E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210--236, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[1] Hence the algorithm of Althofer et al. 2] computes a (2k 1) spanner of size O(n ) which is indeed optimal based on the lower bound mentioned earlier. However, the best known implementation of the algorithm of Althofer et al. 2] for (2k 1) spanner has a running time of O(mn ) Cohen [8] presented a randomized algorithm with O(kmn ) expected running time for computing a spanner of O(kn ) size and slightly larger stretch (2k ) Thorup and Zwick [15] improved the result of Cohen [8] and presented a randomized algorithm for computing a (2k 1) spanner of optimal size in ....

.... algorithm of Althofer et al. 2] for (2k 1) spanner has a running time of O(mn ) Cohen [8] presented a randomized algorithm with O(kmn ) expected running time for computing a spanner of O(kn ) size and slightly larger stretch (2k ) Thorup and Zwick [15] improved the result of Cohen [8], and presented a randomized algorithm for computing a (2k 1) spanner of optimal size in O(kmn ) expected time. All these existing algorithms for computing spanners in undirected weighted graphs require computation of shortest distance information between many pairs of vertices [2] or ....

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput., 28:210--236, 1998.

....of the cheapest path between u; v 2 V . Then, a t spanner of G is a subgraph G (V; E ) where E E and 8u; v 2 V; dG (u; v) t dG (u; v) It should be clear that dG (u; v) dG (u; v) also holds because G is a subgraph of G. Several algorithms to build t spanners are known [5, 7], and we have proposed some speci c construction algorithms for our present metric space application [8] complete G, metric costs, and t 2) The naive construction algorithm is O(n 4 ) time. On euclidean spaces, this drops to O(n log n) Our construction complexity [8] for general metric ....

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. on Computing, 28:210-236, 1998.

....uses O(n log n) space. 1 Introduction The shortest path (SP) problem for weighted graphs with n vertices and m edges is a fundamental problem for which ecient solutions can now be found in any standard algorithms text. The approximation version of this problem has been studied extensively, see [1, 11, 14]. In numerous algorithms, query versions frequently appear as subroutines. In such a query, we are given two vertices and have to compute or approximate the shortest path between them. The latest in a series of results for undirected weighted graphs is by Thorup and Zwick [23] their algorithm ....

E. Cohen. Fast algorithms for constructing t- spanners and paths with stretch t. SIAM Journal on Computing, 28:210-236, 1998.

....constant, such that 0 3. In O(n p n ) time, we can compute a (1; 1 ) approximate stretch factor of G. 5.4. General graphs. In our nal application, we let G be the general class of connected Euclidean graphs. Let G 2 G be any graph with n vertices and m edges. Note that m n 1. Cohen [9] has shown that for any integer 1, and any constant such that 0 1=2, any sequence of (2 (1 ) approximate shortest path queries can be answered in expected time O( m k)n 1= log 2 n) where k is the number of queries. Applying Theorem 4.2 gives the following result. ....

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput., 28:210-236, 1998.

....time bound for Boolean matrix multiplication, obtaining a combinatorial O(n 3 Gammaffl ) time algorithm for the APSP problem would be a major breakthrough. Here we obtain such combinatorial algorithms for the all pairs almost shortest paths (APASP) problem. Awerbuch et al. ABCP93] and Cohen [Coh93] considered the problem of finding stretch t all pairs paths, where t is some fixed constant and a path is of stretch t if its length is at most t times the distance between its endpoints. Cohen [Coh93] improving the results of Awerbuch et al. ABCP93] obtains, for example, an O(n 5=2 ) ....

....for the all pairs almost shortest paths (APASP) problem. Awerbuch et al. ABCP93] and Cohen [Coh93] considered the problem of finding stretch t all pairs paths, where t is some fixed constant and a path is of stretch t if its length is at most t times the distance between its endpoints. Cohen [Coh93] improving the results of Awerbuch et al. ABCP93] obtains, for example, an O(n 5=2 ) time algorithm for finding stretch 4 ffl paths and distances in weighted undirected graphs, for any ffl 0 (all weights from now on are assumed to be positive) She also exhibits a tradeoff between the ....

[Article contains additional citation context not shown here]

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t (extended abstract). In Proceedings of the 34rd Annual IEEE Symposium on Foundations of Computer Science, Palo Alto, California, pages 648--658, 1993.

....constant, such that 0 3. In O(n p n ) time, we can compute a (1; 1 ) approximate stretch factor of G. 5.4 General graphs In our nal application, we let C be the general class of connected Euclidean graphs. Let G 2 C be any graph with n vertices and m edges. Note that m n 1. Cohen [9] has shown that for any integer 1, and any constant such that 0 1=2, any sequence of (2 (1 ) approximate shortest path queries can be answered in expected time O( m k)n 1= log 2 n) where k is the number of queries. Applying Theorem 3 gives the following result. ....

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput., 28:210-236, 1998.

