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## Distributed approximation algorithms for weighted shortest paths

Venue: | In Proc. of the Symp. on Theory of Comp. (STOC |

Citations: | 9 - 1 self |

### Citations

1980 |
Distributed Algorithms
- Lynch
- 1996
(Show Context)
Citation Context ...approximated w.h.p. 1.4 Related Work Unweighted Case. SSSP and APSP are essentially well-understood in the unweighted case. SSSP can be trivially solved in O(D) time using a breadth-first search tree =-=[46, 43]-=-. Frischknecht, Holzer, and Wattenhofer [23, 30] show a (surprising) lower bound of Ω(n/ log n) for computing the diameter of unweighted networks, which implies a lower bound for solving unweighted AP... |

531 | Probability and Computing: Randomized Algorithms and Probabilistic Analysis - Mitzenmacher, Upfal - 2005 |

494 |
On a Routing Problem
- Bellman
- 1958
(Show Context)
Citation Context ...ning times of our algorithms are logarithmic of the largest edge weight. This is in the same spirit as, e.g., [42, 25, 32]. 2 Problems Topology References Time Approximation SSSP General Bellman&Ford =-=[6, 22]-=- Õ(n) exact Lenzen&Patt-Shamir [38]2 Õ(n1/2+1/2k + D) 8kdlog(k + 1)e − 1 this paper Õ(n1/2D1/4 + D) 1 + o(1) (= Õ(n3/4 + D)) Fully-Connected Baswana&Sen [5] Õ(n1/k) 2k − 1 this paper Õ(n1/2) exa... |

450 |
Distributed computing: A locality-sensitive approach
- Peleg
- 2000
(Show Context)
Citation Context ...ts, where B is typically set to log n. This model is often called synchronous CONGEST (or CONGEST(B) if B 6= log n). Time complexity in this model is one of the major studies in distributed computing =-=[46]-=-. Many previous works in this model, including several previous FOCS/STOC papers (e.g. [24, 48, 17, 11, 38]), concern graph problems. Here, we want to learn some topological properties of a network, s... |

271 | Approximate Distance Oracles
- Thorup, Zwick
(Show Context)
Citation Context ...e O(SPDiam(G,w)) time for SSSP. Khan et al. [31] gives a Õ(n ·SPDiam(G,w))-time O(log n)-approximation algorithm via metric tree embeddings [21]. We can also construct Thorup-Zwick distance sketches =-=[54]-=- of size O(kn1/k) and stretch 2k − 1 in Õ(kn1/k · SPDiam(G,w)) time [10]. Since SPDiam(G,w) can be as large as n, these algorithms do not give any improvement to previous algorithms when analyzed in ... |

119 | Packet routing and job-shop scheduling in O(congestion + dilation) steps
- Leighton, Maggs, et al.
- 1994
(Show Context)
Citation Context ...will require to send only Õ(k) messages through each edge, which gives us a hope that we will require only additional Õ(k) time. By a careful paralellization (based on the random delay technique of =-=[36]-=-7), we can solve h-hop SSSP from k sources in Õ(h+ k) time. This is the first tool that we will use later. 6The problem can also be solved by running the distributed version of Bellman-Ford algorithm... |

116 |
Bounds for the quantity of information transmitted by a quantum communication channel
- Holevo
- 1973
(Show Context)
Citation Context ...ess of what Bob sends to her. This fact can be formally proved in many ways (e.g., by using communication complexity lower bounds) and is true even in the quantum setting (see, e.g., Holevo’s theorem =-=[28]-=-). Now, let A be an α(n)-approximation T -time algorithm for weighted APSP. We show that Alice can use A to send her message to Bob using O(T log n) bits, as follows. Construct a graph G consisting of... |

