The One-Way Communication Complexity of Hamming Distance

Consider the following version of the Hamming distance problem for ±1 vec- tors of length n: the promise is that the distance is either at least n + p n or at most n p n, and the goal is to find out which of these two cases occurs. Woodruff (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2004) gave a linear lower bound for the randomized one-way communication complexity of this problem. In this note we give a simple proof of this result. Our proof uses a simple reduction from the indexing problem and avoids the VC-dimension arguments used in the previous paper. As shown by Woodruff (loc. cit.), this implies an W(1/e 2 )-space lower bound for approximating frequency moments within a factor 1+e in the data stream model.

[1]  David P. Woodruff.  Optimal space lower bounds for all frequency moments , 2004, SODA '04.

[2]  Yuan Zhou.  Introduction to Coding Theory , 2010 .

[3]  Noga Alon,et al.  The space complexity of approximating the frequency moments , 1996, STOC '96.

[4]  David P. Woodruff.  Efficient and private distance approximation in the communication and streaming models , 2007 .

[5]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[6]  Noam Nisan,et al.  On Randomized One-round Communication Complexity , 1995, STOC '95.

[7]  Rafail Ostrovsky,et al.  Efficient search for approximate nearest neighbor in high dimensional spaces , 1998, STOC '98.

[8]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[9]  Ziv Bar-Yossef,et al.  Reductions in streaming algorithms, with an application to counting triangles in graphs , 2002, SODA '02.

[10]  Ziv Bar-Yossef,et al.  Information theory methods in communication complexity , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[11]  J. H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[12]  Ilan Newman,et al.  Private vs. Common Random Bits in Communication Complexity , 1991, Inf. Process. Lett..

[13]  Noga Alon,et al.  The Space Complexity of Approximating the Frequency Moments , 1999 .

[14]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .