D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir, "Random walks on weighted graphs and applications to on-line algorithms," J. Assoc. Computing Machinery, vol. 40, no. 3, pp. 421--453, July 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....which are worth investigating for their own interest. Finally, it is reasonable to expect that insights gained through the analysis of Harmonic will help with resolving the question whether there exists a memoryless k competitive randomized algorithm for k servers in arbitrary metric spaces (see [CDRS93] In this paper we prove that Harmonic is 6 competitive for k = 3. This matches the lower bound, and it provides further evidence for the conjectured 2 k(k 1) competitiveness of Harmonic. Our approach is based on the random walk techniques and their relationship to the electrical network ....

Don Coppersmith, Peter G. Doyle, Prabhakar Raghavan, and Marc Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40:421-453, 1993.

....information, but where the aim is to have more knowledge about global properties. This is very useful in computation, where limited resources are used to determine global information. For example, random walks have been used in many applications in computer science, including on line algorithms [9], space efficient algorithms for undirected connectivity [7] approximation algorithms [10, 14] generation of random spanning trees [2, 6, 16] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 18] Frequently, we ask ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40(3):421--453, July 1993.

....II . Laboratoire associ e au CNRS 1 Introduction Random walks have been studied extensively, and have many applications such as generation of random spanning trees[1, 2, 12] token management schemes[8, 11] effective resistance of electrical networks[3, 10, 5] and on line algorithms[4]. In this paper, we consider a connected simple undirected graph G = V; E) together with a positive real valued map w over E, w(e) being called the weight of e. A discrete time random walk (or Markov chain) on G is defined as follows. At each step, a particle, located on a vertex x, moves to a ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40(3):421--453, July 1993.

....for all w, 2 is a lower bound on the competitive ratio of any deterministic on line layered graph traversal algorithm. Thus arbitrary layered graphs are much harder to traverse than those consisting of disjoint paths. Randomized on line algorithms are addressed in several papers including [BLS, RS, CDRS, FKLMSY, BBKTW, KRR]. An oblivious adversary is one who constructs the sequence of events in advance and deals with the sequence optimally. For this adversary model [BLS] and [FKLMSY] give examples where randomization can improve the competitive ratio exponentially. This adversary models a world in which the on line ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random Walks on Weighted Graphs and Applications to On-Line Algorithms. Journal of the ACM, 40, 1993.

....the cost of moving to a vertex from anywhere is the same) 4] a line [4] or a tree [5] Fiat et al. 7] showed that there is an algorithm for the k server problem with a competitive factor that depends only on k. There has also been work on memoryless randomized algorithms for server problems [1, 6, 14]. These algorithms keep no information between requests except the server locations. The randomized algorithm of Coppersmith et al. 6] is k competitive for a large class of graphs. In the uniform k server problem the cost of moving a server from any vertex to any other is one. The paging problem ....

....for the k server problem with a competitive factor that depends only on k. There has also been work on memoryless randomized algorithms for server problems [1, 6, 14] These algorithms keep no information between requests except the server locations. The randomized algorithm of Coppersmith et al. [6] is k competitive for a large class of graphs. In the uniform k server problem the cost of moving a server from any vertex to any other is one. The paging problem is isomorphic to the uniform k server problem. The correspondence between the two problems is as follows: the pages of address space ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 369--378, Baltimore, May 1990. Revised version available as an IBM Research Report.

.... deterministic algorithm for the circle was presented in [15] 3 One of the problems with the known competitive algorithms for the k server problem is that they are not space efficient (the algorithm proved 2k Gamma 1 competitive in this paper is no exception) In order to address this problem, [13] considered memoryless randomized algorithms and showed a competitive ratio of k for the special class of resistive metric spaces. By derandomization, this results in a O(k 2 ) deterministic competitive ratio for resistive or approximately resistive metric spaces (one of them is the circle) ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the Association for Computing Machinery, 40(3):421--53, July 1993.

....stationary chains and observe that tours in reversed time just interleave tours in forward time, so by ergodicity their asymptotic rates are equal. Tetali [31] shows that the cyclic tour property implies reversibility. Cat and mouse game 1 is treated more opaquely in Coppersmith et al. [13], whose deeper results are discussed at yyy. Underlying the use of the optional sampling theorem in game 2 is a general result about optimal stopping, but it s much easier to prove what we want here than to appeal to general theory. Several algorithmic variations on Proposition 2 are discussed in ....

