Seminar on Randomized Algorithms Game-Theoretical Applications ...

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The first step is a simple modification of an AND-OR tree, which replaces all [AND] and [OR] nodes by equivalent elements composed of [NOR] nodes (i.e. an [OR] is a negated [NOR] etc). This yields a NOR tree (which has the same depth due to redundancy of some of the inner nodes). Though identical in terms of functionality, it also possesses a completely homogenous structure, which further facilitates our calculations. Second, we need to specify a probability distribution on values of the leaves. Each leaf is independently set to “1”, with the following probability: 3− p = 2 5 (Note: our peculiar choice of p will be justified by the next equation.) The probability of a [NOR] node’s output being “1” is the probability that both inputs are “0”, i.e.: ⎛ 2 3 − 5 ⎞ ⎛ 2 − 3 − 5 ⎞ ⎛ 5 −1⎞ 3 − 5 (1 − p )(1 − p) = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ − * = = ⎝ 2 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ 2 2 p Now we consider some properties of a deterministic algorithm evaluating our tree. Since we would like to minimize its running time, we make use of the following observation:
For a [NOR] node, the output value will be “0” if one of its children returns “1”. In such cases, the other child (possibly an entire sub-tree) will not influence the result, and therefore, doesn’t need to be inspected. This leads to a notion of Depth-First Pruning algorithms (DFPA). A DFPA essentially functions like a DFS, but it also stops visiting sub-trees of a node once its value has been determined – sub-trees that yield no additional information are “pruned” away. Observe the following proposition: Let T be a NOR tree, with all leaves set to the aforementioned distribution. Let W (T ) denote a minimum, over all deterministic game tree algorithms, of the expected number of steps to evaluate T. Then, there exists a DFP algorithm, whose expected number of steps to evaluate T is W (T ) . Thus, for the purposes of our discussion, we may restrict ourselves to DFPAs. For a DFPA, traversing a NOR tree with n leaves and aforementioned probability distribution, the following holds: Let h be the distance from the leaves to the node in question. Let W (h) denote the expected number of leaves the DFPA will need to inspect in order to evaluate the node. Then: W ( h) = W ( h −1) + (1 − p) ⋅W ( h −1) Here W ( h −1) is the expected number of leaves visited while evaluating one of the sub-trees of the node. The factor ( 1− p) before the second term arises from the fact that the other sub-tree will only be visited if the first sub-tree yielded 0, which will happen with the probability of ( 1− p) . (Note: there is no factor p before the first term, as one might expect, because one of the sub-trees must be visited under all circumstances, i.e. with 100 percent probability.) Now we let h = log 2 n (since we are working with a binary tree), and substitute it into the above equation. The solution of this equation produces the following result: W ( h) ≥ n 0.694
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Seminar on Randomized Algorithms Game-Theoretical Applications ...

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