## Communication complexity of permutation-invariant functions

##### Author(s)

Kamath, Pritish
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##### Other Contributors

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.

##### Advisor

Madhu Sudan.

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Show full item record##### Abstract

Motivated by the quest for a broader understanding of communication complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function f : {0, 1}n x {0, 1}n --> {0, 1, ?} is permutation-invariant if for every bijection [pi]: {1,..., n} --> {1,.. ., n} and every x, y [sum] {0, I}n, it is the case that f (x, y) = f (x[pi], y[pi]). Most of the commonly studied functions in communication complexity are permutation-invariant. For such functions, we present a simple complexity measure (computable in time polynomial in n given an implicit description of f) that describes their communication complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the communication complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as SET-DISJOINTNESS and INDEXING, while complementing them with the relatively lesser-known upper bounds for GAP-INNER-PRODUCT (from the sketching literature) and SPARSE-GAP-INNER-PRODUCT (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in communication complexity after an additive O(log log n) loss.

##### Description

Thesis: S.M. in Computer Science & Engineering, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015. Cataloged from PDF version of thesis. Includes bibliographical references (pages 68-71).

##### Date issued

2015##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Publisher

Massachusetts Institute of Technology

##### Keywords

Electrical Engineering and Computer Science.