.... [10] One can combine the commute formula with the general identities of Chapter 2 to obtain numerous identities relating mean hitting times and resistances, some of which are given (using bare hands proofs instead) in Tetali [30] The connection between Foster s theorem and Lemma 6 was noted in [13]. Section 4. The spectral theory is of course classical. In devising a symmetric matrix one could use i p ij or p ij Gamma1 j instead of 1=2 i p ij Gamma1=2 j there doesn t seem any systematic advantage to a particular choice. I learned the eigentime identity from Andrei Broder who ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. In Proc. 22nd ACM Symp. Theory of Computing, pages 369--378, 1990.

....weaker result was independently obtained by Koutsoupias and Taylor [20] This contrasts with the non weighted case, for which memoryless algorithms exist for any k. For example, the harmonic algorithm is competitive for any k [5, 7] and, for k = 2, a memoryless 2 competitive algorithm is known [9, 14]. Last, we propose a version of the problem in which a request is specified by two points, both of which must be covered by the servers, and the algorithm must decide which server to move to which point. For this version, we show a 9 competitive algorithm and we prove that no better ratio is ....

D. Coppersmith, P. G. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40:421--453, 1993.

....II . y Laboratoire associ e au CNRS 1 Introduction Random walks have been studied extensively, and have many applications such as generation of random spanning trees[1, 2, 12] token management schemes[8, 11] effective resistance of electrical networks[3, 10, 5] and on line algorithms[4]. In this paper, we consider a connected simple undirected graph G = V; E) together with a positive real valued map w over E, w(e) being called the weight of e. A discrete time random walk (or Markov chain) on G is defined as follows. At each step, a particle, located on a vertex x, moves to a ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40(3):421--453, July 1993.

....times for aggregate of sizes from 2 m Gamma1 1 to 2 m , if m is small. If m is large, the times in the entry is the average of times for aggregate of sizes from 2 m Gamma c to 2 m for some constant c. From the two tables, we can construct the cost matrices C s. For example, C ff 1;1 [2 5 ] = 3:51 Theta 10 Gamma6 C ff 1;2 [2 5 ] 2:90 Theta 10 Gamma6 1:09 Theta 10 Gamma3 6:47 Theta 10 Gamma5 C ff 2;1 [2 5 ] 1:81 Theta 10 Gamma4 1:46 Theta 10 Gamma5 3:51 Theta 10 Gamma6 C ff 2;2 [2 5 ] 6:47 Theta 10 Gamma5 sequence ....

.... 1 to 2 m , if m is small. If m is large, the times in the entry is the average of times for aggregate of sizes from 2 m Gamma c to 2 m for some constant c. From the two tables, we can construct the cost matrices C s. For example, C ff 1;1 [2 5 ] 3:51 Theta 10 Gamma6 C ff 1;2 [2 5 ] = 2:90 Theta 10 Gamma6 1:09 Theta 10 Gamma3 6:47 Theta 10 Gamma5 C ff 2;1 [2 5 ] 1:81 Theta 10 Gamma4 1:46 Theta 10 Gamma5 3:51 Theta 10 Gamma6 C ff 2;2 [2 5 ] 6:47 Theta 10 Gamma5 sequence representation time in sec. map list greedy local (ff ....

[Article contains additional citation context not shown here]

Don Coppersmith, Peter Doyle, Prabhakar Raghavan, and Marc Snir. Random walks on weighted graphs and applications to on--line algorithms. Journal of the Association for Computing Machinery, 40(3):421--453, July 1993.

....at each step the random walk moves from the current vertex to a neighbour chosen with probability proportional to the weight of traversed edge. Random walks have been studied extensively, and have numerous applications, including generation of random spanning trees [1, 6, 16, 18] online algorithms[8], space efficient algorithms for undirected connectivity[7] approximation algorithms [9, 14] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 17] The cover time of a random walk on a graph is the number of steps ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40(3):421--453, July 1993.

....for all w, 2 w 2 is a lower bound on the competitive ratio of any deterministic on line layered graph traversal algorithm. Thus arbitrary layered graphs are much harder to traverse than those consisting of disjoint paths. Randomized on line algorithms are addressed in several papers including [BLS, RS, CDRS, FKLMSY, BBKTW, KRR]. An oblivious adversary is one who constructs the sequence of events in advance and deals with the sequence optimally. For this adversary model [BLS] and [FKLMSY] give examples where randomization can improve the competitive ratio exponentially. This adversary models a world in which the on line ....

