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Ambivalent Data Structures For Dynamic 2EdgeConnectivity And k Smallest Spanning Trees
 SIAM J. Comput
, 1991
"... . Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log #(m, n) + min{k 3/2 ,km 1/2 }) time, where m is the number of edges and n the number of vertice ..."
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Cited by 83 (1 self)
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. Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log #(m, n) + min{k 3/2 ,km 1/2 }) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n + k(log n) 3 ) time. Ambivalent data structures are also used to dynamically maintain 2edgeconnectivity information. Edges and vertices can be inserted or deleted in O(m 1/2 ) time, and a query as to whether two vertices are in the same 2edgeconnected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Key words. analysis of algorithms, data structures, embedded planar graph, fully persistent data structures, k smallest spanning trees, minimum spanning tree, online updating, topology tr...
Verification and Sensitivity Analysis Of Minimum Spanning Trees In Linear Time
 SIAM J. Comput
, 1992
"... . Koml'os has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verif ..."
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Cited by 54 (2 self)
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. Koml'os has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verifying a minimum spanning tree. Our algorithm combines the result of Koml'os with a preprocessing and table lookup method for small subproblems and with a previously known almostlineartime algorithm. Additionally, we present an optimal deterministic algorithm and a lineartime randomized algorithm for sensitivity analysis of minimum spanning trees. 1. Introduction. Suppose we wish to solve some problem for which we know in advance the size of the input data, using an algorithm from some welldefined class of algorithms. For example, consider sorting n numbers, when n is fixed in advance, using a binary comparison tree. Given a sufficient amount of preprocessing time and storage space, we ca...
Distributed Verification of Minimum Spanning Trees
 Proc. 25th Annual Symposium on Principles of Distributed Computing
, 2006
"... The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in ..."
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Cited by 19 (17 self)
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The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows ” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (except when W ≤ log n). Both our bounds improve previously known bounds for the problem. Our techniques (both for the lower bound and for the upper bound) may indicate a strong relation between the fields of proof labeling schemes and implicit labeling schemes. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.
Finding the k Smallest Spanning Trees
, 1992
"... We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of ..."
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Cited by 18 (2 self)
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We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of points in the plane can be computed in time O(min(k 2 n + n log n, k 2 + kn log(n/k))). The k best orthogonal spanning trees in the plane can be found in time O(n log n + kn log log(n/k)+k 2 ).
Data Structural Bootstrapping, Linear Path Compression, and Catenable Heap Ordered Double Ended Queues
 SIAM Journal on Computing
, 1992
"... A deque with heap order is a linear list of elements with realvalued keys which allows insertions and deletions of elements at both ends of the list. It also allows the findmin (equivalently findmax) operation, which returns the element of least (greatest) key, but it does not allow a general delet ..."
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Cited by 15 (7 self)
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A deque with heap order is a linear list of elements with realvalued keys which allows insertions and deletions of elements at both ends of the list. It also allows the findmin (equivalently findmax) operation, which returns the element of least (greatest) key, but it does not allow a general deletemin (deletemax) operation. Such a data structure is also called a mindeque (maxdeque) . Whereas implementing mindeques in constant time per operation is a solved problem, catenating mindeques in sublogarithmic time has until now remained open. This paper provides an efficient implementation of catenable mindeques, yielding constant amortized time per operation. The important algorithmic technique employed is an idea which is best described as data structural bootstrapping: We abstract mindeques so that their elements represent other mindeques, effecting catenation while preserving heap order. The efficiency of the resulting data structure depends upon the complexity of a special case of pa...
Nearly Linear Time Minimum Spanning Tree Maintenance for Transient Node Failures
 ALGORITHMICA
, 2004
"... Given a 2node connected, real weighted, and undirected graph G = (V, E), with n nodes and m edges, and given a minimum spanning tree (MST) T = (V, ET) of G, we study the problem of finding, for every node v ∈ V, a set of replacement edges which can be used for constructing an MST of G − v (i.e., ..."
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Cited by 3 (0 self)
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Given a 2node connected, real weighted, and undirected graph G = (V, E), with n nodes and m edges, and given a minimum spanning tree (MST) T = (V, ET) of G, we study the problem of finding, for every node v ∈ V, a set of replacement edges which can be used for constructing an MST of G − v (i.e., the graph G deprived of v and all its incident edges). We show that this problem can be solved on a pointer machine in O(m · α(m, n)) time and O(m) space, where α is the functional inverse of Ackermann’s function. Our solution improves over the previously best known O(min{m · α(n, n), m + n log n}) time bound, and allows us to close the gap existing with the fastest solution for the edgeremoval version of the problem (i.e., that of finding, for every edge e ∈ ET, a replacement edge which can be used for constructing an MST of G − e = (V, E\{e})). Our algorithm finds immediate application in maintaining MSTbased communication networks undergoing temporary node failures. Moreover, in a distributed environment in which nodes are managed by selfish agents, it can be used to design an efficient, truthful mechanism for building an MST.
BIOINSPIRED EVOLUTIONARY METHOD FOR CABLE TRENCH PROBLEM
, 2006
"... Abstract. A bioinspired evolutionary method with DNA is presented for solving a cable trench problem in this paper. The cable trench problem is a combination of the shortest path problem and the minimum spanning tree problem, which makes it difficult to be solved by a conventional computing method. ..."
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Cited by 1 (0 self)
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Abstract. A bioinspired evolutionary method with DNA is presented for solving a cable trench problem in this paper. The cable trench problem is a combination of the shortest path problem and the minimum spanning tree problem, which makes it difficult to be solved by a conventional computing method. DNA computing is applied to overcome the limitation of a siliconbased computer. The numerical values are represented by the fixedlength DNA strands, and the weights are varied by the melting temperatures. Biochemical techniques with DNA thermodynamic properties are used for effective local search of the optimal solution. Keywords: Bioinspired computing, DNA computing, Evolutionary computation, Combinatorial optimization, Cable trench problem
On the Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis
 Transactions of ASME
, 1999
"... A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transform ..."
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Cited by 1 (0 self)
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A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented. Both problems are shown to be NPhard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NPcomplete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems, hence itself an NPhard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed. 1 Introduct...
Improved Algorithms for Replacement Paths Problems in Restricted Graphs
, 2005
"... We present near optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time. ..."
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Cited by 1 (0 self)
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We present near optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time.