Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s,tX, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m²n²), where m is the number of vertices of P and n is the number of points in X.     

Coloring permutation graphs in parallel

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having specific key properties. We propose an efficient parallel algorithm which colors an n-node permutation graph in O(log² n) time using O(n²/log n) processors on the CREW PRAM model. Specifically, given a permutation π we construct a tree T^*[π], which we call coloring-permutation tree, using certain combinatorial properties of π. We show that the problem of coloring a permutation graph is equivalent to finding vertex levels in the coloring-permutation tree.     

An efficient direct approach for computing shortest rectilinear paths among obstacles in a two-layer interconnection model

In this paper, we present a direct approach for routing a shortest rectilinear path between two points among a set of rectilinear obstacles in a two-layer interconnection model that is used for VLSI routing applications. The previously best known direct approach for this problem takes O(nlog²n) time and O(nlogn) space, where n is the total number of obstacle edges. By using integer data structures and an implicit graph representation scheme (i.e., a generalization of the distance table method), we improve the time bound to O(nlog^3/2n) while still maintaining the O(nlogn) space bound. Comparing with the indirect approach for this problem, our algorithm is simpler to implement and is probably faster for a quite large range of input sizes.     

On parallel rectilinear obstacle- avoiding paths

We give improved space and processor complexities for the problem of computing, in parallel, a data structure that supports queries about shortest rectilinear obstacle-avoiding paths in the plane, where the obstacles are disjoint rectangles. That is, a query specifies any source and destination in the plane, and the data structure enables efficient processing of the query. We now can build the data structure with O(n²/log n) CREW PRAM processors, as opposed to the previous O(n²), and with O(n²) space, as opposed to the previous O(n²(log n)²). The time complexity remains unchanged, at O((log n)²). As before, the data structure we compute enables a query to be processed in O(log n) time, by one processor for obtaining a path length, or by O(k/log n) processors for retrieving a shortest path itself, where k is the number of segments on that path. The new ideas that made our improvement possible include a new partitioning scheme of the recursion tree, which is used to schedule the computations performed on that tree. Since a number of other related shortest paths problems are solved using this technique as a subroutine our improvement translates into a similar improvement in the complexities of these problems as well.     

A homotopy equivalence that is not homotopic to a topological embedding

An open subset W of Sⁿ, n 6 or N = 4, and a homotopy equivalence ƒ: S² × S^{n − 4} → W are constructed having the property that ƒ is not homotopic to any topological embedding.     

On a straight-line embedding problem of graphs

Let G be a planar graph with n vertices, v be a specified vertex of G, and P be a set of n points in the Euclidian plane in general position. A straight-line embedding of G onto P is an embedding of G onto whose images of vertices are distinct points in P and whose images of edges are (straight) line segments. In this paper, we classify into five classes those pairs of G and v such that for any P and any p P, G has a straight-line embedding onto P which maps v to p. We then show that all graphs in three of the classes indeed have such an embedding. This result gives a solution to a generalized version of the rooted-tree embedding problem raised by M. Perles.     

ContributionsAn algorithm for the Tutte polynomials of graphs of bounded treewidth

Let k be a fixed, positive integer. We give an algorithm which computes the Tutte polynomial of any graph G of treewidth at most k in time O(n^{2+7 log₂ c}), where c is twice the number of partitions of a set with 3k + 3 elements and n the number of vertices of G.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs

We investigate the complexity of several domination problems on the complements of bounded tolerance graphs and the complements of trapezoid graphs. We describe an O(n² log⁵ n) time and O(n²) space algorithm to solve the domination problem on the complement of a bounded tolerance graph, given a square embedding of that graph. We also prove that domination, connected domination and total domination are all NP-complete on co-trapezoid graphs.     

Consequences of an algorithm for bridged graphs

Chepoi showed that every breadth first search of a bridged graph produces a cop-win ordering of the graph. We note here that Chepoi's proof gives a simple proof of the theorem that G is bridged if and only if G is cop-win and has no induced cycle of length four or five, and that this characterization together with Chepoi's proof reduces the time complexity of bridged graph recognition. Specifically, we show that bridged graph recognition is equivalent to (C₄,C₅)-free graph recognition, and reduce the best known time complexity from O(n⁴) to O(n^3.376).     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

On the P4-components of graphs

Two edges are called P₄-adjacent if they belong to the same P₄ (chordless path on four vertices). P₄-components, in our terminology, are the equivalence classes of the transitive closure of the P₄-adjacency relation. In this paper, new results on the structure of P₄-components are obtained. On the one hand, these results allow us to improve the complexity of orienting P₄-comparability graphs and of recognizing P₄-indifference graphs from O(n⁵) and O(n⁶) to O(m²). On the other hand, by combining the modular decomposition with the substitution of P₄-components, a new unique tree representation for arbitrary graphs is derived which generalizes the homogeneous decomposition introduced by Jamison and Olariu (SIAM J. Discrete Math. 8 (1995) 448–463).     

Area-efficient algorithms for straight-line tree drawings

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment.

Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O(nlogn) (Crescenzi et al., 1992) to O(nloglogn). This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type.

Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O(nloglogn). If T is binary, we can obtain an O(nloglogn)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/logn) bends in total. Secondly, we present O(nloglogn)-area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.     

Maximum number of edges in a critically k-connected graph

A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any vV(G). We show that if n,k are integers with k4 and nk+2, and G is a critically k-connected graph of order n, then |E(G)|n(n−1)/2−p(n−k)+p²/2, where p=(n/k)+1 if n/k is an odd integer and p=n/k otherwise. We also characterize extremal graphs.     

Recognizing graphs without asteroidal triples

We consider the problem of recognizing AT-free graphs. Although there is a simple O(n³) algorithm, no faster method for solving this problem had been known. Here we give three different algorithms which have a better time complexity for graphs which are sparse or have a sparse complement; in particular we give algorithms which recognize AT-free graphs in , , and O(n^2.82+nm). In addition we give a new characterization of graphs with bounded asteroidal number by the help of the knotting graph, a combinatorial structure which was introduced by Gallai for considering comparability graphs.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Numerical approximation of the product of the square root of a matrix with a vector

Given an n×n symmetric positive definite matrix A and a vector , two numerical methods for approximating are developed, analyzed, and computationally tested. The first method applies a Newton iteration to a specific nonlinear system to approximate while the second method applies a step-control method to numerically solve a specific initial-value problem to approximate . Assuming that A is first reduced to tridiagonal form, the first method requires O(n²) operations per iteration while the second method requires O(n) operations per iteration. In contrast, numerical methods that first approximate A^1/2 and then compute generally require O(n³) operations per iteration.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

Triangular range counting query in 2D and its application in finding k nearest neighbors of a line segment

We present an efficient algorithm for finding k nearest neighbors of a query line segment among a set of points distributed arbitrarily on a two dimensional plane. Along the way to finding this algorithm, we have obtained an improved triangular range searching technique in 2D. Given a set of n points, we preprocess them to create a data structure using O(n²) time and space, such that given a triangular query region Δ, the number of points inside Δ can be reported in O(logn) time. Finally, this triangular range counting technique is used for finding k nearest neighbors of a query straight line segment in O(log²n+k) time.