We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly solve a graph problem. We show that the local information cost  presents a natural lower bound on the communication complexity of distributed algorithms. We demonstrate the application of local information cost by deriving a lower bound on the communication complexity of computing a (2t-1)-spanner that consists of at most O(n^{1+ \frac{1 + \epsilon}) edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%%  1 / t^2%DIFDELCMD < } \right)%%%  . Our main result is that any O( \text{poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%%  bits in the CONGEST model under the KT1 assumption , where each node has knowledge of its neighbors' IDs initially . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem ; in fact, our result is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem under the KT1 assumption . A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing spanner algorithms for this setting . In light of the work of King, Kutten, and Thorup (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with \tilde O(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.
We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly 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 \Omega(LIC_\gamma(P)/ \log\tau \log n) bits are required for solving a graph problem P with a \tau-round algorithm that errs with probability at most \gamma. 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 (2t-1)-spanner that consists of at most O(n^{1+ 1/t + \epsilon}) edges, where \epsilon = \Theta %DIFDELCMD < \left( {%%% ( 1 / %DIFDELCMD < } \right)%%% t^2) . Our main result is that any O( poly (n))-time algorithm must send at least \tilde\Omega %DIFDELCMD < \left(1{t^2} %%% ((1/t^2) n ^{1+ 1 / 2t } %DIFDELCMD < \right) %%% ) bits in the CONGEST model under the KT1 assumption . Previously, only a trivial lower bound of \tilde \Omega(n) bits was known for this problem . 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 (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with \tilde O(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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LIC
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We demonstrate the application of
For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that \Omega(
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LIC
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_\gamma(P)/ \log\tau \log n) bits are required for solving a graph problem P with a \tau-round algorithm that errs with probability at most \gamma. 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
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t^2
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, where each node has knowledge of its neighbors' IDs initially
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; in fact, our result is the first nontrivial lower bound on the communication complexity of a sparse subgraph problem under the KT1 assumption
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spanner algorithms for this setting
a distributed spanner algorithm