[Show abstract][Hide abstract] ABSTRACT: An (n, N)–connector of depth d is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length at most d joining each input to its corresponding output. In this paper we consider the problem of construction of sparse depth two connectors with n ≪ N . We use posets of star products and their matching properties to construct such connectors. In particular this gives a simple explicit construction for connectors of size O(N log n/ log log n). Thus our earlier idea to use other posets than the family of subsets of a finite set was successful.
Journal of Combinatorial Theory Series A 11/2006; 113(8). DOI:10.1016/j.jcta.2006.03.009 · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We find the formula for the cardinality of a maximal set of integers from {1,…,n} which does not contain k+1 pairwise coprimes and each integer has a divisor from a specified set of r primes. We also find the explicit formula for this set when r=k+1.
Journal of Combinatorial Theory Series A 11/2006; 113(8):1621-1628. DOI:10.1016/j.jcta.2006.03.015 · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: After Ahlswede introduced identification for source coding he discovered identification entropy and demonstrated that it plays a role analogously to classical entropy in Shannon's noiseless source coding. We give now even more insight into this functional interpreting its two factors
IEEE Transactions on Information Theory 10/2006; 52(9-52):4198 - 4207. DOI:10.1109/TIT.2006.879972 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove that the average error capacity C<sub>q</sub> of a quantum arbitrarily varying channel (QAVC) equals 0 or else the random code capacity lowbarC (Ahlswede's dichotomy). We also establish a necessary and sufficient condition for C<sub>q</sub> > 0
Information Theory, 2006 IEEE International Symposium on; 08/2006
[Show abstract][Hide abstract] ABSTRACT: The two models described in this paper having as ingredients feedback resp. localized errors give possibilities for code constructions not available in the standard model of error correction and also for probabilistic channel models. For the feedback model we present here a coding scheme, which we call the rubber method, because it is based on erasing letters. It is the first scheme achieving the capacity curve for q ges 3. It could be discovered only in the g-ary case for q ges 3, because the letter zero is not used as an information symbol, but solely for error correction. However an extension of the method from using single zeros to blocks of zeros also gives Berlekamp's result - by a different scheme. In the model with feedback and localized errors the help of feedback is addressed. We give an optimal construction for one-error correcting codes with feedback and localized errors
Information Theory, 2006 IEEE International Symposium on; 08/2006
[Show abstract][Hide abstract] ABSTRACT: We continue the investigation of Part I, keep its terminology, and also continue the numbering of sections, equations, theorems
etc.
Consequently we start here with Section 6. As mentioned in Section 4 we present now criteria for a triple (r,t,p) to be k–admissible. Then we consider the f–complexity (extended now to k–ary alphabets) Gk(F)\Gamma_k(\mathcal{F}) of a family F\mathcal{F}. It serves again as a performance parameter of key spaces in cryptography. We give a lower bound for the f–complexity for a family of the type constructed in Part I. In the last sections we explain what can be said about the theoretically
best families F\mathcal{F} with respect to their f–complexity Gk(F)\Gamma_k(\mathcal{F}). We begin with straightforward extensions of the results of [4] for k=2 to general k by using the same Covering Lemma as in [1].
[Show abstract][Hide abstract] ABSTRACT: Let F be a finite family of sets and G(F) be the intersection graph of F (the vertices of G(F) are the sets of family F and the edges of G(F) correspond to intersecting pairs of sets). The transversal number τ(F) is the minimum number of points meeting all sets of F. The independent (stability) number α(F) is the maximum number of pairwise disjoint sets in F. The clique number ω(F) is the maximum number of pairwise intersecting sets in F. The coloring number q(F) is the minimum number of classes in a partition of F into pairwise disjoint sets.
[Show abstract][Hide abstract] ABSTRACT: Consider (X,E)(X,{\mathcal E}), where X is a finite set and E{\mathcal E} is a system of subsets whose union equals X. For every natural number n ∈ℕ define the cartesian products X
n
=∏1
n
X and En=Õ1nE{\mathcal E}_n=\prod_1^n{\mathcal E}. The following problem is investigated: how many sets of En{\mathcal E}_n are needed to cover X
n
? Let this number be denoted by c(n). It is proved that for all n ∈ℕ
exp{Cn} £ c(n) £ exp{Cn+logn+loglog|X|}+1.\exp\{C\cdot n\}\leq c(n)\leq\exp\{Cn+\log n+\log\log|X|\}+1.
A formula for C is given. The result generalizes to the case where X and E{\mathcal E} are not necessarily finite and also to the case of non–identical factors in the product. As applications one obtains estimates
on the minimal size of an externally stable set in cartesian product graphs and also estimates on the minimal number of cliques
needed to cover such graphs.
