On the Distributed Complexity of Large-Scale Graph Computations

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower bounds for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds.

Oblivious stable sorting protocol and oblivious binary search protocol for secure multi-party computation

Multi-party computation (MPC) sorting and searching protocols are frequently used in different databases with varied applications, as in cooperative intrusion detection systems, private computation of set intersection and oblivious RAM. Ivan Damgard et al. have proposed two techniques i.e., bit-decomposition protocol and bit-wise less than protocol for MPC. These two protocols are used as building blocks and have proposed two oblivious MPC protocols. The proposed protocols are based on data-dependent algorithms such as insertion sort and binary search. The proposed multi-party sorting protocol takes the shares of the elements as input and outputs the shares of the elements in sorted order. The proposed protocol exhibits O ( 1 ) constant round complexity and O ( n log n ) communication complexity. The proposed multi-party binary search protocol takes two inputs. One is the shares of the elements in sorted order and the other one is the shares of the element to be searched. If the position of the search element exists, the protocol returns the corresponding shares, otherwise it returns shares of zero. The proposed multi-party binary search protocol exhibits O ( 1 ) round complexity and O ( n log n ) communication complexity. The proposed multi-party sorting protocol works better than the existing quicksort protocol when the input is in almost sorted order. The proposed multi-party searching protocol gives almost the same results, when compared to the general binary search algorithm.

Quantum Alice and silent Bob: Qubit-based Quantum Key Recycling with almost no classical communication

We answer an open question about Quantum Key Recycling (QKR): Is it possible to put the message entirely in the qubits without increasing the number of qubits compared to existing QKR schemes? We show that this is indeed possible. We introduce a prepare-and-measure QKR protocol where the communication from Alice to Bob consists entirely of qubits. As usual, Bob responds with an authenticated one-bit accept/reject classical message. Compared to Quantum Key Distribution (QKD), QKR has reduced round complexity. Compared to previous qubit-based QKR protocols, our scheme has far less classical communication. We provide a security proof in the universal composability framework and find that the communication rate is asymptotically the same as for QKD with one-way postprocessing.

Distributed Approximation Algorithms for Steiner Tree in the CONGESTED CLIQUE

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.

Non-Malleable Zero-Knowledge Arguments with Lower Round Complexity

Round complexity is one of the fundamental problems in zero-knowledge (ZK) proof systems. Non-malleable zero-knowledge (NMZK) protocols are ZK protocols that provide security even when man-in-the-middle adversaries interact with a prover and a verifier simultaneously. It is known that the first constant-round public-coin NMZK arguments for NP can be constructed by assuming the existence of collision-resistant hash functions (Pass, R. and Rosen, A. (2005) New and Improved Constructions of Non-Malleable Cryptographic Protocols. In Gabow, H.N. and Fagin, R. (eds) Proc. 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 2224, 2005, pp. 533542. ACM) and has relatively high round complexity; the first four-round private-coin NMZK arguments for NP can be constructed in the plain model by assuming the existence of one-way functions (Goyal, V., Richelson, S., Rosen, A. and Vald, M. (2014) An Algebraic Approach to Non-Malleability. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 1821, 2014, pp. 4150. IEEE Computer Society and Ciampi, M., Ostrovsky, R., Siniscalchi, L. and Visconti, I. (2017) Delayed-Input Non-Malleable Zero Knowledge and Multi-Party Coin Tossing in Four Rounds. In Kalai, Y. and Reyzin, L. (eds) Theory of Cryptography15th Int. Conf., TCC 2017. Lecture Notes in Computer Science, Baltimore, MD, USA, November 1215, 2017, Part I, Vol. 10677, pp. 711742. Springer). In this paper, we present a six-round public-coin NMZK argument of knowledge system assuming the existence of collision-resistant hash functions and a three-round private-coin NMZK argument system from multi-collision resistance of hash functions assumption in the keyless setting.