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## More algorithms for all-pairs shortest paths in weighted graphs (2007)

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Venue: | In Proceedings of 39th Annual ACM Symposium on Theory of Computing |

Citations: | 74 - 3 self |

### Citations

10600 | Introduction to algorithms
- Cormen, Leiserson, et al.
- 1995
(Show Context)
Citation Context ...textbooks; yet, the complexity of the problem has remained open to this day. For arbitrary dense (directed and undirected) real-weighted graphs with n vertices, the classical Floyd–Warshall algorithm =-=[13]-=- runs in O(n3 ) time. Fredman [17] was the first to realize the possibility of a subcubic algorithm, and since improvements have appeared in a number of papers; Table 1 summarizes the fascinating hist... |

2893 |
The design and analysis of computer algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ..., for arbitrary vertex-weighted graphs. This extends significantly, in a way, the original result of Vassilevska and Williams. 2 APSP for General Graphs in Near O(n 3 / log 2 n) Time It is well known =-=[2]-=- that the APSP problem for an arbitrary real-weighted graph with n vertices can be reduced to the problem of computing the distance product of two arbitrary real-valued, square n × n matrices (also kn... |

2049 |
Computation Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...blem and can be solved in O(n2 ) time for dimensions up to d ≈ log n by using known techniques from computational geometry (specifically, a simple divide-and-conquer algorithm for computing dominance =-=[30]-=-). The distance product of two n × n matrices can then be solved in O(n2 · n/d) = O(n3 / log n) time by performing n/d such rectangular products and taking the element-wise minimum of the resulting ma... |

300 |
Computational Geometry: An Introduction through Randomized Algorithms.
- Mulmuley
- 1994
(Show Context)
Citation Context ...ctually be reduced to O(cn log r) with more care). A list H∆ can be easily generated in O(n/r) time for each of the O(cdrd ) cells ∆. ✷ Remark: By more powerful techniques from computational geometry =-=[12, 26]-=-, the same result (ignoring dependences of the constant factors on d) actually holds for arbitrary hyperplanes with no restrictions on directions. We now describe a new but surprisingly simple way of ... |

89 | Clique partitions, graph compression and speeding-up algorithms, in
- Feder, Motwani
- 1991
(Show Context)
Citation Context ... classical “four-Russians” algorithm [6] from the 70s, with running time O(n3 / log 2 n). 1 (For APSP in undirected unweighted graphs, the previous purely combinatorial algorithm by Feder and Motwani =-=[16]-=- has a worse running time of O(n3 / log n); see also [8] for the sparse graph case.) The most tantalizing question in the area is whether in general the APSP problem could be solved in truly subcubic ... |

89 |
Fast rectangular matrix multiplications and applications.
- HUANG, PAN
- 1998
(Show Context)
Citation Context ...eal-vertex-weighted graphs, which slightly improves our O(n 2.844 ) result. The improvement does not require a new algorithm, but follows just by applying a known improved bound, due to Huang and Pan =-=[20]-=-, for rectangular matrix multiplication (specifically for multiplying the n × n and n × nr matrices in the third paragraph of Theorem 3.2’s proof). Improvements for geometrically weighted graphs for κ... |

85 | All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication,
- Zwick
- 2002
(Show Context)
Citation Context ...his question will remain unanswered here. Nevertheless, in the second part of the paper (Section 3), we will describe new results along these lines for an important class of special cases. Previously =-=[4, 32, 33, 41]-=-, the case of graphs with small integer weights has been the most extensively studied, where the weights lie in the range {1, . . . , c} for a constant c. In this case, Alon, Galil, and Margalit [4] g... |

84 | On the Exponent of the All Pairs Shortest Path Problem,
- Alon, Galil, et al.
- 1997
(Show Context)
Citation Context ...his question will remain unanswered here. Nevertheless, in the second part of the paper (Section 3), we will describe new results along these lines for an important class of special cases. Previously =-=[4, 32, 33, 41]-=-, the case of graphs with small integer weights has been the most extensively studied, where the weights lie in the range {1, . . . , c} for a constant c. In this case, Alon, Galil, and Margalit [4] g... |

84 |
On economic construction of the transitive closure of a directed graph,
- Arlazarov, Dinic, et al.
- 1970
(Show Context)
Citation Context ...an matrix multiplication algorithm known that does not rely on “algebraic” techniques (e.g., as used in Strassen’s or Coppersmith and Winograd’s algorithm) was the classical “four-Russians” algorithm =-=[6]-=- from the 70s, with running time O(n3 / log 2 n). 1 (For APSP in undirected unweighted graphs, the previous purely combinatorial algorithm by Feder and Motwani [16] has a worse running time of O(n3 / ... |

84 |
On the all-pairs-shortest-path problem in unweighted undirected graphs.
- SEIDEL
- 1995
(Show Context)
Citation Context ...his question will remain unanswered here. Nevertheless, in the second part of the paper (Section 3), we will describe new results along these lines for an important class of special cases. Previously =-=[4, 32, 33, 41]-=-, the case of graphs with small integer weights has been the most extensively studied, where the weights lie in the range {1, . . . , c} for a constant c. In this case, Alon, Galil, and Margalit [4] g... |

83 | On range searching with semialgebraic sets,
- Agarwal, Matousek
- 1994
(Show Context)
Citation Context ... . . . , (xm, ym)〉 in O(⌈m/w⌉ log u) time; (c) for a fixed mapping f : {1, . . . , u} → {1, . . . , u}, we can compute the list 〈f(x1), . . . , f(xm)〉 in O(⌈m/w⌉ log u) time; (d) for any given array f=-=[1]-=-, . . . , f[u] ∈ {1, . . . , u}, we can compute the list 〈f[x1], . . . , f[xm]〉 in O(⌈m/w⌉ log(mu) log m + u) time. Proof: (a) Let ¯w = ⌊w/ log u⌋. We first show how to merge two sorted compressed lis... |

79 |
New Bounds on the Complexity of the Shortest Path Problem,
- Fredman
- 1976
(Show Context)
Citation Context ...the problem has remained open to this day. For arbitrary dense (directed and undirected) real-weighted graphs with n vertices, the classical Floyd–Warshall algorithm [13] runs in O(n3 ) time. Fredman =-=[17]-=- was the first to realize the possibility of a subcubic algorithm, and since improvements have appeared in a number of papers; Table 1 summarizes the fascinating history. Notable among the more recent... |

56 | All pairs shortest paths in undirected graphs with integer weights.
- SHOSHAN, ZWICK
- 1999
(Show Context)
Citation Context |

49 | Improved parallel integer sorting without concurrent writing,
- Albers, Hagerup
- 1997
(Show Context)
Citation Context ...ginally slower (costing another log log n factor) and is inspired by the ideas outlined by Han. We first state a few handy subroutines on manipulating compressed lists (some are well known, e.g., see =-=[3]-=- concerning (a)). Lemma 2.3 Given compressed lists X = 〈x1, . . . , xm〉 and Y = 〈y1, . . . , ym〉, where xi, yi ∈ {1, . . . , u}, (a) we can sort X in O(⌈m/w⌉ log u log m) time; (b) we can compute the ... |

46 |
A new upper bound on the complexity of the all pairs shortest path problem
- Takaoka
- 1992
(Show Context)
Citation Context ...39th ACM Sympos. Theory Comput., pages 590–598, 2007. 1time ref. year O(n3 ) Dijkstra/Floyd–Warshall 1959/1962 O(n3 log 1/3 log n/ log 1/3 n) Fredman [17] 1976 O(n3 log 1/2 log n/ log 1/2 n) Takaoka =-=[34]-=- 1991 O(n3 / log 1/2 n) Dobosiewicz [15] 1990 O(n3 log 5/7 log n/ log 5/7 n) Han [18] 2004 O(n3 log 2 log n/ log n) Takaoka [35] 2004 O(n3 log log n/ log n) Takaoka [36] 2005 O(n3 log 1/2 log n/ log n... |

45 | Approximating the stretch factor of Euclidean graphs.
- Narasimhan, Smid
- 2000
(Show Context)
Citation Context ...able interest and have been studied extensively in the literature through the years; for example, see [31] for an early paper about APSP in such graphs. For a more recent example, Narasimhan and Smid =-=[27]-=- investigated the problem of approximating the stretch factor of a Euclidean graph; our result immediately implies a subcubic exact algorithm for the stretch factor. It should be mentioned that the in... |