87 | A sublinear time distributed algorithm for minimum-weight spanning trees,”
- Garay, Kutten, et al.
- 1998
(Show Context)
Citation Context ...ONGEST(B) if B 6= log n). Time complexity in this model is one of the major studies in distributed computing [46]. Many previous works in this model, including several previous FOCS/STOC papers (e.g. =-=[24, 48, 17, 11, 38]-=-), concern graph problems. Here, we want to learn some topological properties of a network, such as minimum spanning tree (MST), minimum cut (mincut), and distances. These problems can be trivially so... |

76 | Computing almost shortest paths
- Elkin
- 2001
(Show Context)
Citation Context ...on-trivial lower bound on networks of diameter one and two [15]. Other Works. While computing shortest paths is among the earliest studied problems in dis5We note that some of these algorithms (e.g., =-=[16, 31]-=-) can actually solve a more general problem called the S-shortest path problem. To avoid confusions, we will focus only on SSSP and APSP. 6 tributed computing, many classic works on this problem conce... |

66 |
Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics
- Fakcharoenphol, Rao
(Show Context)
Citation Context ...mple, Bellman-Ford algorithm [6, 22] can be analyzed to have O(SPDiam(G,w)) time for SSSP. Khan et al. [31] gives a Õ(n ·SPDiam(G,w))-time O(log n)-approximation algorithm via metric tree embeddings =-=[21]-=-. We can also construct Thorup-Zwick distance sketches [54] of size O(kn1/k) and stretch 2k − 1 in Õ(kn1/k · SPDiam(G,w)) time [10]. Since SPDiam(G,w) can be as large as n, these algorithms do not gi... |

66 |
Network Flow Theory
- FORD
- 1956
(Show Context)
Citation Context ...ning times of our algorithms are logarithmic of the largest edge weight. This is in the same spirit as, e.g., [42, 25, 32]. 2 Problems Topology References Time Approximation SSSP General Bellman&Ford =-=[6, 22]-=- Õ(n) exact Lenzen&Patt-Shamir [38]2 Õ(n1/2+1/2k + D) 8kdlog(k + 1)e − 1 this paper Õ(n1/2D1/4 + D) 1 + o(1) (= Õ(n3/4 + D)) Fully-Connected Baswana&Sen [5] Õ(n1/k) 2k − 1 this paper Õ(n1/2) exa... |

65 | Fast Distributed construction of small kdominating sets and applications
- Kutten, Peleg
- 1998
(Show Context)
Citation Context ... constant > 0, where n is the number of nodes and D is the network’s diameter. For example, through decades of extensive research, we now have an algorithm that can find an MST in Õ(n1/2 + D) time =-=[24, 35]-=-, and we know that this running time is tight [48]. This algorithm serves as a building block for several other sublinear-time algorithms (e.g. [55, 50, 25]). It is also natural to ask whether we can ... |

55 |
Design and Analysis of Distributed Algorithms
- Santoro
- 2007
(Show Context)
Citation Context ...), we can solve h-hop SSSP from k sources in Õ(h+ k) time. This is the first tool that we will use later. 6The problem can also be solved by running the distributed version of Bellman-Ford algorithm =-=[46, 43, 52]-=- from every node, but this takes O(n2) time in the worst case. So this is always worse than the trivial algorithm. 7Note that the random delay technique makes the algorithm randomized. Techniques in [... |

45 |
An Unconditional Lower Bound on the Time-Approximation Trade-off for the Distributed Minimum Spanning Tree Problem.
- Elkin
- 2006
(Show Context)
Citation Context ...ONGEST(B) if B 6= log n). Time complexity in this model is one of the major studies in distributed computing [46]. Many previous works in this model, including several previous FOCS/STOC papers (e.g. =-=[24, 48, 17, 11, 38]-=-), concern graph problems. Here, we want to learn some topological properties of a network, such as minimum spanning tree (MST), minimum cut (mincut), and distances. These problems can be trivially so... |

