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On an Online Spanning Tree Problem in Randomly Weighted Graphs
 Combinatorics, Probability and Computing
, 2005
"... This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniform distributed over [0, 1]. An algorithm receives the edges one by one and has to decide immediately whether to includ ..."
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This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniform distributed over [0, 1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves E [ALG] /E [OPT] = O (1) and E [ALG/OPT] = O (1) against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than E [ALG] /E [OPT] =# (log n) if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields E [ALG] /E [OPT] = O (log n) against the strongest imaginable adversary. 1.
On the length of a random minimum spanning tree
, 2013
"... We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→ ∞ E(Ln) = ζ(3) and show that E(Ln) = ζ(3) + c1 c2+o(1) n n4/3 where c1, c2 are expl ..."
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We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→ ∞ E(Ln) = ζ(3) and show that E(Ln) = ζ(3) + c1 c2+o(1) n n4/3 where c1, c2 are explicitly defined constants.
On the Difference of Expected Lengths of Minimum Spanning Trees
, 2008
"... An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele’s formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, w ..."
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Cited by 2 (1 self)
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An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele’s formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, with rate one, and uniform distribution on the interval (0, 1) is shown to be positive and of rate ζ(3)/n. For wheel graphs, precise values of expected lengths are given via calculations of the associated Tutte polynomials.
Average Performance Analysis
, 2006
"... The purpose of measures in algorithm theory is to distinguish between “good ” and “bad ” algorithms. The main drawback of classical worstcase analysis is that one single “bad ” instance decides the performance of an algorithm. Moreover, worstcase instances are often quite artificial and often do n ..."
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Cited by 1 (1 self)
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The purpose of measures in algorithm theory is to distinguish between “good ” and “bad ” algorithms. The main drawback of classical worstcase analysis is that one single “bad ” instance decides the performance of an algorithm. Moreover, worstcase instances are often quite artificial and often do not represent a “realistic ” or “typical ” instance of a problem. In this thesis, we are concerned with an approach that tries to adress this issue: average performance analysis. Consider an optimisation problem and let Alg be an arbitrary (online) algorithm for it. An adversary Adv chooses the distribution D of the input instances out of a fixed class ∆adv of distributions. Let Opt be an optimal algorithm for the considered problem. Then, the average performance ratio apr of the algorithm Alg is defined by alg
Expected Lengths of Minimum Spanning Trees for Nonidentical Edge Distributions
"... o b a b i l i t y Vol. 15 (2010), Paper no. 5, pages 110–141. Journal URL ..."
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o b a b i l i t y Vol. 15 (2010), Paper no. 5, pages 110–141. Journal URL
Minimumcost matching in a random graph with random costs
, 2015
"... Let Gn,p be the standard ErdősRényiGilbert random graph and let Gn,n,p be the random bipartite graph on n + n vertices, where each e ∈ [n]2 appears as an edge independently with probability p. For a graph G = (V,E), suppose that each edge e ∈ E is given an independent uniform exponential rate on ..."
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Let Gn,p be the standard ErdősRényiGilbert random graph and let Gn,n,p be the random bipartite graph on n + n vertices, where each e ∈ [n]2 appears as an edge independently with probability p. For a graph G = (V,E), suppose that each edge e ∈ E is given an independent uniform exponential rate one cost. Let C(G) denote the random variable equal to the length of the minimum cost perfect matching, assuming that G contains at least one. We show that w.h.p. if d = np (log n)2 then w.h.p. E [C(Gn,n,p)] = (1 + o(1))pi26p. This generalises the wellknown result for the case G = Kn,n. We also show that w.h.p. E [C(Gn,p)] = (1 + o(1)) pi2 12p along with concentration results for both types of random graph. 1
unknown title
, 2013
"... The scaling limit of the minimum spanning tree of the complete graph ..."
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Research Showcase @ CMU
, 2015
"... Minimumcost matching in a random bipartite graph with random costs ..."