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26
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 67 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
A Simple Shortest Path Algorithm with Linear Average Time
"... We present a simple shortest path algorithm. If the input lengths are positive and uniformly distributed, the algorithm runs in linear time. The worstcase running time of the algorithm is O(m + n log C), where n and m are the number of vertices and arcs of the input graph, respectively, and C i ..."
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Cited by 39 (7 self)
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We present a simple shortest path algorithm. If the input lengths are positive and uniformly distributed, the algorithm runs in linear time. The worstcase running time of the algorithm is O(m + n log C), where n and m are the number of vertices and arcs of the input graph, respectively, and C is the ratio of the largest and the smallest nonzero arc length.
A New Approach to AllPairs Shortest Paths on RealWeighted Graphs
 Theoretical Computer Science
, 2003
"... We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps ..."
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Cited by 36 (3 self)
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We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.
On dynamic shortest paths problems
 In ESA: Annual European Symposium on Algorithms
, 2004
"... Abstract. We obtain the following results related to dynamic versions of the shortestpaths problem: (i) Reductions that show that the incremental and decremental singlesource shortestpaths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static ..."
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Cited by 34 (2 self)
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Abstract. We obtain the following results related to dynamic versions of the shortestpaths problem: (i) Reductions that show that the incremental and decremental singlesource shortestpaths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static allpairs shortestpaths problem. We also obtain slightly weaker results for the corresponding unweighted problems. (ii) A randomized fullydynamic algorithm for the allpairs shortestpaths problem in directed unweighted graphs with an amortized update time of Õ(m n) and a worst case query time is O(n3/4). (iii) A deterministic O(n2 log n) time algorithm for constructing a (log n)spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining singlesource shortestpaths tree up to a given distance. 1
SingleSource ShortestPaths on Arbitrary Directed Graphs in Linear AverageCase Time
 In Proc. 12th ACMSIAM Symposium on Discrete Algorithms
, 2001
"... The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w ..."
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Cited by 31 (5 self)
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The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m log log n). In the present paper we study the averagecase complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0; 1] and show that it needs linear time O(n + m) with high probability. 1 Introduction The singlesource shortestpath problem (SSSP) is a fundamental and wellstudied combinatorial optimization problem with many practical and theoretical applications [1]. Let G = (V; E) be a directed graph, jV j = n, jEj = m, let s be a distinguished vertex of the graph, and c be a function assigning a n...
Integer Priority Queues with Decrease Key in . . .
 STOC'03
, 2003
"... We consider Fibonacci heap style integer priority queues supporting insert and decrease key operations in constant time. We present a deterministic linear space solution that with n integer keys support delete in O(log log n) time. If the integers are in the range [0,N), we can also support delete i ..."
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Cited by 30 (2 self)
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We consider Fibonacci heap style integer priority queues supporting insert and decrease key operations in constant time. We present a deterministic linear space solution that with n integer keys support delete in O(log log n) time. If the integers are in the range [0,N), we can also support delete in O(log log N) time. Even for the special case of monotone priority queues, where the minimum has to be nondecreasing, the best previous bounds on delete were O((log n) 1/(3−ε) ) and O((log N) 1/(4−ε)). These previous bounds used both randomization and amortization. Our new bounds a deterministic, worstcase, with no restriction to monotonicity, and exponentially faster. As a classical application, for a directed graph with n nodes and m edges with nonnegative integer weights, we get single source shortest paths in O(m + n log log n) time, or O(m + n log log C) ifC is the maximal edge weight. The later solves an open problem of Ahuja, Mehlhorn, Orlin, and
Smoothed Analysis of Three Combinatorial Problems
 Proc. of the 28th Int. Symp. on Mathematical Foundations of Computer Science (MFCS), volume 2747 of Lecture Notes in Computer Science
, 2003
"... Smoothed analysis combines elements over worstcase and average case analysis. For an instance x the smoothed complexity is the average complexity of an instance obtained from x by a perturbation. The smoothed complexity of a problem is the worst smoothed complexity of any instance. Spielman and ..."
