but is achieved without any node knowing the group size. We propose additional mechanisms to achieve balanced view sizes even with highly unbalanced subscription patterns. We present the design, theoretical analysis, and a detailed evaluation of the basic protocol and its refinements. Simulation results show that the reliability guarantees provided by SCAMP are comparable to previous schemes based on global knowledge. The scale of the experiments attests to the scalability of the protocol.
IN ATTEMPTS TO SIMULATE THE SPATIAL DEVELOPMENT OF REAL TRANSPORTATION NETWORKS, THE DEVELOPMENT OF A SIMPLE NETWORK-GENERATING PRINCIPLE THAT APPROXIMATES SOME OF THE MAJOR PROCESSES OCCURING IN REALITY IS IMPORTANT BOTH AS AN ANALYTICAL TOOL AND AS A PLANNING MECHANISM. THE PROBLEM CONSIDERED IN THIS PAPER IS TO ESTABLISH A SET OF ARCS LINKING TOGETHER A GIVEN SET OF VERTICES SUCH THAT THE SUM OF THE SHORTEST DISTANCES (OR AT LEAST TRANSPORT COSTS) BETWEEN EASH PAIR OF VERTICES IS A MINIMUM. NO VERTEX MAY BE OMITTED FROM THE NETWORK, AND ARC-INTERSECTIONS MAY OCCUR ONLY AT THE PREDETERMINED LOCATIONS OF THESE VERTICES. IN ADDITION, AN IMPLIED CONDITION OF THE PROBLEM IS THAT ANY FLOWS THROUGH THE COMPUTED NETWORK ALWAYS INCUR ZERO CONGESTION COSTS. THREE ALTERNATIVE APPROACHES TO THE SOLUTION OF THE OPTIMAL NETWORK PROBLEM ARE CONSIDERED. FIRST, THE POSSIBILITIES OF APPLYING INTEGER PROGRAMMING TO THE SOLUTION OF THE PROBLEM ARE EXAMINED BRIEFLY. SECOND, A BACKTRACK PROGRAMMING SOLUTION ALGORITHM IS PROPOSED THAT DEMANDS LENGTHY ARITHMETICAL OPERATIONS BUT REQUIRES ONLY SMALL AMOUNTS OF STORAGE DURING THE SOLUTION PROCESS. THIRD, TWO APPROXIMATIVE ALGORITHMS ARE PROPOSED. THESE ALGORITHMS REQUIRE ONLY RELATIVELY SMALL AMOUNTS OF ARITHMETIC AND STORAGE, AND APPEAR TO GUARANTEE AT LEAST A GOOD APPROXIMATION TO OPTIMALITY. /TSR&A/
The principal aim of a transportation study is to produce a plan for a future transportation network. This network should satisfy the traffic demands placed upon it and give service to users on the basis of some acceptable criteria within certain budgetary, political and social constraints. Out of a large number of engineering design problems which naturally arise, this dissertation considers a special problem of economic investment in a transportation network. (Author)
This paper presents CONGRESS: a connection-oriented group- address resolution service, and its applications. CONGRESS is an efficient native ATM protocol for resolution and management of multicast group addresses in an ATM WAN. It complements the native ATM multicast mechanisms. CONGRESS resolves multicast group addresses and maintains their membership for applications. It is not designed to handle the applications' data-exchange. Applications can use the resolved addresses returned by CONGRESS, in order to implement a many-to-many communication model. CONGRESS employs hierarchically organized servers in order to be scalable. CONGRESS' hierarchy is naturally mapped onto the ATM private network to network interface peer group hierarchy. CONGRESS communication overhead for management of a single multicast group is linear in the size of the group. Apart from facilitating native ATM multicast applications, CONGRESS can be used for the implementation of an IP multicast 'cut-through' routing over ATM. The cut-through routing paradigm is conceived as one of the most promising techniques for enabling the traditional IP- based communication with QoS. Unfortunately, due to a variety of reasons, the building of scalable IP multicast cut-through protocols is non-trivial. We claim that a multicast address resolution and maintenance service like CONGRESS, can greatly contribute to the development of scalable IP cut-through routing services. A conceptual cut-through routing solution, IP multicast service for non-broadcast access networking technology (IP-SENATE), built on top of CONGRESS and scalable to a large ATM cloud is sketched.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
In the network design problem we are given a weighted undirected graph. We wish to find a subgraph which connects all the original vertices and minimizes the sum of the shortest path weights between all vertex pairs, subject to a budget constraint on the sum of its edge weights. In this note we establish NP-completeness for the network design problem, even for the simple case where all edge weights are equal and the budget restricts the choice to spanning trees. This result justifies the development of enumerative optimization methods and of approximation algorithms, such as those described in a recent paper by R. Dionne and M. Florian.
