Results 11  20
of
47
Heap Games, Numeration Systems and Sequences
 Ann. of Combinatorics
, 1998
"... . We propose and analyze a 2parameter family of 2player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but it is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an effi ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
. We propose and analyze a 2parameter family of 2player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but it is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences and demonstrate its equivalence with the class of sequences defined for the strategy of our games. Keywords: heap games, numeration systems, sequences 1. Example Given a 2player game played on two heaps (piles) of finitely many tokens. There are two types of moves: (I) Take any positive number of tokens from one heap, possibly the entire heap. (II) Take from both heaps, k from one and l from the other, with, say, k l. Then the move is constrained by the condition 0 ! k l ! 2k + 2, which is equivalent to 0 l \Gamma k ! k +2; k ? 0. The player making the last move (after which both heaps are empty) wins, and the opponent ...
Steganographic Communication in Ordered Channels
"... Abstract. In this paper we focus on estimating the amount of information that can be embedded in the sequencing of packets in ordered channels. Ordered channels, e.g. TCP, rely on sequence numbers to recover from packet loss and packet reordering. We propose a formal model for transmitting informati ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. In this paper we focus on estimating the amount of information that can be embedded in the sequencing of packets in ordered channels. Ordered channels, e.g. TCP, rely on sequence numbers to recover from packet loss and packet reordering. We propose a formal model for transmitting information by packetreordering. We present natural and wellmotivated channel models and jamming models including the kdistance permuter, the kbuffer permuter and the kstack permuter. We define the natural informationtheoretic (continuous) game between the channel processes (maxmin) and the jamming process (minmax) and prove the existence of a Nash equilibrium for the mutual information rate. We study the zeroerror (discrete) equivalent and provide errorcorrecting codes with optimal performance for the distancebounded model, along with efficient encoding and decoding algorithms. One outcome of our work is that we extend and complete D. H. Lehmer’s attempt to characterize the number of distance bounded permutations by providing the asymptotically optimal bound this also tightly bounds the first eigenvalue of a related state transition matrix [1]. 1
Combinatorial Representation of Generalized Fibonacci Numbers
, 1991
"... New formulae are presented which express various generalizations of Fibonacci numbers as simple sums of binomial and multinomial coefficients. The equalities are inferred from the special properties of the representations of the integers in certain numeration systems. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
New formulae are presented which express various generalizations of Fibonacci numbers as simple sums of binomial and multinomial coefficients. The equalities are inferred from the special properties of the representations of the integers in certain numeration systems.
FMKZ: an even simpler alphabetindependent FMindex
 Czech Technical University, Prague
, 2006
"... Abstract. In an earlier work [6] we presented a simple FMindex variant, based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The main drawback of using Huffman was its lack of synchronizing properties, forcing us to supply another bit stream ind ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. In an earlier work [6] we presented a simple FMindex variant, based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The main drawback of using Huffman was its lack of synchronizing properties, forcing us to supply another bit stream indicating the Huffman codeword boundaries. In this way, the resulting index needed O(n(H0 +1)) bits of space but with the constant 2 (concerning the main term). There are several options aiming to mitigate the overhead in space, with various effects on the query handling speed. In this work we propose KautzZeckendorf coding as a both simple and practical replacement for Huffman. We dub the new index FMKZ. We also present an efficient implementation of the rank operation, which is the main building brick of the FMKZ. Experimental results show that our index provides an attractive space/time tradeoff in comparison with existing succinct data structures, and in the DNA test it even wins both in search time and space use. An additional asset of our solution is its relative simplicity. 1
The Zeckendorf Decomposition of Certain FibonacciLucas Products." The Fibonacci Quarterly 36.3
"... The decomposition of any positive integer N as a sum of positivesubscripted, distinct, nonconsecutive Fibonacci numbers Fk is commonly referred to as the Zeckendorf decomposition ofN (ZD of N, in brief) [10]. This decomposition is always possible and, apart from the equivalent use of Fx instead of ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
The decomposition of any positive integer N as a sum of positivesubscripted, distinct, nonconsecutive Fibonacci numbers Fk is commonly referred to as the Zeckendorf decomposition ofN (ZD of N, in brief) [10]. This decomposition is always possible and, apart from the equivalent use of Fx instead of F2 (or viceversa), is unique [8].
Recent results and questions in combinatorial game complexities
 NINTH AUSTRALASIAN WORKSHOP ON COMBINATORIAL ALGORITHMS
, 1998
"... Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Isolate the various di ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Isolate the various difficulties separating hard from easy games, and attack them individually. Presentation: Informal; examples of games sampled from various levels illustrate the theory, with emphasis on formulating and motivating new and old research problems.
On Using Patterns in BetaExpansions To Study FibonacciLucas Products
"... The Zeckendorf decomposition of a natural number n is the unique expression of n as a sum of Fibonacci numbers with nonconsecutive indices and with each index greater than 1, where F0 = 0, Fx = 1, and Fi+2 = Ft +Fi+l form the Fibonacci numbers for i> 0 (see [13] and [17], or see [16, pp. 10809]). T ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The Zeckendorf decomposition of a natural number n is the unique expression of n as a sum of Fibonacci numbers with nonconsecutive indices and with each index greater than 1, where F0 = 0, Fx = 1, and Fi+2 = Ft +Fi+l form the Fibonacci numbers for i> 0 (see [13] and [17], or see [16, pp. 10809]). The Zeckendorf decompositions of products of the forms kFm and kLm with
DIAMETERS OF CAYLEY GRAPHS OF CHEVALLEY GROUPS
"... Abstract. We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard twoelement generating set, is at most a constant times n 2 lnk. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard twoelement generating set, is at most a constant times n 2 lnk. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SLn(Z/kZ). We generalize our results to other Chevealley groups over Z/kZ. 1.