9 research outputs found
Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams
In the communication problem (universal relation) [KRW95],
Alice and Bob respectively receive with the promise that
. The last player to receive a message must output an index such
that . We prove that the randomized one-way communication
complexity of this problem in the public coin model is exactly
for failure
probability . Our lower bound holds even if promised
. As a corollary, we obtain
optimal lower bounds for -sampling in strict turnstile streams for
, as well as for the problem of finding duplicates in a stream. Our
lower bounds do not need to use large weights, and hold even if promised
at all points in the stream.
We give two different proofs of our main result. The first proof demonstrates
that any algorithm solving sampling problems in turnstile streams
in low memory can be used to encode subsets of of certain sizes into a
number of bits below the information theoretic minimum. Our encoder makes
adaptive queries to throughout its execution, but done carefully
so as to not violate correctness. This is accomplished by injecting random
noise into the encoder's interactions with , which is loosely
motivated by techniques in differential privacy. Our second proof is via a
novel randomized reduction from Augmented Indexing [MNSW98] which needs to
interact with adaptively. To handle the adaptivity we identify
certain likely interaction patterns and union bound over them to guarantee
correct interaction on all of them. To guarantee correctness, it is important
that the interaction hides some of its randomness from in the
reduction.Comment: merge of arXiv:1703.08139 and of work of Kapralov, Woodruff, and
Yahyazade
Communication Complexity Towards Lower Bounds on Circuit Depth
Karchmer, Raz, and Wigderson [2] discuss the circuit depth complexity of n bit Boolean functions constructed by composing up to d = log n= log log n levels of k = log n bit boolean functions. Any such function is in AC 1 . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires dk 2\Omega (log 2 n= log log n) depth. This would separate AC 1 from NC 1 . They recommend using the communication game characterization of circuit depth [3]. In order to develop techniques for using communications complexity to prove circuit lower bounds, they suggest an intermediate communications complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk \Gamma O(d 2 (k log k) 1=2 ) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring\Omega (k) circuit depth. Altho..
Communication Complexity Towards Lower Bounds on Circuit Depth
Karchmer, Raz, and Wigderson [2] discuss the circuit depth complexity of n bit Boolean functions constructed by composing up to d = log n= log log n levels of k = log n bit boolean functions. Any such function is in AC 1 . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires dk 2\Omega\Gamma446 2 n= log log n) depth. This would separate AC 1 from NC 1 . They recommend using the communication game characterization of circuit depth [3]. In order to develop techniques for using communications complexity to prove circuit lower bounds, they suggest an intermediate communications complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk \Gamma O(d 2 (k log k) 1=2 ) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring\Omega\Gamma k) circuit dept..