Efficient algorithms for reconfiguration in VLSI/WSI arrays

The issue of developing efficient algorithms for reconfiguring processor arrays in the presence of faulty processors and fixed hardware resources is discussed. The models discussed consist of a set of identical processors embedded in a flexible interconnection structure that is configured in the form of a rectangular grid. An array grid model based on single-track switches is considered. An efficient polynomial time algorithm is proposed for determining feasible reconfigurations for an array with a given distribution of faulty processors. In the process, it is shown that the set of conditions in the reconfigurability theorem is not necessary. A polynomial time algorithm is developed for finding feasible reconfigurations in an augmented single-track model and in array grid models with multiple-track switche

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Isotopy classes for 3-periodic net embeddings

Entangled embedded periodic nets and crystal frameworks are defined, along with their {dimension type}, {homogeneity type}, {adjacency depth} and {periodic isotopy type}. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of depth 1 embedded nets with a single vertex quotient graph. Additionally, we classify embeddings of n-fold copies of {pcu} with all connected components in a parallel orientation and n vertices in a repeat unit, and determine their maximal symmetry periodic isotopes. We also introduce the methodology of linear graph knots on the flat 3-torus [0, 1)^3. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases

Geometric Path-Planning Algorithm in Cluttered 2D Environments Using Convex Hulls

Routing or path planning is the problem of finding a collision-free path in an environment usually scattered with multiple objects. Finding the shortest route in a planar (2D) or spatial (3D) environment has a variety of applications such as robot motion planning, navigating autonomous vehicles, routing of cables, wires, and harnesses in vehicles, routing of pipes in chemical process plants, etc. The problem often times is decomposed into two main sub-problems: modeling and representation of the workspace geometrically and optimization of the path. Geometric modeling and representation of the workspace are paramount in any path planning problem since it builds the data structures and provides the means for solving the optimization problem. The optimization aspect of the path planning involves satisfying some constraints, the most important of which is to avoid intersections with the interior of any object and optimizing one or more criteria. The most common criterion in path planning problems is to minimize the length of the path between a source and a destination point of the workspace while other criteria such as minimizing the number of links or curves could also be taken into account. Planar path planning is mainly about modeling the workspace of the problem as a collision-free graph. The graph is, later on, searched for the optimal path using network optimization techniques such as branch-and-bound or search algorithms such as Dijkstra\u27s. Previous methods developed to construct the collision-free graph explore the entire workspace of the problem which usually results in some unnecessary information that has no value but to increase the time complexity of the algorithm, hence, affecting the efficiency significantly. For example, the fastest known algorithm to construct the visibility graph, which is the most common method of modeling the collision-free space, in a workspace with a total of n vertices has a time complexity of order O(n2). In this research, first, the 2D workspace of the problem is modeled using the tessellated format of the objects in a CAD software which facilitates handling of any free-form object. Then, an algorithm is developed to construct the collision-free graph of the workspace using the convex hulls of the intersecting obstacles. The proposed algorithm focuses only on a portion of the workspace involved in the straight line connecting the source and destination points. Considering the worst case that all the objects of the workspace are intersecting, the algorithm yields a time complexity of O(nlog(n/f)), with n being the total number of vertices and f being the number of objects. The collision-free graph is later searched for the shortest path between the two given nodes using a search algorithm known as Dijkstra\u27s