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Shortest paths in directed planar graphs with negative lengths: A linearspace O(n log² n)time algorithm
 PROC. 20TH ANN. ACMSIAM SYMP. DISCRETE ALGORITHMS
, 2009
"... We give an O(n log² n)time, linearspace algorithm that, given a directed planar graph with positive and negative arclengths, and given a node s, finds the distances from s to all nodes. ..."
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We give an O(n log² n)time, linearspace algorithm that, given a directed planar graph with positive and negative arclengths, and given a node s, finds the distances from s to all nodes.
A nearly optimal oracle for avoiding failed vertices and edges
 In Proceedings 41st STOC
, 2009
"... We present an improved oracle for the distance sensitivity problem. The goal is to preprocess a directed graph G =(V,E) with nonnegative edge weights to answer queries of the form: what is the length of the shortest path from x to y that does not go through some failed vertex or edge f. The previou ..."
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We present an improved oracle for the distance sensitivity problem. The goal is to preprocess a directed graph G =(V,E) with nonnegative edge weights to answer queries of the form: what is the length of the shortest path from x to y that does not go through some failed vertex or edge f. The previous best algorithm produces an oracle of size Õ(n 2)thathasanO(1) query time, and an Õ(n2√m)con struction time. It was a randomized Monte Carlo algorithm that worked with high probability. Our oracle also has a constant query time and an Õ(n2) space requirement, but it has an improved construction time of Õ(mn), and it is deterministic. Note that O(1) query, O(n 2) space, and O(mn) construction time is also the best known bound (up to logarithmic factors) for the simpler problem of finding all pairs shortest paths in a weighted, directed graph. Thus, barring improved solutions to the all pairs shortest path problem, our oracle is optimal up to logarithmic factors.
Dualfailure distance and connectivity oracles
 In Proc. of the 20th ACMSIAM Symposium On Discrete Algorithms (SODA
, 2009
"... Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its co ..."
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Cited by 13 (3 self)
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Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its connectivity and distance metric. In this paper we look at the problem of efficiently answering connectivity, distance, and shortest route queries in the presence of two node or link failures. Our data structure uses Õ(n2) space and answers queries in Õ(1) time, which is within a polylogarithmic factor of optimal and nearly matches the singlefailure distance oracles of Demestrescu et al. It may yet be possible to find distance/connectivity oracles capable of handling any fixed number of failures. However, the sheer complexity of our algorithm suggests that moving beyond dualfailures will require a fundamentally different approach to the problem. 1
A nearly optimal algorithm for approximating replacement paths and k shortest simple paths in general graphs
 In Proc. SODA
, 2010
"... Let G = (V, E) be a directed graph with positive edge weights, let s, t be two specified vertices in this graph, and let π(s, t) be the shortest path between them. In the replacement paths problem we want to compute, for every edge e on π(s, t), the shortest path from s to t that avoids e. The naive ..."
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Let G = (V, E) be a directed graph with positive edge weights, let s, t be two specified vertices in this graph, and let π(s, t) be the shortest path between them. In the replacement paths problem we want to compute, for every edge e on π(s, t), the shortest path from s to t that avoids e. The naive solution to this problem would be to remove each edge e, one at a time, and compute the shortest s − t path each time; this yields a running time of O(mn + n 2 log n). Gotthilf and Lewenstein [8] recently improved this to O(mn+n 2 log log n), but no o(mn) algorithms are known. We present the first approximation algorithm for replacement paths in directed graphs with positive edge weights. Given any ɛ ∈ [0, 1), our algorithm returns (1 + ɛ)approximate replacement paths in O(ɛ −1 log 2 n log(nC/c)(m+n log n)) = Õ(m log(nC/c)/ɛ) time, where C is the largest edge weight in the graph and c is the smallest weight. We also present an even faster (1 + ɛ) approximate algorithm for the simpler problem of approximating the k shortest simple s − t paths in a directed graph with positive edge weights. That is, our algorithm outputs k different simple s−t paths, where the kth path we output is a (1 + ɛ) approximation to the actual kth shortest simple s − t path. The running time of our algorithm is O(kɛ −1 log 2 n(m + n log n)) = Õ(km/ɛ). The fastest exact algorithm for this problem has a running time of O(k(mn+n 2 log log n)) = Õ(kmn) [8]. The previous best approximation algorithm was developed by Roditty [15]; it has a stretch of 3/2 and a running time of Õ(km√n) (it does not work for replacement paths). Note that all of our running times are nearly optimal except for the O(log(nC/c)) factor in the replacements paths algorithm. Also, our algorithm can solve the variant of approximate replacement paths where we avoid vertices instead of edges. 1
Solving the replacement paths problem for planar directed graphs in o(n log n) time
 Proc. 21st Ann. ACMSIAM Symp. Discrete Algorithms, 756–765
, 2010
"... In a graph G with nonnegative edge lengths, let P be a shortest path from a vertex s toavertext. We consider the problem of computing, for each edge e on P, the length of a shortest path in G from s to t that avoids e. This is known as the replacement paths problem. We give a linearspace algorithm ..."
