A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....UTS s for directed graphs are of exponential length, Aleliunas et al. AKL 79] prove (nonconstructively) that universal traversal sequences of polynomial length exist for undirected graphs. Upper and lower bounds on the length of UTS s are known for various types of graphs [AAR90, BNBK 89, BRT89, HW89, Tom90] Some authors show how to construct superpolynomial length sequences [BNS89, BNBK 89, Bri87, KPS88, Nis90] and Istrail shows how to construct polynomial length sequences for certain classes of graphs [Ist88, Ist90] If one could construct polynomial length UTS s for arbitrary ....

A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 562--573, Seattle, WA, May 1989. To appear in Journal of Computer and System Sciences. 73

....particular examples. Section 8.1. Proposition 34 is one of the neatest instances of Erdos s Probabilistic Method in Combinatorics , though surprisingly it isn t in the recent book [4] on that subject. Constructing explicit universal traversal sequences is a hard open problem: see Borodin et al. [6] for a survey. Section 8.2. See [10] for a more careful discussion of the issues. The alert reader of our example will have noticed the subtle implication that the reader has written fewer papers than Paul Erdos, otherwise (why ) it would be preferable to do the random walk in the other ....

A. Borodin, W.L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. J. Computer Systems Sci., 45:180--203, 1992.

....time needed for undirected graph st connectivity have also been obtained. These results are particularly strong, because they do not depend on the number of states q. A universal traversal sequence is simply a JAG with an unlimited number of states, but only one pebble. Borodin, Ruzzo, and Tompa [BRT92] prove that on this model, undirected st connectivity requires Omega Gamma m 2 Delta time. Beame, Borodin, Raghavan, Ruzzo, and Tompa [BBRRT90] extend this to Omega Gamma n 2 =p Delta for p pebbles on 3 regular graphs with the restriction that all but one pebble are unmovable. ....

A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, October 1992.

....the bound showing that S 2 Omega Gamma log 2 n log log n log log T ) for any coin flipping probabilistic NNJAG with space S and expected time T . Regarding the time space tradeoff, there are many lower bounds proved for ustcon on various weaker variants of the JAG model [BBR 90, BRT92, CR80] Edmonds [Edm93b] was the first to prove a time space lower bound for ustcon on the regular JAG model (with bounded space) All these results apply to (directed) stcon, which contains ustcon as a special case. However, ustcon appears to be easier than stcon both in terms of space and ....

A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, October 1992.

....that s is connected to t simply by walking a pebble from s to t. Being able to jump a pebble directly to the node containing another pebble also adds a great deal of power to the JAG model that the Turing machine does not have. From lower bounds on JAGs that are not allowed to jump (WAGs) [BRT92, BBRRT90], it can be concluded that jumping increases the model s power significantly, because the model is able to quickly concentrate its limited pebble resources in the subgraph it is currently working on. At first, it might seem unnatural that the JAG model for directed graphs is not allowed to 1 The ....

....for undirected graph st connectivity have also been obtained. These results are particularly strong, because they do not depend on the number of states q. For example, a universal traversal sequence is simply a JAG with an unlimited number of states, but only one pebble. Borodin, Ruzzo, and Tompa [BRT92] prove that on this model, undirected st connectivity requires Omega Gamma m 2 Delta time where the graph has degree 3 d 1 3 n Gamma 2. Beame, Borodin, Raghavan, Ruzzo, and Tompa [BBRRT90] extend this to Omega Gamma n 2 =p Delta time for p pebbles on 3 regular graphs with the ....

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A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, October 1992.

....it to a probabilistic version of the NNJAG. Tompa [17] shows lower bounds on the product of the time and space needed when using certain natural approaches to solve stcon. Many time space lower bounds have been proved for undirected s t connectivity on various weak versions of the JAG model [2, 7, 8]. Edmonds was the first to prove a time space lower bound for ustcon on the unrestricted JAG model [9] The standard algorithms for s t connectivity, breadth and depth first search, run in optimal time Theta(m n) and use Theta(n log n) space. At the other extreme, Savitch s Theorem [15] ....

A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992.

.... authors have considered traversal of undirected regular graphs by a WAG with an unlimited number of states but only the minimum number (one) of pebbles, a model better known as a universal traversal sequence (Aleliunas et al. 2] Alon et al. 3] Bar Noy et al. 4] Borodin, Ruzzo, and Tompa [16], Bridgland [17] Buss and Tompa [19] Istrail [25] Karloff et al. 26] Tompa [33] A result of Borodin, Ruzzo, and Tompa [16] shows that such an automaton requires ) time (on regular graphs with 3n=2 m n =60n) Thus, for the particularly weak version of logarithmic space corresponding ....

