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16
Ambivalent data structures for dynamic 2edgeconnectivity and k smallest spanning trees
 In 32nd Annual Symposium on Foundations of Computer Science FOCS
, 1991
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Verification and Sensitivity Analysis Of Minimum Spanning Trees In Linear Time
 SIAM J. COMPUT
, 1992
"... Komlos 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 verifyi ..."
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Cited by 59 (1 self)
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Komlos 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 Komlos 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.
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 32 (23 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 21 (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 19 (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...
An Improved Algorithm for Biobjective Integer Programs
, 2005
"... A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling ..."
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Cited by 9 (5 self)
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A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented.
Sensitivity Analysis for Shortest Path Problems and Maximum Capacity Path Problems in Undirected Graphs
 MATH. PROGRAM., SER. A
, 2005
"... Let G = (N,A) be an undirected graph with n nodes and m arcs, a designated source node s and a sink node t. This paper addresses sensitivity analysis questions concerning the shortest st path (SP) problem in G and the maximum capacity st path (MCP) problem in G. Suppose that P* is a shortest st p ..."
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Cited by 8 (2 self)
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Let G = (N,A) be an undirected graph with n nodes and m arcs, a designated source node s and a sink node t. This paper addresses sensitivity analysis questions concerning the shortest st path (SP) problem in G and the maximum capacity st path (MCP) problem in G. Suppose that P* is a shortest st path in G with respect to a nonnegative distance vector c. For each arc e A, the lower SP tolerance of an arc e is the minimum nonnegative value that the length of arc e can take (with all other lengths staying fixed) so that P* remains an optimal path. Similarly, the upper SP tolerance of an arc e is the maximum value that the length of arc e can take so that P* remains an optimal path. We show that the problem of finding all upper and lower tolerances of arcs in A can be solved in O(min(n 2 , m log n)) time. Moreover, the problem of finding all tolerances is computationally equivalent to the "Minimum Cost Interval Problem" which we describe as follows. For each i = 1 to m, let [a i , b i ] denote an interval with endpoints in {1, ..., n}, and an associated cost c i . For each k = 1 to n, identify a minimum cost interval [a i , b i ] containing k. Let Q* be the maximum capacity st path in G with respect to capacity vector u. For each arc e A, the lower MCP (resp., upper) tolerance of the arc e is the minimum (resp., maximum) value that the capacity that arc e can take so that Q* remains a maximum capacity path. We show that the problem of finding all upper and lower tolerances of arcs in A can be solved in O(min(n 2 , m log n)) time. Moreover, the problem of finding all tolerances nearly reduces to the "Minimum Cost Interval Problem."
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 5 (1 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.
DataStructural Bootstrapping And Catenable Deques
, 1993
"... The list is a fundamental data structure. It stores a linearly ordered collection of elements and allows access only to the front and rear elements of the list. Catenation can be applied to lists, unifying the rear of one list with the front of another. Absent other requirements, the basic list oper ..."
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Cited by 5 (0 self)
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The list is a fundamental data structure. It stores a linearly ordered collection of elements and allows access only to the front and rear elements of the list. Catenation can be applied to lists, unifying the rear of one list with the front of another. Absent other requirements, the basic list operations, including catenation, have straightforward implementations. If the list has certain secondary properties, however, the operations, particularly catenation, become more difficult. Nondestructive lists
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.