Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Suppose that we are given two independent sets I_b and I_r of a graph such that |I_b|=|I_r|, and imagine that a token is placed on each vertex in |I_b|. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I_b into I_r so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. The reconfiguration problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown, that is, it is in P, NP-complete, or PSPACE-complete in certain conditions. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be NP-complete to find a shortest sequence between them. Namely, finding a shortest sequence between two independent sets can be more difficult than the decision problem of reconfigurability between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. More precisely, we can find a shortest sequence between two independent sets on a graph G in polynomial time if either G is a proper interval graph, a trivially perfect graph, or a caterpillar. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours

The Convex Configurations of "Sei Shonagon Chie no Ita" and Other Dissection Puzzles

The tangram and Sei Shonagon Chie no Ita are popular dissection puzzles consisting of seven pieces. Each puzzle can be formed by identifying edges from sixteen identical right isosceles triangles. It is known that the tangram can form 13 convex polygons. We show that Sei Shonagon Chie no Ita can form 16 convex polygons, propose a new puzzle that can form 19, no 7 piece puzzle can form 20, and 11 pieces are necessary and sufficient to form all 20 polygons formable by 16 identical isosceles right triangles. Finally, we examine the number of convex polygons formable by different quantities of these triangles

The height of random kk-trees and related branching processes

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to clog t, where t is the number of vertices, and c=c(k) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k-->oo the height of both processes is asymptotic to log t/(k log 2)

Sliding Tokens on a Cactus

Given two independent sets I and J of a graph G, imagine that a token (coin) is placed on each vertex in I. Then, the Sliding Token problem asks if one could transforms I to J using a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors, such that the resulting set of vertices where tokens are placed still remains independent. In this paper, we describe a polynomial-time algorithm for solving Sliding Token in case the graph G is a cactus. Our algorithm is designed based on two observations. First, all structures that forbid the existence of a sequence of token slidings between I and J, if exist, can be found in polynomial time. A no-instance may be easily deduced using this characterization. Second, without such forbidden structures, a sequence of token slidings between I and J does exist

A double classification tree search algorithm for index SNP selection

BACKGROUND: In population-based studies, it is generally recognized that single nucleotide polymorphism (SNP) markers are not independent. Rather, they are carried by haplotypes, groups of SNPs that tend to be coinherited. It is thus possible to choose a much smaller number of SNPs to use as indices for identifying haplotypes or haplotype blocks in genetic association studies. We refer to these characteristic SNPs as index SNPs. In order to reduce costs and work, a minimum number of index SNPs that can distinguish all SNP and haplotype patterns should be chosen. Unfortunately, this is an NP-complete problem, requiring brute force algorithms that are not feasible for large data sets. RESULTS: We have developed a double classification tree search algorithm to generate index SNPs that can distinguish all SNP and haplotype patterns. This algorithm runs very rapidly and generates very good, though not necessarily minimum, sets of index SNPs, as is to be expected for such NP-complete problems. CONCLUSIONS: A new algorithm for index SNP selection has been developed. A webserver for index SNP selection is available a

Convex Configurations on Nana-kin-san Puzzle

We investigate a silhouette puzzle that is recently developed based on the golden ratio. Traditional silhouette puzzles are based on a simple tile. For example, the tangram is based on isosceles right triangles; that is, each of seven pieces is formed by gluing some identical isosceles right triangles. Using the property, we can analyze it by hand, that is, without computer. On the other hand, if each piece has no special property, it is quite hard even using computer since we have to handle real numbers without numerical errors during computation. The new silhouette puzzle is between them; each of seven pieces is not based on integer length and right angles, but based on golden ratio, which admits us to represent these seven pieces in some nontrivial way. Based on the property, we develop an algorithm to handle the puzzle, and our algorithm succeeded to enumerate all convex shapes that can be made by the puzzle pieces. It is known that the tangram and another classic silhouette puzzle known as Sei-shonagon chie no ita can form 13 and 16 convex shapes, respectively. The new puzzle, Nana-kin-san puzzle, admits to form 62 different convex shapes