A tight upper bound for the path length of AVL trees

AbstractWe prove that the internal path length of an AVL tree of size N is bounded from above by 1.4404N(log2 N-log2log2N)+O(N) and show that this bound is achieved by an infinite family of AVL trees, each tree of which is not of maximal height. These results carry over to the comparison cost of brother trees

On Monte-Carlo tree search for deterministic games with alternate moves and complete information

We consider a deterministic game with alternate moves and complete information, of which the issue is always the victory of one of the two opponents. We assume that this game is the realization of a random model enjoying some independence properties. We consider algorithms in the spirit of Monte-Carlo Tree Search, to estimate at best the minimax value of a given position: it consists in simulating, successively, $n$ well-chosen matches, starting from this position. We build an algorithm, which is optimal, step by step, in some sense: once the $n$ first matches are simulated, the algorithm decides from the statistics furnished by the $n$ first matches (and the a priori we have on the game) how to simulate the $(n+1)$-th match in such a way that the increase of information concerning the minimax value of the position under study is maximal. This algorithm is remarkably quick. We prove that our step by step optimal algorithm is not globally optimal and that it always converges in a finite number of steps, even if the a priori we have on the game is completely irrelevant. We finally test our algorithm, against MCTS, on Pearl's game and, with a very simple and universal a priori, on the games Connect Four and some variants. The numerical results are rather disappointing. We however exhibit some situations in which our algorithm seems efficient

Two-Way Unary Temporal Logic over Trees

We consider a temporal logic EF+F^-1 for unranked, unordered finite trees. The logic has two operators: EF\phi, which says "in some proper descendant \phi holds", and F^-1\phi, which says "in some proper ancestor \phi holds". We present an algorithm for deciding if a regular language of unranked finite trees can be expressed in EF+F^-1. The algorithm uses a characterization expressed in terms of forest algebras.Comment: 29 pages. Journal version of a LICS 07 pape

A theory of game trees, based on solution trees

In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied

Solution trees as a basis for game tree search

A game tree algorithm is an algorithm computing the minimax value of the root of a game tree. Many algorithms use the notion of establishing proofs that this value lies above or below some boundary value. We show that this amounts to the construction of a solution tree. We discuss the role of solution trees and critical trees in the following algorithms: Principal Variation Search, alpha-beta, and SSS-2. A general procedure for the construction of a solution tree, based on alpha-beta and Null-Window-Search, is given. Furthermore two new examples of solution tree-based algorithms are presented, that surpass alpha-beta, i.e., never visit more nodes than alpha-beta, and often less

Trends in game tree search

This paper deals with algorithms searching trees generated by two-person, zero-sum games with perfect information. The standard algorithm in this field is alpha-beta. We will discuss this algorithm as well as extensions, like transposition tables, iterative deepening and NegaScout. Special attention is devoted to domain knowledge pertaining to game trees, more specifically to solution trees. The above mentioned algorithms implement depth first search. The alternative is best first search. The best known algorithm in this area is Stockman's SSS*. We treat a variant equivalent to SSS* called SSS-2. These algorithms are provably better than alpha-beta, but it needs a lot of tweaking to show this in practice. A variant of SSS-2, cast in alpha-beta terms, will be discussed which does realize this potential. This algorithm is however still worse than NegaScout. On the other hand, applying a similar idea as the one behind NegaScout to this last SSS version yields the best (sequential) game tree searcher known up till now: MTD(f)

A Nice Labelling for Tree-Like Event Structures of Degree 3 (Extended Version)

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. Finally, we exemplify how to use this theorem to construct upper bounds for the labelling number of other event structures of degree 3