<F3.765e+05> D. Coppersmith, P. Doyle, P. Raghavan, and M.<F3.84e+05> Snir,<F3.465e+05> Random walks on weighted graphs and applications to on-line<F3.84e+05> algorithms, J. Assoc. Comput. Mach., 40 (1993), pp. 421--453.

....discovered for specific metric spaces. Specifically, k competitive deterministic on line algorithms for points on a line (Chrobak et al. CKPV] and for points on a tree (Chrobak and Larmore [CL1] Randomized on line algorithms were discovered for resistive graphs (Coppersmith et al. [CDRS]) and points on a circle ( CDRS] and Karp [Kar] A deterministic competitive k server algorithm for the circle was recently discovered (Fiat et al. FRRS] CKPV] also prove that the optimal off line k server problem is equivalent to network flow problems and thus has a polynomial time ....

.... Specifically, k competitive deterministic on line algorithms for points on a line (Chrobak et al. CKPV] and for points on a tree (Chrobak and Larmore [CL1] Randomized on line algorithms were discovered for resistive graphs (Coppersmith et al. CDRS] and points on a circle ([CDRS] and Karp [Kar] A deterministic competitive k server algorithm for the circle was recently discovered (Fiat et al. FRRS] CKPV] also prove that the optimal off line k server problem is equivalent to network flow problems and thus has a polynomial time solution. The [MMS] definition of ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random Walks on Weighted Graphs and Applications to On-line Algorithms. In Proc. of the 22nd Ann. ACM Symp. on Theory of Computing, pages 369--378, May 1990.

.... Theory, Sweden, 1988 [2] Recently, a number of papers in the same vein (with related results) have appear [18, 9, 6] We would like to thank Prabhakar Raghavan who pointed out the relation of the results cited in this paper to the area of amortized analysis of on line algorithms [5] see [8] for further references. We gratefully acknowledge the painstaking attention to detail and the useful organizational suggestions made by the anonymous referees. ....

Coppersmith, D., Doyle, P., Raghavan, P., and Snir, M.; "Random Walks on Weighted Graphs and Applications to On-Line Algorithms," IBM Research Report, IBM T. J. Watson Research Center, Yorktown Heights, 1990.

....information, but where the aim is to have more knowledge about global properties. This is very useful in computation, where limited resources are used to determine global information. For example, random walks have been used in many applications in computer science, including on line algorithms [9], space efficient algorithms for undirected connectivity [7] approximation algorithms [10, 14] generation of random spanning trees [2, 6, 16] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 18] Frequently, we ask for ....

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40(3):421--453, July 1993.

....hence repeated accesses to the aggregate will rarely need the same conversion again. 7 Related and future works The problem of automatic selection of data representations, when approached from a probabilistic framework, is closely related to the problem of random walks in weighted graphs [4] [9] [10] These works often assume the cost matrices are symmetric, and use more complicated techniques to derive tight bounds of the probabilistic schemes involved. The competitive paging problem and its probabilistic solution is related to the data representation selection problem as well [12] ....

Don Coppersmith, Peter Doyle, Prabhakar Raghavan, and Marc Snir. Random walks on weighted graphs and applications to on--line algorithms. Journal of the Association for Computing Machinery, 40(3):421--453, July 1993.

....n) bounds on the competitive ratio of any randomized on line algorithm. All the above considerations pertain to randomized algorithms working against an oblivious adversary, the weakest type of the three introduced in section 3. Against an adaptive on line adversary, Coppersmith et al. [26] proved a lower bound of 2n Gamma 1 for the competitive ratio and gave a simple, memoryless, randomized algorithm for any metric space that is (2n Gamma 1) competitive. The algorithm uses methods for the synthesis of random walks on graphs with positive real costs on the edges. This result ....

....) 4 [37] But there are metric spaces for which Harmonic cannot achieve a competitiveness ratio lower than k(k 1) 2. The most general class of metric spaces for which we know of a k competitive (optimal) algorithm against adaptive on line adversaries follows from the work of Coppersmith et al. [26]; their algorithm works in a class of metric spaces called resistive metric spaces. A lot of research efforts have been focused on finding competitive algorithms for specific values of k. The smallest non trivial value is k = 2. As mentioned above, the algorithm of Manasse et al. 54] is ....

Don Coppersmith, Peter Doyle, Prabhakar Raghavan, and Marc Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40(3):421--453, July 1993.

