Publications (51)14.7 Total impact

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ABSTRACT: We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set \(M\) consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time \({{\mathrm{HT}}}(P,M)\) of any reversible random walk \(P\) on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity \({\hbox {HT}}^{+}(P,M)\) which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk \(P\) and the absorbing walk \(P'\) , whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk \(P\) is not statetransitive. We also provide algorithms in the cases when only approximations or bounds on parameters \(p_M\) (the probability of picking a marked vertex from the stationary distribution) and \({\hbox {HT}}^{+}(P,M)\) are known.Algorithmica 03/2015; DOI:10.1007/s0045301599798 · 0.57 Impact Factor 
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ABSTRACT: We consider the randomized decision tree complexity of the recursive 3majority function. We prove a lower bound of $(1/2\delta) \cdot 2.57143^h$ for the twosidederror randomized decision tree complexity of evaluating height $h$ formulae with error $\delta \in [0,1/2)$. This improves the lower bound of $(12\delta)(7/3)^h$ given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of $(12\delta) \cdot 2.55^h$ given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zeroerror randomized decision tree algorithm that has complexity at most $(1.007) \cdot 2.64944^h$. The previous best known algorithm achieved complexity $(1.004) \cdot 2.65622^h$. The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel "interleaving" of two recursive algorithms. 
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ABSTRACT: We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) pparallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for kSUM. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical pparallel complexity are polynomially related for all total functions when p is not too large. 
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ABSTRACT: This work revisits the study of streaming algorithms where both input and output are data streams. While streaming algorithms with multiple streams have been studied before, such as in the context of sorting, most assumed very nonrestrictive models and thus often had weak lower bounds. For this reason, we consider data streams with restricted access, such as readonly and writeonly streams, as opposed to readwrite streams. We also require streams to be processed in one direction only, and forbid the use of any other external streams. For readwrite streams, we introduce a new complexity measure, the expansion, that is the ratio between the maximal size of the stream during the computation and the input size. We first study the problem of reversing a stream of length n in our models, and give several tight bounds. In the readonly and writeonly model, we show that ppass algorithms need memory space {\Theta}(n/p). But if one of the stream is readwrite, then the complexity falls to {\Theta}(n/p^2) (with some ad ditional restrictions for the lower bound), and to polylog(n) when p = O(log n) if both streams are readwrite. We then study the problem of sorting and give several algorithms with small expansion. Our main sorting algorithm is randomized and has constant expansion, whereas previously known algorithms (without additional external streams) had linear expansion. 
Conference Paper: TimeEfficient quantum walks for 3distinctness
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ABSTRACT: We present two quantum walk algorithms for 3Distinctness. Both algorithms have time complexity $\tilde{O}(n^{5/7})$, improving the previous $\tilde{O}(n^{3/4})$ and matching the best known upper bound for query complexity (obtained via learning graphs) up to log factors. The first algorithm is based on a connection between quantum walks and electric networks. The second algorithm uses an extension of the quantum walk search framework that facilitates quantum walks with nested updates.Proceedings of the 40th international conference on Automata, Languages, and Programming  Volume Part I; 07/2013 
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ABSTRACT: Model checking and testing are two areas with a similar goal: to verify that a system satisfies a property. They start with different hypothesis on the systems and develop many techniques with different notions of approximation, when an exact verification may be computationally too hard. We present some notions of approximation with their logic and statistics backgrounds, which yield several techniques for model checking and testing: Bounded Model Checking, Approximate Model Checking, Approximate BlackBox Checking, Approximate Modelbased Testing and Approximate Probabilistic Model Checking. All these methods guarantee some quality and efficiency of the verification. 
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ABSTRACT: We present an extension to the quantum walk search framework that facilitates quantum walks with nested updates. We apply it to give a quantum walk algorithm for 3Distinctness with query complexity ~O(n^{5/7}), matching the best known upper bound (obtained via learning graphs) up to log factors. Furthermore, our algorithm has time complexity ~O(n^{5/7}), improving the previous ~O(n^{3/4}). 
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ABSTRACT: We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the $O(n^{35/27})$ and $O(n^{9/7})$ learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds. 
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ABSTRACT: We show that the quantum query complexity of detecting if an $n$vertex graph contains a triangle is $O(n^{9/7})$. This improves the previous best algorithm of Belovs making $O(n^{35/27})$ queries. For the problem of determining if an operation $\circ : S \times S \rightarrow S$ is associative, we give an algorithm making $O(S^{10/7})$ queries, the first improvement to the trivial $O(S^{3/2})$ application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constantsized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple highlevel language; our main theorem shows how this highlevel language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms. 
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ABSTRACT: This work is in the line of designing efficient checkers for testing the reliability of some massive data structures. Given a sequential access to the insert/extract operations on such a structure, one would like to decide, a posteriori only, if it corresponds to the evolution of a reliable structure. In a context of massive data, one would like to minimize both the amount of reliable memory of the checker and the number of passes on the sequence of operations. Chu, Kannan and McGregor initiated the study of checking priority queues in this setting. They showed that use of timestamps allows to check a priority queue with a single pass and memory space O(N^(1/2)), up to a polylogarithmic factor. Later, Chakrabarti, Cormode, Kondapally and McGregor removed the use of timestamps, and proved that more passes do not help. We show that, even in the presence of timestamps, more passes do not help, solving a previously open problem. On the other hand, we show that a second pass, but in reverse direction, shrinks the memory space to O((log N)^2), extending a phenomenon the first time observed by Magniez, Mathieu and Nayak for checking wellparenthesized expressions. 
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ABSTRACT: The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently. 
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ABSTRACT: We present three semistreaming algorithms for Maximum Bipartite Matching with one and two passes. Our onepass semistreaming algorithm is deterministic and returns a matching of size at least $1/2+0.005$ times the optimal matching size in expectation, assuming that edges arrive one by one in (uniform) random order. Our first twopass algorithm is randomized and returns a matching of size at least $1/2+0.019$ times the optimal matching size in expectation (over its internal random coin flips) for any arrival order. These two algorithms apply the simple Greedy matching algorithm several times on carefully chosen subgraphs as a subroutine. Furthermore, we present a twopass deterministic algorithm for any arrival order returning a matching of size at least $1/2+0.019$ times the optimal matching size. This algorithm is built on ideas from the computation of semimatchings. 
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ABSTRACT: Let $H$ be a fixed $k$vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the boundederror quantum query complexity of determining if an $n$vertex graph contains $H$ as a subgraph is $O(n^{22/kt})$, where $ t = \max{\frac{k^2 2(m+1)}{k(k+1)(m+1)}, \frac{2k  d  3}{k(d+1)(md+2)}}$. The previous best algorithm of Magniez et al. had complexity $\widetilde O(n^{22/k})$.09/2011; DOI:10.4086/cjtcs.2012.010 
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ABSTRACT: We extend the study of the complexity of finding an $\eps$approximate Nash equilibrium in congestion games from the case of positive delay functions to delays of arbitrary sign. We first prove that in symmetric games with $\alpha$bounded jump the $\eps$Nash dynamic converges in polynomial time when all delay functions are negative, similarly to the case of positive delays. We then establish a hardness result for symmetric games with $\alpha$bounded jump and with arbitrary delay functions: in that case finding an $\eps$Nash equilibrium becomes $\PLS$complete. 
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ABSTRACT: We study the problem of validating XML documents of size $N$ against general DTDs in the context of streaming algorithms. The starting point of this work is a wellknown space lower bound. There are XML documents and DTDs for which $p$pass streaming algorithms require $\Omega(N/p)$ space. We show that when allowing access to external memory, there is a deterministic streaming algorithm that solves this problem with memory space $O(\log^2 N)$, a constant number of auxiliary read/write streams, and $O(\log N)$ total number of passes on the XML document and auxiliary streams. An important intermediate step of this algorithm is the computation of the FirstChildNextSibling (FCNS) encoding of the initial XML document in a streaming fashion. We study this problem independently, and we also provide memory efficient streaming algorithms for decoding an XML document given in its FCNS encoding. Furthermore, validating XML documents encoding binary trees in the usual streaming model without external memory can be done with sublinear memory. There is a onepass algorithm using $O(\sqrt{N \log N})$ space, and a bidirectional twopass algorithm using $O(\log^2 N)$ space performing this task.ACM Transactions on Database Systems 12/2010; DOI:10.1145/2274576.2274581 · 0.75 Impact Factor 
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ABSTRACT: We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk $P$ on the graph. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of the interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not statetransitive, and in the presence of multiple marked vertices. As a consequence we make a progress on an open problem related to the spatial search on the 2Dgrid. Comment: 15 pages 
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ABSTRACT: We study the complexity of validating XML documents against any given DTD in the context of streaming algorithms with external memory. We design a deterministic algorithm that solves this problem with memory space $O(\log^2 N)$, a constant number of auxiliary read/write streams, and $O(\log N)$ total number of passes on the XML document of size $N$ and auxiliary streams. An important intermediate step is the memoryefficient computation of the FCNS encoding of the initial XML document. Then, validity can already be decided in onepass with memory space $O(\sqrt{N\log N})$, and no auxiliary streams. A second but reverse pass makes the memory space collapse to $O(\log^2 N)$. This suggests a systematic use of the FCNS encoding for large XML documents, since, without this encoding, there are DTDs against which validating XML documents requires memory space $\Omega(N/p)$ for any $p$pass streaming algorithm without auxiliary streams, even if randomization is allowed. Last, for the special case of validating XML documents encoding binary trees, we give a deterministic onepass algorithm with memory space $O(\sqrt{N})$, and prove its optimality, up to a multiplicative constant, even if randomization is allowed. 
Conference Paper: Recognizing wellparenthesized expressions in the streaming model.
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ABSTRACT: Motivated by a concrete problem and with the goal of understanding the relationship between the complexity of streaming algorithms and the computational complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with s different types of parenthesis. We present a onepass randomized streaming algorithm for Dyck(2) with space O(√ n log(n)) bits, time per letter polylog(n), and onesided error. We prove that this onepass algorithm is optimal, up to a log(n) factor, even when twosided error is allowed, and conjecture that a similar bound holds for any constant number of passes over the input. Surprisingly, the space requirement shrinks drastically if we have access to the input stream "in reverse". We present a twopass randomized streaming algorithm for Dyck(2) with space O((log n)2), time polylog(n) and onesided error, where the second pass is in the reverse direction. Both algorithms can be extended to Dyck(s) since this problem is reducible to Dyck(2) for a suitable notion of reduction in the streaming model. Except for an extra O(√ log(s)) multiplicative overhead in the space required in the onepass algorithm, the resource requirements are of the same order. For the lower bound, we exhibit hard instances Ascension(m) of Dyck(2) with length Θ(mn). We embed these in what we call a "onepass" communication problem with 2mplayers, where m=~O(n). To establish the hardness of Ascension(m), we prove a direct sum result by following the "information cost" approach, but with a few twists. Indeed, we play a subtle game between public and private coins for Mountain, which corresponds to a primitive instance Ascension(1). This mixture between public and private coins for m results from a balancing act between the direct sum result and a combinatorial lower bound for m.Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 58 June 2010; 01/2010 

