## Abstract

The equality problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the *expected* communication cost (at a worst-case input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the *false negative* error rate separately from the *false positive* error rate. The third is to consider the *information cost* of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for equality: both expected communication complexity and information complexity scale as \(\Theta ({{\mathrm{ilog}}}^{r-1} n)\), where *r* is the number of rounds and \({{\mathrm{ilog}}}^k n = \log \log \cdots \log n\), with *k* logs. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications. As an application of our information cost bounds, we obtain new bounded-round randomized lower bounds for the Intersection problem, in which there are two players who hold subsets \(S,T \subseteq [n]\). In many realistic scenarios, the sizes of *S* and *T* are significantly smaller than *n*, so we impose the constraint that \(|S|, |T| \le k\). We study the minimum number of bits the parties need to communicate in order to compute the entire intersection set \(S \cap T\), using *r* rounds. We show that any *r*-round protocol has information cost (and thus communication cost) \(\Omega (k {{\mathrm{ilog}}}^r k)\) bits. We also give an *O*(*r*)-round protocol achieving \(O(k{{\mathrm{ilog}}}^r k)\) bits, which for \(r = \log ^* k\) gives a protocol with *O*(*k*) bits of communication. This is in contrast to other basic problems such as computing the union or symmetric difference, for which \(\Omega (k\log (n/k))\) bits of communication is required for any number of rounds.

This is a preview of subscription content, log in to check access.

## Notes

- 1.
Throughout this paper we use “\(\log \)” to denote the logarithm to the base 2.

- 2.
We use [

*n*] to denote the set \(\{1, 2, \ldots , n\}\). - 3.
We have replaced the \(\max \) in Klauck’s definition with a sum; this agrees with Klauck’s original definition up to a factor of 2.

- 4.
It is crucial for us to use a strong direct sum theorem of [42] in the lower bound for private-intersection. Unlike generic direct sum and direct product theorems which apply to any function, the strong direct sum of [42] only applies to Equality-type functions but gives a much stronger guarantee in the constant error regime that we study here. This is in contrast with the bounded round direct product theorem of [30, 31] (and other similar results such as [32]), who show that for

*r*-round public-coin randomized information complexity \({{\mathrm{IC}}}^{r, {\text {pub}}}_{1-(1-\varepsilon /2)^{\Omega (k \varepsilon ^2 /r^2)}}(f^k) = \Omega \big (~ ({\varepsilon k}/{r})\cdot (IC^{r,{\text {pub}}}_{\varepsilon }(f) - O({r^2}/{\varepsilon ^2}) ) ~\big )\), where \(\varepsilon > 0\) is arbitrary (the results of [30, 31] are stated in terms of communication complexity but their techniques also imply an information cost lower bound). One cannot apply this theorem to our problem, as one would need to set \(\varepsilon = \Theta (k^{-1/3})\) to obtain our results. Because \({{\mathrm{IC}}}^{r,{\text {pub}}}_{1/k^{1/3}}(\textsc {Equality}) = o(k^{2/3})\) this theorem gives a trivial bound. - 5.
A set of strings is said to be prefix-free if no string in the set is a proper prefix of any other.

- 6.
If (

*x*,*x*) and (*y*,*y*) were in the same rectangle, then so would (*x*,*y*) and (*y*,*x*). Thus, the protocol would err on these inputs. - 7.
For some values of

*k*,*r*, we might have \(r > {{\mathrm{ilog}}}^{r}k\). In fact, it is possible to describe the set of \(c_j\) live coordinates using \(log({k\atopwithdelims ()c_j})\) bits. This sum also telescopes, so it is possible to reduce the*O*(*kr*) cost of describing \(\{c_j\}\) to just*O*(*k*) bits. Thus, the overall cost remains \(O(k {{\mathrm{ilog}}}^{r}k)\).

## References

- 1.
Ablayev, F.: Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theor. Comput. Sci.

**175**(2), 139–159 (1996) - 2.
Ada, A., Chattopadhyay, A., Cook, S.A., Fontes, L., Koucký, M., Pitassi, T.: The hardness of being private. ACM Trans. Comput. Theory

**6**(1), 1:1–1:24 (2014) - 3.
Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. System Sci.

**58**(1), 137–147 (1999). Preliminary version. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 20–29 (1996) - 4.
Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci.

**68**(4), 702–732 (2004) - 5.
Barak, B., Braverman, M., Chen, X., Rao, A.: How to compress interactive communication. SIAM J. Comput.