....geometry graphs and chordal graphs, much efforts has been taken in recent several works [1,6,9,12] The spanner concept has a number of application backgrounds. For example, the sparse spanner of unweighted graphs is used in distributed computing and communication network design [2 5,13 14] Cohen [7] once suggested a randomized parallel algorithm for finding a t spanner with size O(n 1 2 ffl t ) on a weighted graph which needs O( Wmax Wmin fi 2 log 2 n) expected time with O(n 1=fi mfi log 2 n) work on an EREW PRAM, where fi = t= 2 ffl=2) where wt(e) is the weight of edge ....

....finding the shortest path between two vertices [12] Algorithm 3 can be implemented in O(mm mn log n) O(n 4 ) time. The size of G 0 is O(n 1 2= t Gamma1) and the weight is less than ( n t Gamma1 1)wt(MST ) 1] By a considerably improved analysis of this algorithm, Chandra et al. [7] show that the running time of this algorithm is O(n 3 4= t Gamma1) and the weight of G 0 is no more than O(n 2 ffl t Gamma1 wt(MST ) where ffl 0 is an any arbitrarily small constant. The naive parallel version of Algorithm 3 requires O(m Rn log n) time if O(n 2 ) processors are ....

[Article contains additional citation context not shown here]

E. Cohen, Fast algorithms for constructing t-spanners and paths with t, Proc. 34th IEEE Sympo. on Founda. of Comput. Sci., 1993, 648-658.

.... [AP92] deadlock prevention [AKP91] bandwidth management in high speed networks [ACG 90] and database management [BFR92] as well as for classical problems in sequential computing (such as finding small edge cuts in planar graphs [Rao92] and approximate all pairs shortest paths [ABCP93, Coh93] In most of these applications, sparse neighborhood covers yield a polylogarithmic overhead solution to the problem. Thus, in a sense, the impact of efficient sparse neighborhood cover algorithms on distributed network algorithms is analogous to the impact of efficient data structures (like ....

Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. In Proc. 34rd IEEE Symp. on Found. of Comp. Science. IEEE, November 1993. to appear.

....approximate shortest paths. We then use this in conjunction with a scaling technique to find exact shortest paths. In Chapter 4 we use this random sampling technique to construct graphs that approximate the shortest path distances by paths containing a polylogarithmic number of edges. Cohen [13,16] has also used this random sampling technique to find approximate shortest paths in undirected graphs. 1.6 Path detour: A new technique for finding and representing paths that cross a separating path In this section we describe a new technique called path detour that is the basis for our dynamic ....

....than n 1:5 ) Put another way, if the graph is sparse (m = O(n) and one only has n processors available, our algorithm would use the n processors to achieve a speed up of about p n while Spencer s algorithm would achieve a speed up of about n 1=4 . Subsequent work In recent work Cohen [13,16] has given an algorithm to find 1 ffl approximate shortest paths which among other things uses the limited search technique presented in Section 3.3. Her algorithm works in polylogarithmic time and is nearly work optimal. However, her algorithm does not generalize to directed graphs; and there ....

E. Cohen, "Fast algorithms for constructing t-spanners and paths with stretch t," Proc. 34th Annual IEEE Symposium on Foundations of Computer Science (1993).

...., where jd(u; v)j is the length of the shortest u Gamma v path. The approximate all pairs shortest path problem involves a tradeoff of stretch against time short paths with stretch bounded by a constant are computed in time less than it would take to compute exact all pairs shortest paths (see [1, 2, 6, 8, 9, 10]) The compact routing problem considers instead a tradeoff of stretch for space, in the setting where each node locally stores its own routing tables. The stretch of a compact routing algorithm is defined as the maximum stretch over the routes for all pairs of nodes in the network. Clearly if ....

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. on Comput., 28(1):210--236, 1999.

.... a consequence of this paper and the algorithm in [23] down from O(n 2 ) time and space using [11, 23] sequential algorithms for approximating k pairs shortest paths (see below and Section 6 for details) weight and distance preserving graph spanners with O (E) running time and space (see [6, 15], down from O(nE) time using [14] or a combination of [6] and [11] and a distributed fromscratch network synchronizer construction with O (1) messages and space per edge (down from Omega Gamma n) time and space using [10, 11] We also comment that the worst case running times of our new ....

....nodes. Moreover, average degree covers are not sufficient for an efficient solution to the k pairs shortest paths problem, because the imbalance (high and low overlaps) cannot be amortized over many queries. Unlike the algorithms in this paper, other con structions of average degree covers [1, 15] cannot be run for multiple iterations to produce a small maximum degree cover. However, as Afek et al. 1] and Cohen [15] have pointed out, in the special case of the all pairs shortest paths application, an average degree cover suffices and leads to a better tradeoff between running time and ....

[Article contains additional citation context not shown here]

E. Cohen. Fast algorithms for constructing t- spanners and paths with stretch t. these proceedings, 1993.

....number of processors. Our (deterministic) NC algorithm runs in O(log 5 (n) time and uses O(n 2 ) processors. The ( d) decomposition problem is related to the sparse t neighborhood cover problem [8] which has applications to sequential approximation algorithms for all pairs shortest paths [5, 9] and finding small edge cuts in planar graphs [15] We believe the NC algorithm in this paper will also have applications to parallel graph algorithms. 2 The Algorithm In this section, we construct a deterministic NC algorithm for low diameter graph decomposition. This is achieved by modifying an ....

Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. In Proc. 34rd IEEE Symp. on Foundations of Computer Science. IEEE, November 1993. to appear.

.... problem is the so called light, approximate shortest path trees where the goal is to find a tree of small cost that closely approximates the shortest distances from a given single source [20, 11] A more closely related line of research is the extensively studied area of graph spanners [1, 7, 12, 13, 14]: Problem: Mincost d Spanner Instance: A graph G = V; E) with cost function c : E 7 R . Goal: Find a minimum cost set E 0 E of edges such that every pair of vertices is at most a factor d further apart in G 0 = V; E 0 ) than it was in G. Any feasible solution to this problem ....

E. Cohen. Fast Algorithms for constructing t-spanners and paths with stretch t (extended abstract). In Proc. 34th Symp. on Foundation of Computer Science, pp. 648--658, 1993.

....may be longer that the shortest path between the nodes. The compact routing problem instead considers a tradeoff of route lengths for space, in the setting where each node locally stores its own routing tables. Much recent work has been done on fast constructions of approximate shortest paths (see [1, 2, 5, 7, 8, 10]) and on compact routing schemes with sublinear maximum or average space (see [3, 4, 6, 9, 11, 13, 14] for undirected weighted and unweighted graphs. However, there have been no previous schemes that achieved any savings of time or space over the exact schemes for either weighted or unweighted ....

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. on Comput., 28(1):210--236, 1999.

....network design [3,4,5,6,15,16] Peleg and Upfal [16] Awerbuch Bar Noy, Linial and Peleg [4] and Awerbuch and Peleg [5] use it to design efficient routing schemes in distributed networks. Peleg and Ullman [15] also point out that the t spanner is useful to design a synchronizer [3] Whereas Cohen [8] suggests a randomized parallel algorithm for finding a t spanner with size O(n 1 2 ffl t ) on a weighted graph which needs O( Wmax Wmin fi 2 log 2 n) expected time with O(n 1=fi mfi log 2 n) work on an EREW PRAM, where fi = t= 2 ffl=2) where wt(e) is the weight of edge e, W ....

....because U blog tc Gamma1 is a maximal independent set of graph G blog tc Gamma1 . Contradiction. 2 4.2 The algorithm for weighted graphs Though the algorithm in [1] is very simple and efficient, it seems to be inherently sequential. Based on the neighborhood cover of Awerbuch and Peleg [6] Cohen [8] introduces the pairwise cover concept, and presents an efficient sequential algorithm and a randomized parallel algorithm for finding a sparse t spanner in weighted graphs. The basic idea of Cohen s algorithm is to employ a logarithmic number of pairwise covers for different values of W to ....

[Article contains additional citation context not shown here]

E. Cohen, Fast algorithms for constructing t-spanners and paths with t, Proc. 34th IEEE Annual Sympo. on Foundations of Computer Science, 1993, 648-658.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210--236, 1999.

....in directed graphs is again as hard as Boolean matrix multiplication. It is not surprising therefore that the approximation methods used here, and elsewhere, work only for undirected graphs. Algorithms for finding small stretch paths were obtained by Awerbuch, Berger, Cowen and Peleg [8] Cohen [10], and Dor, Halperin and Zwick [15] Awerbuch 1 2 et al. 8] presented an O(mn 64=t kn 32=t ) time algorithm for finding stretch t paths between k specified pairs of vertices. If paths between all pairs of vertices are required, the running time of the algorithm becomes O(mn 64=t n ....

....and Zwick [15] Awerbuch 1 2 et al. 8] presented an O(mn 64=t kn 32=t ) time algorithm for finding stretch t paths between k specified pairs of vertices. If paths between all pairs of vertices are required, the running time of the algorithm becomes O(mn 64=t n 2 32=t ) Cohen [10] improved this result and obtained an O( m k)n 2=t ) time algorithm for obtaining stretch t ffl paths between k specified pairs of vertices, where t is even and ffl 0 is arbitrarily small. This becomes O(n 2 2=t ) if all pairs stretch t ffl paths are wanted. In particular, Cohen ....

[Article contains additional citation context not shown here]

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t (extended abstract). In Proceedings of the 34rd Annual IEEE Symposium on Foundations of Computer Science, Palo Alto, California, pages 648--658, 1993.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210--236, 1999.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210--236, 1999.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Computing, 28:210--236, 1999.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210-236, 1999.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210--236, 1999.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. on Computing, 28:210-236, 1998.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t (extended abstract). In Proceedings of the 34rd Annual IEEE Symposium on Foundations of Computer Science, Palo Alto, California, pages 648--658, 1993.

No context found.

E. Cohen. Fast algorithms for constructing t- spanners and paths with stretch t. SIAM Journal on Computing, 28:210-236, 1999. A preliminary version appeares at the proceedings of FOCS'93.

No context found.

E. Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28:210-236, 1998.

No context found.

E. Cohen, "Fast algorithms for constructing t-spanners and paths with stretch t," 34th IEEE Symposium on Foundations of Computer Science (1993).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC