Ferdinando Cicalese

ALGORITHMS FOR JUMBLED PATTERN MATCHING IN STRINGS

The Parikh vector p(s) of a string s over a finite ordered alphabet Σ = {a1, …, aσ} is defined as the vector of multiplicities of the characters, p(s) = (p1, …, pσ), where pi = |{j | sj = ai}|. Parikh vector q occurs in s if s has a substring t with p(t) = q. The problem of searching for a query q in a text s of length n can be solved simply and worst-case optimally with a sliding window approach in O(n) time. We present two novel algorithms for the case where the text is fixed and many queries arrive over time. The first algorithm only decides whether a given Parikh vector appears in a binary text. It uses a linear size data structure and decides each query in O(1) time. The preprocessing can be done trivially in Θ(n2) time. The second algorithm finds all occurrences of a given Parikh vector in a text over an arbitrary alphabet of size σ ≥ 2 and has sub-linear expected time complexity. More precisely, we present two variants of the algorithm, both using an O(n) size data structure, each of which can be constructed in O(n) time. The first solution is very simple and easy to implement and leads to an expected query time of , where m = ∑i qi is the length of a string with Parikh vector q. The second uses wavelet trees and improves the expected runtime to , i.e., by a factor of log m.

Latency-Bounded Target Set Selection in Social Networks

We study variants of the Target Set Selection problem, first proposed by Kempe et al. In our scenario one is given a graph G = (V,E), integer values t(v) for each vertex v, and the objective is to determine a small set of vertices (target set) that activates a given number (or a given subset) of vertices of G within a prescribed number of rounds. The activation process in G proceeds as follows: initially, at round 0, all vertices in the target set are activated; subsequently at each round r ≥ 1 every vertex of G becomes activated if at least t(v) of its neighbors are active by round r − 1. It is known that the problem of finding a minimum cardinality Target Set that eventually activates the whole graph G is hard to approximate to a factor better than \(O(2^{\log^{1-\epsilon }|V|})\). In this paper we give exact polynomial time algorithms to find minimum cardinality Target Sets in graphs of bounded clique-width, and exact linear time algorithms for trees.

On Approximate Jumbled Pattern Matching in Strings

Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of a Parikh vector q in the text s requires finding a substring t of s with p(t)=q. This can be viewed as the task of finding a jumbled (permuted) version of a query pattern, hence the term Jumbled Pattern Matching. We present several algorithms for the approximate version of the problem: Given a string s and two Parikh vectors u,v (the query bounds), find all maximal occurrences in s of some Parikh vector q such that u≤q≤v. This definition encompasses several natural versions of approximate Parikh vector search. We present an algorithm solving this problem in sub-linear expected time using a wavelet tree of s, which can be computed in time O(n) in a preprocessing phase. We then discuss a Scrabble-like variation of the problem, in which a weight function on the letters of s is given and one has to find all occurrences in s of a substring t with maximum weight having Parikh vector p(t)≤v. For the case of a binary alphabet, we present an algorithm which solves the decision version of the Approximate Jumbled Pattern Matching problem in constant time, by indexing the string in subquadratic time.

Least adaptive optimal search with unreliable tests

We consider the basic problem of searching for an unknown m-bit number by asking the minimum possible number of yes-no questions, when up to a finite number e of the answers may be erroneous. In case the (i+1)th question is adaptively asked after receiving the answer to the ith question, the problem was posed by Ulam and R&enyi and is strictly related to Berlekamps theory of error correcting communication with noiseless feedback. Conversely, in the fully non-adaptive model when all questions are asked before knowing any answer, the problem amounts to finding a shortest e-error correcting code. Let qe(m) be the smallest integer q satisfying Berlekamps bound i=0e()2qm. Then at least qe(m) questions are necessary, in the adaptive, as well as in the non-adaptive model. In the fully adaptive case, optimal searching strategies using exactly qe(m) questions always exist up to finitely many exceptional ms. At the opposite non-adaptive case, searching strategies with exactly qe(m) questions or equivalently, e-error correcting codes with 2m codewords of length qe(m)---are rather the exception, already for e=2, and are generally not known to exist for e>2. In this paper, for each e>1 and all sufficiently large m, we exhibit searching strategies that use a first batch of m non-adaptive questions and then, only depending on the answers to these m questions, a second batch of qe(m)m non-adaptive questions. These strategies are automatically optimal. Since even in the fully adaptive case, qe(m)1 questions do not suffice to find the unknown number, and qe(m) questions generally do not suffice in the non-adaptive case, the results of our paper provide e, fault tolerant searching strategies with minimum adaptiveness and minimum number of tests.

