Ronald I. Becker

A Shifting Algorithm for Min-Max Tree Partitioning

The problem of finding a mm-max partmon of a weJghted tree T with n veruces into q subtrees by means of k = q 1 cuts is considered. A top-down shifting algorithm for this problem ts presented An outhne is given of an efficJent implementatmn of the algorithm wtth complexity O(k3rd(T) + kn), where rd(T) ts the number of edges m the radius of T

Finding the ℓ-core of a tree

An l-core of a tree T = (V,E) with |V|= n, is a path P with length at most l that is central with respect to the property of minimizing the sum of the distances from the vertices in P to all the vertices of T not in P. The distance between two vertices is the length of the shortest path joining them. In this paper we present efficient algorithms for finding the l-core of a tree. For unweighted trees we present an O(nl) time algorithm, while for weighted trees we give a procedure with time complexity of O(nlog2n). The algorithms use two different types of recursive principle in their operation.

Periodic solutions of semilinear equations of evolution of compact type

Abstract The existence of solutions in a weak sense of x ′ + ( A + B ( t , x )) x = f ( t , x ), x (0) = x ( T ) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H ; B ( t , x ) is a bounded linear operator and f ( t , x ) a function with values in H ; for each square integrable ϑ ( t ) the problem with B ( t , ϑ ( t )) and f ( t , ϑ ( t )) in place of B ( t , x ) and f ( t , x ) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.

The shifting algorithm technique for the partitioning of trees

Abstract In this paper we survey a design technique for partitioning on trees. This technique, the shifting algorithm technique, is a top-down greedy technique. A partition of a tree is represented by associating cuts with edges of the tree. The basic operation of the technique is a local transformation called a shift of a cut from an edge to an adjacent edge of the tree. We review several shifting algorithms for different optimization criteria for partitioning. In these algorithms, different shifts and different greedy decisions are utilized. A mathematical framework created for validity proofs of shifting algorithms is introduced. Various applications are outlined.

Shifting algorithms for tree partitioning with general weighting functions

Abstract Recently two shifting algorithms were designed for two optimum tree partitioning problems: The problem of max-min q -partition [4] and the problem of min-max q -partition [1]. In this work we investigate the applicability of these two algorithms to max-min and min-max partitioning of a tree for various different weighting functions. We define the families of basic and invariant weighting functions. It is shown that the first shifting algorithm yields a max-min q -partition for every basic weighting function. The second shifting algorithm yields a min-max q -partition for every invariant weighting function. In addition a modification of the second algorithm yields a min-max q -partition for the noninvariant diameter weighting function.

Max‐min partitioning of grid graphs into connected components

The partitioning of a rectangular grid graph with weighted vertices into p connected components such that the component of smallest weight is as heavy as possible (the max-min problem) is considered. It is shown that the problem is NP-hard for rectangles with at least three rows. A shifting algorithm is given which approximates the optimal solution. Bounds for the relative error are determined under a posteriori hypotheses. A further shifting algorithm is also given which allows for error estimates under a priori hypotheses and for asymptotic error estimates. A similar approach can be taken with the problem of finding the partition whose largest component is as small as possible (the min-max problem). The case of rectangles with two rows has a polynomial algorithm and is dealt with in another paper.

The errors in FFT estimation of the Fourier transform

The problem of determining the error in approximating the Fourier transform by the discrete Fourier transform is studied. Exact formulas for the relative error are established for classes of functions, called canonical-k (k/spl ges/0), and asymptotic error formulas are established for a much wider class of functions, called order-k. The formulas are dependent only on the class and not on the function in the class whose Fourier transform is being approximated, and this facilitates the application of the results.

A shifting algorithm for constrained min-max partition on trees

Abstract Let T = (V,E) be a rooted tree with n edges. We associate nonnegative weight w(v) and size s(v) with each vertex v in V. A q-partition of T into q connected components T1,T2,…,Tq is obtained by deleting k = q−1 edges of T, 1 ≤ k 1. Size-constrained min-max problem: Find a q-partition of T for which WP is a minimum over all partitions P satisfying S(Ti) ≤ M (M > 0). 2. Height-constrained min-max problem: Find a q-partition of T for which WP is a minimum over all partitions P satisfying height h(Ti) ≤ H (H is a positive integer). The first problem is shown to be NP-complete, while a polynomial algorithm is presented for the second problem.

A shifting algorithm for continuous tree partitioning

A central co-ordinator wants to allot the maintenance of a tree-like highway network (such networks are common in coastal regions or in developing countries) among p service units with equal work capacities, so as to achieve maximum workload balance among the p units. For the sake of e2ciency, each unit should be in charge of a connected section of the network (that is, a subtree). A similar problem arises for other kinds of physical networks, such as rail-, pipelineor telephone-networks. If one assumes that the workload for the maintenance of a subtree is proportional to the length of the subtree, then the problem consists in cutting the tree into p subtrees whose lengths di3er as little as possible from each other. There are several di3erent ways to make the latter statement more precise. For example, one may wish to maximize the smallest length of a subtree. If one further requires that no road should be split between two or more units, then it is not hard to see that the above problem can be formulated as follows.

Necessary and sufficient conditions for the simultaneous diagonability of two quadratic forms

Abstract A new necessary and sufficient condition is obtained for the simultaneous diagonability of two quadratic forms in n variables. The condition is such that several known sufficient conditions can easily be derived from it.