Laurent Alonso, Edward M. Reingold, and Rene Schott. The average-case complexity of determining the majority. SIAM Journal on Computing, 26(1):1--14, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be determined in a single quantum query. In fact, our algorithm can be viewed as an XOR decision tree, i.e. a classical decision tree with the additional power of computing the XOR of two input bits at the cost of a single query. The complexity of MAJORITY in this model has been studied before [11, 1, 2], independently of the connection with quantum computation. A tight bound of N 1 w(N) was known [11, 1] We give a simpler proof for the lower bound which generalizes to the case where computing the parity of arbitrarily many input bits is permitted in one query. The lower bound shows that ....

....Our main result is a quantum black box network that computes MAJORITY with zero sided error # using only 2 3 N O( # N log(# 1 log N) queries. For any positive # we construct such a network. The algorithm can be viewed as a randomized variant of an XOR decision tree given by Alonso et al. [2]. We construct an exact randomized XOR decision tree with an expected number of queries of at most 2 3 N 2 log N on any input. We argue that the number of queries is su#ciently concentrated to yield our main result. Alonso et al. 2] show that the average cost of their algorithm over all N bit ....

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Laurent Alonso, Edward M. Reingold, and Rene Schott. The average-case complexity of determining the majority. SIAM Journal on Computing, 26(1):1--14, 1997.

.... Since the uniform distribution puts Omega Gammat = p N) probability on the set of such X, the average case complexity of A is at least Omega Gammaa = p N) Omega Gamma N) Omega Gamma p N ) 2 10 What about the classical average case complexity of MAJORITY Alonso, Reingold, and Schott [2] prove the bound D unif (MAJ) 2N=3 Gamma p 8N=9 O(log N) for deterministic classical computers. We can also prove a linear lower bound for the bounded error classical complexity, using the following lemma: Lemma 7.4 Let Delta 2 f1; p Ng. Any classical bounded error algorithm ....

L. Alonso, E. M. Reingold, and R. Schott. The average-case complexity of determining the majority. SIAM Journal on Computing, 26(1):1--14, 1997.

....times in x. The majority problem is that, given a binary string x, determine a position i such that x i is the majority bit of x, using only bit comparisons. When x has no majority, we must report so. The time complexity for finding the majority has been well studied in the literature (see, e.g. [1, 2, 3, 6, 13]) It is known that, in the worst case, n Gamma (n) bit comparisons are necessary and sufficient [2, 13] where (n) is the number of occurrences of bit 1 in the binary representation of number n. Recently, Alonso, Reingold and Schott [3] studied the average complexity of finding the majority ....

....been well studied in the literature (see, e.g. 1, 2, 3, 6, 13] It is known that, in the worst case, n Gamma (n) bit comparisons are necessary and sufficient [2, 13] where (n) is the number of occurrences of bit 1 in the binary representation of number n. Recently, Alonso, Reingold and Schott [3] studied the average complexity of finding the majority assuming the uniform probability distribution model. Using quite sophisticated arguments based on decision trees, they showed that on the average finding the majority requires at most 2n=3 Gamma p 8n=9 O(log n) comparisons and at least ....

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L. Alonso, E. Reingold and R. Schott, The average-case complexity of determining the majority, SIAM Journal on Computing 26-1, 1997, pp. 1-14.

....which are the X with jXj = N=2 Sigma 1. Since the uniform distribution puts Omega Gammat = p N) probability on the set of such X, the average case complexity of A is at least Omega Gammaa = p N) Omega Gamma N) Omega Gamma p N ) 2 What about the classical average case complexity [ARS97] proves that D unif (MAJ) 2N=3 Gamma p 8N=9 O(log N ) In Theorem 15 we prove that also R unif (MAJ) 2 Omega Gamma N ) Hence quantum is almost quadratically better than classical for this problem. The proof of the following lemma is deferred to the appendix. It shows that ....

L. Alonso, E. M. Reingold, and R. Schott. The average-case complexity of determining the majority. SIAM Journal on Computing, 26(1):1--14, 1997.

....which are the X with jXj = N=2 Sigma 1. Since the uniform distribution puts Omega Gammat = p N) probability on the set of such X, the average case complexity of A is at least Omega Gammaa = p N) Omega Gamma N) Omega Gamma p N ) 2 What about the classical average case complexity [ARS97] proves that D unif (MAJ) 2N=3 Gamma p 8N=9 O(log N ) In Theorem 13 we prove that also R unif (MAJ) 2 Omega Gamma N ) Hence quantum is almost quadratically better than classical for this problem. The proof of the following lemma is deferred to the appendix. It shows that ....

L. Alonso, E. M. Reingold, and R. Schott. The average-case complexity of determining the majority. SIAM Journal on Computing, 26(1):1--14, 1997.

....S T (n) 2 1 n is odd , 6. 2) E S T (n) 1 T (n) n [ E S n [ E S T (n) 1 T (n) n 1 [ P S n = 0 [ O 1 n ( thus E n T (n) E S n [ O(1) It is well known (see the result about the expected margin of victory in the n vote ballot problem in Alonso et al. 1995)) that E S n [ 2n p O(1) 6.3) yielding Lemma 3. # Proposition 8. If r1, E T [ n y r ( O(1) Proof. Proposition 8 is a consequence of: E n t (n) 1 t (n)n [ 1 r 2 n O(1) 6.4) 13 Relation (6.1) holds true, as in the proof of Lemma 3, but the ....

....for the average complexity of the composition and majority problems. Finding the optimal algorithm remains an open problem. A quasi optimal algorithm for the majority problem appears in Section 6: it is quite different from the quasi optimal algorithm for equal different answers, appearing in Alonso et al. 1995) or in Chassaing (1995) In the first of these papers, the average complexity for the majority problem in the uniform case is shown to be at least 2 3 n 8n 9p O(1) 14 and they describe an algorithm achieving an average complexity of 2 3 n 8n 9p O(logn) We see that, in the ....

L. Alonso, E.M. Reingold, and R. Schott, The average-case complexity of determining the majority, to appear in SIAM J. on Computing (1995).

....good and that any chip can test any other. We consider both the worst and average case for this problem, using results of the majority problem as starting point of our investigations. Other algorithmic questions are discussed in the conclusions. 2 Majority Problems The majority problem of [1] [2], and [9] is to determine the majority color in a set of n elements fx 1 ; x 2 ; xn g, each element of which is colored either blue 2 S R H T D I A O J F B K G E C Figure 1: Sample test con guration of fteen chips. We show a link from one chip to another labeled by a smiling face if the ....

....numbers of each color. In the worst case, exactly n (n) questions are necessary and sucient for the majority problem, where, following [6] n) is the number of 1 bits in the binary representation of n. This result was rst proved by Saks and Werman [9] 1] gave a short, elementary proof. [2] proved that any algorithm that correctly determines the majority must on the average use at least 2n 3 r 8n 9 (1) color comparisons, assuming all 2 n distinct colorings of the n elements are equally probable. Furthermore, 2] describes an algorithm that uses an average of 2n 3 r 8n 9 ....

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Alonso, L., E. M. Reingold, and R. Schott, \The average-case complexity of determining the majority," SIAM J. Comput. 26 (1997) 1-14.

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