44 | Distributed verification and hardness of distributed approximation.
- Sarma, Holzer, et al.
- 2011
(Show Context)
Citation Context ...ONGEST(B) if B 6= log n). Time complexity in this model is one of the major studies in distributed computing [46]. Many previous works in this model, including several previous FOCS/STOC papers (e.g. =-=[24, 48, 17, 11, 38]-=-), concern graph problems. Here, we want to learn some topological properties of a network, such as minimum spanning tree (MST), minimum cut (mincut), and distances. These problems can be trivially so... |

36 | Polylog-time and near-linear work approximation scheme for undirected shortest paths,”
- Cohen
- 2000
(Show Context)
Citation Context ... to make sure that one of them will be in a set of n1/2 random nodes (“skeleton nodes”). It was pointed out to us by a STOC 2014 reviewer that our technique might be related to the technique of Cohen =-=[9]-=-. Our technique is indeed closely related to the notion of (d, )-hop set used in [9]: our shortest path diameter technique can be considered as a simple construction of (d, 0) hop set of size O(n2/d)... |

35 | D.: Distributed MST for Constant Diameter Graphs
- Lotker, Patt-Shamir, et al.
- 2006
(Show Context)
Citation Context ...s, an even stronger property also holds; e.g., a peer-to-peer network is usually assumed to be an expander [4].) It is thus of a special interest to develop an algorithm in this setting. For example, =-=[41]-=- studied MST on constant-diameter networks. Das Sarma et al. [12] developed a Õ((`D)1/2)-time algorithm for computing a random walk of length `, which is faster than the trivial O(`)-time algorithm w... |

32 | A Simple and Linear Time Randomized Algorithm for Computing Sparse Spanners
- Baswana, Sen
(Show Context)
Citation Context ...oximation SSSP General Bellman&Ford [6, 22] Õ(n) exact Lenzen&Patt-Shamir [38]2 Õ(n1/2+1/2k + D) 8kdlog(k + 1)e − 1 this paper Õ(n1/2D1/4 + D) 1 + o(1) (= Õ(n3/4 + D)) Fully-Connected Baswana&Sen =-=[5]-=- Õ(n1/k) 2k − 1 this paper Õ(n1/2) exact APSP General Trivial O(m) exact Lenzen&Patt-Shamir [38] Õ(n) O(1) this paper Õ(n) 1 + o(1) Fully-Connected Baswana&Sen [5] Õ(n1/k) 2k − 1 this paper Õ(n1... |

31 |
Provably good routing in graphs: regular arrays
- Raghavan, Thompson
- 1985
(Show Context)
Citation Context ...inted out to us by a STOC 2014 reviewer that this technique is similar to the one used in the PRAM algorithm of KleinSairam [33] which was originally proposed for VLSI routing by Raghavan and Thomson =-=[51]-=-. The main difference between this and our weight approximation technique is that we always round edge weights up while the previous technique has to round the weights up and down randomly (with some ... |

24 | Sub-linear distributed algorithms for sparse certificates and biconnected components
- Thurimella
- 1997
(Show Context)
Citation Context ...rithm that can find an MST in Õ(n1/2 + D) time [24, 35], and we know that this running time is tight [48]. This algorithm serves as a building block for several other sublinear-time algorithms (e.g. =-=[55, 50, 25]-=-). It is also natural to ask whether we can further improve the running time of existing graph algorithms by mean of approximation, e.g., if we allow an algorithm to output a spanning tree that is alm... |

23 | Distributed approximate matching
- Lotker, Patt-Shamir, et al.
(Show Context)
Citation Context ...ignment. We refer to network G with weight assignment w as the weighted network, denoted by (G,w). The weight w(uv) of each edge uv is known only to u and v. As commonly done in the literature (e.g., =-=[32, 42, 35, 24, 25]-=-), we will assume that the maximum weight is poly(n); so, each edge weight can be sent through an edge (link) in one round.1 There are several measures to analyze the performance of such algorithms, a... |

