Ewa Kubicka

An algorithm to find agreement subtrees

Given two binary trees, a largest subtree contained in both of the original trees that has been obtained by pruning vertices is called an agreement subtree. An exact algorithm for finding an agreement subtree is presented.

The agreement metric for labeled binary trees

Let S be a set of n objects. A binary tree of S is a binary tree whose leaves are labeled without repetition from S. The operation of pruning a tree T is that of removing some leaves from T and suppressing all inner vertices of degree 2 which are formed by this deletion. Given two trees T and U, an agreement tree is a tree that can be obtained from T as well as from U by pruning the fewest number of leaves from the two trees. A quadratic algorithm is presented for doing this and two metrics are defined based on agreement trees.

Using graph distance in object recognition

In this paper the concept of the distance between graphs is applied to the problem of recognizing objects. An unknown object is represented by a graph H and from the data base of possible solutions {G1, …, Gn} (also graphs) we select the graph Gi for which the distance to H is the smallest. The graph Gi is the recognition of H. The distance between graphs is defined in terms of edge rotation and deletion. It is shown that this distance defines a metric on the space of all graphs. The bounds for distance between two graphs are given in terms of their sizes and the size of their greatest common subgraph. It is proved that finding a distance between two graphs is an NP-complete problem even for planar graphs. An algorithm based on exhaustive search utilizing the linear algorithm by Hopcroft and Wong for recognizing isomorphism of planar graphs with some stopping criteria to maintain polynomial complexity, is possible.

Steiner intervals in graphs

Abstract LetG be a graph andu, v two vertices ofG. Then the interval fromu tov consists of all those vertices that lie on some shortestu — v path. LetS be a set of vertices in a connected graphG. Then the Steiner distancedG(S) ofS inG is the smallest number of edges in a connected subgraph ofG that containsS. Such a subgraph is necessarily a tree called a Steiner tree forS. The Steiner intervalIG(S) ofS consists of all those vertices that lie on some Steiner tree forS. LetS be ann-set of vertices ofG and suppose thatk ⩽ n. Then thek-intersection interval of S, denoted byIk(S) is the intersection of all Steiner intervals of allk-subsets ofS. It is shown that ifS = {u1, u2, ..., un} is a set ofn ⩾ 2 vertices of a graphG and if the 2-intersection interval ofS is nonempty andxeI2(S), then.d(S) = ∑ni = 1 =1 d(ui, x). It is observed that the only graphs for which the 2-intersection intervals of alln-sets,n ⩾ 4, are nonempty are stars. Moreover, for everyn ⩾ 4, those graphs with the property that the 3-intersection interval of everyn-set is nonempty are completely characterized. In general, ifn = 2k, those graphsG for whichIk(S) is nonempty for everyn-setS ofG are characterized.

The asymptotic plurality rule for molecular sequences

The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n (i.e., each sequence of n bases appearing at an aligned position of n molecules) to a set apl (P) of consensus results (i.e., ambiguity codes) that is a descriptive summary of P. Our main result is to characterize each consensus result X = apl(P) in terms of the frequencies with which the bases in P occur. We then use these characterizations to investigate features (e.g., strong consistency, length independence) of apl that researchers may find useful for the interpretation of apls consensus results.

The irregularity cost or sum of a graph

Abstract Working simultaneously in two teams [1,2], we have independently discovered essentially the same concept and many common results. As expected, each team used its own notation and terminology but the results are easily transformed between the two systems. We plan to publish our full papers separately, but present the results here.

An Efficient Method of Examining all Trees

In this paper, we present a techique for examining all trees of a given order. Our approach is based on the Beyer and Hedetniemi algorithm for generating all rooted trees of a given order and on the Wright, Richmond, Odlyzko and McKay algorithm for generating all free trees of a given order. In the introduction we describe these algorithms. We also give a precise evaluation of the average number of moves it takes to generate a rooted tree, which improves the upper bound given by Beyer and Hedetniemi. In the second section we present a new method of examining all trees which uses these generating algorithms. The last section contains two applications of the method introduced. The main result of the paper is that the average number of steps required by the proposed algorithm to examine a rooted tree is bounded by a constant independent of the order of a tree.

An efficient algorithm for stopping on a sink in a directed graph

Abstract Vertices of an unknown directed graph of order n are revealed one by one in some random permutation. At each point, we know the subgraph induced by the revealed vertices. Our goal is to stop on a sink, a vertex with no out-neighbors. We show that if a sink exists this can be achieved with probability Θ ( 1 / n ) , which is best possible.

Maximizing survival time in a random walk on an interval

ABSTRACT A gambler buys N tokens that enable him to play N rounds of the following game. A symmetric random walk on a discrete interval { − r, …, r} starts from the point 0. The gambler knows only the number of steps made so far, but is unaware of the current position of the walk. Once the walk hits one of the barriers − r or r for the first time in the current round, the round ends with no payoff. The gambler can start a new round by inserting a new token, if there are any tokens left. The gambler can end the game at any time getting the payoff equal to the number of steps made in the current round. We find the optimal stopping strategy for this game and calculate the expected payoff once the optimal strategy is applied.

Total Colorings of Graphs with Minimum Sum of Colors

The total chromatic sum of a graph is the minimum sum of colors (natural numbers) taken over all proper colorings of vertices and edges of a graph. We construct infinite families of graphs for which the minimum number of colors to achieve the total chromatic sum is larger than the total chromatic number.