In this paper, we present near-optimal space bounds for Lpsamplers. Given a stream of updates (additions and subtraction) to the coordinates of an underlying vector x ∈

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Let u, v ∈ R Ω + be positive unit vectors and S ∈ R Ω×Ω + be a symmetric substochastic matrix. For an integer t ≥ 0, let mt = v, S t u , which we view as the heat measured by v after an initial heat configuration u is let to diffuse for t time steps according to S. Since S is entropy improving, one may intuit that mt should not change too rapidly over time. We give the following formalizations of this intuition.We prove that mt+2 ≥ m 1+2/t t , an inequality studied earlier by Blakley and Dixon (also Erdős and Simonovits) for u = v and shown true under the restriction mt ≥ e −4t . Moreover we prove that for any ǫ > 0, a stronger inequality mt+2 ≥ t 1−ǫ · m 1+2/t t holds unless mt+2mt−2 ≥ δm 2 t for some δ that depends on ǫ only. Phrased differently, ∀ǫ > 0,which can be viewed as a truncated log-convexity statement.Using this inequality, we answer two related open questions in complexity theory: Any property tester for k-linearity requires Ω(k log k) queries and the randomized communication complexity of the k-Hamming distance problem is Ω(k log k). Further we show that any randomized parity decision tree computing k-Hamming weight has size exp (Ω(k log k)).

In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k log (r) k) bits over r rounds and errs with very small probability. Here we can take r = log * k to obtain a O(k) total communication log * k-round protocol with exponentially small error probability, improving on the O(k)-bits O(log k)-round constant error probability protocol of Håstad and Wigderson from 1997.In the exist-equal problem, the players receive vectors x, y ∈ [t] n and the goal is to determine whether there exists a coordinate i such that x i = y i . Namely, the exists-equal problem is the OR of n equality problems. Observe that exists-equal is an instance of sparse set disjointness with k = n, hence the protocol above applies here as well, giving an O(n log (r) n) upper bound. Our main technical contribution in this paper is a matching lower bound: we show that when t = Ω(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size Ω(n log (r) n). Our lower bound holds even for super-constant r ≤ log * n, showing that any O(n) bits exists-equal protocol should have log * n − O(1) rounds. Note that the protocol we give errs only with less than polynomially small probability and provides guarantees on the total communication for the harder set disjointness problem, whereas our lower bound holds even for constant error probability protocols and for the easier exists-equal problem with guarantees on the max-communication. Hence our upper and lower bounds match in a strong sense.Our lower bound on the constant round protocols for exists-equal show that solving the OR of n instances of the equality problems requires strictly more than n times the cost of a single instance. To our knowledge this is the first example of such a super-linear increase in complexity.[16]). We note that these lower bounds do not apply to the exists-equal problem, as the input distribution they use generates instances inherently specific to the disjointness problem; furthermore this distribution admits a O(log k) protocol in two rounds. The exists-equal problemIn the equality problem Alice and Bob receive elements x and y of a universe [t] and they have to decide whether x = y. We define the two player communication game existsequal with parameters t and n as follows. Each player is given an n-dimensional vector from [t] n , namely x and y. The value of the game is one if there exists a coordinate i ∈ [n] such that x i = y i , zero otherwise. Clearly, this problem is the OR of n independent instances of the equality problem.The direct sum problem in communication complexity is the study of whether n instances ...