Abstract. The maximum agreement subtree problem is the following. Given two rooted trees whose leaves are drawn from the same set of items (e.g., species), find the largest subset of these items so that the portions of the two trees restricted to these items are isomorphic. We consider the case which occurs frequently in practice, i.e., the case when the trees are binary, and give an O(n log n) time algorithm for this problem.

We present two new algorithms for searching in sorted X+Y +R+S, one based on heaps and the other on sampling. Each of the algorithms runs in time O(n 2 logn) (n being the size of the sorted arrays X, Y , R and S). Hence in each case, by constructing arrays of size n = O(2 s=4 ), we obtain a new algorithm for solving certain NP-Complete problems such as Knapsack on s data items in time equal (up to a constant factor) to the best algorithm currently known. Each of the algorithms is capable of being efficiently implemented in parallel and so solving large instances of these NP-Complete problems fast on coarse-grained distributed memory parallel computers. The parallel version of the heap based algorithm is communication-efficient and exhibits optimal speedup for a number of processors less than n using O(n) space in each one; the sampling based algorithm exhibits optimal speedup for any number of processors up to n using O(n) space in total provided that the architecture is capable of...

We investigate a problem from computational biology: Given a constant size alphabet M with a weight function / : M--> +, find an efficient data structure and query algorithm solving the following problem: For a weight M C + and a string cr over A, decide whether cr contains a substring with weight M (ONE STRING MASS FINDING PROBLEM). If the answer is yes, then we may in addition require a witness, i.e. indices i _ i and ending at position j has weight M. We allow preprocessing of the string, and measure efficiency in two parameters: storage space required for the preprocessed data, and running time of the query algorithm for given M. We are interested in data structures and algorithms requiring subquadratic storage space and sublinear query time, where we measure the input size as the length of the input string. We present two efficient algorithms: LOOKUP solves the problem with O(,) space and (Wg ' loglog,) time; INTERVAL solves the problem for binary alphabets with O0, ) space in O(log,) time. We sketch a third al-gorithm, CLUSTER, which can be adjusted for a space time tradeoff but for which we do not yet have a resource analysis. We introduce a function on weighted strings which is closely related to the analysis of algorithms for the ONE STRING MASS FINDING PROBLEM: The number of different submasses of a weighted string. We present several properties of this function, including upper and lower bounds. Finally, we introduce two more general variants of the problem and sketch how algorithms may be extended for these variants.

We consider the problem of maintaining a binary search tree (BST) that minimizes the average access cost needed to satisfy randomly generated requests. We analyze scenarios in which the accesses are generated according to a vector of fixed probabilities which is unknown. Our approach is statistical. We devise policies for modifying the tree structure dynamically, using rotations of accessed records. The aim is to produce good approximations of the optimal structure of the tree, while keeping the number of rotations as small as possible. The heuristics that we propose achieve a close approximation to the optimal BST, with lower organization costs than any previously studied. We introduce the MOVE ONCE rule. The average access cost to the tree under this rule is shown to equal the value achieved by the common rule Move to the Root (MTR). The advantage of MOVE ONCE over MTR and similar rules is that it relocates each of the items in the tree at most once. We show that the total expected cost of modifying the tree by the MOVE ONCE rule is bounded from above by 2(n +1)H n \Gamma 4n rotations (in a tree with n records), where H n is the nth harmonic number. Extensive experiments show that this value is an over-estimate, and in fact the number of rotations is linear for all the access probability vectors we tested. An approximate analysis is shown to match the experimental results, producing the expected number n \Gamma

We show that multi-splay trees have most of the properties that splay trees have. Specifically, we show that multi-splay trees have the following properties: the access lemma, static optimality, the static finger property, the working set property, and key-independent optimality. Moreover, we prove that multi-splay trees have the deque property, which was conjectured by Tarjan in 1985 for splay trees, but remains unproven despite a significant amount of research toward proving it. Efficiently maintaining and manipulating sets of elements from a totally ordered universe is a fundamental problem in computer science. Specifically, many algorithms need a data structure that can efficiently support at least the following operations: insert, delete, predecessor, and successor, as well as membership testing. A standard data structure that maintains a totally ordered set and

In this thesis, we consider the following problem: Given a strongly connected digraph G = (V, E), where V is the set of vertices and E is the set of edges, “is it a minimal strong connected digraph?”. A reducible edges e is one for which G−e is strongly connected. A minimal strongly connected digraph is one with no reducible edges. Our approach is to apply depth first search on G to generate a depth first search tree and the sets of back, forward, and cross edges. Then we determine if there are any reducible non-tree edges. If not, we then check if there are any reducible tree edges based on an algorithm for finding immediate dominators. We have implemented the algorithm and report experimental results that show the algorithm can handle large digraphs quickly.