42 | On range searching with semialgebraic sets, Discrete Comput - Agarwal, Matouˇsek - 1994 |

41 | All-pairs shortest paths with real weights in O (n 3/log n) time.
- Chan
- 2005
(Show Context)
Citation Context ...algorithm, and since improvements have appeared in a number of papers; Table 1 summarizes the fascinating history. Notable among the more recent results are the O(n3 / log n) algorithm of this author =-=[10]-=-, which is based on a simple geometric approach and does not require explicit table lookup or word tricks; and the O(n3 log 5/4 log n/ log 5/4 n) algorithm of Han [19], which amazingly breaks the O(n3... |

41 | A new approach to all-pairs shortest paths on real-weighted graphs.
- Pettie
- 2004
(Show Context)
Citation Context ...ed, imply an O(n 2 log n + mn) time bound for any graph with m edges; the first term has been lowered to O(n 2 log log n) and O(n 2 α(m, n)) for directed and undirected graphs respectively, by Pettie =-=[28]-=- and Pettie and Ramanchandran [29].) ∗ School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (tmchan@uwaterloo.ca). This work has been supported by NSERC. A preliminary... |

36 |
Efficient partition trees
- Matouˇsek
- 1992
(Show Context)
Citation Context ...ut sums of square roots in the case of Euclidean distances. 3.1 Geometry We begin with another well-known geometric tool, a partition theorem, which plays a central role in range searching. Matouˇsek =-=[25]-=- established the original version; the version we need below follows 9from Agarwal and Matouˇsek’s work on semialgebraic range searching [1]. Lemma 3.1 Let W be a set of c piecewise-algebraic functio... |

30 | Almost tight upper bounds for vertical decompositions in four dimensions
- Koltun
(Show Context)
Citation Context ...e of a function w ∈ W. Let dlin be the linearization dimension [1] (roughly speaking, the number of variables needed to transform the expression w(p, x) + z ≤ a into linear inequalities). It is known =-=[1, 23]-=- that the following value of κ works: κ = { dact if dact ≤ 4 min{⌊(dact + dlin)/2⌋, 2dact − 4} otherwise. For example, suppose that w(·, ·) is the Euclidean distance function. Then dact = d + 1, since... |

27 |
Shortest Paths in Euclidean Graphs
- Sedgewick, Vitter
- 1986
(Show Context)
Citation Context ...bound becomes Õ(n3−(3−ω)/(2d+4) ). Euclidean graphs and geometric graphs of course are of considerable interest and have been studied extensively in the literature through the years; for example, see =-=[31]-=- for an early paper about APSP in such graphs. For a more recent example, Narasimhan and Smid [27] investigated the problem of approximating the stretch factor of a Euclidean graph; our result immedia... |

23 | All-pairs shortest paths for unweighted undirected graphs in o(mn) time.
- Chan
- 2006
(Show Context)
Citation Context ...th running time O(n3 / log 2 n). 1 (For APSP in undirected unweighted graphs, the previous purely combinatorial algorithm by Feder and Motwani [16] has a worse running time of O(n3 / log n); see also =-=[8]-=- for the sparse graph case.) The most tantalizing question in the area is whether in general the APSP problem could be solved in truly subcubic time (O(n 3−δ ) for some specific constant δ > 0), by us... |

19 | R: Regularity Lemmas and Combinatorial Algorithms
- Bansal, Williams
(Show Context)
Citation Context ...nr matrices in the third paragraph of Theorem 3.2’s proof). Improvements for geometrically weighted graphs for κ > 1 also immediately 14follow, but are again very slight. Second, Bansal and Williams =-=[7]-=- have announced a new purely combinatorial algorithm for Boolean matrix multiplication which breaks the O(n 3 / log 2 n) barrier. The running time is O(n 3 log 2 log n/ log 9/4 n). At this point, it i... |