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In a graph G with nonnegative edge lengths, let P be a shortest path from a vertex s toavertext. We consider the problem of computing, for each edge e on P, the length of a shortest path in G from s to t that avoids e. This is known as the replacement paths problem. We give a linearspace algorithm with O(n log n) running time for nvertex planar directed graphs. The previous best time bound was O(n log 2 n). 1
Informational overhead of incentive compatibility
 In: Proceedings of the 9th ACM Conference on Electronic Commerce (EC’08
, 2008
"... In the presence of selfinterested parties, mechanism designers typically aim to achieve their goals (or socialchoice functions) in an equilibrium. In this paper, we study the cost of such equilibrium requirements in terms of communication, a problem that was recently raised by Fadel and Segal [14]. ..."
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Cited by 6 (4 self)
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In the presence of selfinterested parties, mechanism designers typically aim to achieve their goals (or socialchoice functions) in an equilibrium. In this paper, we study the cost of such equilibrium requirements in terms of communication, a problem that was recently raised by Fadel and Segal [14]. While a certain amount of information x needs to be communicated just for computing the outcome of a certain socialchoice function, an additional amount of communication may be required for computing the equilibriumsupporting prices (even if such prices are known to exist). Our main result shows that the total communication needed for this task can be greater than x by a factor linear in the number of players n, i.e., n · x. This is the first known lower bound for this problem. In fact, we show that this result holds even in singleparameter domains (under the common assumption that losing players pay zero). On the positive side, we show that certain classic economic objectives, namely, singleitem auctions and publicgood mechanisms, only entail a small overhead. Finally, we explore the communication overhead in welfaremaximization domains, and initiate the study of the overhead of computing payments that lie in the core of coalitional games. 1
Replacement Paths via Fast Matrix Multiplication
"... Abstract. Let G = (V, E) be a directed edgeweighted graph and let P be a shortest path from s to t in G. The replacement paths problem asks to compute, for every edge e on P, the shortest stot path that avoids e. Apart from approximation algorithms and algorithms for special graph classes, the na ..."
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Abstract. Let G = (V, E) be a directed edgeweighted graph and let P be a shortest path from s to t in G. The replacement paths problem asks to compute, for every edge e on P, the shortest stot path that avoids e. Apart from approximation algorithms and algorithms for special graph classes, the naive solution to this problem – removing each edge e on P one at a time and computing the shortest stot path each time – is surprisingly the only known solution for directed weighted graphs, even when the weights are integrals. In particular, although the related shortest paths problem has benefited from fast matrix multiplication, the replacement paths problem has not, and still required cubic time. For an nvertex graph with integral edgelengths in {−M,..., M}, we give a randomized Õ(Mn1+ 2 3 ω) = O(Mn 2.584) time algorithm that uses fast matrix multiplication and is subcubic for appropriate values of M. In particular, it runs in Õ(n1+ 2 3 ω) time if the weights are small (positive or negative) integers. We also show how to construct a distance sensitivity oracle in the same time bounds. A query (u, v, e) to this oracle requires subquadratic time and returns the length of the shortest utov path that avoids the edge e. In fact, for any constant number of edge failures, we construct a data structure in subcubic time, that answer queries in subquadratic time. Our results also apply for avoiding vertices rather than edges. 1
Computing Replacement Paths in Surface Embedded Graphs
 ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2011
"... Let s and t be vertices in a directed graph G with nonnegative edge weights. The replacement paths problem asks us to compute, for each edge e in G, the length of the shortest path from s to t that does not traverse e. We describe an algorithm that solves the replacement paths problem for directed ..."
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Let s and t be vertices in a directed graph G with nonnegative edge weights. The replacement paths problem asks us to compute, for each edge e in G, the length of the shortest path from s to t that does not traverse e. We describe an algorithm that solves the replacement paths problem for directed graphs embedded on a surface of any genus g in O(gn log n) time, generalizing a recent O(n log n)time algorithm of WulffNilsen for planar graphs [SODA 2010].
Accelerating Dynamic Programming
, 2009
"... Dynamic Programming (DP) is a fundamental problemsolving technique that has been widely used for solving a broad range of search and optimization problems. While DP can be invoked when more specialized methods fail, this generality often incurs a cost in efficiency. We explore a unifying toolkit fo ..."
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Dynamic Programming (DP) is a fundamental problemsolving technique that has been widely used for solving a broad range of search and optimization problems. While DP can be invoked when more specialized methods fail, this generality often incurs a cost in efficiency. We explore a unifying toolkit for speeding up DP, and algorithms that use DP as subroutines. Our methods and results can be summarized as follows. – Acceleration via Compression. Compression is traditionally used to efficiently store data. We use compression in order to identify repeats in the table that imply a redundant computation. Utilizing these repeats requires a new DP, and often different DPs for different compression schemes. We present the first provable speedup of the celebrated Viterbi algorithm (1967) that is used for the decoding and training of Hidden Markov Models (HMMs). Our speedup relies on the compression of the HMM’s observable sequence. – Totally Monotone Matrices. It is well known that a wide variety of DPs can be reduced to the problem of finding row minima in totally monotone matrices. We introduce this scheme in the context of planar graph problems. In particular, we show that planar graph problems