A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a non-negative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is order-preserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1

We design a succinct full-text index based on the idea of Huffman-compressing the text and then applying the Burrows-Wheeler transform over it. The resulting structure can be searched as an FM-index, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structure. On a text of length n with zeroorder entropy H0, our index needs O(n(H0 + 1)) bits of space, without any significant dependence on σ. The average search time for a pattern of length m is O(m(H0 + 1)), under reasonable assumptions. Each position of a text occurrence can be located in worst case time O((H0 + 1)log n), while any text substring of length L can be retrieved in O((H0 + 1)L) average time in addition to the previous worst case time. Our index provides a relevant space/time tradeoff between existing succinct data structures, with the additional interest of being easy to implement. We also explore other coding variants alternative to Huffman and exploit their synchronization properties. Our experimental results on various types of texts show that our indexes are highly competitive in the space/time tradeoff map.

Let \Sigma and \Delta be finite alphabets, and let f be a map from \Sigma to \Delta. Then the deterministic automaticity of f , A f (n), is defined to be the size of the minimum finitestate machine that correctly computes f on all inputs of size n. A similar definition applies to languages L. We denote the nondeterministic analogue (for languages L) of automaticity by NL (n). In a previous paper, J. Shallit and Y. Breitbart examined the properties of this measure of descriptional complexity in the case j\Sigmaj 2. In this paper, we continue the study of automaticity, focusing on the case where j\Sigmaj = 1. Research supported in part by DMS-9206784. y Research supported in part by a grant from NSERC. Partial support under NSF Grant DCR 920-8639 and the Wisconsin Alumni Research Foundation. We prove that A f (n) n + 1 \Gamma blog ` nc, where ` = j\Deltaj. We also prove that A f (n) ? n \Gamma 2 log ` n \Gamma 2 log ` log ` n for almost all functions f . In the nondetermi...

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 need 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. 1. Motivation and Introduction Let A = fA 1 ; A 2 ; \Delta \Delta \Delta ; An g be a finite set of elements, called cleartext elements, to be encoded by a static uniquely decipherable (UD) code. For notational ease, we use the term `code' as abbreviation for `set of codewords'; the corresponding encoding and decoding algorithms are always either given or clear from the context. A code i...

Let 0 < α < β be two irrational numbers satisfying 1/α +1/β = 1. Then the sequences a ′ n = ⌊nα⌋, b ′ n = ⌊nβ⌋, n ≥ 1, are complementary: 1 ≤ i < n}, n ≥ 1 over Z≥1, thus a ′ n satisfies: a ′ n = mex1{a ′ i, b ′ i (mex1(S), the smallest positive integer not in the set S). Suppose that c = β − α is an integer. Then b ′ n = a ′ n + cn for all n ≥ 1. We define the following generalization of sequences a ′ n, b ′ n: Let c, n0 ∈ Z≥1, and let X ⊂ Z≥1 be an arbitrary finite set. Let an = mex1(X ∪{ai, bi: 1 ≤ i < n}), bn = an +cn, n ≥ n0. Let sn = a ′ n −an. We show that no matter how we pick c, n0 and X, from some point on the shift sequence sn assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractal-like pattern, which we describe. This work was motivated by a generalization of Wythoff’s game to N ≥ 3 piles.

. We propose and analyze a 2-parameter family of 2-player 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 2-player 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 ...

Abstract. In an earlier work [6] we presented a simple FM-index variant, based on the idea of Huffman-compressing the text and then applying the Burrows-Wheeler 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 Kautz-Zeckendorf coding as a both simple and practical replacement for Huffman. We dub the new index FM-KZ. We also present an efficient implementation of the rank operation, which is the main building brick of the FM-KZ. 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

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.

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 packet-reordering. We present natural and well-motivated channel models and jamming models including the kdistance permuter, the k-buffer permuter and the k-stack permuter. We define the natural information-theoretic (continuous) game between the channel processes (max-min) and the jamming process (min-max) and prove the existence of a Nash equilibrium for the mutual information rate. We study the zero-error (discrete) equivalent and provide error-correcting codes with optimal performance for the distance-bounded 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

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