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37
Index Coding with Side Information
, 2006
"... Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0, 1} n and wishes to broadcast a single message so that each receiver Ri c ..."
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Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0, 1} n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G on n nodes; Ri knows the bits of x in the positions {j  (i, j) is an edge of G}. G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEX codes for {0, 1} n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd antiholes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but is shown not to be tight.
Exponential separation of quantum and classical oneway communication complexity
 SIAM J. Comput
"... Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is t ..."
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Abstract. We give the first exponential separation between quantum and boundederror randomized oneway communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum oneway communication complexity of HMn is O(log n), yet any randomized oneway protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for oneway communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum oneway communication complexity remains O(log n) and that the 0error randomized oneway communication complexity is Ω(n). We prove that any randomized linear oneway protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The
Prior entanglement, message compression and privacy in quantum communication
 In IEEE Conference on Computational Complexity
, 2005
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Efficient Algorithms Using The Multiplicative Weights Update Method
, 2006
"... Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more eff ..."
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Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In this thesis, we explore applications of the Multiplicative Weights method in the design of efficient algorithms for various optimization problems. This method, which was repeatedly discovered in quite diverse fields, is an algorithmic technique which maintains a distribution on a certain set of interest, and updates it iteratively by multiplying the probability mass of elements by suitably chosen factors based on feedback obtained by running another algorithm on the distribution. We present a single metaalgorithm which unifies all known applications of this method in a common framework. Next, we generalize the method to the setting of symmetric matrices rather than real numbers. We derive the following applications of the resulting Matrix Multiplicative Weights algorithm: 1. The first truly general, combinatorial, primaldual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for obtaining O(plog n) approximations to various graph partitioning problems, such as Sparsest Cut, Balanced Separator in both directed and undirected weighted graphs, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the AlonRoichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3) time deterministic O(log n) approximation algorithm for the quantum hypergraph covering problem. 4. An alternative proof of a result of Aaronson that the flfatshattering dimension of quantum states on n qubits is O ( nfl2).
On Rounds in Quantum Communication
 IEEE Transactions on Information Theory
, 2000
"... We investigate the power of interaction in two player quantum communication protocols. Our main result is a roundscommunication hierarchy for the pointer jumping function f k . We show that f k needs quantum communication n) if Bob starts the communication and the number of rounds is limited to k ( ..."
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We investigate the power of interaction in two player quantum communication protocols. Our main result is a roundscommunication hierarchy for the pointer jumping function f k . We show that f k needs quantum communication n) if Bob starts the communication and the number of rounds is limited to k (for any constant k). Trivially, if Alice starts, O(k log n) communication in k rounds suces. The lower bound employs a result relating the relative von Neumann entropy between density matrices to their trace distance and uses a new measure of information.
Quantum information and the PCP theorem
 In FOCS
, 2005
"... We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial ..."
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Cited by 15 (1 self)
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We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial in n. This shows how to go around HolevoNayak’s Theorem, using ArthurMerlin proofs. We use the new representation to prove the following results: 1. Interactive proofs with quantum advice: We show that the class QIP/qpoly contains all languages. That is, for any language L (even nonrecursive), the membership x ∈ L (for x of length n) can be proved by a polynomialsize quantum interactive proof, where the verifier is a polynomialsize quantum circuit with working space initiated with some quantum state ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2. PCP with only one query: We show that the membership x ∈ SAT (for x of length n) can be proved by a logarithmicsize quantum state Ψ〉, together with a polynomialsize classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state Ψ 〉 the verifier only needs to read one block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum lowdegreetest that may be interesting in its own right.
Recognizing WellParenthesized Expressions in the Streaming Model
"... 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 pres ..."
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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)
Quantum Network Coding
, 2006
"... Since quantum information is continuous, its handling is sometimes surprisingly harder than the classical counterpart. A typical example is cloning; making a copy of digital information is straightforward but it is not possible exactly for quantum information. The question in this paper is whether o ..."
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Since quantum information is continuous, its handling is sometimes surprisingly harder than the classical counterpart. A typical example is cloning; making a copy of digital information is straightforward but it is not possible exactly for quantum information. The question in this paper is whether or not quantum network coding is possible. Its classical counterpart is another good example to show that digital information flow can be done much more efficiently than conventional (say, liquid) flow. Our answer to the question is similar to the case of cloning, namely, it is shown that quantum network coding is possible if approximation is allowed, by using a simple network model called Butterfly. In this network, there are two flow paths, s1 to t1 and s2 to t2, which shares a single bottleneck channel of capacity one. In the classical case, we can send two bits simultaneously, one for each path, in spite of the bottleneck. Our results for quantum network coding include: (i) We can send any quantum state ψ1 〉 from s1 to t1 and ψ2 〉 from s2 to t2 simultaneously with a fidelity strictly greater than 1/2. (ii) If one of ψ1 〉 and ψ2 〉 is classical, then the fidelity can be improved to 2/3. (iii) Similar improvement is also possible if ψ1 〉 and ψ2 〉 are restricted to only a finite number of (previously known) states. This allows us to design an interesting protocol which can send two classical bits from s1 to t1 (similarly from s2 to t2) but only one of them should be recovered.