Showing papers on "Communication complexity published in 2023"

On the Amortized Communication Complexity of Byzantine Broadcast

Abstract: Designing an efficient solution for Byzantine broadcast is an important problem for many distributed computing and cryptographic tasks. There have been many attempts to achieve sub-quadratic communication complexity in several directions, both in theory and practice, all with pros and cons. This paper initiates the study of another attempt: improving the amortized communication complexity of multi-shot Byzantine broadcast. Namely, we try to improve the average cost when we have sequential multiple broadcast instances. We present a protocol that achieves optimal amortized linear complexity under an honest majority. Our core technique is to efficiently form a network for disseminating the sender's message by keeping track of dishonest behaviors over multiple instances. We also generalize the technique for the dishonest majority to achieve amortized quadratic communication complexity.

On the Message Complexity of Fault-Tolerant Computation: Leader Election and Agreement

Abstract: This paper investigates the message complexity of two fundamental problems, leader election and agreement in the crash-fault synchronous and fully-connected distributed network. We present randomized (Monte Carlo) algorithms for both the problems and also show non-trivial lower bounds on the message complexity. Our algorithms achieve sublinear message complexity in the so-called implicit version of the two problems when tolerating more than a constant fraction of the faulty nodes. In comparison to the state-of-art, our results improved and extended the works of [Gilbert-Kowalski, SODA'10] (which studied only the agreement problem) in several directions. Specifically, our algorithms tolerate any number of faulty nodes up to $(n -\operatorname{polylog}n)$ . The message complexity (and also the time complexity) of our algorithms is optimal (up to a $\operatorname{polylog}n$ factor). Further, our algorithm works in anonymous networks, where nodes do not know each other. To the best of our knowledge, these are the first sub-linear results for both the leader election and the agreement problem in the crash-fault distributed networks.

Distributed MIS with Low Energy and Time Complexities

Abstract: We present randomized distributed algorithms for the maximal independent set problem (MIS) that, while keeping the time complexity nearly matching the best known, reduce the energy complexity substantially. These algorithms work in the standard CONGEST model of distributed message passing with $O(\log n)$ bit messages. The time complexity measures the number of rounds in the algorithm. The energy complexity measures the number of rounds each node is awake; during other rounds, the node sleeps and cannot perform any computation or communications. Our first algorithm has an energy complexity of $O(\log\log n)$ and a time complexity of $O(\log^2 n)$. Our second algorithm is faster but slightly less energy-efficient: it achieves an energy complexity of $O(\log^2 \log n)$ and a time complexity of $O(\log n \cdot \log\log n \cdot \log^* n)$. Thus, this algorithm nearly matches the $O(\log n)$ time complexity of the state-of-the-art MIS algorithms while significantly reducing their energy complexity from $O(\log n)$ to $O(\log^2 \log n)$.

A Near-Optimal Deterministic Distributed Synchronizer

Abstract: We provide the first deterministic distributed synchronizer with near-optimal time complexity and message complexity overheads. Concretely, given any distributed algorithm $\mathcal{A}$ that has time complexity $T$ and message complexity $M$ in the synchronous message-passing model (subject to some care in defining the model), the synchronizer provides a distributed algorithm $\mathcal{A}'$ that runs in the asynchronous message-passing model with time complexity $T \cdot poly(\log n)$ and message complexity $(M+m)\cdot poly(\log n)$. Here, $n$ and $m$ denote the number of nodes and edges in the network, respectively. The synchronizer is deterministic in the sense that if algorithm $\mathcal{A}$ is deterministic, then so is algorithm $\mathcal{A}'$. Previously, only a randomized synchronizer with near-optimal overheads was known by seminal results of Awerbuch, Patt-Shamir, Peleg, and Saks [STOC 1992] and Awerbuch and Peleg [FOCS 1990]. We also point out and fix some inaccuracies in these prior works. As concrete applications of our synchronizer, we resolve some longstanding and fundamental open problems in distributed algorithms: we get the first asynchronous deterministic distributed algorithms with near-optimal time and message complexities for leader election, breadth-first search tree, and minimum spanning tree computations: these all have message complexity $\tilde{O}(m)$ message complexity. The former two have $\tilde{O}(D)$ time complexity, where $D$ denotes the network diameter, and the latter has $\tilde{O}(D+\sqrt{n})$ time complexity. All these bounds are optimal up to logarithmic factors. Previously all such near-optimal algorithms were either restricted to the synchronous setting or required randomization.