[Show abstract][Hide abstract] ABSTRACT: Denote by Ω={1,...,n} an n–element set. For all
A,B Î \binomWkA,B\in\binom{\Omega}k, the k–element subsets of Ω, define the relation ~ as follows:
A~B iff A and B have a common shadow, i.e. there is a
C Î \binomWk-1C\in\binom{\Omega}{k-1} with C ⊂A and C ⊂B. For fixed integer α, our goal is to find a family A{\mathcal A} of k–subsets with size α, having as many as possible ~–relations for all pairs of its elements. For k=2 this was achieved by Ahlswede and Katona [2] many years ago.
[Show abstract][Hide abstract] ABSTRACT: To a large extent the present work is far from being conclusive, instead, new directions of research in combinatorial extremal theory are started. Also questions concerning generalizations are immediately noticeable. The incentive came from problems in several fields such as Algebra, Geometry, Probability, Information and Complexity Theory. Like several basic combinatorial problems they may play a role in other fields. For scenarios of interplay we refer also to [R. Ahlswede, “Advances on extremal problems in number theory and combinatorics”, in: C. Casacuberta et al. (eds.), 3rd European congress of mathematics (ECM) Volume I. Basel: Birkhäuser. Prog. Math. 201, 147–175 (2001; Zbl 1094.11001)].
[Show abstract][Hide abstract] ABSTRACT: We analyze wire-tape channels with secure feedback from the legitimate receiver. We present a lower bound on the transmission
capacity (Theorem 1), which we conjecture to be tight and which is proved to be tight (Corollary 1) for Wyner’s original (degraded)
wire-tape channel and also for the reversely degraded wire-tape channel for which the legitimate receiver gets a degraded
version from the enemy (Corollary 2).
Somewhat surprisingly we completely determine the capacities of secure common randomness (Theorem 2) and secure identification
(Theorem 3 and Corollary 3). Unlike for the DMC, these quantities are different here, because identification is linked to
non-secure common randomness.
[Show abstract][Hide abstract] ABSTRACT: We investigate non--binary error correcting codes with noiseless feedback, localized errors, or both. It turns out that the Hamming bound is a central concept. For block codes with feedback we present here a coding scheme based on an idea of erasions, which we call the {bf rubber method}. It gives an optimal rate for big error correcting fraction $tau$ ($>{1over q}$) and infinitely many points on the Hamming bound for small $tau$. We also consider variable length codes with all lengths bounded from above by $n$ and the end of a word carries the symbol $Box$ and is thus recognizable by the decoder. For both, the $Box$-model with feedback and the $Box$-model with localized errors, the Hamming bound is the exact capacity curve for $tau <1/2.$ Somewhat surprisingly, whereas with feedback the capacity curve coincides with the Hamming bound also for $1/2leq tau leq 1$, in this range for localized errors the capacity curve equals 0. Also we give constructions for the models with both, feedback and localized errors. @InProceedings{ahlswede_et_al:DSP:2006:784, author = {Rudolf Ahlswede and Christian Deppe and Vladimir Lebedev}, title = {Non--binary error correcting codes with noiseless feedback, localized errors, or both}, booktitle = {Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, year = {2006}, editor = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, number = {06201}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2006/784}, annote = {Keywords: Error-correcting codes, localized errors, feedback, variable length codes} }
[Show abstract][Hide abstract] ABSTRACT: In the last century together with Levon Khachatrian we established a diametric theorem in Hamming space H<sup>n</sup>=(X<sup>n</sup>,d H ). Now we contribute a diametric theorem for such spaces, if they are endowed with the group structure G<sup>n</sup>=<sup>n</sup>Σ 1 G, the direct sum of a group G on X={0,1,...,q-1}, and as candidates are considered subgroups of G<sup>n</sup>. For all finite groups G, every permitted distance d, and all n≥d subgroups of G<sup>n</sup>with diameter d have maximal cardinality q<sup>d</sup>. Other extremal problems can also be studied in this setting.
Information Theory Workshop, 2006. ITW '06 Punta del Este. IEEE; 04/2006
[Show abstract][Hide abstract] ABSTRACT: In Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0⩽t, k⩽n determine or estimate the maximum number of (0,1)-vectors in a k-dimensional subspace of the Euclidean n-space Rn, such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the (0,1)-vectors of the same weight ω.The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used.In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21]. We also consider a diametric problem under dimension constraint.In Part II, we study the restricted case and solve the problem for t=1 and k<2ω, and also for any fixed 1⩽t⩽ω and k large.
