In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piecebypiece, our model presents one of the players, Bob, his input piecebypiece, and has the players Alice and Bob cooperate to compute a result each time before the next piece is revealed to Bob. This model has a closer and more natural correspondence to dynamic data structures than classic communication models do, and hence presents a new perspective on data structures.
We first present a tight lower bound for the online set intersection problem in the online communication model, demonstrating a general approach for proving online communication lower bounds. The online communication model prevents a batching trick that classic communication complexity allows, and yields a stronger lower bound. We then apply the online communication model to prove data structure lower bounds for two dynamic data structure problems: the Group Range problem and the Dynamic Connectivity problem for forests. Both of the problems admit a worst case O(logn)time data structure. Using online communication complexity, we prove a tight cellprobe lower bound for each: spending o(logn) (even amortized) time per operation results in at best an exp(−δ2 n) probability of correctly answering a (1/2+δ)fraction of the n queries.
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Space and computationallyefficient set reconciliation via parity bitmap sketch (PBS)
Set reconciliation is a fundamental algorithmic problem that arises in many networking, system, and database applications. In this problem, two large sets A and B of objects (bitcoins, files, records, etc.) are stored respectively at two different networkconnected hosts, which we name Alice and Bob respectively. Alice and Bob communicate with each other to learn A Δ B , the difference between A and B , and as a result the reconciled set A ∪ B. Current set reconciliation schemes are based on either invertible Bloom filters (IBF) or errorcorrection codes (ECC). The former has a low computational complexity of O(d) , where d is the cardinality of A Δ B , but has a high communication overhead that is several times larger than the theoretical minimum. The latter has a low communication overhead close to the theoretical minimum, but has a much higher computational complexity of O(d 2 ). In this work, we propose Parity Bitmap Sketch (PBS), an ECCbased set reconciliation scheme that gets the better of both worlds: PBS has both a low computational complexity of O(d) just like IBFbased solutions and a low communication overhead of roughly twice the theoretical minimum. A separate contribution of this work is a novel rigorous analytical framework that can be used for the precise calculation of various performance metrics and for the nearoptimal parameter tuning of PBS.
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 NSFPAR ID:
 10296507
 Date Published:
 Journal Name:
 Proceedings of the VLDB Endowment
 Volume:
 14
 Issue:
 4
 ISSN:
 21508097
 Page Range / eLocation ID:
 458 to 470
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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