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86
Approximate distance oracles
, 2004
"... Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
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Cited by 273 (9 self)
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Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
More algorithms for allpairs shortest paths in weighted graphs
 In Proceedings of 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
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Cited by 75 (3 self)
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In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of smallintegerweighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in realvertexweighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the allpairs lightest shortest path problem for smallintegerweighted graphs. 1
Fast Sparse Matrix Multiplication
, 2004
"... Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multi ..."
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Cited by 53 (3 self)
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Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multiplication algorithm, on the other hand, may need to perform #(mn) operations to accomplish the same task. For , the new algorithm performs an almost optimal number of only n operations. For m the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n ) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices.
Allpairs shortest paths with real weights in O(n³ / log n) time
 PROC. OF THE 9TH WADS, LECTURE NOTES IN COMPUTER SCIENCE 3608
, 2005
"... We describe an O(n³ / log n) ..."
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 41 (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.
A new algorithm for optimal 2constraint satisfaction and its implications
 Theoretical Computer Science
, 2005
"... Abstract. We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX2CSP and MIN2CSP), which gives the first exponential improvement over the trivial algorithm. More precisely, th ..."
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Cited by 40 (6 self)
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Abstract. We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX2CSP and MIN2CSP), which gives the first exponential improvement over the trivial algorithm. More precisely, the runtime bound is a constant factor improvement in the base of the exponent: the algorithm can count the number of optima in MAX2SAT and MAXCUT instances in O(m 3 2 ωn/3) time, where ω < 2.376 is the matrix product exponent over a ring. When constraints have arbitrary weights, there is a (1 + ɛ)approximation with roughly the same runtime, modulo polynomial factors. Our construction shows that improvement in the runtime exponent of either kclique solution (even when k = 3) or matrix multiplication over GF(2) would improve the runtime exponent for solving 2CSP optimization. Our approach also yields connections between the complexity of some (polynomial time) high dimensional search problems and some NPhard problems. For example, if there are sufficiently faster algorithms for computing the diameter of n points in ℓ1, then there is an (2 − ɛ) n algorithm for MAXLIN. These results may be construed as either lower bounds on the highdimensional problems, or hope that better algorithms exist for the corresponding hard problems. 1
A new algorithm for optimal constraint satisfaction and its implications
 Alexander D. Scott) Mathematical Institute, University of Oxford
, 2004
"... We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX2CSP and MIN2CSP), which gives the first exponential improvement over the trivial algorithm; more precisely, it is a cons ..."
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Cited by 38 (1 self)
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We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX2CSP and MIN2CSP), which gives the first exponential improvement over the trivial algorithm; more precisely, it is a constant factor improvement in the base of the runtime exponent. In the case where constraints have arbitrary weights, there is a (1 + ǫ)approximation with roughly the same runtime, modulo polynomial factors. Our algorithm may be used to count the number of optima in MAX2SAT and MAXCUT instances in O(m 3 2 ωn/3) time, where ω < 2.376 is the matrix product exponent over a ring. This is the first known algorithm solving MAX2SAT and MAXCUT in provably less than c n steps in the worst case, for some c < 2; similar new results are obtained for related problems. Our main construction may also be used to show that any improvement in the runtime exponent of either kclique solution (even when k = 3) or matrix multiplication over GF(2) would improve the runtime exponent for solving 2CSP optimization. As a corollary, we prove that an n o(k)time kclique algorithm implies SNP ⊆ DTIME[2 o(n)], for any k(n) ∈ o ( √ n / log n). Further extensions of our technique yield connections between the complexity of some (polynomial time) high dimensional geometry problems and that of some general NPhard problems. For example, if there are sufficiently faster algorithms for computing the diameter of n points in ℓ1, then there is an (2 −ǫ) n algorithm for MAXLIN. Such results may be construed as either lower bounds on these highdimensional problems, or hope that better algorithms exist for more general NPhard problems. 1
Improved Dynamic Reachability Algorithms for Directed Graphs
, 2002
"... We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the ..."
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Cited by 35 (3 self)
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We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the transitive closure of a directed graph, through an arbitrary sequence of edge deletions, in O(mn) total expected time, essentially the time needed for computing the transitive closure of the initial graph. Such a result was previously known only for acyclic graphs.
On the Difficulty of Some Shortest Path Problems
, 2003
"... We prove superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with nonnegative edge weights, and a shortest path P = {e_1, e_2, ..., e_p} ..."
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Cited by 32 (8 self)
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We prove superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with nonnegative edge weights, and a shortest path P = {e_1, e_2, ..., e_p} between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e_i. We show that the replacement paths problem requires &Omega;(m&radic;n) time in the worst case whenever m = O(n&radic;n). This also establishes a similar...
Faster algorithms for approximate distance oracles and allpairs small stretch paths
 In 47th Annual IEEE Symp. on Foundations of Computer Science (FOCS
, 2006
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