Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [n] and Bob ends up with a set y ⊆ [n], such that (x, y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω(n) communication even to get within statistical distance 1 − β^n of the target distribution. Previously, Ambainis, Schulman, TaShma, Vazirani, and Wigderson (FOCS 1998) proved that Ω(√n) communication is required to get within some constant statistical distance ε > 0 of the uniform distribution over all pairs of disjoint sets of size √n.
more »
« less
A Lower Bound for Sampling Disjoint Sets
Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [ n ] and Bob ends up with a set y ⊆ [ n ], such that ( x , y ) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω ( n ) communication even to get within statistical distance 1− β n of the target distribution. Previously, Ambainis, Schulman, TaShma, Vazirani, and Wigderson (FOCS 1998) proved that Ω (√ n ) communication is required to get within some constant statistical distance ɛ > 0 of the uniform distribution over all pairs of disjoint sets of size √ n .
more »
« less
 Award ID(s):
 1942742
 NSFPAR ID:
 10275818
 Date Published:
 Journal Name:
 ACM Transactions on Computation Theory
 Volume:
 12
 Issue:
 3
 ISSN:
 19423454
 Page Range / eLocation ID:
 1 to 13
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


We study the role of interaction in the Common Randomness Generation (CRG) and Secret Key Generation (SKG) problems. In the CRG problem, two players, Alice and Bob, respectively get samples X1, X2, . . . and Y1, Y2, . . . with the pairs (X1, Y1), (X2, Y2), . . . being drawn independently from some known probability distribution μ. They wish to communicate so as to agree on L bits of randomness. The SKG problem is the restriction of the CRG problem to the case where the key is required to be close to random even to an eavesdropper who can listen to their communication (but does not have access to the inputs of Alice and Bob). In this work, we study the relationship between the amount of communication and the number of rounds of interaction in both the CRG and the SKG problems. Specifically, we construct a family of distributions μ = μr,n,L, parametrized by integers r, n and L, such that for every r there exists a constant b = b(r) for which CRG (respectively SKG) is feasible when (Xi, Yi) ~ μr,n,L with r + 1 rounds of communication, each consisting of O(log n) bits, but when restricted to r/2 − 2 rounds of interaction, the total communication must exceed Ω(n/ logb(n)) bits. Prior to our work no separations were known for r ≥ 2.more » « less

In a recent work (Ghazi et al., SODA 2016), the authors with Komargodski and Kothari initiated the study of communication with contextual uncertainty, a setup aiming to understand how efficient communication is possible when the communicating parties imperfectly share a huge context. In this setting, Alice is given a function f and an input string x, and Bob is given a function g and an input string y. The pair (x,y) comes from a known distribution mu and f and g are guaranteed to be close under this distribution. Alice and Bob wish to compute g(x,y) with high probability. The lack of agreement between Alice and Bob on the function that is being computed captures the uncertainty in the context. The previous work showed that any problem with oneway communication complexity k in the standard model (i.e., without uncertainty, in other words, under the promise that f=g) has publiccoin communication at most O(k(1+I)) bits in the uncertain case, where I is the mutual information between x and y. Moreover, a lower bound of Omega(sqrt{I}) bits on the publiccoin uncertain communication was also shown. However, an important question that was left open is related to the power that public randomness brings to uncertain communication. Can Alice and Bob achieve efficient communication amid uncertainty without using public randomness? And how powerful are publiccoin protocols in overcoming uncertainty? Motivated by these two questions:  We prove the first separation between privatecoin uncertain communication and publiccoin uncertain communication. Namely, we exhibit a function class for which the communication in the standard model and the publiccoin uncertain communication are O(1) while the privatecoin uncertain communication is a growing function of n (the length of the inputs). This lower bound (proved with respect to the uniform distribution) is in sharp contrast with the case of publiccoin uncertain communication which was shown by the previous work to be within a constant factor from the certain communication. This lower bound also implies the first separation between publiccoin uncertain communication and deterministic uncertain communication. Interestingly, we also show that if Alice and Bob imperfectly share a sequence of random bits (a setup weaker than public randomness), then achieving a constant blowup in communication is still possible.  We improve the lowerbound of the previous work on publiccoin uncertain communication. Namely, we exhibit a function class and a distribution (with mutual information I approx n) for which the oneway certain communication is k bits but the oneway publiccoin uncertain communication is at least Omega(sqrt{k}*sqrt{I}) bits. Our proofs introduce new problems in the standard communication complexity model and prove lower bounds for these problems. Both the problems and the lower bound techniques may be of general interest.more » « less

