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Register a new userThe 1.375 Approximation Algorithm for Sorting by Transpositions Can Run in $O(nlog n)$ Time
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Added by
Masud Hasan
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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Sorting a Permutation by Transpositions (SPbT) is an important problem in Bioinformtics. In this paper, we improve the running time of the best known approximation algorithm for SPbT. We use the permutation tree data structure of Feng and Zhu and improve the running time of the 1.375 Approximation Algorithm for SPbT of Elias and Hartman to $O(nlog n)$. The previous running time of EH algorithm was $O(n^2)$.
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Massive sizes of real-world graphs, such as social networks and web graph, impose serious challenges to process and perform analytics on them. These issues can be resolved by working on a small summary of the graph instead . A summary is a compressed version of the graph that removes several details, yet preserves its essential structure. Generally, some predefined quality measure of the summary is optimized to bound the approximation error incurred by working on the summary instead of the whole graph. All known summarization algorithms are computationally prohibitive and do not scale to large graphs. In this paper we present an efficient randomized algorithm to compute graph summaries with the goal to minimize reconstruction error. We propose a novel weighted sampling scheme to sample vertices for merging that will result in the least reconstruction error. We provide analytical bounds on the running time of the algorithm and prove approximation guarantee for our score computation. Efficiency of our algorithm makes it scalable to very large graphs on which known algorithms cannot be applied. We test our algorithm on several real world graphs to empirically demonstrate the quality of summaries produced and compare to state of the art algorithms. We use the summaries to answer several structural queries about original graph and report their accuracies.
We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(min{m, frac{n^{2.5}ln n}{d}})$ expected time, an expression that is bounded by $tilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(min{m, frac{n^2ln^3 n}{d}})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((nln n)^{1.5})$. To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(nln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $tilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $tilde{O}(min{m, frac{n^2}{d} })$ time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Halls theorem ``witnesses for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n times n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $tilde{O}(m + n^{1.5})$.
Tree comparison metrics have proven to be an invaluable aide in the reconstruction and analysis of phylogenetic (evolutionary) trees. The path-length distance between trees is a particularly attractive measure as it reflects differences in tree shape as well as differences between branch lengths. The distance equals the sum, over all pairs of taxa, of the squared differences between the lengths of the unique path connecting them in each tree. We describe an $O(n log n)$ time for computing this distance, making extensive use of tree decomposition techniques introduced by Brodal et al. (2004).
In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V,E)$ with costs on edges, along with a connectivity requirement $r(u,v)$ for each pair $u,v$ of vertices. The goal is to find a minimum-cost subset $E^*$ of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are $r(u,v)$ edge-disjoint paths for every pair $u, v$ of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an $O(k^3 log n)$-approximation for this problem, where $k$ denotes the maximum connectivity requirement, and $n$ denotes the number of vertices. We also give a simple proof of the recently discovered $O(k^2 log n)$-approximation result for the single-source version of vertex-connectivity SNDP. We note that in both cases, our analysis in fact yields slightly better guarantees in that the $log n$ term in the approximation guarantee can be replaced with a $log tau$ term where $tau$ denotes the number of distinct vertices that participate in one or more pairs with a positive connectivity requirement.
One of the most fundamental results in combinatorial optimization is the polynomial-time 3/2-approximation algorithm for the metric traveling salesman problem. It was presented by Christofides in 1976 and is well known as the Christofides algorithm. Recently, some authors started calling it Christofides-Serdyukov algorithm, pointing out that it was published independently in the USSR in 1978. We provide some historic background on Serdyukovs findings and a translation of his article from Russian into English.
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