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The Range of Topological Effects on Communication

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9135))

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Abstract

We continue the study of communication cost of computing functions when inputs are distributed among k processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges.

Chattopadhyay, Radhakrishnan and Rudra (FOCS’14) recently initiated the study of the effect of topology of the network on the total communication cost using tools from \(L_1\) embeddings. Their techniques provided tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. This work addresses two other kinds of natural functions. We show that for a large class of natural functions like Set-Disjointness the communication cost is essentially n times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like \(\text {ED} \circ \text {XOR}\) and \(\text {XOR} \circ \text {ED}\), the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs.

To obtain our results, we use some new tools in addition to ones used in Chattopadhyay et al. These include (i) viewing the communication constraints via a linear program; (ii) using tools from the theory of tree embeddings to prove topology sensitive direct sum results that handle the case of composed functions and (iii) representing the communication constraints of certain problems as a family of collection of multiway cuts, where each multiway cut simulates the hardness of computing the function on the star topology.

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Authors and Affiliations

  1. Tata Institute of Fundamental Research, Mumbai, India

    Arkadev Chattopadhyay

  2. University at Buffalo, SUNY, Buffalo, NY, USA

    Atri Rudra

Authors
  1. Arkadev Chattopadhyay
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  2. Atri Rudra
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Correspondence to Atri Rudra .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    MagnĂşs M. HalldĂłrsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Chattopadhyay, A., Rudra, A. (2015). The Range of Topological Effects on Communication. In: HalldĂłrsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47666-6_43

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  • DOI: https://doi.org/10.1007/978-3-662-47666-6_43

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