Communication complexity of permutation-invariant functions

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.

Department dc:contributor.department

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

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2015

Author dc:creator

• Kamath, Pritish

Advisor dc:contributor.advisor

• Madhu Sudan.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/99861

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/99861