**42**(3), 1327–1363 (2013). Preliminary version. In: Proceedings of the 41st Annual ACM Symposium on the Theory of Computing, pp. 67–76 (2010) - 6.
Braverman, M.: Interactive information complexity. In: Proceedings of the 44th Annual ACM Symposium on the Theory of Computing, pp. 505–524 (2012)

- 7.
Braverman, M., Ellen, F., Oshman, R., Pitassi, T., Vaikuntanathan, V.: A tight bound for set disjointness in the message-passing model. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pp. 668–677 (2013)

- 8.
Braverman, M., Garg, A., Pankratov, D., Weinstein, O.: From information to exact communication. In: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, pp. 151–160 (2013)

- 9.
Braverman, M., Moitra, A.: An information complexity approach to extended formulations. In: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, pp. 161–170 (2013)

- 10.
Braverman, M., Rao, A.: Information equals amortized communication. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 748–757 (2011)

- 11.
Braverman, M., Rao, A., Weinstein, O., Yehudayoff, A.: Direct product via round-preserving compression. In: Proceedings of the 40th International Colloquium on Automata, Languages and Programming, pp. 232–243 (2013)

- 12.
Brody, J., Chakrabarti, A., Kondapally, R.: Certifying equality with limited interaction. Technical Report TR12-153, ECCC (2012)

- 13.
Buhrman, Harry, Garcıa-Soriano, David, Matsliah, Arie, de Wolf, Ronald: The non-adaptive query complexity of testing k-parities. Chic. J. Theor. Comput. Sci.

**6**, 1–11 (2013) - 14.
Chakrabarti, A., Cormode, G., Kondapally, R., McGregor, A.: Information cost tradeoffs for augmented index and streaming language recognition. SIAM J. Comput.

**42**(1), 61–83 (2013). Preliminary version. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 387–396 (2010) - 15.
Chakrabarti, A., Khot, S., Sun, X.: Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In: Proceedings of the 18th Annual IEEE Conference on Computational Complexity, pp. 107–117 (2003)

- 16.
Chakrabarti, A., Kondapally, R.: Everywhere-tight information cost tradeoffs for augmented index. In: Proceedings of the 15th International Workshop on Randomization and Approximation Techniques in Computer Science, pp. 448–459 (2011)

- 17.
Chakrabarti, A., Shi, Y., Wirth, A., Yao, A.C.: Informational complexity and the direct sum problem for simultaneous message complexity. In: Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 270–278 (2001)

- 18.
Chakrabarti, A., Shubina, A.: Nearly private information retrieval. In: Proceedings of the 32nd International Symposium on Mathematical Foundations of Computer Science, volume 4708 of Lecture Notes in Computer Science, pp. 383–393 (2007)

- 19.
Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, Hoboken (2006)

- 20.
Dasgupta, A., Kumar, R., Sivakumar, D.: Sparse and lopsided set disjointness via information theory. In: Proceedings of the 16th International Workshop on Randomization and Approximation Techniques in Computer Science, vol. 7409, pp. 517–528 (2012)

- 21.
Datar, M., Muthukrishnan, S.: Estimating rarity and similarity over data stream windows. In: Proceedings of the 10th Annual European Symposium on Algorithms, pp. 323–334 (2002)

- 22.
Fagin, R., Naor, M., Winkler, P.: Comparing information without leaking it. Commun. ACM

**39**(5), 77–85 (1996) - 23.
Feder, T., Kushilevitz, E., Naor, M., Nisan, N.: Amortized communication complexity. SIAM J. Comput.

**24**(4), 736–750 (1995). Preliminary version. In: Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, pp. 239–248 (1991) - 24.
Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with \(O(1)\) worst case access time. J. ACM

**31**(3), 538–544 (1984). Preliminary version. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 165–169 (1982) - 25.
Freedman, M.J., Nissim, K., Pinkas, B.: Efficient private matching and set intersection. Advances in Cryptology-EUROCRYPT

**2004**, 1–19 (2004) - 26.
Freivalds, R.: Probabilistic machines can use less running time. In: IFIP Congress, pp. 839–842 (1977)

- 27.
Gronemeier, A.: Asymptotically optimal lower bounds on the NIH-multi-party information complexity of the AND-function and disjointness. In: Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science, pp. 505–516 (2009)

- 28.
Harsha, P., Jain, R., McAllester, D., Radhakrishnan, J.: The communication complexity of correlation. In: Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, pp. 10–23 (2007)

- 29.
Håstad, J., Wigderson, A.: The randomized communication complexity of set disjointness. Theory Comput.

