M. Santha. On the Monte Carlo Boolean decision tree complexity of read-once formulae. Random Structures and Algorithms, 6(1):75--88, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for nondeterministic complexity. For decision tree complexity, a notable feature of our approach is a unified treatment of the randomized complexity with zero error and two sided error; this is in contrast to [23, 13, 14] who handle only the zero error randomized complexity. Subsequently, Santha [24] enhanced their techniques in a non trivial manner to show that R # (f) 1 2#)R for any balanced read once formula f . Our technique is also general enough to apply to well balanced monotone formulas with threshold gates and of arbitrary fanin. 1.4 Organization Section 2 contains the ....

M. Santha. On the Monte Carlo Boolean decision tree complexity of read-once formulae. Random Structures and Algorithms, 6(1):75--88, 1995.

....Carlo complexity is universal if we identify bounds di ering by a multiplicative constant. We present also the method by which the exact lower bounds for the Monte Carlo complexity can be proved (Theorem 3. 4) Examples of the exact lower bounds for the Monte Carlo complexity were established in [8]. In the second part of the paper we compare di erent types of randomized algorithms to evaluate the so called read once formulae. A read once formula is a formula having connectivities ; of any fanin and having exactly one occurrence of each its variable. The problem of evaluating read once ....

....if an algorithm A does not ask the value of x and outputs a constant then C (A; 0 and S (e f (A; 1 2 . If A evaluates x and outputs its value then C (A; 1 and S (e f (A; 2 . 7 This example can be extended to h ; Si needed to obtain the lower bound by Santha [8] for Monte Carlo complexity of read once formulae. Proof of Theorem 3.3. It suces, by Theorem 3.4, to show that for any 0 1 max ; S min A2D C (A; S [e f (A; 2 sup min A2D =2 (f) C (A; Let us take arbitrary 0 1. We will prove that for all ; ....

M. Santha. On the Monte Carlo boolean decision tree complexity of readonce formulae. In 6th Annual Conference on Structure in Complexity Theory, pages 180-187, 1991.

....classical randomized algorithm, on the worst case error criterion. The best classical Las Vegas algorithm (Las Vegas here means stochastic runtime, but guaranteed to give the correct answer) is quite simple, and Saks and Wigderson [22] showed it was the optimal algorithm for any depth tree. Santha [23] showed, for read once Boolean functions, that no classical Monte Carlo algorithm with all error probabilities below p can take fewer than (1 xp)Q queries, where Q is the time taken by the optimal Las Vegas algorithm, and x = 1, 2 as the error is one or two sided. This is just the trivial ....

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

.... trees where the mistake occurs in only one direction (it might output 0 instead of the real value 1 but not the other way around) then it is called 1 way error computation (the corresponding complexity measure is denoted by C R1 ) For further information we refer the reader to [38] 29] and [34]. The main question is this: how much can we save by adding the extra power of randomization 2.2. Boolean decision trees. First we mention some basic inequalities on the relation between deterministic and randomized complexity. 5 Theorem 2.2.1. M. Blum [3] For any Boolean function f p C ....

M. Santha, On the Monte Carlo Boolean Decision Tree Complexity of Read-Once Formulae, manuscript, 1991.

....the value of that sub formula before probing any variable that appears in another part of the formula. This is the reason for the use of a bottom up induction given in the next subsection, whose single step (the shrinking lemma) consists of a global statement on the formula. Interestingly, Santha [San91] developed a proof for a similar problem that uses a top down induction and need not use Yao s lemma. In the proof of the shrinking lemma (sub section 3.3) we carefully define a distribution on inputs and a set of decision trees that enable reducing the lemma s statement into a statement ....

M. Santha. On the Monte Carlo Boolean decision tree complexity of read-once formulae. In Proc. 6th Structure in Complexity Theory Conf., 1991. To appear.

....Carlo complexity is universal if we identify bounds differing by a multiplicative constant. We present also the method by which the exact lower bounds for the Monte Carlo complexity can be proved (Theorem 3. 4) Examples of the exact lower bounds for the Monte Carlo complexity were established in [8]. In the second part of the paper we compare different types of randomized algorithms to evaluate the so called read once formulae. A read once formula is a formula having connectivities ; of any fanin and having exactly one occurrence of each its variable. The problem of evaluating read once ....

....value of x and outputs a constant then C (A; 0 and S Delta (e f (A; oe) Gamma ) 1 Gamma 2 . If A evaluates x and outputs its value then C (A; 1 and S Delta (e f (A; oe) Gamma ) Gamma2 . This example can be extended to h ; oe; Si needed to obtain the lower bound by Santha [8] for Monte Carlo complexity of read once formulae. Proof of Theorem 3.3. It suffices, by Theorem 3.4, to show that for any 0 1 max ;oe;S min A2D i C (A; S Delta [e f (A; oe) Gamma ] j 2 sup min A2D =2 (f) C (A; Let us take arbitrary 0 1. We will prove that ....

M. Santha. On the Monte Carlo boolean decision tree complexity of readonce formulae. In 6th Annual Conference on Structure in Complexity Theory, pages 180--187, 1991.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC