M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180--187, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... was Ro ( f) O (Q2 (f)2 Qo (f)2) due to de Wolf [23] It is worth remarking that we also obtain Ro (f) O (Rs (f) deg (f) log ) for total f, since no relation between Ro and R2 better than Ro (f) O (R2 (f)3) is currently known (although no asymptotic gap between Ro and R2 is known either [17]) 6. Asymptotic Gaps Having related RC (f) and QC (f) to other query com plexity measures in Section 5, in what follows we seek the largest possible asymptotic gaps among the measures. In particular, Section 6.1 gives a total f for which RC ( f) O (C ( f) 97 ) and hence C ( f) O (QC ....

....V chooses an input variable to query as follows Let X be the claimed input, and let K = 91 gt i (Xi) Let Io = i:gt 1(Xi) O andI1 : i:gt l(Xi) l . With probability p, V chooses an i 6 I1 unifoly at random; otherwise A chooses an i 6 Io unifoly at random Here p is as follows K [0,12] 13 14 15 16 [17,29] p 0 1 5 4 1 17 12 12 17 Once is chosen, V repeats the procede for Xi, and continues recursively in this manner until reaching a variable 5 to query. e can check that if St (X) St (Y) then St (Xi) St ( with probability at least 1 17. Hence x5 5 with probability at least 1 17 t, and nc (a, ....

M. Santha. On the Monte-Carlo decision tree complexity of read-once formulae, Random Structures and Algorithms 6(1):75-87, 1995.

....of a randomized decision tree that represents f with a probability of error at most 1 3. Nisan showed that R 0 (f) D(f) and R 2 (f) # (D(f) 16] On the other hand, the best known separation between deterministic and randomized query complexity is R 0 (f) R 2 (f) D(f) 0.753. [19, 20], for f an AND OR tree with two children per node. Whether better separations are possible is a long standing open question, and one that might be fruitfully investigated with computer analysis . Unfortunately, though, we do not know how to compute R 0 (f) or R 2 (f) in polynomial time without ....

M. Santha, On the Monte Carlo decision tree complexity of read-once formulae, Random Structures and Algorithms 6:1, pp. 75--87, 1995.

....its value. It can be shown that this algorithm always gives the correct answer with expected number of queries O(n ff ) where ff = log( 1 p 33) 4) 0:7537 : Saks and Wigderson [38] showed that this is asymptotically optimal for zero error algorithms for this function, and Santha [39] proved the same for bounded error algorithms. Thus we have D(f) n = Theta(R 2 (f) 1:3: Open problem 3 What is the biggest gap between D(f) and R 2 (f) 5.3 Quantum As in the classical case, deg(f) and g deg(f) give lower bounds on quantum query complexity. The next lemma from [3] ....

M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180--187, 1991.

....its value. It can be shown that this algorithm always gives the correct answer with expected number of queries O(n ff ) where ff = log( 1 p 33) 4) 0:7537 : Saks and Wigderson [SW86] showed that this is asymptotically optimal for zero error algorithms for this function, and Santha [San91] proved the same for boundederror algorithms. Thus we have D(f) n = Theta(R 2 (f) 1:3: Open problem 3 What is the biggest gap between D(f) and R 2 (f) 5.3 Quantum As in the classical case, deg(f) and g deg(f) give lower bounds on quantum query complexity. The next lemma from [BBC ....

M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180-- 187, 1991.

....our zero error setting with Q 0 (f) 2 O(N 1=2 1=d ) queries. The classical lower bound follows from combining two known results. First, an AND OR tree of depth d on N variables has R 0 (f) N=2 d [20, Theorem 2. 1] see also [37] Second, for such trees we have R 2 (f) 2 Omega Gamma R 0 (f) [39]. Hence R 2 (f) 2 Omega Gamma N) 2 This analysis is not quite optimal. It gives only trivial bounds for d = 2, but a more refined analysis shows that we can also get speed ups for such 2 level trees: Theorem 9 Let f be the AND of N 1=3 ORs of N 2=3 variables each. Then Q 0 (f) 2 Theta(N ....

M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180--187, 1991.

....setting with Q 0 (f) 2 O(N 1=2 1=d ) queries. The classical lower bound follows from combining two known results. First, an AND OR tree of depth d on N variables has R 0 (f) N=2 d [HNW93, Theorem 2. 1] see also [SW86] Second, for such trees we have R 2 (f) 2 Omega Gamma R 0 (f) San91] Hence R 2 (f) 2 Omega Gamma N ) 2 This analysis is not quite optimal. It gives only trivial bounds for d = 2, but a more refined analysis shows that we can also get speed ups for such 2 level trees: Theorem 9 Let f be the AND of N 1=3 ORs of N 2=3 variables each. 3 Then Q 0 (f) 2 ....

M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180-- 187, 1991.

....gap between D(f) and R(f) The best known separation is for complete binary AND OR trees, where D(f) N and R(f) 2 Theta(N 0:753: and it has been conjectured that this is the best separation possible. This holds both for zero error randomized trees [29] and for bounded error trees [30]. i B i f0; 1g which sets variables according to X , is a certificate for X of size bs(f) 2 . Firstly, if C were not an f(X) certificate then let X 0 be an input that agrees with C, such that f(X 0 ) 6= f(X) Let X 0 = X B b 1 . Now f is sensitive to B b 1 on X and B b 1 is ....

M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180--187, 1991.

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M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180--187, 1991.

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SANTHA, M. 1991. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference. IEEE Computer Society Press, Los Alamitos, Calif., pp. 180--187.

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M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 180--187, 1991.

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, 4, 383-387. Santha, M. 1991. On the Monte Carlo decision tree complexity of read-once formulae. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference (1991), pp.

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