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LIPIcs, Volume 281, DISC 2023, Complete Volume
No full textLIPIcs, Volume 281, DISC 2023, Complete Volum
Front Matter, Table of Contents, Preface, Conference Organization
No full textFront Matter, Table of Contents, Preface, Conference Organizatio
The Communication Complexity of Set Intersection Under Product Distributions
Full text linkWe consider a multiparty setting where k parties have private inputs X_1,…,X_k ⊆ [n] and wish to compute the intersection ⋂_{ =1}^k X_ of their sets, using as little communication as possible. This task generalizes the well-known problem of set disjointness, where the parties are required only to determine whether the intersection is empty or not. In the worst-case, it is known that the communication complexity of finding the intersection is the same as that of solving set disjointness, regardless of the size of the intersection: the cost of both problems is Ω(n log k + k) bits in the shared blackboard model, and Ω (nk) bits in the coordinator model.
In this work we consider a realistic setting where the parties' inputs are independent of one another, that is, the input is drawn from a product distribution. We show that this makes finding the intersection significantly easier than in the worst-case: only Θ̃((n^{1-1/k} (H(S) + 1)^{1/k}) + k) bits of communication are required, where {H}(S) is the Shannon entropy of the intersection S. We also show that the parties do not need to know the exact underlying input distribution; if we are given in advance O(n^{1/k}) samples from the underlying distribution μ, we can learn enough about μ to allow us to compute the intersection of an input drawn from μ using expected communication Θ̃((n^{1-1/k}[|S|]^{1/k}) + k), where |S| is the size of the intersection
Lower Bounds for Subgraph Detection in the CONGEST Model
Full text linkIn the subgraph-freeness problem, we are given a constant-sized graph H, and wish to de- termine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs.
The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle Ck of length k ≥ 4, and constructing a graph H ̃ from Ck by replacing each edge {i, (i + 1) mod k} of the cycle with a connected graph Hi, subject to some constraints on the graphs H_{0}, . . . , H_{k−1}. In this case we obtain a polynomial lower bound for the new graph H ̃, depending on the size of the shortest cycle in H ̃ passing through the vertices of the original k-cycle
A Distributed Algorithm for Directed Minimum-Weight Spanning Tree
Full text linkIn the directed minimum spanning tree problem (DMST, also called minimum weight arborescence), the network is given a root node r, and needs to construct a minimum-weight directed spanning tree, rooted at r and oriented outwards. In this paper we present the first sub-quadratic DMST algorithms in the distributed CONGEST network model, where the messages exchanged between the network nodes are bounded in size. We consider three versions: a model where the communication links are bidirectional but can have different weights in the two directions; a model where communication is unidirectional; and the Congested Clique model, where all nodes can communicate directly with each other.
Our algorithm is based on a variant of Lovász' DMST algorithm for the PRAM model, and uses a distributed single-source shortest-path (SSSP) algorithm for directed graphs as a black box. In the bidirectional CONGEST model, our algorithm has roughly the same running time as the SSSP algorithm; using the state-of-the-art SSSP algorithm, we obtain a running time of O~(min(sqrt{nD},sqrt{n}D^{1/4} + n^{3/5} +D)) rounds for the bidirectional communication case.
For the unidirectional communication model we give an O~(n) algorithm, and show that it is nearly optimal. And finally, for the Congested Clique, our algorithm again matches the best known SSSP algorithm: it runs in O~(n^{1/3}) rounds.
On the negative side, we adapt an observation of Chechik in the sequential setting to show that in all three models, the DMST problem is at least as hard as the (s,t)-shortest path problem. Thus, in terms of round complexity, distributed DMST lies between single-source shortest path and (s,t)-shortest path
Truthful Information Dissemination in General Asynchronous Networks
Full text linkWe give a protocol for information dissemination in asynchronous networks of rational players, where each player may have its own desires and preferences as to the outcome of the protocol, and players may deviate from the protocol if doing so achieves their goals. We show that under minimalistic assumptions, it is possible to solve the information dissemination problem in a truthful manner, such that no participant has an incentive to deviate from the protocol we design. Our protocol works in any asynchronous network, provided the network graph is at least 2-connected. We complement the protocol with two impossibility results, showing that 2-connectivity is necessary, and also that our protocol achieves optimal bit complexity.
As an application, we show that truthful information dissemination can be used to implement a certain class of communication equilibria, which are equilibria that are typically reached by interacting with a trusted third party. Recent work has shown that communication equilibria can be implemented in synchronous networks, or in asynchronous, complete networks; we show that in some useful cases, our protocol yields a lightweight mechanism for implementing communication equilibria in any 2-connected asynchronous network
On Polynomial Time Local Decision
Full text linkThe field of distributed local decision studies the power of local network algorithms, where each network can see only its own local neighborhood, and must act based on this restricted information. Traditionally, the nodes of the network are assumed to have unbounded local computation power, and this makes the model incomparable with centralized notions of efficiency, namely, the classes and NP. In this work we seek to bridge this gap by studying local algorithms where the nodes are required to be computationally efficient: we introduce the classes PLD and NPLD of polynomial-time local decision and non-deterministic polynomial-time local decision, respectively, and compare them to the centralized complexity classes and NP, and to the distributed classes LD and NLD, which correspond to local deterministic and non-deterministic decision, respectively.
We show that for deterministic algorithms, requiring both computational and distributed efficiency is likely to be more restrictive than either requirement alone: if the nodes do not know the network size, then PLD ⊊ LD ∩ holds unconditionally; if the network size is known to all nodes, then the same separation holds under a widely believed complexity assumption (UP ∩ coUP ≠ ). However, when nondeterminism is introduced, this distinction vanishes, and NPLD = NLD ∩ NP. To complete the picture, we extend the classes PLD and NPLD into a hierarchy akin to the centralized polynomial hierarchy, and we characterize its connections to the centralized polynomial hierarchy and to the distributed local decision hierarchy of Balliu, D'Angelo, Fraigniaud, and Olivetti
Fully Local Succinct Distributed Arguments
Full text linkDistributed certification is a proof system for detecting illegal network states or improper execution of distributed algorithms. A certification scheme consists of a proving algorithm, which assigns a certificate to each node, and a verification algorithm where nodes use these certificates to decide whether to accept or reject. The system must ensure that all nodes accept if and only if the network is in a legal state, adhering to the principles of completeness and soundness. The main goal is to design a scheme where the verification process is local and the certificates are succinct, while using as efficient as possible proving algorithm.
In cryptographic proof systems, the soundness requirement is often relaxed to computational soundness, where soundness is guaranteed only against computationally bounded adversaries. Computationally sound proof systems are called arguments. Recently, Aldema Tshuva, Boyle, Cohen, Moran, and Oshman (TCC 2023) showed that succinct distributed arguments can be used to enable any polynomially bounded distributed algorithm to certify its execution with polylogarithmic-length certificates. However, their approach required a global communication phase, adding O(D) communication rounds in networks of diameter D, which limits its applicability to local algorithms.
In this work, we give the first construction of a fully local succinct distributed argument system, where the prover and the verifier are both local. We show that a distributed algorithm that runs in R rounds, has polynomial local computation, and messages of B bits each can be compiled into a self-certifying algorithm that runs in R + polylog(n) rounds and sends messages of size B + polylog(n), with certificates of length polylog(n). This construction has several applications, including self-certification for local algorithms, ongoing certification of long-lived algorithms, and efficient local mending of the certificates when the network changes
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