.... number (one) of pebbles, a model better known as a universal traversal sequence (Aleliunas et al. 2] Alon et al. 3] Bar Noy et al. 4] Borodin, Ruzzo, and Tompa [16] Bridgland [17] Buss and Tompa [19] Istrail [25] Karloff et al. 26] Tompa [33] A result of Borodin, Ruzzo, and Tompa [16] shows that such an automaton requires ) time (on regular graphs with 3n=2 m n =60n) Thus, for the particularly weak version of logarithmic space corresponding to the case P = 1, a quadratic lower bound on time is known. The known algorithms and the lower bounds for universal traversal ....

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A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992.

....into good bounds on the time required by certain simple undirected graph traversal algorithms running in very limited space. In particular, determining good lower bounds on U (d; n) is a prerequisite to proving timespace tradeoffs for traversing undirected graphs. See Borodin, Ruzzo, and Tompa [5] for more detailed discussion. Proving such lower bounds is the emphasis of this paper. The current best upper and lower bounds on U (d; n) are summarized in Table 1. See Borodin, Ruzzo, and Tompa [5] for more background. Prior to the current work, the best lower bound for d = 2 was U(2; n) ....

....to proving timespace tradeoffs for traversing undirected graphs. See Borodin, Ruzzo, and Tompa [5] for more detailed discussion. Proving such lower bounds is the emphasis of this paper. The current best upper and lower bounds on U (d; n) are summarized in Table 1. See Borodin, Ruzzo, and Tompa [5] for more background. Prior to the current work, the best lower bound for d = 2 was U(2; n) Omega# n log n) due to Bar Noy et al. 4] This bound is improved in Corollary 3.5 to U(2; n) Omega# n log 4 6 ) Omega# n 1:29 ) For 3 d n=3 0 2, the best previous lower bound was U ....

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A. Borodin, W. L. Ruzzo, and M. Tompa, Lower bounds on the length of universal traversal sequences, in Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, Seattle, WA, May 1989, pp. 562--573. To appear in Journal of Computer and System Sciences.

....into good bounds on the time required by certain simple undirected graph traversal algorithms running in very limited space. In particular, determining good lower bounds on U(d; n) is a prerequisite to proving time space tradeoffs for traversing undirected graphs. See Borodin, Ruzzo, and Tompa [4] for more detailed discussion. Proving such lower bounds is the emphasis of this paper. The current best upper and lower bounds on U(d; n) are summarized in Table 1. See Borodin, Ruzzo, and Tompa [4] for more background. Prior to the current work, the best lower bound for d = 2 was U(2; n) ....

....to proving time space tradeoffs for traversing undirected graphs. See Borodin, Ruzzo, and Tompa [4] for more detailed discussion. Proving such lower bounds is the emphasis of this paper. The current best upper and lower bounds on U(d; n) are summarized in Table 1. See Borodin, Ruzzo, and Tompa [4] for more background. Prior to the current work, the best lower bound for d = 2 was U(2; n) Omega# n 1:33 ) due to Coppersmith (see Tompa [7] This bound is improved in Corollary 8 to U(2; n) Omega# n log 5 10 ) Omega# n 1:43 ) 4 Table 1: Best known bounds on length of ....

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A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992.

....into good bounds on the time required by certain simple undirected graph traversal algorithms running in very limited space. In particular, determining good lower bounds on U(d; n) is a prerequisite to proving time space tradeoffs for traversing undirected graphs. See Borodin, Ruzzo, and Tompa [4] for more detailed discussion. Proving such lower bounds is the emphasis of this paper. The current best upper and lower bounds on U(d; n) are summarized in Table 1. See Borodin, Ruzzo, and Tompa [4] for more background. Prior to the current work, the best lower bound for d = 2 was U(2; n) ....

....to proving time space tradeoffs for traversing undirected graphs. See Borodin, Ruzzo, and Tompa [4] for more detailed discussion. Proving such lower bounds is the emphasis of this paper. The current best upper and lower bounds on U(d; n) are summarized in Table 1. See Borodin, Ruzzo, and Tompa [4] for more background. Prior to the current work, the best lower bound for d = 2 was U(2; n) Omega Gamma n 1:33 ) due to Coppersmith (see Tompa [7] This bound is improved in Corollary 7 to U(2; n) Omega Gamma n log 5 10 ) Omega Gamma n 1:43 ) 2 Table 1: Best Known Bounds on ....

