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2012

Volume 41, Issue 3 (partial)


Almost-Optimal Gossip-Based Aggregate Computation

Jen-Yeu Chen and Gopal Pandurangan

SIAM J. Comput. 41, pp. 455-483 (29 pages)

Online Publication Date: May 03, 2012

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Motivated by applications to modern networking technologies, there has been interest in designing efficient gossip-based protocols for computing aggregate functions. While gossip-based protocols provide robustness due to their randomized nature, reducing the message and time complexity of these protocols is also of paramount importance in the context of resource-constrained networks such as sensor and peer-to-peer networks. We present provably time-optimal efficient gossip-based algorithms for aggregate computation with almost optimal message complexity. Given an $n$-node network, our algorithms guarantee that all the nodes can compute the common aggregates (such as Max, Min, Average, Sum, and Count) of their values in optimal $O(\log n)$ time and using $O(n \log \log n)$ messages. Our result improves on the algorithm of Kempe, Dobra, and Gehrke [Proceedings of the IEEE Annual Symposium on Foundations of Computer Science, 2003, pp. 482–491] that is time-optimal but uses $O(n \log n)$ messages, as well as on the algorithm of Kashyap et al. [Proceedings of Symposium on Principles of Database Systems, 2006, pp. 308–317] that uses $O(n \log \log n)$ messages but is not time-optimal (takes $O(\log n \log \log n)$ time). Furthermore, we show that our algorithms can be used to improve gossip-based aggregate computation in sparse communication networks, such as in peer-to-peer networks. The main technical ingredient of our algorithm is a technique called distributed random ranking (DRR) that can be useful in other applications as well. DRR gives an efficient distributed procedure to partition the network into a forest of (disjoint) trees of small size. Since the size of each tree is small, aggregates within each tree can be efficiently obtained at their respective roots. All the roots then perform a uniform gossip algorithm on their local aggregates to reach a distributed consensus on the global aggregates. Our algorithms are non-address-oblivious. In contrast, we show a lower bound of $\Omega(n\log n)$ on the message complexity of any address-oblivious algorithm for computing aggregates. This shows that non-address-oblivious algorithms are needed to obtain significantly better message complexity. Our lower bound holds regardless of the number of rounds taken or the size of the messages used. Our lower bound is the first nontrivial lower bound for gossip-based aggregate computation and also gives the first formal proof that computing aggregates is strictly harder than rumor spreading in the address-oblivious model.

Multiparty Communication Complexity and Threshold Circuit Size of $\ensuremath{\sfAC}^0$

Paul Beame and Trinh Huynh

SIAM J. Comput. 41, pp. 484-518 (35 pages)

Online Publication Date: May 08, 2012

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We prove an $n^{\Omega(1)}/4^k$ lower bound on the randomized $k$-party communication complexity of depth 4 $\ensuremath {\sf AC}^0$ functions in the number-on-forehead (NOF) model for up to $\Theta(\log n)$ players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any $\ensuremath {\sf AC}^0$ function for $\omega(\log\log n)$ players. For nonconstant $k$ the bounds are larger than all previous lower bounds for any $\ensuremath {\sf AC}^0$ function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of $\ensuremath {\sf AC}^0$ by $\ensuremath {\sf MAJ\circ SYM\circ AND}$ circuits, showing that the well-known quasi-polynomial simulations of $\ensuremath {\sf AC}^0$ by such circuits due to Allender (1989) and Yao (1990) are qualitatively optimal,-1pt even for formulas of small constant depth. We also exhibit a depth 5 formula in ${\ensuremath {\sf NP}^{cc}_k}-{\ensuremath {\sf BPP}^{cc}_k}$ for $k$ up to $\Theta(\log n)$ and derive $\Omega(2^{\sqrt{\log n}/\sqrt{k}})$ lower bound on the randomized $k$-party NOF communication complexity of set disjointness for up to $\Theta(\log^{1/3} n)$ players, which is significantly larger than the $O(\log\log n)$ players allowed in the best previous lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and 4 $\ensuremath {\sf AC}^0$ functions.

On the Inherent Sequentiality of Concurrent Objects

Faith Ellen, Danny Hendler, and Nir Shavit

SIAM J. Comput. 41, pp. 519-536 (18 pages)

Online Publication Date: May 17, 2012

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We present $\Omega(n)$ lower bounds on the worst case time to perform a single instance of an operation in any nonblocking implementation of a large class of concurrent data structures shared by $n$ processes. Time is measured by the number of stalls a process incurs as a result of contention with other processes. For standard data structures such as counters, stacks, and queues, our bounds are tight. The implementations considered may apply any primitives to a base object. No upper bounds are assumed on either the number of base objects or their size.
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