**3**(1), 211–219 (2007) - 30.
Jain, R.: New strong direct product results in communication complexity. Electron. Colloq. Comput. Complex. (ECCC)

**18**, 24 (2011) - 31.
Jain, R., Pereszlényi, A., Yao, P.: A direct product theorem for the two-party bounded-round public-coin communication complexity. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pp. 167–176 (2012)

- 32.
Jain, R., Sen, P., Radhakrishnan, J.: Optimal direct sum and privacy trade-off results for quantum and classical communication complexity. CoRR (2008). arXiv:0807.1267

- 33.
Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discret. Math.

**5**(4), 547–557 (1992) - 34.
Kane, D.M., Nelson, J., Woodruff, D.P.: On the exact space complexity of sketching and streaming small norms. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1161–1178 (2010)

- 35.
Kerenidis, I., de Wolf, R.: Exponential lower bound for 2-query locally decodable codes. J. Comput. Syst. Sci.

**69**(3), 395–420 (2004). Preliminary version. In: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, pp. 106–115 (2003) - 36.
Klauck, H.: On quantum and approximate privacy. Theory Comput. Syst.

**37**(1), 221–246 (2004). Preliminary version. In: Proceedings of the 19th International Symposium on Theoretical Aspects of Computer Science, pp. 335-346 (2002) - 37.
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)

- 38.
Kushilevitz, E., Weinreb, E.: The communication complexity of set-disjointness with small sets and 0-1 intersection. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 63–72 (2009)

- 39.
Magniez, F., Mathieu, C., Nayak, A.: Recognizing well-parenthesized expressions in the streaming model. In: Proceedings of the 41st Annual ACM Symposium on the Theory of Computing, pp. 261–270 (2010)

- 40.
Mehlhorn, K., Schmidt, E.M.: Las Vegas is better than determinism in VLSI and distributed computing (extended abstract). In: Proceedings of the 14th Annual ACM Symposium on the Theory of Computing, pp. 330–337 (1982)

- 41.
Miltersen, P.B., Nisan, N., Safra, S., Wigderson, A.: On data structures and asymmetric communication complexity. J. Comput. Syst. Sci.

**57**(1), 37–49 (1998). Preliminary version. In: Proceedings of the 27th Annual ACM Symposium on the Theory of Computing, pp. 103–111 (1995) - 42.
Molinaro, M., Woodruff, D., Yaroslavtsev, G.: Beating the direct sum theorem in communication complexity with implications for sketching. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (2013)

- 43.
Naor, M., Pinkas, B.: Oblivious polynomial evaluation. SIAM J. Comput.

**35**(5), 1254–1281 (2006) - 44.
Nayak, A.: Optimal lower bounds for quantum automata and random access codes. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, pp. 124–133 (1999)

- 45.
Newman, I.: Private vs. common random bits in communication complexity. Inf. Process. Lett.

**39**(2), 67–71 (1991) - 46.
Pǎtraşcu, M.: Unifying the landscape of cell-probe lower bounds. SIAM J. Comput.

**40**(3), 827–847 (2011) - 47.
Phillips, J.M., Verbin, E., Zhang, Q.: Lower bounds for number-in-hand multiparty communication complexity, made easy. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 486–501 (2012)

- 48.
Razborov, A.: On the distributional complexity of disjointness. Theor. Comput. Sci.

**106**(2), 385–390 (1992). Preliminary version. In: Proceedings of the 17th International Colloquium on Automata, Languages and Programming, pp. 249–253 (1990) - 49.
Saglam, M., Tardos, G.: On the communication complexity of sparse set disjointness and exists-equal problems. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pp. 678–687 (2013)

- 50.
Yao, A.C.: Probabilistic computations: Towards a unified measure of complexity. In: Proceedings of the 18th Annual IEEE Symposium on Foundations of Computer Science, pp. 222–227 (1977)

- 51.
Yao, A.C.: Some complexity questions related to distributive computing. In: Proceedings of the 11th Annual ACM Symposium on the Theory of Computing, pp. 209–213 (1979)

## Author information

### Affiliations

### Corresponding author

## Additional information

This is a full version containing results from papers “Certifying Equality With Limited Interaction” (RANDOM’14) and “Beyond Set Disjointness: The Communication Complexity of Finding the Intersection” (PODC’14) by the same authors.

Part of this work was done while G.Y. was an intern at IBM Research, Almaden. G.Y. was also supported by the Warren Center Fellowship at the University of Pennsylvania and the Institute Postdoctoral Fellowship at Brown University, ICERM.

## Rights and permissions

## About this article

### Cite this article

Brody, J., Chakrabarti, A., Kondapally, R. *et al.* Certifying Equality With Limited Interaction.
*Algorithmica* **76, **796–845 (2016). https://doi.org/10.1007/s00453-016-0163-6

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Communication complexity
- Information theory
- Lower bounds
- Privacy
- Big data
- Round complexity
- Distributed computing