Self-intersection elimination in metamorphosis of two-dimensional curves

H :[0, 1]× ³→ ³, where H(t, r) for t=0 and t=1 are two given planar curves C ₁(r) and C ₂(r). The first t parameter defines the time of fixing the intermediate metamorphosis curve. The locus of H(t, r) coincides with the ruled surface between C ₁(r) and C ₂(r), but each isoparametric curve of H(t, r) is self-intersection free. The second algorithm suits morphing operations of planar curves. First, it constructs the best correspondence of the relative parameterizations of the initial and final curves. Then it eliminates the remaining self-intersections and flips back the domains that self-intersect.     

The Intersection of Two Ringed Surfaces and Some Related Problems

We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪_uC^u generated by a moving circle. Given two ringed surfaces ∪_uC^u₁ and ∪_vC^v₂, we formulate the condition C^u₁ ∩ C^v₂ ≠ ∅ (i.e., that the intersection of the two circles C^u₁ and C^v₂ is nonempty) as a bivariate equation λ(u, v)=0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution of λ(u, v)=0 to the intersection point C^u₁ ∩ C^v₂. Thus it is trivial to construct the intersection curve once we have computed the zero-set of λ(u, v)=0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation λ(u, v)=0 is reduced to a decomposable form, f(u)=g(v) or 6f(u)−g(v)6=|r(u)|, which can be solved more efficiently than the general case.     

Constrained approximation of rational Bézier curves based on a matrix expression of its end points continuity condition

For high order interpolations at both end points of two rational Bézier curves, we introduce the concept of C^(v,u)-continuity and give a matrix expression of a necessary and sufficient condition for satisfying it. Then we propose three new algorithms, in a unified approach, for the degree reduction of Bézier curves, approximating rational Bézier curves by Bézier curves and the degree reduction of rational Bézier curves respectively; all are in L₂ norm and C^(v,u)-continuity is satisfied. The algorithms for the first and second problems can get the best approximation results, and for the third one, resorting to the steepest descent method in numerical optimization obtains a series of degree reduced curves iteratively with decreasing approximation errors. Compared to some well-known algorithms for the degree reduction of rational Bézier curves, such as the uniformizing weights algorithm, canceling the best linear common divisor algorithm and shifted Chebyshev polynomials algorithm, the new one presented here can give a better approximation error, do multiple degrees of reduction at a time and preserve high order interpolations at both end points.     

A Quadratic Algorithm for Finding Next-to-Shortest Paths in Graphs

Given an edge-weighted undirected graph G and two prescribed vertices u and v, a next-to-shortest (u,v)-path is a shortest (u,v)-path amongst all (u,v)-paths having length strictly greater than the length of a shortest (u,v)-path. In this paper, we deal with the problem of computing a next-to-shortest (u,v)-path. We propose an O(n²){\mathcal{O}}(n^{2}) time algorithm for solving this problem, which significantly improves the bound of a previous one in O(n³){\mathcal{O}}(n^{3}) time where n is the number of vertices in G.     

Not necessarily proper total colourings which are adjacent vertex distinguishing

Let G be a simple graph with no isolated edge. Let f be a total colouring of G which is not necessarily proper. f is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u, v of G, we have C(u)≠C(v), where C(u) denotes the set of colours of u and its incident edges under f. The minimum number of colours required for an adjacent vertex distinguishing not necessarily proper total colouring of G is called the adjacent vertex distinguishing not necessarily proper total chromatic number. Seven kinds of adjacent vertex distinguishing not necessarily proper total colourings are introduced in this paper. Some results of adjacent vertex distinguishing not necessarily proper total chromatic numbers are obtained and some conjectures are also proposed.     

A Simple Optimal Parallel Algorithm for a Core of a Tree

A core of a tree T = (V, E) is a path in T which minimizes ∑_vVd(v, P), where d(v, P), the distance from a vertex v to path P, is defined as min_uPd(v, u). We present an optimal parallel algorithm to find a core of T in O(log n) time using O(n/log n) processors on an EREW PRAM machine, where n is the number of vertices of tree T.     

