Ulrich Tamm

Lattice paths not touching a given boundary

Abstract Lattice paths are enumerated which first touch a periodic boundary at time n. Following a probabilistic method introduced by Gessel, for period length 2 formulae are obtained for a wide class of boundaries. This allows to give the generating function for paths not crossing or touching the diagonal cx=2y for odd c and to obtain a closed formula similar to the ballot numbers for the sum of the entries of two two-dimension arrays related to these boundaries.

Splittings of cyclic groups and perfect shift codes

A splitting (M,S) of an additive Abelian group G consists of a set of integers M and a subset S/spl sub/G such that every nonzero element g/spl isin/G can be uniquely written as m/spl middot/h for some m/spl isin/M and h/spl isin/S. Splittings M={/spl plusmn/1,/spl middot//spl middot//spl middot/,/spl plusmn/k} correspond to perfect k-shift codes used in the analysis of run-length-limited codes correcting single peak shifts. We shall determine the set S for splittings of cyclic groups Z/sub p/, p prime, by M={1,a,/spl middot//spl middot//spl middot/,a/sup r/,b,/spl middot//spl middot//spl middot/,b/sup s/} and M={/spl plusmn/1,/spl plusmn/a,/spl middot//spl middot//spl middot/,/spl plusmn/a/sup r/,/spl plusmn/b,/spl middot//spl middot//spl middot/,/spl plusmn/b/sup s/}. This yields new conditions on the existence of perfect 3- and 4-shift codes. Further, it can be shown that splittings of Z/sub p/ by {/spl plusmn/1,/spl plusmn/2,/spl plusmn/3} exist exactly if Z/sub p/ is also split by {1,2,3}.

Communication complexity in lattices

The communcation complexity of functions defined in lattices is bounded from above and below, hereby generalizing former results of Lovasz [1] and Ahlswede and Cai [2]. Especially in geometric lattices, upper and lower bound often differ by at most one bit.

Communication Complexity of Sum-Type Functions Invariant under Translation

The communication complexity of a function f denotes the number of bits that two processors have to exchange in order to compute f(x, y), when each processor knows one of the variables x and y, respectively. In this paper the deterministic communication complexity of sum-type functions, such as the Hamming distance and the Lee distance, is examined. Here f: X × X ? G, where X is a finite set and G is an Abelian group, and the sum-type function fn:Xn × Xn ? G is defined by fn((x1, ..., xn), (y1, ..., yn)) = ?ni=1f(xi, yi) Since the functions examined are also translation-invariant, their function matrices are simultaneously diagonalizable and the corresponding eigenvalues can be calculated. This allows to apply a rank lower bound for the communication complexity. The best results are obtained for G = Z/2Z. For prime numbers |X| in this case the communication complexity of all non-trivial sum-type functions is determined exactly. Exact results are also obtained for the parity of the Hamming distance and the parity of the Lee distance. For the Hamming distance and the Lee distance exact results are only obtained for special parameters n and |X|.

On perfect 3-shift N-designs

A necessary condition is given for the existence of a perfect 3-shift N-design in a group of order 6N+1. Further, some examples for perfect shift designs in cyclic groups Z/sub 6N+1/ are presented.

Deterministic communication complexity of set intersection

Abstract In this paper the communication complexity C(mn) of the cardinality of set intersection, mn say, will be determined up to one bit: n + ⌈log2(n + 1)⌉ − 1 ⩽ C(mn) ⩽ n + ⌈log2(n + 1)⌉. The proof for the lower bound can also be applied to a larger class of “sum-type” functions sharing the property that f(0,y) = f(x,0) = 0 for all possible x,y. Furthermore, using Krafts inequality for prefix codes, it is possible to find a communication protocol, which for n = 2t, t ⩾ 2, assumes the lower bound. The upper bound is assumed for n = 2 t − 1, t ϵ N .

Communication Complexity of Functions on Direct Sums

The paper surveys direct sum methods in communication complexity, mostly concentrating on the results obtained by several authors in the research group of Rudolf Ahlswede in Bielefeld. Lower bound techniques are investigated which behave multiplicatively for functions defined on direct sums of sets. Applications, as the exact or asymptotic determination of the communication complexity and the comparison of bounding techniques are discussed.

On a problem of Berlekamp

At the 3rd Waterloo Conference on Combinatorics Berlekamp presented the following combinatorial problem, which originally arose in his studies on burst-error correcting convolution codes Berlekamp, ER (1963). The problem will be illustrated with example. We shall analyze an equivalent lattice path model and use a probablistic approach due to Gessel, I (1986).

Still another rank determination of set intersection matrices with an application in communication complexity

Abstract The set-intersection function gives the cardinality of the intersection of two sets. In order to obtain a lower bound for the communication complexity of this function, the rank of the corresponding characteristic function-value matrices was calculated in [1] and [2]. In this note, the rank of these matrices is determined by another method of proof, which makes use of a factorization into a product of an upper and a lower triangular matrix.

Size of downsets in the pushing order and a problem of Berlekamp

The shifting technique is a useful tool in extremal set theory. It was successfully used and developed by Levon Khachatrian to obtain many significant results. The shifting operation also referred to as pushing gives rise to a partial order called pushing order. Here we consider the problem of determination of the size of special downsets in this order. For the analysis, the pushing order will be expressed isomorphically in terms of lattice paths and of majorization of sequences. In the case that the sequences under consideration are periodic the generating function for the numbers arising in an old combinatorial problem due to Berlekamp will be determined.