17 | Finding a maximum weight triangle in n 3−δ time, with applications
- Vassilevska, Williams
(Show Context)
Citation Context ... recently, vertex-weighted graphs have been the subject of several papers (where the weight of a path/cycle is defined as the sum of the weights of its vertices). At STOC’06, Vassilevska and Williams =-=[37]-=- showed that a problem related to APSP, namely, the minimum-weight triangle problem, can be solved in truly subcubic time, namely, O(n 2.688 ) time, for arbitrary real vertex weights. Subsequently, Va... |

16 | A shortest path algorithm for real-weighted undirected graphs
- Pettie, Ramachandran
(Show Context)
Citation Context ...e bound for any graph with m edges; the first term has been lowered to O(n 2 log log n) and O(n 2 α(m, n)) for directed and undirected graphs respectively, by Pettie [28] and Pettie and Ramanchandran =-=[29]-=-.) ∗ School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (tmchan@uwaterloo.ca). This work has been supported by NSERC. A preliminary version appeared in Proc. 39th AC... |

16 | A slightly improved sub-cubic algorithm for the all pairs shortest paths problem with real edge lengths - Zwick - 2004 |

16 | Improved algorithms for all pairs shortest paths. - Han - 2004 |

15 | Dynamic subgraph connectivity with geometric applications.
- CHAN
- 2002
(Show Context)
Citation Context ...ance method [24]. The combination of geometric range searching techniques with fast matrix multiplication is rather interesting, although examples of such combinations have appeared before (e.g., see =-=[9, 21, 22]-=-). Instead of the partition theorem, it is possible to prove Theorem 3.2 alternatively using a cutting lemma, as in the previous section, along with randomization. (Known proofs of the partition theor... |

15 |
A faster algorithm for the all-pairs shortest path problem and its application
- Takaoka
- 2004
(Show Context)
Citation Context ... n/ log 1/3 n) Fredman [17] 1976 O(n3 log 1/2 log n/ log 1/2 n) Takaoka [34] 1991 O(n3 / log 1/2 n) Dobosiewicz [15] 1990 O(n3 log 5/7 log n/ log 5/7 n) Han [18] 2004 O(n3 log 2 log n/ log n) Takaoka =-=[35]-=- 2004 O(n3 log log n/ log n) Takaoka [36] 2005 O(n3 log 1/2 log n/ log n) Zwick [42] 2004 O(n3 / log n) Chan [10] 2005 O(n3 log 5/4 log n/ log 5/4 n) Han [19] 2006 O(n3 log 3 log n/ log 2 n) this pape... |

12 | Counting colors in boxes
- Kaplan, Rubin, et al.
- 2007
(Show Context)
Citation Context ...ance method [24]. The combination of geometric range searching techniques with fast matrix multiplication is rather interesting, although examples of such combinations have appeared before (e.g., see =-=[9, 21, 22]-=-). Instead of the partition theorem, it is possible to prove Theorem 3.2 alternatively using a cutting lemma, as in the previous section, along with randomization. (Known proofs of the partition theor... |

12 | Colored intersection searching via sparse rectangular matrix multiplication
- Kaplan, Sharir, et al.
- 2006
(Show Context)
Citation Context ...ance method [24]. The combination of geometric range searching techniques with fast matrix multiplication is rather interesting, although examples of such combinations have appeared before (e.g., see =-=[9, 21, 22]-=-). Instead of the partition theorem, it is possible to prove Theorem 3.2 alternatively using a cutting lemma, as in the previous section, along with randomization. (Known proofs of the partition theor... |

12 | Finding the smallest H-subgraph in real weighted graphs and related problems
- Vassilevska, Williams, et al.
- 2006
(Show Context)
Citation Context ...o APSP, namely, the minimum-weight triangle problem, can be solved in truly subcubic time, namely, O(n 2.688 ) time, for arbitrary real vertex weights. Subsequently, Vassilevska, Williams, and Yuster =-=[38]-=- improved this bound to O(n 2.575 ), and Czumaj and Lingas [14] further to O(n 2.376 ). Related problems have also been considered (like finding fixed subgraphs besides triangles), but we are not awar... |