21 | Distributed approximation: a survey.
- Elkin
- 2004
(Show Context)
Citation Context ...k’s diameter in O(n3/4 +D) time [30, 47]. The question whether distributed approximation algorithms can help improving the time complexity of computing shortest paths was raised a decade ago by Elkin =-=[15]-=-. It is surprising that, despite so much progress on other problems in the last decade, the problem of computing shortest paths is still widely open, especially when we want a small approximation guar... |

19 | Networks cannot compute their diameter in sublinear time.
- Frischknecht, Holzer, et al.
- 2012
(Show Context)
Citation Context ...Case. SSSP and APSP are essentially well-understood in the unweighted case. SSSP can be trivially solved in O(D) time using a breadth-first search tree [46, 43]. Frischknecht, Holzer, and Wattenhofer =-=[23, 30]-=- show a (surprising) lower bound of Ω(n/ log n) for computing the diameter of unweighted networks, which implies a lower bound for solving unweighted APSP. This lower bound holds even for (3/2− )-app... |

18 |
Towards robust and efficient computation in dynamic peer-to-peer networks. Arxiv preprint
- Augustine, Pandurangan, et al.
- 1108
(Show Context)
Citation Context ...e.g., ad hoc networks and peer-to-peer networks) D is small (usually Õ(1)). (In some networks, an even stronger property also holds; e.g., a peer-to-peer network is usually assumed to be an expander =-=[4]-=-.) It is thus of a special interest to develop an algorithm in this setting. For example, [41] studied MST on constant-diameter networks. Das Sarma et al. [12] developed a Õ((`D)1/2)-time algorithm f... |

17 |
Mathematics for the analysis of algorithms,
- Greene, Knuth
- 1990
(Show Context)
Citation Context ... selected nodes |V (G′)| of size α will “hit” a simple path of length at least cn log n/|V (G′), for some constant c, with high probability. To the best of our knowledge, this fact was first shown in =-=[26]-=- and has been used many times in dynamic graph algorithms (e.g. [13] and references there in). The following fact appears as Theorem 36.5 in [13] (attributed to [56]). Fact 4.4 (Ullman and Yannakakis ... |

17 | R.: Tight Bounds for Parallel Randomized Load Balancing: Extended Abstract
- Lenzen, Wattenhofer
- 2011
(Show Context)
Citation Context .... The case of fully-connected networks is an extreme case where D = 1. This special setting captures, e.g., overlay and peer-to-peer networks, and has received a considerable attention recently (e.g. =-=[40, 39, 45, 37, 14, 7]-=-). Obviously, this model gives more power to algorithms since every node can directly communicate with all other nodes; for example, MST can be constructed in O(log log n) time [40], as opposed to the... |

16 | Optimal distributed all pairs shortest paths and applications.
- Holzer, Wattenhofer
- 2012
(Show Context)
Citation Context ...roblems in sublinear time by sacrifying a small approximation factor; e.g., we can (2 + )-approximate mincut in Õ(n1/2 + D) time [25] and (3/2)-approximate the network’s diameter in O(n3/4 +D) time =-=[30, 47]-=-. The question whether distributed approximation algorithms can help improving the time complexity of computing shortest paths was raised a decade ago by Elkin [15]. It is surprising that, despite so ... |

12 | An all pair shortest paths distributed algorithm using 2n2 messages
- Haldar
- 1997
(Show Context)
Citation Context ... and convergence. When faced with the bandwidth constraint, the time complexities of these algorithms become higher than the trivial O(m)-time algorithm; e.g., BellmanFord algorithm and algorithms in =-=[1, 27, 2]-=- require Ω(n2) time. To the best of our knowledge, there is still no exact distributed algorithm for APSP that is faster than the trivial O(m)-time algorithm6, except for the special case of BHC netwo... |

