Cooperative Estimation via Altruism

A novel approach, based on the notion of altruism, is presented to cooperative estimation in a system comprising two information-sharing estimators. The underlying assumption is that the system's global mission can be accomplished even if only one of the estimators achieves satisfactory performance. The notion of altruism motivates a new definition of cooperative estimation optimality that generalizes the common definition of minimum mean square error optimality. Fundamental equations are derived for two types of altruistic cooperative estimation problems, corresponding to heterarchical and hierarchical setups. Although these equations are hard to solve in the general case, their solution in the Gaussian case is straightforward and only entails the largest eigenvalue of the conditional covariance matrix and its corresponding eigenvector. Moreover, in that case the performance improvement of the two altruistic cooperative estimation techniques over the conventional (egoistic) estimation approach is shown to depend on the problem's dimensionality and statistical distribution. In particular, the performance improvement grows with the dispersion of the spectrum of the conditional covariance matrix, rendering the new estimation approach especially appealing in ill-conditioned problems. The performance of the new approach is demonstrated using a numerical simulation study.Comment: 14 pages, 9 figure

On The Multiparty Communication Complexity of Testing Triangle-Freeness

In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the $\textit{coordinator}$ model, where we have $k$ players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or $\textit{far}$ from triangle-free? For general communication protocols, we show that $\tilde{O}(k(nd)^{1/4}+k^2)$ bits are sufficient to test triangle-freeness in graphs of size $n$ with average degree $d$ (the degree need not be known in advance). For $\textit{simultaneous}$ protocols, where there is only one communication round, we give a protocol that uses $\tilde{O}(k \sqrt{n})$ bits when $d = O(\sqrt{n})$ and $\tilde{O}(k (nd)^{1/3})$ when $d = \Omega(\sqrt{n})$; here, again, the average degree $d$ does not need to be known in advance. We show that for average degree $d = O(1)$, our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free.Comment: To Appear in PODC 201

Experiments with a Galton board

Galton boards have been used for over a half-century as a tool to illustrate the formation of Gaussian shaped distributions as well as the Central Limit Theorem. Here, the Galton board was used to study the spontaneous percolation of a particle through an ordered array of rigid scatterers. The apparatus that was designed and fabricated provided a means to release 1/8 diameter spheres one at a time in a controlled and precise manner at any location on the board. The three experimental variables used in these experiments were the particle material, the release height, and the board tilt. angle. The data, consisting of residence time and exit location, were analyzed and the relationship between statistical values and parameter settings was found to be as follows: (1) standard deviation of the radial displacement increased with release height and was unaffected by board angle, (2) average residence time increased with release height and decreased with board angle, (3) standard deviation of the residence time increased with release height, (4) average axial velocity was unaffected by release height and increased with board angle, and (5) standard deviation of the axial velocity increased with a decrease of release height and increased with an increase in board angle. From an analysis of the data, it can be inferred that the motion of particles on the Galton board is governed by a diffusional mechanism

On the Computational Power of Radio Channels

Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

Tight Bounds for Set Disjointness in the Message Passing Model

In a multiparty message-passing model of communication, there are $k$ players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work \cite{phillips12,woodruff12,woodruff13}, strong lower bounds of the form $\Omega(n \cdot k)$ were obtained for several functions in the message-passing model; however, a lower bound on the classical Set Disjointness problem remained elusive. In this paper, we prove tight lower bounds of the form $\Omega(n \cdot k)$ for the Set Disjointness problem in the message passing model. Our bounds are obtained by developing information complexity tools in the message-passing model, and then proving an information complexity lower bound for Set Disjointness. As a corollary, we show a tight lower bound for the task allocation problem \cite{DruckerKuhnOshman} via a reduction from Set Disjointness