Finding Undercut-Free Parting Directions for Polygons with Curved Edges

2005 ◽  
Vol 6 (1) ◽  
pp. 60-68 ◽  
Author(s):  
Sara McMains ◽  
Xiaorui Chen
Keyword(s):  
Opposite Direction ◽  
Efficient Algorithm ◽  
Time And Space ◽  
Normal Graph

We consider the problem of whether a given geometry can be molded in a two-part, rigid, reusable mold with opposite removal directions. We describe an efficient algorithm for solving the opposite direction moldability problem for a 2D “polygon” bounded by edges that may be either straight or curved. We introduce a structure, the normal graph of the polygon, that represents the range of normals of the polygon’s edges, along with their connectivity. We prove that the normal graph captures the directions of all lines corresponding to feasible parting directions. Rather than building the full normal graph, which could take time O(nlogn) for a polygon bounded by n possibly curved edges, we build a summary structure in O(n) time and space, from which we can determine all feasible parting directions in time O(n).

Determining Moldability and Parting Directions for Polygons With Curved Edges

2004 ◽  
Author(s):  
Sara McMains ◽  
Xiaorui Chen
Keyword(s):  
Opposite Direction ◽  
Efficient Algorithm ◽  
Time And Space ◽  
Normal Graph

We consider the problem of whether a given geometry can be molded in a two part, rigid, reusable mold with opposite removal directions. We describe an efficient algorithm for solving the opposite direction moldability problem for a 2D “polygon” bounded by edges that may be either straight or curved. We introduce a structure, the normal graph of the polygon, that represents the range of normals of the polygon’s edges, along with their connectivity. We prove that the normal graph captures the directions of all lines corresponding to feasible parting directions. Rather than building the full normal graph, which could take time O(n log n) for a polygon bounded by n possibly curved edges, we build a summary structure in O(n) time and space, from which we can determine all feasible parting directions in time O(n).

scholarly journals Disimplicial arcs, transitive vertices, and disimplicial eliminations

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Martiniano Eguia ◽  
Francisco Soulignac
Keyword(s):  
Efficient Algorithm ◽  
Bipartite Graphs ◽  
Space Complexity ◽  
Time And Space ◽  
Graph Classes ◽  
International Audience ◽  
Time And Space Complexity ◽  
Finite Posets

International audience In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A <i>diclique</i> of a digraph is a pair $V$ &rarr; $W$ of sets of vertices such that $v$ &rarr; $w$ is an arc for every $v$ &isin; $V$ and $w$ &isin; $W$. An arc $v$ &rarr; $w$ is <i>disimplicial</i> when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.

A TIME AND SPACE EFFICIENT ALGORITHM FOR MINIMIZING COVER AUTOMATA FOR FINITE LANGUAGES

2003 ◽  
Vol 14 (06) ◽  
pp. 1071-1086 ◽  
Author(s):  
HEIKO KÖRNER
Keyword(s):  
Data Structure ◽  
Programming Language ◽  
Finite Automaton ◽  
Efficient Algorithm ◽  
Previous Method ◽  
Deterministic Finite Automaton ◽  
Time And Space ◽  
Language L ◽  
Finite Language ◽  
The Common

A deterministic finite automaton (DFA) [Formula: see text] is called a cover automaton (DFCA) for a finite language L over some alphabet Σ if [Formula: see text], with l being the length of some longest word in L. Thus a word w ∈ Σ* is in L if and only if |w| ≤ l and [Formula: see text]. The DFCA [Formula: see text] is minimal if no DFCA for L has fewer states. In this paper, we present an algorithm which converts an n–state DFA for some finite language L into a corresponding minimal DFCA, using only O(n log n) time and O(n) space. The best previously known algorithm requires O(n2) time and space. Furthermore, the new algorithm can also be used to minimize any DFCA, where the best previous method takes O(n4) time and space. Since the required data structure is rather complex, an implementation in the common programming language C/C++ is also provided.

Efficient outerplanarity testing

1979 ◽  
Vol 2 (1) ◽  
pp. 261-275
Author(s):  
Maciej M. Sysło ◽  
Masao Iri
Keyword(s):  
Efficient Algorithm ◽  
Time And Space ◽  
Outerplanar Graphs ◽  
Depth First Search

This paper describes an efficient algorithm for finding whether a graph G has an outerplanar embedding in the plane. The algorithm is a realization of an inductive characterization of outerplanar graphs and uses depth-first search for coding a structure of a graph which is represented by adjacency lists. The algorithm has O(V) time and space bounds, where V is the number of vertices in G.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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