David Eisenstat

A simple population protocol for fast robust approximate majority

We describe and analyze a 3-state one-way population protocol for approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of xs, ys and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least ω(√n log n). This protocol has the additional property of tolerating Byzantine behavior in o(√n) of the agents, making it the first known population protocol that tolerates Byzantine agents. Turning to the register machine construction from [2], we apply the 3-state approximate majority protocol and other techniques to speed up the per-step parallel time overhead of the simulation from O(log4 n) to O(log2 n). To increase the robustness of the phase clock at the heart of the register machine, we describe a consensus version of the phase clock and present encouraging simulation results; its analysis remains an open problem.

Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs

We give simple linear-time algorithms for two problems in planar graphs: max st-flow in directed graphs with unit capacities, and multiple-source shortest paths in undirected graphs with unit lengths.

Lower bounds on learning random structures with statistical queries

We show that random DNF formulas, random log-depth decision trees and random deterministic finite acceptors cannot be weakly learned with a polynomial number of statistical queries with respect to an arbitrary distribution on examples.

The VC dimension of k-fold union

The known O(dklogk) bound on the VC dimension of k-fold unions or intersections of a given concept class with VC dimension d is shown to be asymptotically tight.

Low-contention data structures

We consider the problem of minimizing contention in static dictionary data structures, where the contention on each cell is measured by the expected number of probes to that cell given an input that is chosen from a distribution that is not known to the query algorithm (but that may be known when the data structure is built). When all positive queries are equally probable, and similarly all negative queries are equally probable, we show that it is possible to construct a data structure using linear space s, a constant number of queries, and with contention O(1/s) on each cell, corresponding to a nearly-flat load distribution. All of these quantities are asymptotically optimal. For arbitrary query distributions, the lack of knowledge of the query distribution by the query algorithm prevents perfect load leveling in this case: we present a lower bound, based on VC-dimension, that shows that for a wide range of data structure problems, achieving contention even within a polylogarithmic factor of optimal requires a cell-probe complexity of Ω(log log n).

k-Fold unions of low-dimensional concept classes

We show that 2 is the minimum VC dimension of a concept class whose k-fold union has VC dimension @W(klogk).

Storage capacity of labeled graphs

We consider the question of how much information can be stored by labeling the vertices of a connected undirected graph G using a constant-size set of labels, when isomorphic labelings are not distinguishable. An exact information-theoretic bound is easily obtained by counting the number of isomorphism classes of labelings of G, which we call the information-theoretic capacity of the graph. More interesting is the effective capacity of members of some class of graphs, the number of states distinguishable by a Turing machine that uses the labeled graph itself in place of the usual linear tape. We show that the effective capacity equals the information-theoretic capacity up to constant factors for trees, random graphs with polynomial edge probabilities, and bounded-degree graphs.