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We consider distributed iterative algorithms for the averaging problem over timevarying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for time-varying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels. *

Given a linear system, we consider the problem of finding a small set of variables to affect with an input so that the resulting system is controllable. We show that this problem is NP-hard; indeed, we show that even approximating the minimum number of variables that need to be affected within a multiplicative factor of c log n is NP-hard for some positive c. On the positive side, we show it is possible to find sets of variables matching this inapproximability barrier in polynomial time. This can be done by a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs demonstrate this heuristic almost always succeeds at findings the minimum number of variables.1 One may also consider variations in which we search for a matrix B ′ ∈ R n×n renderingẋ = Ax + B ′ u controllable while seeking to minimize either the number of nonzero entries of B ′ , or the number of rows of B ′ with a nonzero entry (representing the number of components of the system affected). However, for any such matrix B ′ , we can easily construct a diagonal matrix B rendering the system controllable without increasing the number of nonzero entries, or the number of rows with a nonzero entry: indeed, if the i'th row of B ′ contains a nonzero entry, we simply set Bii = 1, and else we set Bii = 0. Consequently, both of these variations are easily seen to be equivalent to the problem of finding the sparsest diagonal matrix.

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