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A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992. 10

.... authors have considered traversal of undirected regular graphs by a JAG with an unlimited number of states but only the minimum number (one) of pebbles, a model better known as a universal traversal sequence (Aleliunas et al. 2] Alon et al. 3] Bar Noy et al. 4] Borodin, Ruzzo, and Tompa [18], Bridgland [19] Buss and Tompa [21] Istrail [34] Karloff et al. 37] Tompa [49] A result of Borodin, Ruzzo, and Tompa [18] shows that such an automaton requires Omega Gamma m 2 ) time (on regular graphs with 3n=2 m n 2 =6 Gamma n) Thus, for the particularly weak version of ....

.... number (one) of pebbles, a model better known as a universal traversal sequence (Aleliunas et al. 2] Alon et al. 3] Bar Noy et al. 4] Borodin, Ruzzo, and Tompa [18] Bridgland [19] Buss and Tompa [21] Istrail [34] Karloff et al. 37] Tompa [49] A result of Borodin, Ruzzo, and Tompa [18] shows that such an automaton requires Omega Gamma m 2 ) time (on regular graphs with 3n=2 m n 2 =6 Gamma n) Thus, for the particularly weak version of logarithmic space corresponding to the case P = 1, a quadratic lower bound on time is known. The known algorithms and the lower bounds ....

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A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992.

....bound is also tight for the chain. There are also graphs for which none of the bounds above is tight. Let G be a family of labeled d regular graphs on n vertices. Let U(G) denote the length of the shortest universal traversal sequence for all the labeled graphs in G. See Aleliunas et al. 1979 or Borodin et al. 1992 for definitions. Let R(G) denote the maximum resistance between any pair of vertices in any graph in G. Theorem 2.6. U(G) 4 o(1) mR(G) log 2 (njGj) Proof. The proof is by a probabilistic argument similar to that in Aleliunas et al. 1979) Given a labeled graph G 2 G; let v be a ....

....log n) when d = bn=2c. Applying Theorem 2. 6 we see that the length of universal traversal sequences for d regular graphs, for any d bn=2c, is O(n 3 log n) This bound was previously known to hold only for cliques (d = n Gamma 1) Interestingly, lower bounds for universal traversal sequences (Borodin et al. 1992) are Omega Gamma n 4 ) for linear d n=3 Gamma 2. Thus, length of universal traversal sequences also declines somewhere between d = n=3 Gamma 2 and d = bn=2c; whether there is a sharp threshold at d = bn=2c as in the case of cover time is unknown. 4. Resistance and Eigenvalues Let G be a ....

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A. Borodin, W. L. Ruzzo, and M. Tompa, Lower bounds on the length of universal traversal sequences. J. Comput. System Sci. 45(2) (1992), 180--203.

....is solvable by a deterministic logarithmic space algorithm, and perhaps showing this would be an easier first step towards the main goal. In fact, considerable effort has been expended on this step, for example in studying and attempting to constructively generate universal traversal sequences [1, 2, 3, 4, 8, 9, 12, 13, 14, 15, 19, 25]. Alternatively, if deterministic and nondeterministic classes are distinct, then ustcon is a likely candidate for a problem that will separate the classes. In either case, its complexity is of interest. Settling the deterministic space complexity of ustcon is a very difficult open problem. A ....

A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 562--573, Seattle, WA, May 1989. To appear in Journal of Computer and System Sciences.

.... by a JAG with an unlimited number of states but only the minimum number (one) of pebbles, a model better known as a universal traversal sequence (Aleliunas et al. 2] Alon et al. 3] Bar Noy et al. 4] Bridgland [15] Istrail [25] Karloff et al. 28] A result of Borodin, Ruzzo, and Tompa [14] shows that such an automaton requires Omega0 m 2 ) time (on regular graphs with 3n=2 m n 2 =6 0 n) Thus, for the particularly weak version of logarithmic space corresponding to the case P = 1, a quadratic lower bound on time is known. The known algorithms and the lower bounds for ....

....JAG with strong jumping that has 1 active pebble and P 0 1 unmovable pebbles. If M determines st nonconnectivity for all 3 regular symmetrically labeled graphs, then M requires time Omega0 n 2 =P ) Proof: The proof generalizes the main lower bound technique introduced by Borodin et al. [14]. Assume without loss of generality that n is a multiple of 4. If not, set aside 6 vertices in a 3regular connected component containing neither s nor t. We define a family of n vertex graphs, each formed by joining two copies of an n=2 vertex graph H by switching some combination of edge ....

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A. Borodin, W. L. Ruzzo, and M. Tompa. Lower bounds on the length of universal traversal sequences. Journal of Computer and System Sciences, 45(2):180--203, Oct. 1992.

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