Aviezri S. Fraenkel

Computing a perfect strategy for n × n chess requires time exponential in n

Abstract It is proved that a natural generalization of chess to an n × n board is complete in exponential time. This implies that there exist chess positions on an n × n chessboard for which the problem of determing who can win from that position requires an amount of time which is at least exponential in n .

How Many Squares Can a String Contain

All our words (strings) are over afixedalphabet. A square is a subword of the formuu=u2, whereuis a nonempty word. Two squares aredistinctif they are of different shape, not just translates of each other. A worduisprimitiveifucannot be written in the formu=vjfor somej?2. A squareu2withuprimitive isprimitive rooted. LetM(n) denote the maximum number of distinct squares,P(n) the maximum number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduceM(n) 0, andP(n)=n?o(n) for infinitely manyn.

How to Beat Your Wythoff Games' Opponent on Three Fronts

1. Wythoff Games. Let a be a positive integer. Given two piles of tokens, two players move alternately. The moves are of two types: a player may remove any positive number of tokens from a single pile, or he may take from both piles, say k (> 0) from one and 1 (> 0) from the other, provided that I k 11 0, is an N-position for every a; the Next player moves to (0, 0) and wins. For a = 2, the position (1, 3) is a P-position: if Next moves to (0, 3), (0,2) or (0, 1), then Previous, using a move of the first type, moves to (0, 0) and wins. If Next moves to (1, 2) or to (1, 1), then Previous, using a move of the second type, can again move to (0, 0). The set of all P-positions is denoted by P, and the set of all N-positions by N.

Robust universal complete codes for transmission and compression

Abstract Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variable-length codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and needs not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several “real-life” examples.

Complexity of problems in games, graphs and algebraic equations

Abstract We prove NP-hardness of six families of naturally defined, interesting board games. Some of them are “only just hard” in the sense that slight variations of them are polynomial. We further prove NP-completeness of two problems on digraphs which are related to game strategies; and NP-completeness and NP-hardness respectively of two classical problems of abstract algebra concerning the existence of solutions of algebraic equations. Also these problems were suggested by an investigation in combinatorial game theory.

Novel Compression of Sparse Bit-Strings — Preliminary Report

New methods for the compression of large sparse binary strings are presented. They are based on various new numeration systems in which the lengths of zero-block runs are represented. The basis elements of these systems, together with the non-zero blocks, are assigned Huffman codes. Experiments run on bit-maps of the Responsa Retrieval Project, and for comparison on randomly generated maps and on a digitized picture, yield compressions superior to previously known methods.

The bracket function and complementary sets of integers

The following result is well known (as usual, [x] denotes the integral part of x) : (A) Let α and β be positive irrational numbers satisfying 1 Then the sets [nα], [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers] see, e.g. ( 1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16 ). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoffs game, with or without proof. It appears that Beatty ( 4 ) was the originator of the problem. The theorem has a converse, and the following holds: (B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

Improved hierarchical bit-vector compression in document retrieval systems

The “concordance” of an information retrieval system can often be stored in form of bit-maps, which are usually very sparse and should be compressed. Hierarchical bit-vector compression consists of partitioning a vector <italic>v<subscrpt>i</subscrpt></italic> into equi-sized blocks, constructing a new bit-vector <italic>v<subscrpt>i</subscrpt></italic>+1 which points to the non-zero blocks in <italic>v<subscrpt>i</subscrpt></italic>, dropping the zero-blocks of <italic>v<subscrpt>i</subscrpt></italic>, and repeating the process for <italic>v<subscrpt>i</subscrpt></italic>+1. We refine the method by pruning some of the tree branches if they ultimately point to very few documents; these document numbers are then added to an appended list which is compressed by the prefix-omission technique. The new method was thoroughly tested on the bit-maps of the Responsa Retrieval Project, and gave a relative improvement of about 40% over the conventional hierarchical compression method.

Complexity, appeal and challenges of combinatorial games

Studying the precise nature of the complexity of games enables gamesters to attain a deeper understanding of the difficulties involved in certain new and old open game problems, which is a key to their solution. For algorithmicians, such studies provide new interesting algorithmic challenges, Substantiations of these assertions are illustrated on hand of many sample games, leading to a definition of the tractability, polynomiality and efficiency of subsets of games. In particular, there are tractable games that need not be polynomial, polynomial games that need not be efficient. We also define and explore the nature of the subclasses PlayGames and MathGames.

Complementing and exactly covering sequences

Abstract The following is proved (in a slightly more general setting): Let α1, …, αm be positive real, γ1, …, γm real, and suppose that the system [nαi + γi], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then α i α j is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all αi but one are integers; (iii) m ⩾ 5, two of the αi are integers, at least one of them prime; (iv) m ⩾ 5 and αn ⩽ 2n for n = 1, 2, …, m − 4. For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m − 1 sequences.