Ami Paz

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n1-2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: triangle and 4-cycle counting in O(n0.158) rounds, improving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], a (1 + o(1))-approximation of all-pairs shortest paths in O(n0.158) rounds, improving upon the ~O (n1/2)-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n0.158) rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

Counting-based impossibility proofs for renaming and set agreement

Renaming and set agreement are two fundamental sub-consensus tasks. In the M-renaming task, processes start with names from a large domain and must decide on distinct names in a range of size M; in the k-set agreement task, processes must decide on at most k of their input values. Renaming and set agreement are representatives of the classes of colored and colorless tasks, respectively. This paper presents simple proofs for key impossibility results for wait-free computation using only read and write operations: n processes cannot solve (n−1)-set agreement, and, if n is a prime power, n processes cannot solve (2n−2)-renaming. Our proofs consider a restricted set of executions, and combine simple operational properties of these executions with elementary counting arguments, to show the existence of an execution violating the tasks requirements. This makes the proofs easier to understand, verify, and hopefully, extend.

A (2 + ϵ )-approximation for maximum weight matching in the semi-streaming model

We present a simple deterministic single-pass (2 + ϵ)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5 + ϵ). Our algorithm uses O(n log2 n) space for constant values of ϵ. It relies on a variation of the local-ratio theorem, which may be of independent interest in the semi-streaming model.

Distributed Construction of Purely Additive Spanners

This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts.

Counting-based impossibility proofs for set agreement and renaming

Set agreement and renaming are two tasks that allow processes to coordinate, even when agreement is impossible. In k -set agreement, n processes must decide on at most k of their input values. While n -set agreement is trivially wait-free solvable by each process deciding on its input, ( n - 1 ) -set agreement is not wait-free solvable. In M -renaming, processes must decide on distinct names in a range of size M . For any number n of processes, ( 2 n - 1 ) -renaming is wait-free solvable, but surprisingly, ( 2 n - 2 ) -renaming is wait-free solvable if and only if n is not a prime power; the only previous lower bound on the number of names necessary for renaming, when n is not a prime power, is n + 1 . In adaptive renaming, M decreases when the number p of participants in the execution decreases. It is known that ( 2 p - 1 ) -adaptive renaming is wait-free solvable, while ( 2 p - ? p n - 1 ? ) -adaptive renaming is not.This paper presents counting-based proofs for the above mentioned impossibility results: n processes can wait-free solve neither ( n - 1 ) -set agreement nor ( 2 p - ? p n - 1 ? ) -adaptive renaming; if n is a prime power, n processes cannot wait-free solve ( 2 n - 2 ) -renaming. For an arbitrary number of processes, we give a lower bound for renaming, by reduction from renaming for a different number of processes, and relying on the distribution of prime numbers.Our proofs combine simple operational properties of a restricted set of executions with elementary counting arguments to show the existence of an execution violating the tasks conditions. This makes the proofs easier to understand, verify, and, we hope, extend. We consider the wait-free solvability of tasks in shared memory systems.Only combinatorial and number-theoretic arguments are used. ( n - 1 ) -set agreement is not wait-free solvable.Adaptive ( 2 n - 2 ) -renaming and strong symmetry breaking are not wait-free solvable.Nonadaptive ( 2 n - 2 ) -renaming and weak symmetry breaking are not wait-free solvable, when n is the power of a prime number.

Approximate proof-labeling schemes

We study a new model of verification of boolean predicates over distributed networks. Given a network configuration, the proof-labeling scheme (PLS) model defines a distributed proof in the form of a label that is given to each node, and all nodes locally verify that the network configuration satisfies the desired boolean predicate by exchanging labels with their neighbors. The proof size of the scheme is defined to be the maximum size of a label.

Distributed construction of purely additive spanners

This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts. We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners. The standard reductions from set-disjointness and equality seem unsuitable for this task because no specific edge needs to be removed from the graph. Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.

Approximate Proof-Labeling Schemes.

We study a new model of verification of boolean predicates over distributed networks. Given a network configuration, the proof-labeling scheme (PLS) model defines a distributed proof in the form of a label that is given to each node, and all nodes locally verify that the network configuration satisfies the desired boolean predicate by exchanging labels with their neighbors. The proof size of the scheme is defined to be the maximum size of a label.

Algebraic methods in the congested clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an