12 |
A parallel randomized approximation scheme for shortest paths
- Klein, Sairam
- 1992
(Show Context)
Citation Context ...s an important role in his recent breakthrough [8]. Also, it was recently pointed out to us by a STOC 2014 reviewer that this technique is similar to the one used in the PRAM algorithm of KleinSairam =-=[33]-=- which was originally proposed for VLSI routing by Raghavan and Thomson [51]. The main difference between this and our weight approximation technique is that we always round edge weights up while the ... |

12 |
Shortest-path queries in static networks.
- Sommer
- 2014
(Show Context)
Citation Context ... show that solving the single source shortest path problem on a network (G,w) can be reduced to the same problem on a certain type of an overlay network, usually known as a landmark or skeleton (e.g. =-=[53, 38]-=-). In general, an overlay network G′ is a virtual network of nodes and logical links that is built on top of an underlying real network G; i.e., V (G′) ⊂ V (G) and an edge in G′ (a “virtual edge”) cor... |

11 | Minimum-weight spanning tree construction in O(log log n) communication rounds
- Lotker, Patt-Shamir, et al.
- 2005
(Show Context)
Citation Context .... The case of fully-connected networks is an extreme case where D = 1. This special setting captures, e.g., overlay and peer-to-peer networks, and has received a considerable attention recently (e.g. =-=[40, 39, 45, 37, 14, 7]-=-). Obviously, this model gives more power to algorithms since every node can directly communicate with all other nodes; for example, MST can be constructed in O(log log n) time [40], as opposed to the... |

10 | Optimal deterministic routing and sorting on the congested clique
- Lenzen
(Show Context)
Citation Context .... The case of fully-connected networks is an extreme case where D = 1. This special setting captures, e.g., overlay and peer-to-peer networks, and has received a considerable attention recently (e.g. =-=[40, 39, 45, 37, 14, 7]-=-). Obviously, this model gives more power to algorithms since every node can directly communicate with all other nodes; for example, MST can be constructed in O(log log n) time [40], as opposed to the... |

10 |
Vitaly Rubinovich. A Near-Tight Lower Bound on the Time Complexity of Distributed Minimum-Weight Spanning Tree Construction
- Peleg
- 2000
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Citation Context |

10 | Distributed algorithms for ultrasparse spanners and linear size skeletons
- Pettie
(Show Context)
Citation Context ... setting k = log n/ log log n, we have an O(n log n)-time O(log n/ log logn)-approximation algorithm (we need O(k2) to construct a spanner and O(kn1+1/k) = O(n log n) to aggregate it). The spanner of =-=[49]-=- can also be used to get a linear-time (2O(log ∗ n) log n)-approximation algorithm in the unweighted case. In general, it is not clear how to use graph sparsification for computing shortest paths sinc... |

9 | Fast routing table construction using small messages: extended abstract
- Lenzen, Patt-Shamir
- 2013
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8 | Distributed minimum cut approximation
- Ghaffari, Kuhn
- 2013
(Show Context)
Citation Context ...rithm that can find an MST in Õ(n1/2 + D) time [24, 35], and we know that this running time is tight [48]. This algorithm serves as a building block for several other sublinear-time algorithms (e.g. =-=[55, 50, 25]-=-). It is also natural to ask whether we can further improve the running time of existing graph algorithms by mean of approximation, e.g., if we allow an algorithm to output a spanning tree that is alm... |

8 | The round complexity of distributed sorting: extended abstract
- Patt-Shamir, Teplitsky
- 2011
(Show Context)
Citation Context |

7 | Danupon Nanongkai, Gopal Pandurangan, and Prasad Tetali. Efficient distributed random walks with applications - Sarma - 2010 |

