Streaming algorithms for language recognition problems

We study the complexity of the following problems in the streaming model. Membership testing for \DLIN We show that every language in \DLIN\ can be recognised by a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass $O(n/p)$ space algorithm. We show that these algorithms are optimal. Membership testing for \LL$(k)$ For languages generated by \LL$(k)$ grammars with a bound of $r$ on the number of nonterminals at any stage in the left-most derivation, we show that membership can be tested by a randomized one-pass $O(r\log n)$ space algorithm with inverse polynomial (in $n$) one-sided error. Membership testing for \DCFL We show that randomized algorithms as efficient as the ones described above for \DLIN\ and \LL(k) (which are subclasses of \DCFL) cannot exist for all of \DCFL: there is a language in \VPL\ (a subclass of \DCFL) for which any randomized p-pass algorithm with error bounded by $\epsilon < 1/2$ must use $\Omega(n/p)$ space. Degree sequence problem We study the problem of determining, given a sequence $d_1, d_2,..., d_n$ and a graph $G$, whether the degree sequence of $G$ is precisely $d_1, d_2,..., d_n$. We give a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error probability. We show that our algorithms are optimal. Our randomized algorithms are based on the recent work of Magniez et al. \cite{MMN09}; our lower bounds are obtained by considering related communication complexity problems

Exponential Separation of Quantum Communication and Classical Information

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.Comment: v1, 36 pages, 3 figure

Quantum Information Complexity and Amortized Communication

We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum channels, and then investigate the properties of such quantities. These are the fully quantum generalizations of the analogous quantities for bipartite classical functions that have found many applications recently, in particular for proving communication complexity lower bounds. Our definition is strongly tied to the quantum state redistribution task. Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite channel on a given state. This generalizes a result of Braverman and Rao to quantum protocols, and even strengthens the classical result in a bounded round scenario. Also, this provides an analogue of the Schumacher source compression theorem for interactive quantum protocols, and answers a question raised by Braverman. We also discuss some potential applications to quantum communication complexity lower bounds by specializing our definition for classical functions and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new evidence suggesting that the bounded round quantum communication complexity of the disjointness function is \Omega (n/M + M), for M-message protocols. This would match the best known upper bound.Comment: v1, 38 pages, 1 figur

Calculs sur des grosses données (algorithmes de streaming et communication entre deux joueurs)

Dans cette thèse on considère deux modèles de calcul qui abordent des problèmes qui se posent lors du traitement des grosses données. Le premier modèle est le modèle de streaming. Lors du traitement des grosses données, un accès aux données de façon aléatoire est trop couteux. Les algorithmes de streaming ont un accès restreint aux données: ils lisent les données de façon séquentielle (par passage) une fois ou peu de fois. De plus, les algorithmes de streaming utilisent une mémoire d'accès aléatoire de taille sous-linéaire dans la taille des données. Le deuxième modèle est le modèle de communication. Lors du traitement des données par plusieurs entités de calcul situées à des endroits différents, l'échange des messages pour la synchronisation de leurs calculs est souvent un goulet d'étranglement. Il est donc préférable de minimiser la quantité de communication. Un modèle particulier est la communication à sens unique entre deux participants. Dans ce modèle, deux participants calculent un résultat en fonction des données qui sont partagées entre eux et la communication se réduit à un seul message. On étudie les problèmes suivants: 1) Les couplages dans le modèle de streaming. L'entrée du problème est un flux d'arêtes d'un graphe G=(V,E) avec n=|V|. On recherche un algorithme de streaming qui calcule un couplage de grande taille en utilisant une mémoire de taille O(n polylog n). L'algorithme glouton remplit ces contraintes et calcule un couplage de taille au moins 1/2 fois la taille d'un couplage maximum. Une question ouverte depuis longtemps demande si l'algorithme glouton est optimal si aucune hypothèse sur l'ordre des arêtes dans le flux est faite. Nous montrons qu'il y a un meilleur algorithme que l'algorithme glouton si les arêtes du graphe sont dans un ordre uniformément aléatoire. De plus, nous montrons qu'avec deux passages on peut calculer un couplage de taille strictement supérieur à 1/2 fois la taille d'un couplage maximum sans contraintes sur l'ordre des arêtes. 2) Les semi-couplages en streaming et en communication. Un semi-couplage dans un graphe biparti G=(A,B,E) est un sous-ensemble d'arêtes qui couple tous les sommets de type A exactement une fois aux sommets de type B de façon pas forcement injective. L'objectif est de minimiser le nombre de sommets de type A qui sont couplés aux même sommets de type B. Pour ce problème, nous montrons un algorithme qui, pour tout 00 nous montrons qu'il y a un protocole presque optimal de communication avec coût de communication Ô(kd) tel que les déplacements des points effectués par Bob aboutissent à un facteur d'approximation de O(d) par rapport aux meilleurs déplacements de d points.In this PhD thesis, we consider two computational models that address problems that arise when processing massive data sets. The first model is the Data Streaming Model. When processing massive data sets, random access to the input data is very costly. Therefore, streaming algorithms only have restricted access to the input data: They sequentially scan the input data once or only a few times. In addition, streaming algorithms use a random access memory of sublinear size in the length of the input. Sequential input access and sublinear memory are drastic limitations when designing algorithms. The major goal of this PhD thesis is to explore the limitations and the strengths of the streaming model. The second model is the Communication Model. When data is processed by multiple computational units at different locations, then the message exchange of the participating parties for synchronizing their calculations is often a bottleneck. The amount of communication should hence be as little as possible. A particular setting is the one-way two-party communication setting. Here, two parties collectively compute a function of the input data that is split among the two parties, and the whole message exchange reduces to a single message from one party to the other one. We study the following four problems in the context of streaming algorithms and one-way two-party communication: (1) Matchings in the Streaming Model. We are given a stream of edges of a graph G=(V,E) with n=|V|, and the goal is to design a streaming algorithm that computes a matching using a random access memory of size O(n polylog n). The Greedy matching algorithm fits into this setting and computes a matching of size at least 1/2 times the size of a maximum matching. A long standing open question is whether the Greedy algorithm is optimal if no assumption about the order of the input stream is made. We show that it is possible to improve on the Greedy algorithm if the input stream is in uniform random order. Furthermore, we show that with two passes an approximation ratio strictly larger than 1/2 can be obtained if no assumption on the order of the input stream is made. (2) Semi-matchings in Streaming and in Two-party Communication. A semi-matching in a bipartite graph G=(A,B,E) is a subset of edges that matches all A vertices exactly once to B vertices, not necessarily in an injective way. The goal is to minimize the maximal number of A vertices that are matched to the same B vertex. We show that for any 00, we show that there is an almost tight randomized protocol with communication cost Ô(kd) such that Bob's adjustments lead to an O(d)-approximation compared to the k best possible adjustments that Bob could make.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

Quantum conditional mutual information and approximate Markov chains

A state on a tripartite quantum system $A \otimes B \otimes C$ forms a Markov chain if it can be reconstructed from its marginal on $A \otimes B$ by a quantum operation from $B$ to $B \otimes C$. We show that the quantum conditional mutual information $I(A: C | B)$ of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.Comment: v3: 31 pages, corrected error in dimension factor for the application to squashed entanglement (Corollary D.3) and added some clarifications and remarks. v2: 30 pages, improved introduction and added application to squashed entanglemen