Radiocoloring in planar graphs: Complexity and approximations

The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters, by exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function such that |Φ(u)-Φ(v)|2, when u,v are neighbors in G, and |Φ(u)-Φ(v)|1 when the distance of u,v in G is two. The number of discrete frequencies and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the radiocoloring problem for general graphs is hard to approximate (unless NP=ZPP) within a factor of n^1/2-ε (for any ), where n is the number of vertices of the graph. However, when restricted to some special cases of graphs, the problem becomes easier. We prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(nΔ) time algorithm (|V|=n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where Δ the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with λ colors, in the case where λ4Δ+50.     

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

Let G=(V,E,w) be a directed graph, where w:V→ℝ is a weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u,v the capacity from u to v, denoted by c(u,v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem the task is to find the capacities for all ordered pairs of vertices. Our main result is an O(n ^2.575) time algorithm for APBP. The exponent is derived from the exponent of fast matrix multiplication.     

The super-connected property of recursive circulant graphs

In a graph G, a k-container C_k(u,v) is a set of k disjoint paths joining u and v. A k-container C_k(u,v) is k∗-container if every vertex of G is passed by some path in C_k(u,v). A graph G is k∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1?k?m. In this paper, we prove that the recursive circulant graphs G(2^m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35-62], are super-connected if and only if m≠2.     

Adaptive robust control of nonholonomic systems with stochastic disturbances

Based on the Brockett’s necessary condition for feedback asymptotic stabilization[1], nonholonomic systems fail to be stabilized at the origin by any static continuous state feedback though they are open loop controllable. There are two novel approaches …     

Improved algorithm for finding next-to-shortest paths

We study the problem of finding the next-to-shortest paths in a weighted undirected graph. A next-to-shortest (u,v)-path is a shortest (u,v)-path amongst (u,v)-paths with length strictly greater than the length of the shortest (u,v)-path. The first polynomial algorithm for this problem was presented in [I. Krasikov, S.D. Noble, Finding next-to-shortest paths in a graph, Inform. Process. Lett. 92 (2004) 117-119]. We improve the upper bound from O(n³m) to O(n³).     

Sorting a sequence of strong kings in a tournament

A king in a tournament is a player who beats any other player directly or indirectly. According to the existence of a king in every tournament, Wu and Sheng [Inform. Process. Lett. 79 (2001) 297-299] recently presented an algorithm for finding a sorted sequence of kings in a tournament of size n, i.e., a sequence of players u₁,u₂,…,u_n such that u_i→u_i+1 (u_i beats u_i+1) and u_i is a king in the sub-tournament induced by {u_i,u_i+1,…,u_n} for each i=1,2,…,n−1. With each pair u,v of players in a tournament, let b(u,v) denote the number of third players used for u to beat v indirectly. Then, a king u is called a strong king if the following condition is fulfilled: if v→u then b(u,v)>b(v,u). In the sequel, we will show that the algorithm proposed by Wu and Sheng indeed generates a sorted sequence of strong kings, which is more restricted than the previous one.     

Improved Approximation Algorithm for Convex Recoloring of Trees

A pair (T,C) of a tree T and a coloring C is called a colored tree. Given a colored tree (T,C) any coloring C′ of T is called a recoloring of T. Given a weight function on the vertices of the tree the recoloring distance of a recoloring is the total weight of recolored vertices. A coloring of a tree is convex if for any two vertices u and v that are colored by the same color c, every vertex on the path from u to v is also colored by c. In the minimum convex recoloring problem we are given a colored tree and a weight function and our goal is to find a convex recoloring of minimum recoloring distance. The minimum convex recoloring problem naturally arises in the context of phylogenetic trees. Given a set of related species the goal of phylogenetic reconstruction is to construct a tree that would best describe the evolution of this set of species. In this context a convex coloring corresponds to perfect phylogeny. Since perfect phylogeny is not always possible the next best thing is to find a tree which is as close to convex as possible, or, in other words, a tree with minimum recoloring distance. We present a (2+ε)-approximation algorithm for the minimum convex recoloring problem, whose running time is O(n ²+n(1/ε)²4^1/ε ). This result improves the previously known 3-approximation algorithm for this NP-hard problem. We also present an algorithm for computing an optimal convex recoloring whose running time is , where n ^* is the number of colors that violate convexity in the input tree, and Δ is the maximum degree of vertices in the tree. The parameterized complexity of this algorithm is O(n ²+n⋅k⋅2 ^k ).     

