Uniform-cost inverse absolute and vertex center location problems with edge length variations on trees

This article considers the inverse absolute and the inverse vertex 1-center location problems with uniform cost coefficients on a tree network T with n+1 vertices. The aim is to change (increase or reduce) the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified vertex s becomes an absolute (or a vertex) 1-center under the new edge lengths. First an O(nlogn) time method for solving the height balancing problem with uniform costs is described. In this problem the height of two given rooted trees is equalized by decreasing the height of one tree and increasing the height of the second rooted tree at minimum cost. Using this result a combinatorial O(nlogn) time algorithm is designed for the uniform-cost inverse absolute 1-center location problem on tree T. Finally, the uniform-cost inverse vertex 1-center location problem on T is investigated. It is shown that the problem can be solved in O(nlogn) time if all modified edge lengths remain positive. Dropping this condition, the general model can be solved in O(r_vnlogn) time where the parameter r_v is bounded by ⌈n/2⌉. This corrects an earlier result of Yang and Zhang.     

An O(mn log (nU)) time algorithm to solve the feasibility problem

The phase I maximum flow and most positive cut methods are used to solve the feasibility problem. Both of these methods take one maximum flow computation. Thus the feasibility problem can be solved using maximum flow algorithms. Let n and m be the number of nodes and arcs, respectively. In this paper, we present an algorithm to solve the feasibility problem with integer lower and upper bounds. The running time of our algorithm is O(mn log (nU)), where U is the value of maximum upper bound. Our algorithm improves the O(m² log (nU))-time algorithm in [12]. Hence the current algorithm improves the running time in [12] by a factor of n. Sleator and Goldberg’s algorithm is one of the well-known maximum flow algorithms, which runs in O(mn log n) time, see [5]. Under similarity assumption [11], our algorithm runs in O(mn log n) time, which is the running time of Sleator and Goldberg’s algorithm. The merit of our algorithm is that, in the case of infeasibility of the given network, it not only diagnoses infeasibility but also presents some information that is useful to modeler in estimating the maximum cost of adjusting the infeasible network.     

On the number of queries necessary to identify a permutation

Let p and q be two permutations over {1, 2,…, n}. We denote by m(p, q) the number of integers i, 1 ≤ i ≤ n, such that p(i) = q(i). For each fixed permutation p, a query is a permutation q of the same size and the answer a(q) to this query is m(p, q). We investigate the problem of finding the minimum number of queries required to identify an unknown permutation p. A polynomial-time algorithm that identifies a permutation of size n by O(n · log₂n) queries is presented. The lower bound of this problem is also considered. It is proved that the problem of determining the size of the search space created by a given set of queries and answers is #P-complete. Since this counting problem is essential for the analysis of the lower bound, a complete analysis of the lower bound appears infeasible. We conjecture, based on some preliminary analysis, that the lower bound is Ω(n · log₂n).     

Optimal algorithms to compute the closure of a set of iso-rectangles

Three varieties of the closure of a set of iso-(oriented) rectangles, i.e., rectilin-early-oriented rectangles, are introduced. These are uni-directional, diagonal, and rectangular closure. First a strong decomposition theorem for diagonal closure in terms of uni-directional closure is proved. Then time and space optimal algorithms to compute uni-directional and diagonal closure, each running in O(nlogn) time and O(n) space, are described. An O(nlogn) time and space algorithm for rectangular closure is also described. The algorithm for diagonal closure has applications in database concurrency control: an O(nlogn) time and O(n) space algorithm for testing for safety and detecting deadlocks in locked transaction systems is obtained.     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

An improved approximation algorithm for requirement cut

This note presents improved approximation guarantees for the requirement cut problem: given an n-vertex edge-weighted graph G=(V,E), and g groups of vertices X₁,…,X_g⊆V with each group X_i having a requirement r_i between 0 and |X_i|, the goal is to find a minimum cost set of edges whose removal separates each group X_i into at least r_i disconnected components. We give a tight Θ(logg) approximation ratio for this problem when the underlying graph is a tree, and show how this implies an O(logk⋅logg) approximation ratio for general graphs, where .     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

Large tactical configurations

It is shown that if r = 2, then for all m, n, h ≥ 3 and all “large enough” R, W such that mR = nW, there is a tactical configuration of rank 2 girth g = 2h, and degrees m and n on sets of cardinalities R and W. We also show that if r ≥ 3 then for all h and all compatible degree sets N = {n(i, j)≥3} and all large enough numbers R(1), R(2),…, R(r) satisfying R(i)n(i, j) = R(j)n(j, i) there is a tactical configuration of rank r, girth h, and degrees N on set with cardinalities R(1), R(2),…, R(r).     

