P. Beame, A. Borodin, P. Raghavan, W. L. Ruzzo, and M. Tompa. A time-space tradeoff for undirected graph traversal by walking automata. SIAM J. Comput., 28(3):1051--1072 (electronic), 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in any way. Others have studied the issue of lower bounds for time space tradeoffs for USTCON in a more general (yet still structured) setting. The models on which lower bounds are proved are known as JAG models ( Jumping Automata on Graphs ) and there are several versions of these models. In [BBRRT], a lower bound of Omega Gamma n 2 ) is proved for a model that allows for an initial phase of distributing landmarks, but requires that the pebble takes deterministic walks on the graph, rather than random walks. Lower bounds on other variants of the JAG model are presented in [BBRRT, ....

....In [BBRRT] a lower bound of Omega Gamma n 2 ) is proved for a model that allows for an initial phase of distributing landmarks, but requires that the pebble takes deterministic walks on the graph, rather than random walks. Lower bounds on other variants of the JAG model are presented in [BBRRT, edmonds]. In particular, Edmonds [edmonds] allows for randomized algorithms, and proves that when the number of pebbles is sublogarithmic, the time to decide USTCON is superlinear (on the JAG model of computation) The main part of our paper is concerned with the analysis of the short term behavior of ....

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P. Beame, A. Borodin, P. Raghavan, W. Ruzzo, and M. Tompa. "Time-space tradeoffs for undirected graph traversal". In Proc. of 31st Annual Symposium on Foundations of Computer Science, 429--438, St. Louis, MO, October 1990.

....revisiting it, and keeping the stack or queue of pebble names in its state. Furthermore, as Savitch [31] shows, a JAG with the additional power to move a pebble from vertex i to vertex i 1 can simulate an arbitrary Turing machine on directed graphs. Even without this extra feature, we have shown [10] that JAGs are as powerful as Turing machines for the purposes of solving undirected graph problems (our main focus) Cook and Rackoff define the time T used by a JAG to be the number of pebble moves, and the space to be S = P log 2 n log 2 Q, where P is the number of pebbles and Q the number ....

....achievability of logarithmic space and linear time when m = O(n) since the families of regular graphs mentioned above have degree d = 1) and hence m = n) see Sections 3 and 4. We prove upper and lower bounds for undirected graph problems on other variants of the JAG in a companion paper [10]. Following the preliminary appearance of these results, Edmonds [21] proved a much stronger result for traversing undirected graphs, and Barnes and Edmonds [6] and Edmonds and Poon [22] proved even more dramatic tradeoffs for traversing directed graphs. 2. Walking Automata for Graphs The ....

P. W. Beame, A. Borodin, P. Raghavan, W. L. Ruzzo, and M. Tompa. Time-space tradeoffs for undirected graph traversal by graph automata. Information and Computation, 130(2):101--129, Nov. 1996.

....next neighbor. In time 2(d) this has touched all 2(d) connecting edges incident to that entry vertex, which was impossible in the construction above. 22 4. Open Problem The obvious important problem is to strengthen and generalize these lower bounds. Following an earlier version of this paper [9], Edmonds [21] proved a much stronger time space tradeoff on general JAGs: for every z 2, a JAG with at most 28z log n log log n pebbles and at most 2 states requires time n 1 2 Omega1 log n) log log n) to traverse 3 regular graphs. The ultimate goal might be to prove that ST = Omega# ....

P. W. Beame, A. Borodin, P. Raghavan, W. L. Ruzzo, and M. Tompa. Time-space tradeoffs for undirected graph traversal. In Proceedings 31st Annual Symposium on Foundations of Computer Science, pages 429--438, St. Louis, MO, Oct. 1990. IEEE.

....reason why our upper bound is the best possible. We thus hope that this work will spark interest in proving a time space tradeoff for USTCON, even in a restricted model of space bounded computation such as the JAGs of Cook and Rackoff [5] For a restricted version of the JAG model, Beame et al. [2] have shown that space p implies time #.n 2 = p log n for bounded degree graphs. Acknowledgement We are very grateful to Lyle Ramshaw for a thorough reading of the manuscript and many useful comments and corrections. ....

P. Beame, A. Borodin, P. Raghavan, W.L. Ruzzo, and M. Tompa. Time-Space Tradeoffs for Undirected Graph Traversal. In 31st Annual Symposium on Foundations of Computer Science, pages 429--438, St. Louis, Missouri, October 1990.

....another. This nonjumping model is closer to the one studied by Blum and Sakoda [13] Blum and Kozen [12] and Hemmerling [30] We will distinguish this nonjumping variant by referring to it as a WAG walking automaton for graphs . Following the preliminary appearance of some of these results [10], Edmonds [26] proved a much stronger result for traversing undirected graphs than that proved in [9] and Barnes and Edmonds [6] and Edmonds and Poon [27] proved even more dramatic tradeoffs for traversing directed graphs. The results described above have the strength that they hold independent ....

....of Feige [28] based on Broder et al. 20] and Barnes and Feige [7] cannot be made both errorless and substantially faster. We also showed that our lower bound is tight. The obvious important problem is to strengthen and generalize these lower bounds. Following an earlier version of this paper [10], Edmonds [26] proved a time space tradeoff on general JAGs: for every z 2, a JAG with at most 1 28z log n log log n pebbles and at most 2 log z n states requires time n Delta 2 Omega Gamma9144 n) log log n) to traverse 3 regular graphs. The ultimate goal might be to prove that ST ....

P. W. Beame, A. Borodin, P. Raghavan, W. L. Ruzzo, and M. Tompa. Time-space tradeoffs for undirected graph traversal. In Proceedings 31st Annual Symposium on Foundations of Computer Science, pages 429--438, St. Louis, MO, Oct. 1990. IEEE.

....are movable. In fact, our proof does extend to give a nonlinear lower bound when some motion of the pebbles is allowed, but the bound degenerates when the pebbles are allowed to move with complete freedom. Such models are surprisingly powerful; see Section 3. Nevertheless, in a companion paper [9] we prove a lower bound on a model with freely moving pebbles, but without the ability to jump one pebble to another. This nonjumping model is closer to the one studied by Blum and Sakoda [13] Blum and Kozen [12] and Hemmerling [30] We will distinguish this nonjumping variant by referring to it ....

....and Hemmerling [30] We will distinguish this nonjumping variant by referring to it as a WAG walking automaton for graphs . Following the preliminary appearance of some of these results [10] Edmonds [26] proved a much stronger result for traversing undirected graphs than that proved in [9], and Barnes and Edmonds [6] and Edmonds and Poon [27] proved even more dramatic tradeoffs for traversing directed graphs. The results described above have the strength that they hold independent of the magnitude of Q, the number of states. Presumably the bounds can be strengthened by also ....

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P. W. Beame, A. Borodin, P. Raghavan, W. L. Ruzzo, and M. Tompa. A time-space tradeoff for undirected graph traversal by walking automata. SIAM Journal on Computing. To appear.

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P. Beame, A. Borodin, P. Raghavan, W. L. Ruzzo, and M. Tompa. A time-space tradeoff for undirected graph traversal by walking automata. SIAM J. Comput., 28(3):1051--1072 (electronic), 1999.

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P. Beame, A. Borodin, P. Raghavan, W. Ruzzo, and M. Tompa, "TimeSpace Tradeoffs for Undirected Graph Traversal," Proceedings of the 31st Annual IEEE Found. of Comp. Sci. Symp. , pages 429-438, St. Louis, MO, October 1990. 19

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P. Beame, A. Borodin, P. Raghavan, W. Ruzzo, and M. Tompa, "TimeSpace Tradeoffs for Undirected Graph Traversal," Proceedings of the 31st Annual IEEE Found. of Comp. Sci. Symp. (1990), 429-438.

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Paul Beame, Allan Borodin, Prabhakar Raghavan, Walter Ruzzo, and Martin Tompa. Time-space tradeoffs for undirected graph traversal. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pages 429--430, 1990.

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