On Polynomial Time Constructions of Minimum Height Decision Tree

A decision tree T in B_m:={0,1}^m is a binary tree where each of its internal nodes is labeled with an integer in [m]={1,2,...,m}, each leaf is labeled with an assignment a in B_m and each internal node has two outgoing edges that are labeled with 0 and 1, respectively. Let A subset {0,1}^m. We say that T is a decision tree for A if (1) For every a in A there is one leaf of T that is labeled with a. (2) For every path from the root to a leaf with internal nodes labeled with i_1,i_2,...,i_k in[m], a leaf labeled with a in A and edges labeled with xi_{i_1},...,xi_{i_k}in {0,1}, a is the only element in A that satisfies a_{i_j}=xi_{i_j} for all j=1,...,k. Our goal is to write a polynomial time (in n:=|A| and m) algorithm that for an input A subseteq B_m outputs a decision tree for A of minimum depth. This problem has many applications that include, to name a few, computer vision, group testing, exact learning from membership queries and game theory. Arkin et al. and Moshkov [Esther M. Arkin et al., 1998; Mikhail Ju. Moshkov, 2004] gave a polynomial time (ln |A|)- approximation algorithm (for the depth). The result of Dinur and Steurer [Irit Dinur and David Steurer, 2014] for set cover implies that this problem cannot be approximated with ratio (1-o(1))* ln |A|, unless P=NP. Moshkov studied in [Mikhail Ju. Moshkov, 2004; Mikhail Ju. Moshkov, 1982; Mikhail Ju. Moshkov, 1982] the combinatorial measure of extended teaching dimension of A, ETD(A). He showed that ETD(A) is a lower bound for the depth of the decision tree for A and then gave an exponential time ETD(A)/log(ETD(A))-approximation algorithm and a polynomial time 2(ln 2)ETD(A)-approximation algorithm. In this paper we further study the ETD(A) measure and a new combinatorial measure, DEN(A), that we call the density of the set A. We show that DEN(A) <=ETD(A)+1. We then give two results. The first result is that the lower bound ETD(A) of Moshkov for the depth of the decision tree for A is greater than the bounds that are obtained by the classical technique used in the literature. The second result is a polynomial time (ln 2)DEN(A)-approximation (and therefore (ln 2)ETD(A)-approximation) algorithm for the depth of the decision tree of A. We then apply the above results to learning the class of disjunctions of predicates from membership queries [Nader H. Bshouty et al., 2017]. We show that the ETD of this class is bounded from above by the degree d of its Hasse diagram. We then show that Moshkov algorithm can be run in polynomial time and is (d/log d)-approximation algorithm. This gives optimal algorithms when the degree is constant. For example, learning axis parallel rays over constant dimension space

Efficient Algorithms for Battleship

We consider an algorithmic problem inspired by the Battleship game. In the variant of the problem that we investigate, there is a unique ship of shape $S \subset Z^2$ which has been translated in the lattice $Z^2$. We assume that a player has already hit the ship with a first shot and the goal is to sink the ship using as few shots as possible, that is, by minimizing the number of missed shots. While the player knows the shape $S$, which position of $S$ has been hit is not known. Given a shape $S$ of $n$ lattice points, the minimum number of misses that can be achieved in the worst case by any algorithm is called the Battleship complexity of the shape $S$ and denoted $c(S)$. We prove three bounds on $c(S)$, each considering a different class of shapes. First, we have $c(S) \leq n-1$ for arbitrary shapes and the bound is tight for parallelogram-free shapes. Second, we provide an algorithm that shows that $c(S) = O(\log n)$ if $S$ is an HV-convex polyomino. Third, we provide an algorithm that shows that $c(S) = O(\log \log n)$ if $S$ is a digital convex set. This last result is obtained through a novel discrete version of the Blaschke-Lebesgue inequality relating the area and the width of any convex body.Comment: Conference version at 10th International Conference on Fun with Algorithms (FUN 2020

Point Probe Decision Trees for Geometric Concept Classes

A fundamental problem in model-based computer vision is that of identifying to which of a given set of concept classes of geometric models an observed model belongs. Considering a &quot;probe&quot; to be an oracle that tells whether or not the observed model is present at a given point in an image, we study the problem of computing efficient strategies (&quot;decision trees&quot;) for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine in which class the observed model belongs. We prove a hardness result and give strategies that obtain decision trees whose height is within a log factor of optimal. These results grew out of discussions that began in a series of workshops on Geometric Probing in Computer Vision, sponsored by the Center for Night Vision and Electro-Optics, Fort Belvoir, Virginia, and monitored by the U.S. Army Research Office. The views, opinions, and/or findings contained in this report are those of the authors and should not be con..