Preemptive scheduling of two uniform parallel machines to minimize total tardiness

We consider the problem of preemptive scheduling n jobs on two uniform parallel machines. All jobs have equal processing requirements. For each job we are given its due date. The objective is to find a schedule minimizing total tardiness ∑T_i. We suggest an O(n log n) algorithm to solve this problem.     

Some simple in-place merging algorithms

The theoretical presentation and analysis is given for two families of simple in-place merging algorithms and their limiting cases. The first family merges stably inO(k·n) time andO(n ^1/k ) additional space with a limiting case running inO(n logn) time and constant space. The second family merges unstably inO (k ·n) time andO(log ^k n) space with a limiting case running inO(nG(n)) time and constant space. HereG(n) is the leastk such thatF(k) n whereF(0)=1 andF(i)=2 ^F(i–1) fori1. Each algorithm gives rise to a corresponding merge sort.     

Improved algorithms for path partition and related problems

We study the L_∞ path partition problem: given a path of n weighted vertices and an integer k, remove k−1 edges from the path so that the maximum absolute deviation of the weights of the resulting k sub-paths from their mean is minimized. Previously, the best algorithm solves this problem in O(nklogk) time. We present an O(nk) time algorithm. We also give improved solutions for two related problems: the L_d path partition problem and the web proxies placement problem.     

Sampling to provide or to bound: With applications to fully dynamic graph algorithms

In dynamic graph algorithms the following provide-or-bound problem has to be solved quickly: Given a set S containing a subset R and a way of generating random elements from S testing for membership in R, either (i) provide an element of R, or (ii) give a (small) upper bound on the size of R that holds with high probability. We give an optimal algorithm for this problem. This algorithm improves the time per operation for various dynamic graph algorithms by a factor of O(log n). For example, it improves the time per update for fully dynamic connectivity from O(log³n) to O(log²n). © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 369–379 (1997)     

Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

In the present paper we present the tensor-product approximation of a multidimensional convolution transform discretized via a collocation-projection scheme on uniform or composite refined grids. Examples of convolving kernels are provided by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, and with x∈R^d. For piecewise constant elements on the uniform grid of size n^d, we prove quadratic convergence O(h²) in the mesh parameter h=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O(h³). A fast algorithm of complexity O(dR₁R₂nlogn) is described for tensor-product convolution on uniform/composite grids of size n^d, where R₁,R₂ are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order O(h²) and O(h³); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on an n×n×n grid in the range n≤16384.     

An Improved Algorithm for the Traveler′s Problem

In a paper in Journal of Algorithms13 (1992), 148-160, Hirschberg and Larmore introduced the traveler′s problem as a subroutine for constructing the B-tree. They gave an O(n^5/3 log^1/3n) time algorithm for solving the traveler′s problem of size n. In this paper, we improve their time bound to O(n^3/2 log n). As a consequence, we build a B-tree in O(n^3/2 log²n) time as compared to the O(n^5/3 log^4/3n) time algorithm of Hirschberg and Larmore.     

Optimally Cutting a Surface into a Disk

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n ^O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log² g)-approximation of the minimum cut graph in O(g ² n log n) time.     

Fast broadcasting and gossiping in radio networks

We establish an O(nlog²n) upper bound on the time for deterministic distributed broadcasting in multi-hop radio networks with unknown topology. This nearly matches the known lower bound of Ω(nlogn). The fastest previously known algorithm for this problem works in time O(n^3/2). Using our broadcasting algorithm, we develop an O(n^3/2log²n) algorithm for gossiping in the same network model.     

Finding a minimum independent dominating set in a permutation graph

We give a linear time reduction of the problem of finding a minimum independent dominating set in a permutation graph, into that of finding a shortest maximal increasing subsequence. We then give an O(n log²n)-time algorithm for solving the second (and hence the first) problem. This improves on the O(n³)-time algorithm given in [4] for solving the problem of finding a minimum independent dominating set in a permutation graph.     

A Leibniz differentiation formula for positive operators

Is is shown that for n→+∞ the Leibnizian combination L′_n(fg)−fL′_n(g)−gL′_n(f) converges uniformly to zero on a compact interval W if the positive operators L_n belong to a certain class (including Bernstein, Gauss-Weierstrass and many others), and if the moduli of continuity of f,g satisfy ω_W(f;h)ω_W(g;h)=o(h) as h→0+. A counterexample shows that Lipschitz conditions are not appropriate to bring about a second-order version of this formula.     

On the connectivity and superconnected graphs with small diameter

In this paper, first we prove that any graph G is 2-connected if diam(G)≤g−1 for even girth g, and for odd girth g and maximum degree Δ≤2δ−1 where δ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G)≤g−2 satisfies that (i) G is 5-connected for even girth g and Δ≤2δ−5, and (ii) G is super-κ for odd girth g and Δ≤3δ/2−1.