A set G of points on a terrain, also known as an x-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem.

A set G of points on a 1.5-dimensional terrain, also known as an x-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem.

BACKGROUND: Improving the accuracy and efficiency of motif recognition is an important computational challenge that has application to detecting transcription factor binding sites in genomic data. Closely related to motif recognition is the CONSENSUS STRING decision problem that asks, given a parameter d and a set of l-length strings S = {s1, ..., sn}, whether there exists a consensus string that has Hamming distance at most d from any string in S. A set of strings S is pairwise bounded if the Hamming distance between any pair of strings in S is at most 2d. It is trivial to determine whether a set is pairwise bounded, and a set cannot have a consensus string unless it is pairwise bounded. We use CONSENSUS STRING to determine whether or not a pairwise bounded set has a consensus. Unfortunately, CONSENSUS STRING is NP-complete. The lack of an efficient method to solve the CONSENSUS STRING problem has caused it to become a computational bottleneck in MCL-WMR, a motif recognition program capable of solving difficult motif recognition problem instances. RESULTS: We focus on the development of a method for solving CONSENSUS STRING quickly with a small probability of error. We apply this heuristic to develop a new motif recognition program, sMCL-WMR, which has impressive accuracy and efficiency. We demonstrate the performance of sMCL-WMR in detecting weak motifs in large data sets and in real genomic data sets, and compare the performance to other leading motif recognition programs. In our preliminary discussion of our CONSENSUS STRING algorithm we give insight into the issue of sampling pairwise bounded sets, and discuss its relevance to motif recognition. CONCLUSION: Our novel heuristic gives birth to a state of the art program, sMCL-WMR, that is capable of detecting weak motifs in data sets with a large number of strings. sMCL-WMR is orders of magnitude faster than its predecessor MCL-WMR and is capable of solving previously unsolved synthetic motif recognition problems. Lastly, sMCL-WMR shows impressive accuracy in detecting transcription factor binding sites in the genomic data and used in the assessment of Tompa et al.

A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in $\mathbb{R}^d$. We show that, uniformly over all such data sets S, the expected height of the hyperplane tree is not worse than that of the k-d tree or the ordinary one-dimensional random binary search tree, and that, for any fixed $d\ge3$, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one.