A Distributed Algorithm for Finding All Best Swap Edges of a Minimum Diameter Spanning Tree

Gfeller, Beat; Santoro, Nicola; Widmayer, Peter

doi:10.1007/978-3-540-75142-7_22

Cited by 11 publications

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“…As a final remark, we note that in [2], Gfeller et al study the problem of finding the optimal swap edges of a minimum spanning tree having minimum diameter: They provide a distributed algorithm that already works in linear time. The general technique presented here can be also adapted to this case.…”

Section: (A))mentioning

confidence: 98%

“…Related Work and Our Contribution. In [2,9], several different criteria for defining the "best" swap edge for a tree edge e have been considered. In each case, the best swap edge for e is that swap edge e for which some penalty function F is minimized.…”

Section: (A))mentioning

confidence: 99%

“…A number of studies have been done for this problem, both for the sequential [1][2][3][4][5][6] and distributed [7][8][9][10] In this paper, we consider the all best swap edges problem in the distributed setting. We are given a positively weighted 2-edge connected network X of processes, where w(x, y) denotes the weight of any edge {x, y} of X, together with a spanning tree T of X, rooted at a process r. Suppose that all communication between processes is routed through T .…”

Section: Introductionmentioning

confidence: 99%

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Linear Time Distributed Swap Edge Algorithms

Datta

Larmore

Pagli

et al. 2013

Lecture Notes in Computer Science

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Abstract. In this paper, we consider the all best swap edges problem in a distributed environment. We are given a 2-edge connected positively weighted network X, where all communication is routed through a rooted spanning tree T of X. If one tree edge e = {x, y} fails, the communication network will be disconnected. However, since X is 2-edge connected, communication can be restored by replacing e by non-tree edge e , called a swap edge of e, whose ends lie in different components of T − e. Of all possible swap edges of e, we would like to choose the best, as defined by the application. The all best swap edges problem is to identify the best swap edge for every tree edge, so that in case of any edge failure, the best swap edge can be activated quickly. There are solutions to this problem for a number of cases in the literature. A major concern for all these solutions is to minimize the number of messages. However, especially in fault-transient environments, time is a crucial factor. In this paper we present a novel technique that addresses this problem from a time perspective; in fact, we present a distributed solution that works in linear time with respect to the height h of T for a number of different criteria, while retaining the optimal number of messages. To the best of our knowledge, all previous solutions solve the problem in O(h 2 ) time in the cases we consider.

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Section: (A))mentioning

confidence: 98%

Section: (A))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear Time Distributed Swap Edge Algorithms

Datta

Larmore

Pagli

et al. 2013

Lecture Notes in Computer Science

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“…The distributed version of the problem was solved in [9] using O(max{n * , m}) messages of constant size, where n * is the size of the transitive closure of the tree, when the edges are directed towards the node initiating the computation.…”

Section: Our Contributionsmentioning

confidence: 99%

“…For temporary network failures, the second approach is much better suited than the first, because it is more efficient to use a swap edge for the duration of the failure, so that we can quickly revert back to the original spanner T , once the fault has been repaired. Furthermore, this approach needs only a very small adjustment of routing tables and has therefore attracted research attention in recent years for simpler spanning trees, under the name of "on-the-fly rerouting" [5,7,9]. As an aside, note also that an entirely new optimal tree spanner might not only require a total replacement of all routing table entries, but is in addition NP-hard to find.…”

Section: Introductionmentioning

confidence: 99%

Computing Best Swaps in Optimal Tree Spanners

Das

Gfeller

Widmayer

2008

Algorithms and Computation

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Abstract. In a densely connected communication network, represented by a graph G with nonnegative edge-weights, it is often advantageous to route all communication on a sparse, spanning subnetwork, typically a spanning tree of G. With the communication overhead in mind, we consider a spanning tree T of G which guarantees that for any two nodes, their distance in T is at most k times their distance in G, where k, called the stretch, is as small as possible. Such a spanning tree which minimizes the stretch is called an optimal tree spanner, and it can be used for efficient routing. However, for a communication tree, the failure of an edge is catastrophic; it disconnects the tree. Functionality can be restored by connecting both parts of the tree with another edge, while leaving the two parts themselves untouched. In situations where the failure can be repaired rapidly, such a quick fix is preferred over the recomputation of an entirely new optimal tree spanner, because it is much closer to the previous solution and hence requires far fewer adjustments in the routing scheme. We are therefore interested in the problem of finding for any possibly failing edge in the spanner T a best swap edge to replace it. The objective here is naturally to minimize the stretch of the new tree. We show how all these best swap edges can be computed in total time O(m 2 log n) in graphs with arbitrary nonnegative edge weights. For graphs with unit weight edges (also called unweighted graphs), we present an O(n 3 ) time algorithm. Furthermore, we present a distributed algorithm for computing the best swap for each edge in the spanner.

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Faster Swap Edge Computation in Minimum Diameter Spanning Trees

Gfeller

2008

Algorithms - ESA 2008

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Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum travel time of messages. When a transient failure disables an edge of the MDST, the network is disconnected, and a temporary replacement edge must be chosen, which should ideally minimize the diameter of the new spanning tree. Preparing for the failure of any edge of the MDST, the all-best-swaps (ABS) problem asks for finding the best swap for every edge of the MDST. Given a 2-edgeconnected weighted graph G = (V, E), where |V | = n and |E| = m, we solve the ABS problem in O (m log n) time and O(m) space, thus considerably improving upon the decade-old previously best solution, which requires O(n √ m) time and O(m) space, for m = o n 2 / log 2 n . IntroductionFor communication in computer networks, often only a subset of the available connections is used to communicate at any given time. If all nodes are connected using the smallest possible number of links, the subset forms a spanning tree of the network. When an edge in a communication tree fails, routing information becomes wrong and message transmission is interrupted. For transient failures that are expected to be repaired quickly, the idea of online point-of-failure rerouting has gained popularity recently [2,4, 6,8]: Instead of changing a lot of routing information, only one alternative (so-called swap) edge is used to reconnect the disconnected parts of the tree. For the corresponding change in routing information to be fast, a swap edge for each failing edge needs to be readily available, as the result of an earlier computation. Among all possible swap links for a failing edge, one should choose a best swap link, that is, a swap edge which reconnects the two disconnected parts of the tree in such a way that the resulting swap tree is best w.r.t. some objective. We show in the following that the common computation of all best swaps (ABS) has the further advantage of gaining efficiency (against computing swap edges individually), because dependencies between the computations for different failing edges can be exploited.In this paper, we are interested in using a Minimum Diameter Spanning Tree (MDST) as the communication tree, i.e., a tree that minimizes the largest distance between any pair of nodes, thus minimizing the worst case length of any transmission path, even if edge lengths are not uniform. Consequentially, a best swap edge in our case minimizes the diameter of the resulting swap tree. Interestingly, this choice of swapping against adjusting the entire tree even comes at a moderate loss in diameter: The swap tree diameter is at most a factor of 5/2 larger than the diameter of an entirely adjusted tree [9].Related Work. During the last decade, the ABS problem has been investigated for spanning trees with various objectives [1,8,9,11,13,15].Computing all best swaps of a MDST was one of the first swap problems that were studied. In [9], an algorithm for this problem is given which requires O...

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A Distributed Algorithm for Finding All Best Swap Edges of a Minimum Diameter Spanning Tree

Cited by 11 publications

References 12 publications

Linear Time Distributed Swap Edge Algorithms

Linear Time Distributed Swap Edge Algorithms

Computing Best Swaps in Optimal Tree Spanners

Faster Swap Edge Computation in Minimum Diameter Spanning Trees

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