On the Communication Complexity Methodology for Proving Lower Bounds on the Query Complexity of Property Testing

Abstract

We consider the methodology for proving lower bounds on the query complexity of property testing via communication complexity, which was put forward by Blais, Brody, and Matulef (Computational Complexity, 2012). They provided a restricted formulation of their methodology (via “simple combining operators”) and also hinted towards a more general formulation, which we spell out in this paper.

A special case of the general formulation proceeds as follows: In order to derive a lower bound on testing the property $\varPi$, one presents a mapping F of pairs of inputs $(x,y)\in \{0,1\}^{n+n}$ for a two-party communication problem $\varPsi$ to $\ell (n)$-bit long inputs for $\varPi$ such that $(x,y)\in \varPsi$ implies $F(x,y)\in \varPi$ and $(x,y)\not \in \varPsi$ implies that F(x, y) is far from $\varPi$. Let $f_i(x,y)$ be the $i^\mathrm{th}$ bit of F(x, y), and suppose that B is an upper bound on the (deterministic) communication complexity of each $f_i$ and that C is a lower bound on the randomized communication complexity of $\varPsi$. Then, testing $\varPi$ requires at least C/B queries.

The foregoing formulation is generalized by considering randomized protocols (with small error) for computing the $f_i$’s. In contrast, the restricted formulation (via “simple combining operators”) requires that each $f_i(x,y)$ be a function of $x_i$ and $y_i$ only, and uses $B=2$ for the straightforward computation of $f_i$.

We show that the general formulation cannot yield significantly stronger lower bounds than those that can be obtained by the restricted formulation. Nevertheless, we advocate the use of the general formulation, because we believe that it is easier to work with. Following Blais et al., we also describe a version of the methodology for nonadaptive testers and one-way communication complexity.

Appendix: Generalization to Multi-Party Communication

The formulation presented in Sect. 3 generalizes easily to the model of multi-party communication. The treatment is quite oblivious of the details of the model; for example, it does not matter if one considers the standard model of “input on the forehead” or to the more natural model in which each party gets a part of the input (with no overlap). (These variations can be captured by the promise problems that the parties wish to solve.) The exact way in which the parties communicate is also not crucial, at least as long as the number of parties (denoted m) is small. For simplicity, we consider here a broadcast model, where in each communication round there is a single designated sender (determined by the transcript of the communication so far).

In light of the above, we consider m-party communication protocols in which the local input of the $j^\mathrm{th}$ party is denoted $x^{(j)}$. Let ${\langle {A^{(1)}(x^{(1)}),...,A^{(m)}(x^{(m)})}\rangle }(r)$ denote the (joint) output of the m parties, when the $j^\mathrm{th}$ party uses strategy $A^{(j)}$ and gets input $x^{(j)}$, and all parties have free access to the shared randomness r. Considering promise problems $\varPsi =(P,S)$ such that $P,S\subseteq \{0,1\}^{m\cdot n}$, Definition 2.1 extends naturally; that is, the $\eta$ -error communication complexity of $\varPsi$, denoted $\mathtt{CC}_\eta (\varPsi )$, is the minimum communication complexity of all m-protocols that solve $\varPsi$ with error at most $\eta$.

Theorem A.1

(Theorem 3.1, generalized to m-party protocols): Let $\varPsi =(P,S)$ be a promise problem such that $P,S\subseteq \{0,1\}^{m\cdot n}$, and let $\varPi \subseteq \{0,1\}^\ell$ be a property, and $\epsilon ,\eta >0$. Suppose that the mapping $F:\{0,1\}^{m\cdot n}\rightarrow \{0,1\}^\ell$ satisfies the following two conditions:

• 1. For every $(x^{(1)},...,x^{(m)})\in P\cap S$, it holds that $F(x^{(1)},...,x^{(m)})\in \varPi$.

• 2. For every $(x^{(1)},...,x^{(m)})\in P\setminus S$, it holds that $F(x^{(1)},...,x^{(m)})$ is $\epsilon$-far from $\varPi$.