We consider the problem of selecting an optimal traffic network in its simplest form, where there are no congestion costs. The superadditive effect of deleted links on the objective function is pointed out and used to develop some implicit enumeration procedures for this problem. Some efforts are devoted to evaluate these algorithms, and to show how to stop the computations with an "acceptable" approximate solution when computations are taking too much time.
The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.
Given a set of nodes which may represent cities and a set of requirements which may represent the number of telephone calls between and , the problem is to build a spanning tree connecting these n nodes such that the total cost of communication of the spanning tree is a minimum among all spanning trees. The cost of communication for a pair of nodes is multiplied by the sum of the distances of arcs which form the unique path connecting and in the spanning tree. Summing over all pairs of nodes, we have the total cost of communication of the spanning tree. Note that the problem is different from the minimum spanning tree problem solved by Kruskal and Prim.
Given an undirected graph with a nonnegative weight on each edge, the shortest total path length spanning tree problem is to find a spanning tree of the graph such that the total path length summed over all pairs of the vertices is minimized. In this paper, we present several approximation algorithms for this problem. Our algorithms achieve approximation ratios of 2, 15/8, and 3/2 in time O(n2+f(G)),O(n3), and O(n4) respectively, in which f(G) is the time complexity for computing all-pairs shortest paths of the input graph G and n is the number of vertices of G. Furthermore, we show that the approximation ratio of (4/3+ε) can be achieved in polynomial time for any constant ε>0.
Drugs and other chemical compounds are often modeled as polygonal
shapes, where each vertex represents an atom
of the molecule, and covalent bonds between atoms are represented by edges between
the corresponding vertices. This polygonal shape derived from a chemical
compound is often called its molecular graph, and can be a
path, a tree, or in general a graph. An indicator defined over this molecular graph,
the Wiener index, has been shown to be strongly correlated to various
chemical properties of the compound.
The Wiener index conjecture for trees states that for any integer n (except for
a finite set), one can find a tree with Wiener index n.
This conjecture has been open for quite some time, and many authors have presented incremental progress
on this problem. In this paper we present further progress towards proving this conjecture—through
the design of efficient algorithms, we show that enumerating all possible
trees to verify this conjecture (as done by all the previous approaches) is not necessary,
but instead searching in a small special family
of trees suffices, thus achieving the first polynomial (in n) time algorithm to verify the
conjecture up to integer n.
More precisely, we
(i) present an infinite family of trees and prove various properties of these trees,
(ii) show that a large number of integers, up to at least 108
(compared with the previous best 104) are representable as Wiener indices of trees in this
succinct family,
(iii) provide several efficient algorithms for computing trees with given Wiener indices, and
(iv) implement our algorithms and experimentally show that their performance is asymptotically
much better than their theoretical worst-case upper bound.
Given a set of points and distances between them, a basic problem in network design calls for selecting a graph connecting them at a minimum total routing cost, that is, the sum over all pairs of points of the length of their shortest path in the graph. In this paper, we describe some branch-and-bound algorithms for the exact solution of a relevant special case arising when the graph has to be a tree. One of the enhancements to our algorithms is the use of “LP shortcutting,” which we introduce as a general-purpose technique for speeding up the search. Besides network design, we show how trees of small routing cost find useful application in computational biology, where they can be used to determine good alignments of genomic sequences. This leads to a novel alignment heuristic that we analyze in our computational section. © 2002 Wiley Periodicals, Inc.
