A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easy-to-verify proof) that the particular output has not been compromised by a bug. A user of a certifying program P (= the implementation of a certifying algorithm) inputs x, receives an output y and a certificate w, and then checks, either manually or by use of a checking program, that w proves th...

This paper considers the optimal control problem of minimizing control effort subject to multiple performance constraints on output covariance matrices Yi of the form Yi ≤ Y i, where Y i is given. The contributions of this paper are a set of conditions that characterize global optimality, and an iterative algorithm for finding a solution to the optimality conditions. This iterative algorithm is...

We now look at the list decoding algorithm in the breakthrough work of Sudan [2]. To motivate the algorithm, recall that in the previous algorithm, in order to prove that Step 2 works, we defined a polynomial R(X) △ = Q(X,P (X)). In particular, this implied that deg(R) ≤ degX(Q) + k · degY (Q) (and we had to set t > degX(Q) + k · degY (Q)). One shortcoming of this approach is that the maximum d...

The problem is finding a general, robust, and efficient mechanism for estimating a conditional probability P (y|x) where robustness and efficiency are measured using techniques from learning reductions. In particular, suppose we have access to a binary regression oracle B which has two interfaces—one for specifying training information and one for testing. Training information is specified as B...

An absent (or forbidden) word of a word y is a word that does not occur in y. It is then called minimal if all its proper factors occur in y. There exist linear-time and linear-space algorithms for computing all minimal absent words of y (Crochemore et al., 1998, Belazzougui et al., 2013, Barton et al., 2014). Minimal absent words are used for data compression (Crochemore et al., 2000, Ota and ...

We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 . error ratio can be given for the problem unless P s NP. We present an O n time algorithm for the problem, where n is the number of nodes in the graph. A...

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ξ = F (x), η = P (x) y+Q(x)] where ξ ∂/∂x+η ∂/∂y is the symmetry generator is presented. To these linear symmetries one can associate an ODE class which embraces all first order ODEs mappable into separable through linear transformations {t = f(x), u = p(x) y + q(x)}. This single ODE class includes ...

In the String-to-String Correction problem we are given two strings x and y and a positive integer k, and are asked whether it is possible to transform y into x with at most k single character deletions and adjacent character exchanges. We present a first simple fixed-parameter algorithm for Stringto-String Correction that runs in O∗(2k). Moreover, we present a search tree algorithm that exhibi...

We present a simple linear time combinatorial algorithm for the Betweenness problem. In the Betweenness problem, we are given a set of vertices and betweenness constraints. Each betweenness constraint of the form x {y, z} requires that vertex x lies between vertices y and z. Our goal is to find a linear ordering of vertices that maximizes the number of satisfied constraints. In 1995, Chor and S...

Given data in the form of a collection of (x,y) pairs of real numbers, the symbolic function identification problem is to find a functional model of the form y = f(x) that fits the data. This paper describes a system for solution of symbolic function identification problems that combines a genetic algorithm and the Levenberg-Marquardt nonlinear regression algorithm. The genetic algorithm uses a...