Then, $\mathtt{Q}_\eta (\epsilon ,\varPi )\ge \mathtt{CC}_{2\eta }(\varPsi )/B$, where $B=\max _{i\in [\ell ]}\{\mathtt{CC}_{\eta /n}(f_i)\}$ and $f_i(x^{(1)},...,x^{(m)})$ is the $i^\mathrm{th}$ bit of $F(x^{(1)},...,x^{(m)})$. Furthermore, if $B=\max _{i\in [\ell ]}\{\mathtt{CC}_{0}(f_i)\}$, then $\mathtt{Q}_\eta (\epsilon ,\varPi )\ge \mathtt{CC}_{\eta }(\varPsi )/B$.

Theorem A.1 is proved by a straightforward generalization of the proof of Theorem 3.1; that is, we merely replace “two” by “m” (and everything goes through). We believe that this generalization further clarifies the ideas underlying the proof of Theorem 3.1 by presenting them in a slightly more abstract form.

Proof:

The following description applies to any communication model in which all parties obtain the output produced by the protocol. Given an $\epsilon$-tester with error $\eta$ for $\varPi$ and communication protocols for the $f_i$’s, we present a protocol for solving $\varPsi$. The key idea is that, using their shared randomness, the parties (holding the inputs $x^{(1)},...,x^{(m)}$, respectively) can emulate the execution of the $\epsilon$-tester, while providing it with virtual access to $F(x^{(1)},...,x^{(m)})$. Specifically, when the tester queries the $i^\mathrm{th}$ bit of the oracle, the parties provide it with the value of $f_i(x^{(1)},...,x^{(m)})$ by first executing the corresponding communication protocol. Details follow.

The protocol for $\varPsi$ proceeds as follows: On local input $x^{(j)}$ and shared randomness $r=(r_0,r_1,...,r_\ell )\in (\{0,1\}^*)^{\ell +1}$, the $j^\mathrm{th}$ party invokes the $\epsilon$-tester on randomness $r_0$, and answers the tester’s queries by interacting with the other parties. That is, each of the parties invokes a local copy of the tester’s program, but all copies are invoked on the same randomness, and are fed with identical answers to their (identical) queries. When the tester issues a query $i\in [\ell ]$, the parties compute the value of $f_i(x^{(1)},...,x^{(m)})$ by using the corresponding communication protocol, and feed $f_i(x^{(1)},...,x^{(m)})$ to (their local copy of) the tester. Specifically, denoting the latter protocol (i.e., sequence of strategies) by $(A^{(1)}_i,...,A^{(m)}_i)$, the parties answer with ${\langle {A^{(1)}_i(x^{(1)}),...,A^{(m)}_i(x^{(m)})}\rangle }(r_i)$. When the tester halts, each party outputs the output it has obtained from (its local copy of) the tester.

We stress that the above description is oblivious to the details of the communication model, as long as in this model all parties obtain the output produced by the protocol.²⁷ Indeed, the description presented in the proof of Theorem 3.1 is merely a special case (which corresponds to the standard model of two-party computation), and the analysis of the general case (omitted here) is identical to the analysis of the special case presented in the proof of Theorem 3.1.    $\blacksquare$

On the Potential Usefulness of the Generalization. Tom Gur has pointed out that the generalization to multi-party communication complexity allows additional flexibility for the design of reductions. To illustrate the point, he suggested the proof outlined below, which refers to a multi-party communication complexity model in which parties obtain non-overlapping inputs and communication is by individual point-to-point channels.

Theorem A.2

(a property testing (encoded) version of the frequency moment problem of [2]):²⁸ For $k(n)=n/2$ and $\ell (n)=n^{1+o(1)}$, let $\mathcal{F}$ be a finite field of size n, and $C:\mathcal{F}^{k(n)} \rightarrow \mathcal{F}^{\ell (n)}$ be a $\mathcal{F}$-linear code of constant relative distance, denoted $\delta$. For any sequence $x=(x_1,...,x_k)\in \mathcal{F}^k$ and $v\in \mathcal{F}$, let $\#_v(x)$ denote the number of occurrences of v in x; that is, $\#_v(x)=|\{i\in [k]:x_i\!=\!v\}|$. For any constant $c>1$, let

$$\begin{aligned} \varPi= & {} \left\{ C(x): x\in \mathcal{F}^{k(n)} \wedge \sum _{v\in \mathcal{F}}\#_v(x)^c = k(n)\right\} \end{aligned}$$

(12)

$$\begin{aligned} \varPi '= & {} \left\{ C(x): x\in \mathcal{F}^{k(n)} \wedge \sum _{v\in \mathcal{F}}\#_v(x)^c \le 2k(n)\right\} _. \end{aligned}$$

(13)

Then, distinguishing inputs in $\varPi$ from inputs that are $\delta$-far from $\varPi '$ requires $\varOmega (\ell (n)^{1-(7/c)})$ queries.