....the capacities are arbitrary We have a generalization of Rand that we believe is O( p n ) competitive. When the capacities are arbitrary, the main difficulty is that there can be many non disjoint augmenting paths for a new connect request. Our generalization of Rand defines a resistive network [CDR] of the augmenting paths and switches a path proportional to the current that would flow through it when a unit voltage is placed at the new connect request point. We expect that the online algorithms discussed in this paper will do better when the capacities of the stations are larger, and the ....

D. Coppersmith, P. Doyle, P. Raghavan, M. Snir, "Random Walks on Weighted Graphs and Applications to On-line Algorithms," Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (May 1990, and 369--378), Also appears as IBM Research Report RC 15840.

....constant between 3 and 6 for two servers. For k = 3, Berman et al. [1] proved that HARMONIC is competitive, but its competitive constant was not given. For general k, the solution is known only for some specific metric spaces. In [3] a k competitive algorithm is given for trees. Coppersmith et al. [5] gave a randomized k competitive algorithm for a broad class of metric spaces, that also includes trees. The known 2 competitive deterministic algorithms for two servers from [4, 8] are very spaceand time consuming. If the given metric space has m points, then the algorithms need O(m) space and ....

....j ; r) Note that ffl 1 ffl 2 = js 1 s 2 j. Let us consider the following randomized algorithm: Algorithm RANDOM SLACK: Move s i to the request point r with probability p i = ffl j js 1 s 2 j , where j 6= i. P. Raghavan pointed out to us that this algorithm is equivalent to the one given in [5], and therefore we obtain that RANDOM SLACK is 2 competitive. Thus it appears that balancing the accumulated slack of a server, instead of the total work, should give a better algorithm than BALANCE2. Below we show that this is indeed true. Algorithm BALANCE SLACK: Let e i denote the cumultative ....

[Article contains additional citation context not shown here]

D.Coppersmith, P.G.Doyle, P.Raghavan, M.Snir, Random walks on weighted graphs and applications to on-line algorithms, manuscript.

....k = 2 in [5, 2] and recently Fiat et al. [4] announced the result that for each k there is a c competitive algorithm, where c is a large constant. Some research has been done for some restricted metric spaces. It is known that there is a k competitive algorithm for trees [1] Coppersmith et al. [3] have presented a randomized k competitive algorithm for so called resistive spaces. The reader is referred to [1, 2, 3, 4, 5] for more information about the problem and current progress. 2 Benevolent Adversary Very often the server problem is stated in terms of a game between our k servers and ....

....where c is a large constant. Some research has been done for some restricted metric spaces. It is known that there is a k competitive algorithm for trees [1] Coppersmith et al. [3] have presented a randomized k competitive algorithm for so called resistive spaces. The reader is referred to [1, 2, 3, 4, 5] for more information about the problem and current progress. 2 Benevolent Adversary Very often the server problem is stated in terms of a game between our k servers and the adversary s k servers. In this game the adversary makes the requests and serves them with his servers, trying to maximize ....

D.Coppersmith, P.G.Doyle, P. Raghavan and M. Snir, Random walks on weighted graphs and applications to on-line algorithms, Proceedings 22nd ACM STOC (1990) pp. 369--378.

....of the theory of computing, as well as an elegant subject for mathematical analysis. The most notable, perhaps, of recent applications has been to polynomial time randomized approximation algorithms e.g. 16, 10, 12] but there have also been uses in space complexity [4] on line algorithms [9], and distributed computing [15] There are at least three graph parameters of the form expected number of steps of a random walk before X happens which have been extensively studied. The mixing time is the least t such that the distribution of the walk at step t is approximately (within ....

D. Coppersmith, P. Doyle, P. Raghavan and M. Snir, Random walks on weighted graphs and applications to on-line algorithms, J. ACM 40 #3 (1993), 421-453.

No context found.

Don Coppersmith, Peter G. Doyle, Prabhakar Raghavan, and Marc Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40:421-453, 1993.

No context found.

D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir, "Random walks on weighted graphs and applications to on-line algorithms," J. Assoc. Computing Machinery, vol. 40, no. 3, pp. 421--453, July 1993.

No context found.

D. Coppersmith, P. G. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs and applications to on-line algorithms. Journal of the ACM, 40:421-453, 1993.

No context found.

D. Coppersmith, P. G. Doyle, P. Raghavan, M. Snir, Random walks on weighted graphs and applications to on-line algorithms, Proc. 22nd Annual ACM Symp. on Theory of Computing (1990) 369-378.

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