Certifying Equality With Limited Interaction

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.

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Notes

  1. 1.

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

  2. 2.

    We use [n] to denote the set \(\{1, 2, \ldots , n\}\).

  3. 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. 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. 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. 6.

    If (xx) and (yy) were in the same rectangle, then so would (xy) and (yx). Thus, the protocol would err on these inputs.

  7. 7.

    For some values of kr, 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 \(O(k {{\mathrm{ilog}}}^{r}k)\).

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Correspondence to Grigory Yaroslavtsev.

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.

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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

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Keywords

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