M. Tompa. Lower bounds on universal traversal sequences for cycles and other low degree graphs. SIAM Journal on Computing, 21(6):1153--1160, Dec. 1992. 25  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 graphs in ....

M. Tompa. Lower bounds on universal traversal sequences for cycles and higher degree graphs. Technical Report 90-07-02, Department of Computer Science and Engineering, University of Washington, July 1990. To appear in SIAM Journal on Computing.

....on general graphs. Below we examine another concept, universal traversal sequences, for which the best known bounds on regular graphs for d greater than 2 are the same for cubic graphs as they are for other d greater than 3. We begin with some preliminary definitions following the discussion in [73]. Cook introduced universal traversal sequences as a simple method for traversing graphs (see [1] They are defined below. Definition 8.5 [73] Let G = d; n) be the set of all connected, d regular, n vertex, edge labeled, undirected graphs G = V; E) Edges are labeled as follows: for every edge ....

....greater than 2 are the same for cubic graphs as they are for other d greater than 3. We begin with some preliminary definitions following the discussion in [73] Cook introduced universal traversal sequences as a simple method for traversing graphs (see [1] They are defined below. Definition 8. 5 [73] Let G = d; n) be the set of all connected, d regular, n vertex, edge labeled, undirected graphs G = V; E) Edges are labeled as follows: for every edge fu; vg 2 E there are two labels l u;v and l v;u with the property that, for every u 2 V; fl u;v j fu; vg 2 Eg = f0; 1; d Gamma 1g. ....

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M. Tompa. Lower bounds on universal traversal sequences for cycles and higher degree graphs. Technical Report 90-07-02, University of Washington, July 1990. To appear in SIAM Journal on Computing.

....1.48 ) and for UTSs for d regular graphs of n vertices where 3dn 17 1 from W(d 2 1.43 n 2.43 ) to W(d 2 1.48 n 2. 48 ) 179 1 Introduction Reflecting sequences are variants of universal traversal sequences (UTSs) and were introduced by Tompa in proving a lower bound on the length of UTSs [12]. The study of UTSs is motivated by the complexity of graph traversal. Good bounds on the length of UTSs translate into good bounds on the time complexity of certain undirected graph traversal algorithms running in very limited space. Aleliunas et al. 1] proved the existence of polynomial length ....

....traversal sequence (UTS) for G(d,n) if U traverses every G G(d,n) starting at any vertex in G. Let U(d,n) denote the length of a shortest UTS for non empty G(d,n) and define U(d,n) U(d,n 1) in case G(d,n) is empty. The lower and upper bounds on U(d,n) for various ranges of d were studied in [1, 2, 3, 5, 8, 10, 12]. Prior to the current work, the best lower bounds on U(d,n) for d=2 and for 3 d n 17 1 were U(2,n) W(n log 5 10 ) and U(d,n) W(d 2 log 5 10 n 1 log 5 10 ) due to a personal communication [6] These lower bounds are improved in this paper to U(2,n) W(n log 7 17.82 ) and U(d,n) ....

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M. Tompa. Lower bounds on universal traversal sequences for cycles and higher degree graphs. Department of Computer Science and Engineering, University of Washington, Technical Report 90-07-02. July 1990.

....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. Tompa. Lower bounds on universal traversal sequences for cycles and higher degree graphs. Technical Report 90-07-02, Department of Computer Science and Engineering, University of Washington, July 1990.

.... 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 = 1, a quadratic lower bound on time is known. The known algorithms ....

M. Tompa. Lower bounds on universal traversal sequences for cycles and other low degree graphs. SIAM Journal on Computing, 21(6):1153--1160, Dec. 1992. 25

....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 universal traversal sequences Bound Relevant Range Source U(d; n) O(n 3 ) d = 2 Aleliunas [1] U(d; n) O(dn 3 log n) 3 d n=2 0 1 Kahn et al. ....

.... This paper U(d; n) Omega# d 2 n 2 ) n log 10 2 d n=3 0 2 Borodin et al. 4] U(d; n) Omega# n 2 ) n=3 0 2 d Alon et al. 3] For 3 d n=3 0 2, the best previous lower bound was U(d; n) Omega# d 0:67 n 2:33 d 2 n 2 ) the first term due to Coppersmith (see Tompa [7]) and the second term due to Borodin, Ruzzo, and Tompa [4] This bound is improved in Corollary 9 to U(d; n) Omega# d 20log 5 10 n 1 log 5 10 d 2 n 2 ) Omega# d 0:57 n 2:43 d 2 n 2 ) The value of d at which the second term begins to dominate is d = n log 10 2 n 0:3 ....

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M. Tompa. Lower bounds on universal traversal sequences for cycles and other low degree graphs. SIAM Journal on Computing, 21(6):1153--1160, Dec. 1992. 14

....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 Length of Universal Traversal Sequences Bound Relevant Range Source U(d; n) O(n 3 ) d = 2 Aleliunas [1] U(d; n) O(dn 3 log n) 3 d n=2 Gamma ....

.... d 2 n 2 ) n log 10 2 d n=3 Gamma 2 Borodin et al. 4] U(d; n) Omega Gamma n 2 ) n=3 Gamma 2 d Alon et al. 3] For 3 d n=3 Gamma 2, the best previous lower bound was U(d; n) Omega Gamma d 0:67 n 2:33 d 2 n 2 ) the first term due to Coppersmith (see Tompa [7]) and the second term due to Borodin, Ruzzo, and Tompa [4] This bound is improved in Corollary 8 to U(d; n) Omega Gamma d 2 Gammalog 5 10 n 1 log 5 10 d 2 n 2 ) Omega Gamma d 0:57 n 2:43 d 2 n 2 ) The value of d at which the second term begins to dominate is d = n ....

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M. Tompa. Lower bounds on universal traversal sequences for cycles and other low degree graphs. SIAM Journal on Computing, 21(6):1153--1160, Dec. 1992. 11

.... 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 corresponding to the case P = 1, a quadratic lower bound on time is known. ....

M. Tompa. Lower bounds on universal traversal sequences for cycles and other low degree graphs. SIAM Journal on Computing, 21(6):1153--1160, Dec. 1992.

....M never visits more vertices. 2 Using similar techniques, Theorem 22 proves that the previous result in fact holds for any even length ff such that ff is not a universal traversal sequence for all labeled (n=2) cycles. For instance, it holds for any ff whose length is even and O(n 1:29 ) Tompa [36]) Theorem 22: For any ff 2 f0; 1g 3 of even length, any even integer n, and any integer k, if ff is not a universal traversal sequence for all labeled (n=2) cycles, then ff k is not a universal traversal sequence for all labeled n cycles. Proof: Since ff is not a universal traversal sequence ....

M. Tompa. Lower bounds on universal traversal sequences for cycles and higher degree graphs. SIAM Journal on Computing, 21(6):1153--1160, Dec. 1992.

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