33 research outputs found

    Rounds vs Communication Tradeoffs for Maximal Independent Sets

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    We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are nn players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex -- this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that when rr rounds of interaction are allowed, at least one player needs to communicate Ω(n1/20r+1)\Omega(n^{1/20^{r+1}}) bits. In particular, with logarithmic bandwidth, finding an MIS requires Ω(loglogn)\Omega(\log\log{n}) rounds. This lower bound can be compared with the algorithm of Ghaffari, Gouleakis, Konrad, Mitrovi\'c, and Rubinfeld [PODC 2018] that solves MIS in O(loglogn)O(\log\log{n}) rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. To prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of edge-sharing between players. Finally, we discuss several implications of our results to multi-round (adaptive) distributed sketching algorithms, broadcast congested clique, and to the welfare maximization problem in two-sided matching markets.Comment: Full version of the paper in FOCS 2022, 44 page

    Optimal Multi-Pass Lower Bounds for MST in Dynamic Streams

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    The seminal work of Ahn, Guha, and McGregor in 2012 introduced the graph sketching technique and used it to present the first streaming algorithms for various graph problems over dynamic streams with both insertions and deletions of edges. This includes algorithms for cut sparsification, spanners, matchings, and minimum spanning trees (MSTs). These results have since been improved or generalized in various directions, leading to a vastly rich host of efficient algorithms for processing dynamic graph streams. A curious omission from the list of improvements has been the MST problem. The best algorithm for this problem remains the original AGM algorithm that for every integer p1p \geq 1, uses n1+O(1/p)n^{1+O(1/p)} space in pp passes on nn-vertex graphs, and thus achieves the desired semi-streaming space of O~(n)\tilde{O}(n) at a relatively high cost of O(lognloglogn)O(\frac{\log{n}}{\log\log{n}}) passes. On the other hand, no lower bounds beyond a folklore one-pass lower bound is known for this problem. We provide a simple explanation for this lack of improvements: The AGM algorithm for MSTs is optimal for the entire range of its number of passes! We prove that even for the simplest decision version of the problem -- deciding whether the weight of MSTs is at least a given threshold or not -- any pp-pass dynamic streaming algorithm requires n1+Ω(1/p)n^{1+\Omega(1/p)} space. This implies that semi-streaming algorithms do need Ω(lognloglogn)\Omega(\frac{\log{n}}{\log\log{n}}) passes. Our result relies on proving new multi-round communication complexity lower bounds for a variant of the universal relation problem that has been instrumental in proving prior lower bounds for single-pass dynamic streaming algorithms. The proof also involves proving new composition theorems in communication complexity, including majority lemmas and multi-party XOR lemmas, via information complexity approaches

    Characterizing the Multi-Pass Streaming Complexity for Solving Boolean CSPs Exactly

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    Noisy Radio Network Lower Bounds via Noiseless Beeping Lower Bounds

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    Protecting Single-Hop Radio Networks from Message Drops

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    Single-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the n participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic message drops (i.e., erasures), where in every round, the message received by each party is erased (replaced by ?) with some small constant probability, independently. Our main result is a constant rate coding scheme, allowing one to run protocols designed to work over the (noiseless) SHRN model over the SHRN model with erasures. Our scheme converts any protocol ? of length at most exponential in n over the SHRN model to a protocol ?\u27 that is resilient to constant fraction of erasures and has length linear in the length of ?. We mention that for the special case where the protocol ? is non-adaptive, i.e., the order of communication is fixed in advance, such a scheme was known. Nevertheless, adaptivity is widely used and is known to hugely boost the power of wireless channels, which makes handling the general case of adaptive protocols ? both important and more challenging. Indeed, to the best of our knowledge, our result is the first constant rate scheme that converts adaptive protocols to noise resilient ones in any multi-party model

    Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs

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    For a directed graph G with n vertices and a start vertex u_start, we wish to (approximately) sample an L-step random walk over G starting from u_start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be ??(n ? L). Prior to our work, a better space complexity was only known with O?(?L) passes. We essentially settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is ??(n ? ?L), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses ??(n ? L^{1/p}) space. In addition, we show a similar ??(n ? ?L) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph
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