M. Bridgland, Universal traversal sequences for paths and cycles, Journal of Algorithms, 8(3):395-404, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....labels, such that the path guided by this sequence visits all edges of any graph. It is known that, with high probability, a sequence of length d log n) chosen uniformly at random, guides a walk in any d regular (connected) graph of n nodes. Explicit UTS are known for 2 regular graphs (cf. [6, 12, 13, 19, 21]) for 3 regular graphs (cf. 4, 18, 23] for cliques (cf. 2, 20] and for expanders (cf. 17] Some of these sequences can be constructed in log space, and hence can produce perpetual exploration with compact memory. However, without the a priori knowledge of n, non of these constructions ....

M. Bridgland, Universal traversal sequences for paths and cycles, Journal of Algorithms, 8(3):395-404, 1987.

.... 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 to the case P = ....

M. F. Bridgland. Universal traversal sequences for paths and cycles. Journal of Algorithms, 8(3):395--404, 1987.

....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 graphs in space O(log n) this would give an O(log n) space algorithm for ustcon, since a UTS can easily be ....

M. F. Bridgland. Universal traversal sequences for paths and cycles. Journal of Algorithms, 8(3):395--404, 1987.

.... 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 logarithmic space ....

M. F. Bridgland. Universal traversal sequences for paths and cycles. Journal of Algorithms, 8(3):395--404, 1987.

....traversal sequence of angles: We have shown the existence of a polynomial length universal sequence of angles (UTSA) for gridded polygons. However we do not know how to find one. The similar question for graphs is also wide open, with the only exceptions (known to us) being paths and cycles [Bridgland 1987], Bar Noy, Borodin, Karchemer, Linial Werman 1989] Intuitively, one may think that finding a UTSA in our case is easier, since the robot is assumed to have a kind of compass , while in the UTS problem for graphs, edges are arbitrarily ordered. 5. The minimum memory needed to deterministically ....

M.F. Bridgland, "Universal Traversal Sequences for Paths and Cycles," J. of Alg., 8, (1987), pp.395-404.

....traversal sequence of angles: We have shown the existence of a polynomial length universal sequence of angles (UTSA) for gridded polygons. However we do not know how to find one. The similar question for graphs is also wide open, with the only exceptions (known to us) being paths and cycles [9], 8] Intuitively, one may think that finding a UTSA in our case is easier, since the robot is assumed to have a kind of compass , while in the UTS problem for graphs, edges are arbitrarily ordered. 6 Acknowledgement We would like to thank David Aldous for his advice with respect to the ....

M.F. Bridgland, "Universal Traversal Sequences for Paths and Cycles," J. of Alg., 8, (1987), pp.395-404.

....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 ....

M. F. Bridgland. Universal traversal sequences for paths and cycles. Journal of Algorithms, 8(3):395--404, 1987.

.... Several 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] 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 ....

M. F. Bridgland. Universal traversal sequences for paths and cycles. Journal of Algorithms, 8(3):395--404, 1987.

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M. Bridgland, Universal traversal sequences for paths and cycles, Journal of Algorithms, 8(3):395-404, 1987.

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