7 |
tri again”: Finding triangles and small subgraphs in a distributed setting - (extended abstract).
- ”tri
- 2012
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5 |
A fast distributed shortest path algorithm for a class of hierarchically clustered data network
- Antonio, Huang, et al.
(Show Context)
Citation Context ... O(m)-time algorithm6, except for the special case of BHC network, whose topology is structured as a balanced hierarchy of clusters. In this special case, the problem can be solved in O(n log n)-time =-=[3]-=-. For the related problem of computing network’s diameter and girth, many results are known in the unweighted case but none is previously known for the weighted case. Peleg, Roditty, and Tal [47] show... |

5 | Super-fast distributed algorithms for metric facility location
- Berns, Hegeman, et al.
- 2012
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5 |
Dahlia Malkhi, Gopal Pandurangan, and Kunal Talwar. Efficient distributed approximation algorithms via probabilistic tree embeddings
- Khan, Kuhn
(Show Context)
Citation Context ...e Definition 3.8 for details). It naturally arises in the analysis of several algorithms. For example, Bellman-Ford algorithm [6, 22] can be analyzed to have O(SPDiam(G,w)) time for SSSP. Khan et al. =-=[31]-=- gives a Õ(n ·SPDiam(G,w))-time O(log n)-approximation algorithm via metric tree embeddings [21]. We can also construct Thorup-Zwick distance sketches [54] of size O(kn1/k) and stretch 2k − 1 in Õ(k... |

5 |
Elad Tal. Distributed algorithms for network diameter and girth
- Peleg, Roditty
- 2012
(Show Context)
Citation Context ...roblems in sublinear time by sacrifying a small approximation factor; e.g., we can (2 + )-approximate mincut in Õ(n1/2 + D) time [25] and (3/2)-approximate the network’s diameter in O(n3/4 +D) time =-=[30, 47]-=-. The question whether distributed approximation algorithms can help improving the time complexity of computing shortest paths was raised a decade ago by Elkin [15]. It is surprising that, despite so ... |

4 |
Maintaining shortest paths under deletions in weighted directed graphs
- Bernstein
(Show Context)
Citation Context ...ynamic data structure context, Bernstein has independently used a similar weight rounding technique to construct a bounded-hop data structure, which plays an important role in his recent breakthrough =-=[8]-=-. Also, it was recently pointed out to us by a STOC 2014 reviewer that this technique is similar to the one used in the PRAM algorithm of KleinSairam [33] which was originally proposed for VLSI routin... |

4 |
Efficient algorithms for constructing (1+epsilon, beta)-spanners in the distributed and streaming models
- Elkin, Zhang
(Show Context)
Citation Context ...s one of the main motivations to study distributed algorithms for graph sparsification. These algorithms5 have either super-linear time or large approximation guarantees. For example, Elkin and Zhang =-=[19]-=- present an algorithm for the unweighted case based on a sparse spanner that takes (very roughly) O(nξ) time and gives (1 + )-approximate solution, for small constants ξ and . The algorithm is also ... |

3 |
Some shortest path algorithms with decentralized information and communication requirements. Automatic Control
- Abram, Rhodes
- 1982
(Show Context)
Citation Context ... and convergence. When faced with the bandwidth constraint, the time complexities of these algorithms become higher than the trivial O(m)-time algorithm; e.g., BellmanFord algorithm and algorithms in =-=[1, 27, 2]-=- require Ω(n2) time. To the best of our knowledge, there is still no exact distributed algorithm for APSP that is faster than the trivial O(m)-time algorithm6, except for the special case of BHC netwo... |

3 |
Hartmut Klauck, Danupon Nanongkai, and Gopal Pandurangan. Can quantum communication speed up distributed computation?
- Elkin
- 2014
(Show Context)
Citation Context ... the trivial O(`)-time algorithm when D is small. In the same spirit, our algorithm is faster than previous algorithms. Moreover, in this case our running time matches the lower bound of Ω̃(n1/2 + D) =-=[11, 18]-=-, which holds even for any algorithm with poly(n) approximation ratio; thus, our result settles the status of SSSP for this case. Additionally, since the same lower bound also holds in the quantum set... |