Journal of Combinatorial Theory Series A 04/2006; 113(3-113):483-519. DOI:10.1016/j.jcta.2005.04.009 · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The goals of this seminar have been (1) to identify and match recently developed methods to specific tasks and data sets in a core of application areas; next, based on feedback from the specific applied domain, (2) to fine tune and personalize those applications, and finally (3) to identify and tackle novel combinatorial and algorithmic problems, in some cases all the way to the development of novel software tools. @InProceedings{ahlswede_et_al:DSP:2006:792, author = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, title = {06201 Executive Summary -- Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, booktitle = {Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, year = {2006}, editor = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, number = {06201}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2006/792}, annote = {Keywords: Data compression, pattern matching, pattern discovery, search, sorting, molecular biology, reconstruction, genome rearrangements} }
[Show abstract][Hide abstract] ABSTRACT: From 15.05.06 to 20.05.06, the Dagstuhl Seminar 06201 ``Combinatorial and Algorithmic Foundations of Pattern and Association Discovery'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. @InProceedings{ahlswede_et_al:DSP:2006:787, author = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, title = {06201 Abstracts Collection -- Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, booktitle = {Combinatorial and Algorithmic Foundations of Pattern and Association Discovery}, year = {2006}, editor = {Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein}, number = {06201}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2006/787}, annote = {Keywords: Data compression, pattern matching, pattern discovery, search, sorting, molecular biology, reconstruction, genome rearrangements} }
[Show abstract][Hide abstract] ABSTRACT: Shannon (1948) has shown that a source (U,P,U)({\mathcal {U}},P,U) with output U satisfying Prob (U=u)=P
u
, can be encoded in a prefix code C={cu:u Î U} Ì {0,1}*{\mathcal{C}}=\{c_u:u\in{\mathcal {U}}\}\subset\{0,1\}^* such that for the entropy
H(P)=åu Î U-pulogpu £ åpu|| cu|| £ H(P)+1, H(P)=\sum\limits_{u\in{\mathcal {U}}}-p_u\log p_u\leq\sum p_u|| c_u|| \leq H(P)+1,
where || c
u
|| is the length of c
u
.
We use a prefix code C\mathcal{C} for another purpose, namely noiseless identification, that is every user who wants to know whether a u
(u Î U)(u\in{\mathcal {U}}) of his interest is the actual source output or not can consider the RV C with C=cu=(cu1,...,cu || cu ||)C=c_u=(c_{u_1},\dots,c_{u || c_u ||}) and check whether C=(C
1,C
2,...) coincides with c
u
in the first, second etc. letter and stop when the first different letter occurs or when C=c
u
. Let LC(P,u)L_{\mathcal{C}}(P,u) be the expected number of checkings, if code C\mathcal{C} is used.
Our discovery is an identification entropy, namely the function
HI(P)=2(1-åu Î UPu2).H_I(P)=2\left(1-\sum\limits_{u\in{\mathcal {U}}}P_u^2\right).
We prove that LC(P,P)=åu Î UPuL_{\mathcal{C}}(P,P)=\sum\limits_{u\in{\mathcal {U}}}P_u
LC(P,u) ³ HI(P)L_{\mathcal{C}}(P,u)\geq H_I(P) and thus also that
L(P)=minCmaxu Î ULC(P,u) ³ HI(P) L(P)=\min\limits_{\mathcal{C}}\max\limits_{u\in{\mathcal {U}}}L_{\mathcal{C}}(P,u)\geq H_I(P)
and related upper bounds, which demonstrate the operational significance of identification entropy in noiseless source coding
similar as Shannon entropy does in noiseless data compression.
Also other averages such as
[`(L)]C(P)=\frac1|U| åu Î ULC(P,u)\bar L_{\mathcal{C}}(P)=\frac1{|{\mathcal {U}}|} \sum\limits_{u\in{\mathcal {U}}}L_{\mathcal{C}}(P,u) are discussed in particular for Huffman codes where classically equivalent Huffman codes may now be different.
We also show that prefix codes, where the codewords correspond to the leaves in a regular binary tree, are universally good
for this average.
General Theory of Information Transfer and Combinatorics; 01/2006
[Show abstract][Hide abstract] ABSTRACT: We introduce a new aspect of the influence of the information-theoretical methods on the statistical theory. The procedures
of the probability distributions identification for K(≥1) random objects each having one from the known set of M(≥2) distributions are studied. N-sequences of discrete independent random variables represent results of N observations for each of K objects. On the base of such samples decisions must be made concerning probability distributions of the objects. For N ® ¥N \longrightarrow \infty the exponential decrease of the test’s error probabilities is considered. The reliability matrices of logarithmically asymptotically
optimal procedures are investigated for some models and formulations of the identification problems. The optimal subsets of
reliabilities which values may be given beforehand and conditions guaranteeing positiveness of all the reliabilities are investigated.
General Theory of Information Transfer and Combinatorics; 01/2006
[Show abstract][Hide abstract] ABSTRACT: We consider codes over the alphabet Q={0,1,..,q-1}intended for the control of unidirectional errors of level l. That is, the transmission channel is such that the received word cannot contain both a component larger than the transmitted one and a component smaller than the transmitted one. Moreover, the absolute value of the difference between a transmitted component and its received version is at most l. We introduce and study q-ary codes capable of correcting all unidirectional errors of level l. Lower and upper bounds for the maximal size of those codes are presented. We also study codes for this aim that are defined by a single equation on the codeword coordinates(similar to the Varshamov-Tenengolts codes for correcting binary asymmetric errors). We finally consider the problem of detecting all unidirectional errors of level l.
[Show abstract][Hide abstract] ABSTRACT: In order to put the present model and our results into the right perspectives we describe first key steps in multiuser source coding theory.
General Theory of Information Transfer and Combinatorics; 01/2006