Multiple string (sequence) alignment is a difficult and important problem in computational biology, where it is central in two related tasks: finding highly conserved subregions or embedded patterns of a set of biological sequences (strings of DNA, RNA or amino acids), and inferring the evolutionary history of a set of taxa from their associated biological sequences. Several precise measures have been proposed for evaluating the goodness of a multiple alignment, but no efficient methods are known which compute the optimal alignment for any of these measures in any but small cases. In this paper, we consider two previously proposed measures, and give two computationaly efficient multiple alignment methods (one for each measure) whose deviation from the optimal value is guaranteed to be less than a factor of two. This is the novel feature of thse methods. but the methods have additional virtues as well. For both methods, the guaranteed bounds are much smaller than two when the number of strings is small (1.33 for three strings of any length); for one of the methods we give a related randomized method which is much faster and which gives, with high probability, multiple alignments with fairly small error bounds; and for the other measure, the method given yields a non-obvious lower bound on the optimal alignment.
Several distributed peer-to-peer applications require weakly-consistent knowledge of process group membership information at all participating processes. SWIM is a generic software module that offers this service for large scale process groups. The SWIM effort is motivated by the unscalability of traditional heart-beating protocols, which either impose network loads that grow quadratically with group size, or compromise response times or false positive frequency w.r.t. detecting process crashes. This paper reports on the design, implementation and performance of the SWIM sub-system on a large cluster of commodity PCs. Unlike traditional heart beating protocols, SWIM separates the failure detection and membership update dissemination functionalities of the membership protocol. Processes are monitored through an efficient peer-to-peer periodic randomized probing protocol. Both the expected time to first detection of each process failure, and the expected message load per member do not vary with group size. Information about membership changes, such as process joins, drop-outs and failures, is propagated via piggybacking on ping messages and acknowledgments. This results in a robust and fast infection style (also epidemic or gossip-style) of dissemination. The rate of false failure detections in the SWIM system is reduced by modifying the protocol to allow group members to suspect a process before declaring it as failed - this allows the system to discover and rectify false failure detections. Finally, the protocol guarantees a deterministic time bound to detect failures. Experimental results from the SWIM prototype are presented. We discuss the extensibility of the design to a WAN-wide scale.
The use of a high bandwidth multicast is becoming widespread in today's network applications. Many of these applications use multicast groups with dynamic membership such as multi-media conferencing, multi-media broadcasting and multi-media distributed data bases. ATM UNI 3.1 and 4.0 protocols offer the point-to-multipoint connection type that enables multicast over native ATM. Point-to-multipoint connections may be utilized for efficient implementation of multicast groups. In both ATM UNI 3.1 and 4.0, however, explicit information about the end-points (members of a multicast group) participating in the multicast connection is required at a connection set up time. Unfortunately, there is no standard mechanism that facilitates the maintenance of such group membership information. In this document we present for the first time a CONnection-oriented Group-address RESolution Service (congress). congress incorporates a protocol for efficient maintenance and propagation of group membership ...
A reliable multicast protocol ensures that all of the intended recipients of a message m that do not fail eventually deliver m. For example, consider the reliable multicast protocol of 11, and consider a message m, sent by process p
1, that is intended to be delivered by p
1, p
2, and p
3. We impose a directed spanning tree on these processes that is rooted at the message source. For example, for m we could have the directed spanning tree p
1 → p
2 → p
3. The message m propagates down this spanning tree and acknowledgments of the receipt of m propagate back up the tree. A leaf process in this tree delivers m when it receives m, and a non-leaf process delivers m when it gets the acknowledgment for m from all of its children. If a non-leaf process (say, p
1) does not get an acknowledgment for m from one of its children (here, p
2), then it removes the child from the tree and “adopts” that child’s children (here, p
3). The process sends m to the newly-adopted children and continues the broadcast. A similar monitoring and adoption approach is used to recover from the failure of the root of the tree.
Algorithme incrémental pour le maintien d’un arbre de connexion
- N Thibault
- C Laforest
Migration de processus linux sous i-cluster
- I I Nlong
- J M Denneulin
Selecting an optimal traffic network
- S Stairs
- S. Stairs
Efficient methods for multiple sequence alignment with guaranteed error bounds
- D Gusfield
- D. Gusfield