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ABSTRACT: Motivated by a concrete problem and with the goal of understanding the sense in which the complexity of streaming algorithms is related to the complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with $s$ different types of parenthesis. We present a onepass randomized streaming algorithm for Dyck(2) with space $\Order(\sqrt{n}\log n)$, time per letter $\polylog (n)$, and onesided error. We prove that this onepass algorithm is optimal, up to a $\polylog n$ factor, even when twosided error is allowed. For the lower bound, we prove a direct sum result on hard instances by following the "information cost" approach, but with a few twists. Indeed, we play a subtle game between public and private coins. This mixture between public and private coins results from a balancing act between the direct sum result and a combinatorial lower bound for the base case. Surprisingly, the space requirement shrinks drastically if we have access to the input stream in reverse. We present a twopass randomized streaming algorithm for Dyck(2) with space $\Order((\log n)^2)$, time $\polylog (n)$ and onesided error, where the second pass is in the reverse direction. Both algorithms can be extended to Dyck(s) since this problem is reducible to Dyck(2) for a suitable notion of reduction in the streaming model. Comment: 20 pages, 5 figures
Publication Stats
918  Citations  
14.70  Total Impact Points  
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Institutions

2010–2015

Paris Diderot University
Lutetia Parisorum, ÎledeFrance, France


2002–2013

French National Centre for Scientific Research
Lutetia Parisorum, ÎledeFrance, France


1970–2008

Université ParisSud 11
Orsay, ÎledeFrance, France