Indeed, it follows that testing $\varPi$ requires query complexity $\varOmega (\ell (n)^{1-(7/c)})$, but this (and, in fact, a stronger $\varOmega (n/\log n)$ lower bound) can be proved by reduction from a two-party communication complexity problem (i.e., $\mathrm DISJ$).²⁹ In contrast, Theorem A.2 refers to a doubly-relaxed decision problem, where one level of relaxation is the approximation of the norm (captured by the gap between $\varPi$ and $\varPi '$) and the second is the standard property testing relaxation (captured by the gap between $\varPi '$ and $\delta$-far from $\varPi '$). Such doubly-relaxed problems have been often considered in the property testing literature (cf., e.g., [1, 20]), starting with [18]. The following proof, which adapts a proof of [2] (which in turn refers to streaming algorithms), relies on a reduction from a multi-party communication problem. As is the case with its streaming original [2], it is not clear whether Theorem A.2 can be proved by reduction from a two-party communication problem.

Proof:

We shall use a reduction from the following multi-party communication problem, denoted (m, t)-$\mathrm{DISJ}_n$. In this problem, there are m parties, each holding a t-subset of [n], and the problem is to distinguish the case that the subsets are pairwise disjoint from the case that the intersection of all subsets is non-empty. By [2], if $n \ge 2mt-m+1$, then the communication complexity of (m, t)-$\mathrm{DISJ}_n$ (in the point-to-point channels model) is $\varOmega (t/m^3)$.³⁰

We set $m=n^{1/c}$ and $t=n/2m$ (so that $n=2mt$), and represent the input of the $j^\mathrm{th}$ party by a sequence $x^{(j)}\in \mathcal{F}^t$. Recall that $|\mathcal{F}|=n$ and $k(n)=n/2=mt$. Now, we let $F(x^{(1)},...,x^{(m)}) = C(x^{(1)}\cdots x^{(m)})$, which equals $\sum _{j\in [m]} C(0^{(j-1)t}x^{(j)}0^{(m-j)t})$ by the $\mathcal{F}$-linearity. Hence, each bit of $F(x^{(1)},...,x^{(m)})$ can be computed (in this communication model) by communicating $m^2\log _2n$ bits (i.e., each party sends a single field elements to each of the other parties). Note that if $x=(x^{(1)},...,x^{(m)})$ is a yes-instance of (m, t)-$\mathrm{DISJ}_n$ then $\sum _{v\in \mathcal{F}}\#_v(x)^c = mt = k(n)$, since each element that occurs in x occurs in it exactly once (i.e., in one of the $x^{(j)}$’s), which means that $F(x^{(1)},...,x^{(m)})$ is in $\varPi$. On the other hand, if $(x^{(1)},...,x^{(m)})$ is a no-instance of (m, t)-$\mathrm{DISJ}_n$ then $\sum _{v\in \mathcal{F}}\#_v(x)^c > m^c = n = 2k(n)$, since at least one element occurs m times (i.e., in all the $x^{(j)}$’s), which means that $F(x^{(1)},...,x^{(m)})$ is not in $\varPi '$, and so it is $\delta$-far from any codeword in $\varPi '$ (since it is itself a codeword).

Applying Theorem A.1,³¹ it follows that the query complexity of the promise problem of distinguishing $\varPi$ from the set of $\ell (n)$-long sequences that are $\delta$-far from $\varPi '$ is lower bounded by $\varOmega (t/m^3)/(m^2\log n)$, which equals $\varOmega (n/(m^6\log n)) = \varOmega (n^{1-6(1+o(1))/c})$. Using $n=\ell (n)^{1/(1+o(1))}$, the claim follows.    $\blacksquare$