TY - GEN
T1 - Being fast means being chatty
T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
AU - Robinson, Peter
N1 - Funding Information:
∗DepartmentofComputerScience, CityUniversityofHong Kong; email: peter.robinson@cityu.edu.hk. The work described inthispaperwaspartiallysupportedbyagrantfromtheResearch Grants CounciloftheHongKongSpecialAdministrativeRegion, China [Project No. CityU 11213620], as well as by a grant from the City University of Hong Kong [Project No. 7200639/CS].
Publisher Copyright:
Copyright © 2021 by SIAM
PY - 2021
Y1 - 2021
N2 - We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to solve a graph problem. We show that the local information cost (LIC) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that Ω ( logLICτγlog(P)n) bits are required for solving a graph problem P with a τ-round algorithm that errs with probability at most γ. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a multiplicative spanner with stretch 2t − 1 that consists of at most O(n1+ 1 t +ε) edges, where ε = O(1/t2). Our main result is that any O(poly(n))-time algorithm must send at least Ω-( t12 n1+1/2t) bits in the CONGEST model under the KT1 assumption. Previously, only a trivial lower bound of Ω(-n) bits was known for this problem; in fact, this is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem in this setting. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with Õ(n) communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.
AB - We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to solve a graph problem. We show that the local information cost (LIC) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that Ω ( logLICτγlog(P)n) bits are required for solving a graph problem P with a τ-round algorithm that errs with probability at most γ. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a multiplicative spanner with stretch 2t − 1 that consists of at most O(n1+ 1 t +ε) edges, where ε = O(1/t2). Our main result is that any O(poly(n))-time algorithm must send at least Ω-( t12 n1+1/2t) bits in the CONGEST model under the KT1 assumption. Previously, only a trivial lower bound of Ω(-n) bits was known for this problem; in fact, this is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem in this setting. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with Õ(n) communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth.
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M3 - Conference contribution
AN - SCOPUS:85105280198
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 2105
EP - 2120
BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
A2 - Marx, Daniel
PB - Association for Computing Machinery
Y2 - 10 January 2021 through 13 January 2021
ER -