3 | Notions of connectivity in overlay networks
- Emek, Fraigniaud, et al.
- 2012
(Show Context)
Citation Context ... is a virtual network of nodes and logical links that is built on top of an underlying real network G; i.e., V (G′) ⊂ V (G) and an edge in G′ (a “virtual edge”) corresponds to a path in G (see, e.g., =-=[20]-=-). Its implementation is usually abstracted as a routing scheme that maps virtual edges to underlying routes. However, for the purpose of this paper, we do not need a routing scheme but will need the ... |

3 |
and Mihalis Yannakakis. High-probability parallel transitive-closure algorithms
- Ullman
- 1991
(Show Context)
Citation Context ...e k ≥ n log n/α. Let i1 < i2 < . . . < it be such that vi1 , . . . , vit are nodes in P ∩V (G′). Let i0 = 0 (i.e., vi0 = u). We note the following simple fact, which is very easy and well-known (e.g. =-=[56]-=-). We provide its proof here only for completeness. Lemma 4.3 (Bound on the number of hops between two landmarks in a path). For any j, ij − ij−1 ≤ n log n/α, with probability at least 1− 2−βn, for so... |

2 |
and Gopal Pandurangan. A fast distributed approximation algorithm for minimum spanning trees
- Khan
(Show Context)
Citation Context ...ignment. We refer to network G with weight assignment w as the weighted network, denoted by (G,w). The weight w(uv) of each edge uv is known only to u and v. As commonly done in the literature (e.g., =-=[32, 42, 35, 24, 25]-=-), we will assume that the maximum weight is poly(n); so, each edge weight can be sent through an edge (link) in one round.1 There are several measures to analyze the performance of such algorithms, a... |

1 |
Gopal Pandurangan. Efficient computation of distance sketches in distributed networks
- Sarma, Dinitz
- 2012
(Show Context)
Citation Context ...))-time O(log n)-approximation algorithm via metric tree embeddings [21]. We can also construct Thorup-Zwick distance sketches [54] of size O(kn1/k) and stretch 2k − 1 in Õ(kn1/k · SPDiam(G,w)) time =-=[10]-=-. Since SPDiam(G,w) can be as large as n, these algorithms do not give any improvement to previous algorithms when analyzed in terms of n and D. One crucial component of our algorithms involves reduci... |

1 |
Distance Computation, Information Dissemination, and Wireless Capacity in Networks
- Holzer
- 2013
(Show Context)
Citation Context ...ered is when nodes can talk to any other node in one time unit. This can be thought of as a special case of APSP on fully-connected networks where edge weights are either 1 or ∞. In this case, Holzer =-=[29]-=- shows that SSSP can be solved in Õ(n1/2) time3. Name-Dependent Routing Scheme. For the weighted SSSP and APSP on general networks, the best known results follow from the recent algorithm for computi... |

1 |
Tight bounds for distributed minimum-weight spanning tree verification
- Kor, Korman, et al.
(Show Context)
Citation Context ...This question has generated a research in the direction of distributed approximation algorithms which has become fruitful in the recent years. On the negative side, Das Sarma et al. [11] (building on =-=[48, 17, 34]-=-) show that MST and a dozen other problems, including mincut and computing the distance between two nodes, cannot be computed faster than Õ(n1/2 +D) in the synchronous CONGEST model even when we allo... |

1 |
Ramakrishna Thurimella. Fast computation of small cuts via cycle space sampling
- Pritchard
- 2011
(Show Context)
Citation Context ...rithm that can find an MST in Õ(n1/2 + D) time [24, 35], and we know that this running time is tight [48]. This algorithm serves as a building block for several other sublinear-time algorithms (e.g. =-=[55, 50, 25]-=-). It is also natural to ask whether we can further improve the running time of existing graph algorithms by mean of approximation, e.g., if we allow an algorithm to output a spanning tree that is alm... |