The notorious middle two levels problem is to nd a Hamilton cycle in the middle two levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known result, due to Moews and Reid 11] in 1990, was that M 2k+1 is Hamiltonian for all positive k through k = 11. We show that if a Hamilton path between two ...

Given a point s and a set of h pairwise disjoint polygonal obstacles with a total of n vertices in the plane, after the free space is triangulated, we present an O(n + h log h) time and O(n) space algorithm for building a data structure (called shortest path map) of size O(n) such that for any query point t, the length of the L1 shortest obstacle-avoiding path from s to t can be reported in O(l...

A hierarchical memetic algorithm (MA) is proposed for the path planning and formation control of swarm robots. The proposed algorithm consists of a global path planner (GPP) and a local motion planner (LMP). The GPP plans a trajectory within the Voronoi diagram (VD) of the free space. An MA with a non‐random initial population plans a series of configu...

In this paper we discuss the concept of space-time diagrams as a representation of the execution of an application, and then give a method, based on critical path analysis, for calculating nontrivial upper bounds on the potential parallelism, known as the average parallelism, a complement to the speedup and efficiency.

We present the first algorithm to compute the geodesic Fréchet distance between two polygonal curves in a plane with polygonal obstacles, where distances between points are measured as the length of a shortest path between them. Using shortest path structures that we call dynamic and static spotlights, we efficiently construct and propagate reachability information through the free space diagra...

We present techniques for fast motion planning by using discrete approximations of generalized Voronoi diagrams, computed with graphics hardware. Approaches based on this diagram computation are applicable to both static and dynamic environments of fairly high complexity. We compute a discrete Voronoi diagram by rendering a three-dimensional distance mesh for each Voronoi site. The sites can be...

In the first part of this work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth power of a path ordered by inclusion. For h = 1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself. In the second part we consi...

This paper investigates geometric and algorithmic properties of the Voronoi diagram with a transportation network on the Euclidean plane. With a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm + m + nm log n) algorithm to compute the Voronoi diagram with a transportatio...

We consider a variation of Voronoi diagram, or time-based Voronoi diagram, for a set S of points in the presence of transportation lines or highways in the plane. A shortest time-distance path from a query point to any given point in S is a path that takes the least travelling time. The travelling speeds and hence travelling times of the subpaths along the highways and in the plane are differen...

Traditionally, the word "abracadabra" was encrypted onto amulets and other magical paraphernalia to help ward off evil. George Polya ([8], [9]) provided the cryptic form of this word shown in Figure 1 and asked how many ways abracadabra can be spelled out using this diagram. If we replace the diagram in Figure 1 with the grid shown in Figure 2 where letters in the original diagram are placed at...