Tight Network Topology Dependent Bounds on Rounds of Communication

Abstract

We prove tight network topology dependent bounds on the round complexity of computing well studied k-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing, we fix the function and then vary the underlying network topology. This complements the recent such results on total communication that have received some attention. We also present some applications to distributed graph computation problems.Our main contribution is a proof technique that allows us to reduce the problem on a general graph topology to a relevant two-party communication complexity problem. However, unlike many previous works that also used the same high level strategy, we do not reason about a twoparty communication problem that is induced by a cut in the graph. To 'stitch' back the various lower bounds from the two party communication problems, we use the notion of timed graph that has seen prior use in network coding. Our reductions use some tools from Steiner tree packing and multi-commodity flow problems that have a delay constraint.

Topology Dependent Bounds For FAQs

“…Bounds. All our lower bounds follow from known lower bounds on the well-studied TRIBES function (see [18] and references therein) in two-party communication complexity literature. To this end, we rst consider an arbitrary TRIBES instance of a speci c size and show that it can be reduced to a suitable two-party BCQ instance.…”

“…We remark that the existing technique of [18] that 'stitches' the lower bounds induced by cuts can only give a lower bound of Ω(N ): for any edge (P i , P i+1 ) on the path, we can only prove a lower bound of Ω(N ) on the number of bits that need to be exchanged between P i and P i+1 , since if su ces for P i to send the product A k A k −1 · · · A 1 x to P i+1 . e lower bound given by [18] is then the minimum number of rounds needed to make sure that Ω(N ) bits are exchanged between {P 0 , P 1 , · · · , P i } and {P i+1 , P i+2 , · · · , P k +1 } for every i, which can only be Ω(N ). However, this analysis does not capture a very simple fact: P i needs to know A k A k −1 · · · A 1 x before it can be sent to P i+1 .…”

“…We denote this reduction succinctly by TRIBES m, N ≤ BCQ H, N . Finally, we generalize our results from the two-party se ing to general G using ideas from [18,19]. We crucially exploit the fact that our lower bounds are for worst-case assignment of input functions to players in G and show a very speci c class of assignments that achieves the required lower bound.…”

“…18 17 Note that N here is the dimension of the matrices as opposed to the number of non-zero entries used in the previous sections. 18 Note that the trivial algorithm of sending all inputs to a single player takes Ω(k N 2 ) rounds. P 6.1. e FAQ-SS problem from (5) can be computed in O(kN ) rounds.…”

“…In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate eries (FAQs) of [36] in a synchronous distributed network under the model considered by Cha opadhyay et al [18,19]. For ease of exposition, we consider the FAQ-SS problem [7,36,46] i.e., FAQ with a single semiring (called Marginalize a Product Function in [3]), which is a special case of the general FAQ problem (de ned in Section 5).…”