Composing Power Series Over a Finite Ring in Essentially Linear Time

Fix a finite commutative ringR. Letuandvbe power series overR, withv(0) = 0. This paper presents an algorithm that computes the firstnterms of the compositionu(v), given the firstnterms ofuandv, inn^1 + o(1)ring operations. The algorithm is very fast in practice whenRhas small characteristic.     

Periods in Extensions of Words

Let π(w) denote the minimum period of the word w,let w be a primitive word with period π(w) < |w|, and let z be a prefix of w. It is shown that if π(wz) = π(w), then |z| < π(w) − gcd (|w|, |z|). Detailed improvements of this result are also proven. Finally, we show that each primitive word w has a conjugate w′ = vu, where w = uv, such that π(w′) = |w′| and |u| < π(w). As a corollary we give a short proof of the fact that if u,v,w are words such that u ² is a prefix of v ², and v ² is a prefix of w ², and v is primitive, then |w| > 2|u|.     

Continuous point projection to planar freeform curves using spiral curves

We present an efficient algorithm for projecting a continuously moving query point to a family of planar freeform curves. The algorithm is based on the one-sided Hausdorff distance from the trajectory curve (of the query point) to the planar curves. Using a bounding volume hierarchy (BVH) of the planar curves, we estimate an upper bound [`(h)]\overline{h} of the one-sided Hausdorff distance and eliminate redundant curve segments when they are more than distance [`(h)]\overline{h} away from the trajectory curve. Recursively subdividing the trajectory curve and repeating the same elimination procedure to the BVH of the remaining curves, we can efficiently determine where to project the moving query point. The explicit continuous point projection is then interpreted as a curve reparameterization problem, for which we propose a few simple approximation techniques. Using several experimental results, we demonstrate the effectiveness of the proposed approach.     

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u,v∈V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC(u,v)=max{dC(x,y)|x,y∈V(C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to |V(G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices u,v∈V(G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG(u,v),3}?l?|V(G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249-1251] proved that the n-dimensional Augmented cube, AQn, is pancyclic in the sense that a cycle of length l exists, 3?l?|V(AQn)|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n?2.     

FitConnect: Connecting Noisy 2D Samples by Fitted Neighbourhoods

We propose a parameter‐free method to recover manifold connectivity in unstructured 2D point clouds with high noise in terms of the local feature size. This enables us to capture the features which emerge out of the noise. To achieve this, we extend the reconstruction algorithm HNN‐Crust , which connects samples to two (noise‐free) neighbours and has been proven to output a manifold for a relaxed sampling condition. Applying this condition to noisy samples by projecting their k‐nearest neighbourhoods onto local circular fits leads to multiple candidate neighbour pairs and thus makes connecting them consistently an NP‐hard problem. To solve this efficiently, we design an algorithm that searches that solution space iteratively on different scales of k. It achieves linear time complexity in terms of point count plus quadratic time in the size of noise clusters. Our algorithm FitConnect extends HNN‐Crust seamlessly to connect both samples with and without noise, performs as local as the recovered features and can output multiple open or closed piecewise curves. Incidentally, our method simplifies the output geometry by eliminating all but a representative point from noisy clusters. Since local neighbourhood fits overlap consistently, the resulting connectivity represents an ordering of the samples along a manifold. This permits us to simply blend the local fits for denoising with the locally estimated noise extent. Aside from applications like reconstructing silhouettes of noisy sensed data, this lays important groundwork to improve surface reconstruction in 3D. Our open‐source algorithm is available online.     

The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

A Hamiltonian cycle C=? u ₁, u ₂, …, u _n(G), u ₁ ? with n(G)=number of vertices of G, is a cycle C(u ₁; G), where u ₁ is the beginning and ending vertex and u _i is the ith vertex in C and u _i ≠u _j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C ₁, C ₂, …, C _k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B _n )=n?1 if n≥4, where B _n is the n-dimensional bubble-sort graph.     

Radio number of ladder graphs

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f: V(G)→{0, 1, 2, …, k} with the following satisfied for all vertices u and v:|f(u)?f(v)|≥diam (G)?d _G (u, v)+1, where d _G (u, v) is the distance between u and v in G. In this paper, we determine the radio number of ladder graphs.