Andrew Drucker

On the power of the congested clique model

We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors. It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(nε) time for any ε > 0. The broadcast version of the congested clique is the well-known multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model.

The communication complexity of distributed task allocation

We consider a distributed task allocation problem in which <i>m</i> players must divide a set of <i>n</i> tasks between them. Each player <i>i</i> receives as input a set <i>X_i</i> of tasks such that the union of all input sets covers the task set. The goal is for each player to output a subset <i>Y_i</i> ⊆ <i>X_i</i>, such that the outputs (<i>Y₁</i>,...,<i>Y_m</i>) form a partition of the set of tasks. The problem can be viewed as a distributed one-shot variant of the well-known <i>k</i>-server problem, and we also show that it is closely related to the problem of finding a rooted spanning tree in directed broadcast networks. We study the communication complexity and round complexity of the task allocation problem. We begin with the classical two-player communication model, and show that the randomized communication complexity of task allocation is Ω(<i>n</i>), even when the set of tasks is known to the players in advance. For the multi-player setting with <i>m</i> = <i>O</i>(<i>n</i>) we give two upper bounds in the shared-blackboard model of communication. We show that the problem can be solved in <i>O</i>(log <i>n</i>) rounds and <i>O</i>(<i>n</i> log <i>n</i>) total bits for arbitrary inputs; moreover, if for any set <i>X</i> of tasks, there are at least α|<i>X</i>| players that have at least one task from <i>X</i> in their inputs, then <i>O</i>((1/α + log <i>m</i>)log <i>n</i>) rounds suffice even if each player can only write <i>O</i>(log <i>n</i>) bits on the blackboard in each round. Finally, we extend our results to the case where the players communicate over an arbitrary directed communication graph instead of a shared blackboard. As an application of these results, we also consider the related problem of constructing a directed spanning tree in strongly-connected directed networks and we show lower and upper bounds for that problem.

New Limits to Classical and Quantum Instance Compression

Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given ...

Advice coins for classical and quantum computation

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states can be sensitive to arbitrarily small changes in a coins bias. This contrasts with classical probabilistic finite automata, whose sensitivity to changes in a coins bias is bounded by a classic 1970 result of Hellman and Cover. Despite this finding, we are able to bound the power of advice coins for space-bounded classical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin, of languages decidable by classical and quantum polynomial-space machines with advice coins. Our main theorem is that both classes coincide with PSPACE/poly. Proving this result turns out to require substantial machinery. We use an algorithm due to Neff for finding roots of polynomials in NC; a result from algebraic geometry that lower-bounds the separation of a polynomials roots; and a result on fixed-points of superoperators due to Aaronson and Watrous, originally proved in the context of quantum computing with closed timelike curves.

Limitations of Lower-Bound Methods for the Wire Complexity of Boolean Operators

We study the circuit complexity of Boolean operators, i.e., collections of Boolean functions defined over a common input. Our focus is the well-studied model in which arbitrary Boolean functions are allowed as gates, and in which a circuits complexity is measured by its depth and number of wires. We show sharp limitations of several existing lowerbound methods for this model. First, we study an information-theoretic lower-bound method due to Cherukhin, that yields bounds of form Ωd(n · λd-1(n)) on the number of wires needed to compute cyclic convolutions in depth d ≥ 2. This was the first improvement over the lower bounds provided by the well-known superconcentrator technique (for d = 2, 3 and for even d ≥ 4). Cherukhins method was formalized by Jukna as a general lower-bound criterion for Boolean operators, the “Strong Multiscale Entropy” (SME) property. It seemed plausible that this property could imply significantly better lower bounds by an improved analysis. However, we show that this is not the case, by exhibiting an explicit operator with the SME property that is computable in depth d with O(n · λd-1(n)) wires, for d = 2, 3 and for even d ≥ 6. Next, we show limitations of two simpler lower-bound criteria given by Jukna: the “entropy method” for general operators, and the “pairwise-distance method” for linear operators. We show that neither method gives super-linear lower bounds for depth 3. In the process, we obtain the first known polynomial separation between the depth-2 and depth-3 wire complexities for an explicit operator. We also continue the study (initiated by Jukna) of the complexity of “representing” a linear operator by bounded-depth circuits, a weaker notion than computing the operator.