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Cited by 23 (1 self)
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Smoothed analysis combines elements over worstcase and average case analysis. For an instance x the smoothed complexity is the average complexity of an instance obtained from x by a perturbation. The smoothed complexity of a problem is the worst smoothed complexity of any instance. Spielman and Teng introduced this notion for continuous problems. We apply the concept to combinatorial problems and study the smoothed complexity of three classical discrete problems: quicksort, lefttoright maxima counting, and shortest paths. This opens a vast eld of nice analyses (using for example generating functions in the discrete case) which should lead to a better understanding of complexity landscapes of algorithms.
Shortest Path Algorithms: Engineering Aspects
 In Proc. ESAAC ’01, Lecture Notes in Computer Science
, 2001
"... We review shortest path algorithms based on the multilevel bucket data structure [6] and discuss the interplay between theory and engineering choices that leads to e#cient implementations. Our experimental results suggest that the caliber heuristic [17] and adaptive parameter selection give an ..."
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Cited by 22 (3 self)
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We review shortest path algorithms based on the multilevel bucket data structure [6] and discuss the interplay between theory and engineering choices that leads to e#cient implementations. Our experimental results suggest that the caliber heuristic [17] and adaptive parameter selection give an e#cient algorithm, both on typical and on hard inputs, for a wide range of arc lengths.
Computing Shortest Paths with Comparisons and Additions
 SODA
, 2002
"... We present an undirected allpairs shortest paths (APSP) algorithm which runs on a pointer machine in time O(mnot(m, n)) while making O(ran log a(m, n)) comparisons and additions, where m and n are the number of edges and vertices, respectively, and a(ra, n) is Tarjan's inverseAckermann funct ..."
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Cited by 20 (7 self)
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We present an undirected allpairs shortest paths (APSP) algorithm which runs on a pointer machine in time O(mnot(m, n)) while making O(ran log a(m, n)) comparisons and additions, where m and n are the number of edges and vertices, respectively, and a(ra, n) is Tarjan's inverseAckermann function. This improves upon all previous comparison & additionbased APSP algorithms when the graph is sparse, i.e., when m = o(n log n). At the heart of our APSP algorithm is a new singlesource shortest paths algorithm which runs in time O(ma(m,n) + nloglogr) on a pointer machine, where r is the ratio of the maximumtominimum edge length. So long as r < 2 '~°(a) this algorithm is faster than any implementation of Dijkstra's classical algorithm in the comparisonaddition model. For directed graphs we give an O(ra + nlogr)time comparison & additionbased SSSP algorithm on a pointer machine. Similar algorithms assuming integer weights or the RAM model were given earlier.
Parallel Shortest Path Algorithms for Solving . . .
, 2006
"... We present an experimental study of the single source shortest path problem with nonnegative edge weights (NSSP) on largescale graphs using the ∆stepping parallel algorithm. We report performance results on the Cray MTA2, a multithreaded parallel computer. The MTA2 is a highend shared memory s ..."
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Cited by 15 (3 self)
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We present an experimental study of the single source shortest path problem with nonnegative edge weights (NSSP) on largescale graphs using the ∆stepping parallel algorithm. We report performance results on the Cray MTA2, a multithreaded parallel computer. The MTA2 is a highend shared memory system offering two unique features that aid the efficient parallel implementation of irregular algorithms: the ability to exploit finegrained parallelism, and lowoverhead synchronization primitives. Our implementation exhibits remarkable parallel speedup when compared with competitive sequential algorithms, for lowdiameter sparse graphs. For instance, ∆stepping on a directed scalefree graph of 100 million vertices and 1 billion edges takes less than ten seconds on 40 processors of the MTA2, with a relative speedup of close to 30. To our knowledge, these are the first performance results of a shortest path problem on realistic graph instances in the order of billions of vertices and edges.