Optimal Coding with One Asymmetric Error: Below the Sphere Packing Bound

Ulam and REnyi asked what is the minimum number of yes-no questions needed to find an unknown m-bit number x, if up to l of the answers may be erroneous/mendacious. For each l it is known that, up to only finitely many exceptional m, one can find x asking Berlekamps minimum number ql(m) of questions, i.e., the smallest integer q satisfying the sphere packing bound for error-correcting codes. The Ulam-REnyi problem amounts to finding optimal error-correcting codes for the binary symmetric channel with noiseless feedback, first considered by Berlekamp. In such concrete situations as optical transmission, error patterns are highly asymmetric--in that only one of the two bits can be distorted. Optimal error-correcting codes for these asymmetric channels with feedback are the solutions of the half-lie variant of the Ulam-REnyi problem, asking for the minimum number of yes-no questions needed to find an unknown m-bit number x, if up to l of the negative answers may be erroneous/mendacious. Focusing attention on the case l = 1; in this self-contained paper we shall give tight upper and lower bounds for the half-lie problem. For infinitely many ms our bounds turn out to be matching, and the optimal solution is explicitly given, thus strengthening previous estimates by Rivest, Meyer et al.

Optimal strategies against a liar

Abstract We consider the following scenario: There are two individuals, say Q (Questioner) and R (Responder), involved in a search game. Player R chooses a number, say x , from the set S={1,…,M} . Player Q has to find out x by asking questions of type: “which one of the sets A 1 ,A 2 ,…,A q , does x belong to?”, where the sets A 1 ,…,A q constitute a partition of S . Player R answers “ i ” to indicate that the number x belongs to A i . We are interested in the least number of questions player Q has to ask in order to be always able to correctly guess the number x , provided that R can lie at most e times. The case e=0 obviously reduces to the classical q -ary search, and the necessary number of questions is [ log q M] . The case q=2 and e⩾1 has been widely studied, and it is generally referred to as Ulams game. In this paper we consider the general case of arbitrary q⩾2 . Under the assumption that player R is allowed to lie at most twice throughout the game, we determine the minimum number of questions Q needs to ask in order to successfully search for x in a set of cardinality M=q i , for any i⩾1 . As a corollary, we obtain a counterexample to a recently proposed conjecture of Aigner, for the case of an arbitrary number of lies. We also exactly solve the problem when player R is allowed to lie a fixed but otherwise arbitrary number of times e , and M=q i , with i not too large with respect to q . For the general case of arbitrary M , we give fairly tight upper and lower bounds on the number of the necessary questions.

Rota-Metropolis cubic logic and Ulam-Renyi games

In their paper [43] Rota and Metropolis considered the partially ordered set F n of all nonempty faces of the n-cube [0, 1] n for each n = 1, 2,…, equipped with the following operation: (⊔) taking the supremum A⊔ B of any two faces A and B of F n , together with the following two partially defined operations: (⊓) taking the set-theoretic intersection A ⊓ B of any two intersecting faces A and B of F n , and (Δ) when a face A is contained in another face B, taking the antipode Δ (B, A) of A in B.

Optimal Binary Search with Two Unreliable Tests and Minimum Adaptiveness

What is the minimum number of yes-no questions needed to find an m bit number x in the set S = {0,1,...,2m -1} if up to l answers may be erroneous/false ? In case when the (t+1)th question is adaptively asked after receiving the answer to the tth question, the problem, posed by Ulam and Renyi, is a chapter of Berlekamps theory of error-correcting communication with feedback. It is known that, with finitely many exceptions, one can find x asking Berlekamps minimum number ql(m) of questions, i.e., the smallest integer q such that 2q ? 2m((q l)+(q l-1)+...+(q 2)+q+1). At the opposite, nonadaptive extreme, when all questions are asked in a unique batch before receiving any answer, a search strategy with ql(m) questions is the same as an l-error correcting code of length ql(m) having 2m codewords. Such codes in general do not exist for l > 1: Focusing attention on the case l = 2, we shall show that, with the exception of m = 2 and m = 4, one can always find an unknown m bit number x ? S by asking q2(m) questions in two nonadaptive batches. Thus the results of our paper provide shortest strategies with as little adaptiveness/interaction as possible.

Graphs of separability at most 2

We introduce graphs of separability at mostk as graphs in which every two non-adjacent vertices are separated by a set of at most k other vertices. Graphs of separability at most k arise in connection with the Parsimony Haplotyping problem from computational biology. For k@?{0,1}, the only connected graphs of separability at most k are complete graphs and block graphs, respectively. For k>=3, graphs of separability at most k form a rich class of graphs containing all graphs of maximum degree k. We prove several characterizations of graphs of separability at most 2, which generalize complete graphs, cycles and trees. The main result is that every connected graph of separability at most 2 can be constructed from complete graphs and cycles by pasting along vertices or edges, and vice versa, every graph constructed this way is of separability at most 2. The structure theorem has nice algorithmic implications-some of which cannot be extended to graphs of higher separability-however certain optimization problems remain intractable on graphs of separability 2. We then characterize graphs of separability at most 2 in terms of minimal forbidden induced subgraphs and minimal forbidden induced minors. Finally, we discuss the possibilities of extending these results to graphs of higher separability.

Supermodularity and subadditivity properties of the entropy on the majorization lattice

We prove that the entropy is a supermodular and subadditive function on the lattice of all n-dimensional probability distributions, ordered according to the partial order relation defined by majorization among vectors.