10 |
A more efficient algorithm for the min-plus multiplication
- Dobosiewicz
- 1990
(Show Context)
Citation Context ...90–598, 2007. 1time ref. year O(n3 ) Dijkstra/Floyd–Warshall 1959/1962 O(n3 log 1/3 log n/ log 1/3 n) Fredman [17] 1976 O(n3 log 1/2 log n/ log 1/2 n) Takaoka [34] 1991 O(n3 / log 1/2 n) Dobosiewicz =-=[15]-=- 1990 O(n3 log 5/7 log n/ log 5/7 n) Han [18] 2004 O(n3 log 2 log n/ log n) Takaoka [35] 2004 O(n3 log log n/ log n) Takaoka [36] 2005 O(n3 log 1/2 log n/ log n) Zwick [42] 2004 O(n3 / log n) Chan [10... |

9 | Finding a heaviest triangle is not harder than matrix multiplication
- Czumaj, Lingas
- 2007
(Show Context)
Citation Context ...ved in truly subcubic time, namely, O(n 2.688 ) time, for arbitrary real vertex weights. Subsequently, Vassilevska, Williams, and Yuster [38] improved this bound to O(n 2.575 ), and Czumaj and Lingas =-=[14]-=- further to O(n 2.376 ). Related problems have also been considered (like finding fixed subgraphs besides triangles), but we are not aware of any nontrivial results for cycles of length beyond 3, or f... |

9 |
Computing Dominances
- Matouˇsek
- 1991
(Show Context)
Citation Context ... 1 into r sublists of size O(n/r) in the obvious way). The resulting Õ(n(3+ω)/2 ) algorithm bear superficial resemblance with some previous Õ(n (3+ω)/2 ) algorithms, like Matouˇsek’s dominance method =-=[24]-=-. The combination of geometric range searching techniques with fast matrix multiplication is rather interesting, although examples of such combinations have appeared before (e.g., see [9, 21, 22]). In... |

8 | All pairs lightest shortest paths - Zwick - 1999 |

6 | Fly cheaply: on the minimum fuel consumption problem - Chan, Efrat - 2001 |

6 | An O(n3 log log / log n) time algorithm for the all-pairs shortest path problem
- Takaoka
(Show Context)
Citation Context ... 1/2 log n/ log 1/2 n) Takaoka [34] 1991 O(n3 / log 1/2 n) Dobosiewicz [15] 1990 O(n3 log 5/7 log n/ log 5/7 n) Han [18] 2004 O(n3 log 2 log n/ log n) Takaoka [35] 2004 O(n3 log log n/ log n) Takaoka =-=[36]-=- 2005 O(n3 log 1/2 log n/ log n) Zwick [42] 2004 O(n3 / log n) Chan [10] 2005 O(n3 log 5/4 log n/ log 5/4 n) Han [19] 2006 O(n3 log 3 log n/ log 2 n) this paper 2007 Table 1: APSP algorithms for gener... |

6 | Efficient algorithms on sets of permutations, dominance, and real-weighted apsp
- Yuster
- 2009
(Show Context)
Citation Context ...number of times and using color-coding [5] to prevent nonsimple cycles. 4 Final Remarks We briefly mention two recent results that have appeared after the initial version of this paper. First, Yuster =-=[39]-=- has announced an O(n 2.842 ) time bound for APSP in real-vertex-weighted graphs, which slightly improves our O(n 2.844 ) result. The improvement does not require a new algorithm, but follows just by ... |

5 |
An O(n 3 (log log n/ log n) 5/4 ) time algorithm for all pairs shortest paths
- Han
- 2008
(Show Context)
Citation Context ... log n) algorithm of this author [10], which is based on a simple geometric approach and does not require explicit table lookup or word tricks; and the O(n3 log 5/4 log n/ log 5/4 n) algorithm of Han =-=[19]-=-, which amazingly breaks the O(n3 / log n) barrier by exploiting sophisticated word-packing tricks (implementable by table lookups), and is the best result known to date. (We have ignored APSP algorit... |

2 | An O(n3(log log n/ log n)5/4) time algorithm for all pairs shortest paths - Han - 2006 |

1 | On economical construction of the transitive closure of a directed graph - ACM - 1995 |

1 | Finding a maximum weight triangle in n3-ffi time, with applications - Vassilevska, Williams - 2006 |