Tauman Kalai, Yael (Ed.)We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm N on the space ℝⁿ. Here, Alice and Bob hold two vectors v,u such that ‖v‖_N ≤ 1 and ‖u‖_{N^*} ≤ 1, where N^* is the dual norm. The goal is to compute their inner product ⟨v,u⟩ up to an ε additive term. The problem is denoted by IP_N, and generalizes important previously studied problems, such as: (1) Computing the expectation 𝔼_{x∼𝒟}[f(x)] when Alice holds 𝒟 and Bob holds f is equivalent to IP_{𝓁₁}. (2) Computing v^TAv where Alice has a symmetric matrix with bounded operator norm (denoted S_∞) and Bob has a vector v where ‖v‖₂ = 1. This problem is complete for quantum communication complexity and is equivalent to IP_{S_∞}. We systematically study IP_N, showing the following results, near tight in most cases: 1) For any symmetric norm N, given ‖v‖_N ≤ 1 and ‖u‖_{N^*} ≤ 1 there is a randomized protocol using 𝒪̃(ε^{6} log n) bits of communication that returns a value in ⟨u,v⟩±ε with probability 2/3  we will denote this by ℛ_{ε,1/3}(IP_N) ≤ 𝒪̃(ε^{6} log n). In a special case where N = 𝓁_p and N^* = 𝓁_q for p^{1} + q^{1} = 1, we obtain an improved bound ℛ_{ε,1/3}(IP_{𝓁_p}) ≤ 𝒪(ε^{2} log n), nearly matching the lower bound ℛ_{ε, 1/3}(IP_{𝓁_p}) ≥ Ω(min(n, ε^{2})). 2) One way communication complexity ℛ^{→}_{ε,δ}(IP_{𝓁_p}) ≤ 𝒪(ε^{max(2,p)}⋅ log n/ε), and a nearly matching lower bound ℛ^{→}_{ε, 1/3}(IP_{𝓁_p}) ≥ Ω(ε^{max(2,p)}) for ε^{max(2,p)} ≪ n. 3) One way communication complexity ℛ^{→}_{ε,δ}(N) for a symmetric norm N is governed by the distortion of the embedding 𝓁_∞^k into N. Specifically, while a small distortion embedding easily implies a lower bound Ω(k), we show that, conversely, nonexistence of such an embedding implies protocol with communication k^𝒪(log log k) log² n. 4) For arbitrary origin symmetric convex polytope P, we show ℛ_{ε,1/3}(IP_{N}) ≤ 𝒪(ε^{2} log xc(P)), where N is the unique norm for which P is a unit ball, and xc(P) is the extension complexity of P (i.e. the smallest number of inequalities describing some polytope P' s.t. P is projection of P').more » « less

We present decidability results for a subclass of “noninteractive” simulation problems, a wellstudied class of problems in information theory. A noninteractive simulation problem is specified by two distributions P(x, y) and Q(u, v): The goal is to determine if two players, Alice and Bob, that observe sequences Xn and Y n respectively where {(Xi, Yi)}n i=1 are drawn i.i.d. from P(x, y) can generate pairs U and V respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to Q(u, v). Even when P and Q are extremely simple: e.g., P is uniform on the triples {(0, 0), (0, 1), (1, 0)} and Q is a “doubly symmetric binary source”, i.e., U and V are uniform ±1 variables with correlation say 0.49, it is open if P can simulate Q. In this work, we show that whenever P is a distribution on a finite domain and Q is a 2 × 2 distribution, then the noninteractive simulation problem is decidable: specifically, given δ > 0 the algorithm runs in time bounded by some function of P and δ and either gives a noninteractive simulation protocol that is δclose to Q or asserts that no protocol gets O(δ)close to Q. The main challenge to such a result is determining explicit (computable) convergence bounds on the number n of samples that need to be drawn from P(x, y) to get δclose to Q. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds.more » « less