Average Awake Complexity of MIS and Matching

Abstract: Chatterjee, Gmyr, and Pandurangan [PODC 2020] recently introduced the notion of awake complexity for distributed algorithms, which measures the number of rounds in which a node is awake. In the other rounds, the node is sleeping and performs no computation or communication. Measuring the number of awake rounds can be of significance in many settings of distributed computing, e.g., in sensor networks where energy consumption is of concern. In that paper, Chatterjee et al. provide an elegant randomized algorithm for the Maximal Independent Set (MIS) problem that achieves an $O(1)$ node-averaged awake complexity. That is, the average awake time among the nodes is $O(1)$ rounds. However, to achieve that, the algorithm sacrifices the more standard round complexity measure from the well-known $O(\log n)$ bound of MIS, due to Luby [STOC'85], to $O(\log^{3.41} n)$ rounds. Our first contribution is to present a simple randomized distributed MIS algorithm that, with high probability, has $O(1)$ node-averaged awake complexity and $O(\log n)$ worst-case round complexity. Our second, and more technical contribution, is to show algorithms with the same $O(1)$ node-averaged awake complexity and $O(\log n)$ worst-case round complexity for $(1+\varepsilon)$-approximation of maximum matching and $(2+\varepsilon)$-approximation of minimum vertex cover, where $\varepsilon$ denotes an arbitrary small positive constant.

Matching upper bounds on symmetric predicates in quantum communication complexity

Abstract: In this paper, we focus on the quantum communication complexity of functions of the form $f \circ G = f(G(X_1, Y_1), \ldots, G(X_n, Y_n))$ where $f: \{0, 1\}^n \to \{0, 1\}$ is a symmetric function, $G: \{0, 1\}^j \times \{0, 1\}^k \to \{0, 1\}$ is any function and Alice (resp. Bob) is given $(X_i)_{i \leq n}$ (resp. $(Y_i)_{i \leq n}$). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of $f \circ G$ is $O(Q(f)\mathrm{QCC}_\mathrm{E}(G))$ when the parties are allowed to use shared entanglement, where $Q(f)$ is the query complexity of $f$ and $\mathrm{QCC}_\mathrm{E}(G)$ is the exact communication complexity of $G$. In this paper, we first show that the same statement holds without shared entanglement, which generalizes their result. Based on the improved result, we next show tight upper bounds on $f \circ \mathrm{AND}_2$ for any symmetric function $f$ (where $\textrm{AND}_2 : \{0, 1\} \times \{0, 1\} \to \{0, 1\}$ denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov~[Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.

Communication complexity of entanglement assisted multi-party computation

Abstract: We consider a quantum and classical version multi-party function computation problem with $n$ players, where players $2, \dots, n$ need to communicate appropriate information to player 1, so that a ``generalized'' inner product function with an appropriate promise can be calculated. The communication complexity of a protocol is the total number of bits that need to be communicated. When $n$ is prime and for our chosen function, we exhibit a quantum protocol (with complexity $(n-1) \log n$ bits) and a classical protocol (with complexity $(n-1)^2 (\log n^2$) bits). In the quantum protocol, the players have access to entangled qudits but the communication is still classical. Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols.

Distributed MIS with Low Energy and Time Complexities

Abstract: We present randomized distributed algorithms for the maximal independent set problem (MIS) that, while keeping the time complexity nearly matching the best known, reduce the energy complexity substantially. These algorithms work in the standard CONGEST model of distributed message passing with O(log n) bit messages. The time complexity measures the number of rounds in the algorithm. The energy complexity measures the number of rounds each node is awake; during other rounds, the node sleeps and cannot perform any computation or communications. Our first algorithm has an energy complexity of O(log log n) and a time complexity of O(log2 n). Our second algorithm is faster but slightly less energy-efficient: it achieves an energy complexity of O(log2 log n) and a time complexity of O(log n · log log n · log* n). Thus, this algorithm nearly matches the O(log n) time complexity of the state-of-the-art MIS algorithms while significantly reducing their energy complexity from O(log n) to O(log2 log n).

Patterned non-determinism in communication complexity

Abstract: Abstract We define and study the model of patterned non-determinism in bipartite communication complexity, denoted by PNP X↔Y . It generalises the known models UP X↔Y and FewP X↔Y through relaxing the constraints on the witnessing structure of the underlying NP X↔Y -protocol. It is shown that for the case of total functions PNP X↔Y equals P X↔Y (similarly to UP X↔Y and FewP X↔Y ). Moreover, the corresponding exhaustive witness-searching problem – determining the full set of witnesses that lead to the acceptance of a given input pair – also has an efficient deterministic protocol. Structurally, the possibility of efficient exhaustive PNP X↔Y -search summarises the above results and can be stated like this: if f 1 , . . . . ,f m are bipartite total Boolean functions with efficient deterministic protocols, then for every input (x, y) the set {i|f i (x, y) = ⊤} can be found by a deterministic protocol of cost poly-logarithmic in n and the total number of such sets for these f i ’s . Finally, the possibility of efficient exhaustive PNP X↔Y -search is used to analyse certain three-party communication regime (under the “number in hand” input partition): The corresponding three-party model is shown to be as strong qualitatively as the weakest among its two-party amplifications obtained by allowing free communication between a pair of players.

Fault-Tolerant Consensus in Quantum Networks

Abstract: Fault-tolerant consensus is about reaching agreement on some of the input values in a limited time by non-faulty autonomous processes, despite of failures of processes or communication medium. This problem is particularly challenging and costly against an adaptive adversary with full information. Bar-Joseph and Ben-Or (PODC'98) were the first who proved an absolute lower bound $\Omega(\sqrt{n/\log n})$ on expected time complexity of consensus in any classic (i.e., randomized or deterministic) message-passing network with $n$ processes succeeding with probability $1$ against such a strong adaptive adversary crashing processes. Seminal work of Ben-Or and Hassidim (STOC'05) broke the $\Omega(\sqrt{n/\log n})$ barrier for consensus in classic (deterministic and randomized) networks by employing quantum computing. They showed an (expected) constant-time quantum algorithm for a linear number of crashes $t

Unbounded Quantum Advantage in One-Way Strong Communication Complexity of a Distributed Clique Labelling Relation

Abstract: We investigate the one-way zero-error classical and quantum communication complexities for a class of relations induced by a distributed clique labelling problem. We consider two variants: 1) the receiver outputs an answer satisfying the relation - the traditional communication complexity of relations (CCR) and 2) the receiver has non-zero probabilities of outputting every valid answer satisfying the relation (equivalently, the relation can be fully reconstructed), that we denote the strong communication complexity of the relation (S-CCR). We prove that for the specific class of relations considered here when the players do not share any resources, there is no quantum advantage in the CCR task for any graph. On the other hand, we show that there exist, classes of graphs for which the separation between one-way classical and quantum communication in the S-CCR task grow with the order of the graph, specifically, the quantum complexity is $O(1)$ while the classical complexity is $\Omega(\log m)$. Secondly, we prove a lower bound (that is linear in the number of cliques) on the amount of shared randomness necessary to overcome the separation in the scenario of fixed restricted communication and connect this to the existence of Orthogonal Arrays. Finally, we highlight some applications of this task to semi-device-independent dimension witnessing as well as to the detection of Mutually Unbiased Bases.

Cooperative Multi-Agent Reinforcement Learning: Asynchronous Communication and Linear Function Approximation

Abstract: We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where multiple agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that enable asynchronous communication while ensuring the advantage of cooperation with low communication overhead. With linear function approximation, we prove that our algorithm enjoys an $\tilde{\mathcal{O}}(d^{3/2}H^2\sqrt{K})$ regret with $\tilde{\mathcal{O}}(dHM^2)$ communication complexity, where $d$ is the feature dimension, $H$ is the horizon length, $M$ is the total number of agents, and $K$ is the total number of episodes. We also provide a lower bound showing that a minimal $\Omega(dM)$ communication complexity is required to improve the performance through collaboration.

Structural Complexity of Rational Interactive Proofs

Abstract: This is the full version of a paper submitted to the Computability in Europe (CiE 2023) conference, with all proofs omitted there. In 2012 P. D. Azar and S. Micali introduced a new model of interactive proofs, called "Rational Interactive Proofs". In this model the prover is neither honest nor malicious, but rational in terms of maximizing his expected reward. In this article we explore the connection of this area with classic complexity results. In the first part of this article we revise the ties between the counting hierarchy and the hierarchy of constant-round rational proofs. We prove that a polynomial-time machine with oracle access to DRMA[k] decides exactly languages in DRMA[k], a coincidence unknown for levels of the counting hierarchy. In the second part we study communication complexity of single-round rational proofs. We show that the class defined by logarithmic-communication single-round rational proofs coincides with PP. We also show that single-round rational protocols that treat problems in Parity-P as black-box samplers of a random variable require at least a linear number of bits of communication.

Threshold Private Set Intersection with Better Communication Complexity

Abstract: Given $$\ell $$ parties with sets $$X_1, \dots , X_\ell $$ of size n, we would like to securely compute the intersection $$\cap _{i=1}^\ell X_i$$ , if it is larger than $$n-t$$ for some threshold t, without revealing any other additional information. It has previously been shown (Ghosh and Simkin, Crypto 2019) that this function can be securely computed with a communication complexity that only depends on t and in particular does not depend on n. For small values of t, this results in protocols that have a communication complexity that is sublinear in the size of the inputs. Current protocols either rely on fully homomorphic encryption or have an at least quadratic dependency on the parameter t. In this work, we construct protocols with a quasilinear dependency on t from simple assumptions like additively homomorphic encryption and oblivious transfer. All existing approaches, including ours, rely on protocols for computing a single bit, which indicates whether the intersection is larger than $$n-t$$ without actually computing it. Our key technical contribution, which may be of independent interest, takes any such protocol with secret shared outputs and communication complexity $$\mathcal {O}(\lambda \ell {{\,\textrm{poly}\,}}(t))$$ , where $$\lambda $$ is the security parameter, and transforms it into a protocol with communication complexity $$\mathcal {O}(\lambda ^2 \ell t {{\,\textrm{polylog}\,}}(t))$$ .

Spectral approach to the communication complexity of multi-party key agreement

Abstract: In multi-party key agreement protocols it is assumed that the parties are given correlated input data and should agree on a common secret key so that the eavesdropper cannot obtain any information on this key by listening to the communications between the parties. We consider the one-shot setting, when there is no ergodicity assumption on the input data. It is known that the optimal size of the secret key can be characterized in terms of the mutual information between different combinations of the input data sets, and the optimal key can be produced with the help of the omniscience protocol. However, the optimal communication complexity of this problem remains unknown. We show that the communication complexity of the omniscience protocol is optimal, at least for some complexity profiles of the input data, in the setting with restricted interaction between parties (the simultaneous messages model). We also provide some upper and lower bounds for communication complexity for other communication problems. Our proof technique combines information-theoretic inequalities and the spectral method.

Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation

Abstract: We study the robust communication complexity of maximum matching. Edges of an arbitrary $n$-vertex graph $G$ are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph $G$. We specifically study the best approximation ratio achievable via protocols where Alice communicates only $\widetilde{O}(n)$ bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a $(2/3+\epsilon_0 \sim .667)$-approximation for a small constant $0 < \epsilon_0 < 10^{-18}$, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) $(2/3 \sim .666)$-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) $(5/6 \sim .833)$-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein's protocol is tight.

Byzantine Consensus is Θ(n2): The Dolev-Reischuk Bound is Tight even in Partial Synchrony!

Abstract: Abstract The Dolev-Reischuk bound says that any deterministic Byzantine consensus protocol has (at least) quadratic communication complexity in the worst case. While it has been shown that the bound is tight in synchronous environments, it is still unknown whether a consensus protocol with quadratic communication complexity can be obtained in partial synchrony. Until now, the most efficient known solutions for Byzantine consensus in partially synchronous settings had cubic communication complexity (e.g., HotStuff, binary DBFT). This paper closes the existing gap by introducing SQuad, a partially synchronous Byzantine consensus protocol with quadratic worst-case communication complexity. In addition, SQuad is optimally-resilient and achieves linear worst case latency complexity. The key technical contribution underlying SQuad lies in the way we solve view synchronization, the problem of bringing all correct processes to the same view with a correct leader for sufficiently long. Concretely, we present RareSync, a view synchronization protocol with quadratic communication complexity and linear latency complexity, which we utilize in order to obtain SQuad.