Genus distributions for two classes of graphs

The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. It is proved that the genus distribution of any member of either class is strongly unimodal. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their cellular orientable imbeddings in the sphere

Genus Distributions for Two Classes of Graphs

The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their polygonal orientable imbeddings in the sphere

Hierarchy for Imbedding-Distribution Invariants of a Graph

Most existing papers about graph imbeddings ale concerned with the determination of minimum genus, and various others have been devoted to maximum genus or to highly symmetric imbeddings of special graphs. An entirely different viewpoint is now presented, in which one seeks distributional information about the huge family of all cellular imbeddings of a graph into all closed surfaces, instead of focusing on just one imbedding or on the existence of imbeddings into just one surface. The distribution of imbeddings admits a hierarchically ordered class of computable invariants, each of which partitions the set of all graphs into much finer subcategories than the subcategories corresponding to minimum genus or to any other single imbedding surface. Quite low in this hierarchy are invariants such as the average genus, taken over all cellular imbeddings, and the average region size, where "region size" means the number of edge traversals required to complete a tour of a region boundary. Further up in the hierarchy is the multiset of duals of a graph. At an intermediate level are the "imbedding polynomials." The hierarchy is explored, and several specific calculations of the values of some of the invariants are provided. The main results are concerned with the amount of work needed to derive one invariant from another, when possible, and with principles for computing the algebraic effect of adding an edge or of forming a composition of two graphs

Fast Planning Through Planning Graph Analysis

We introduce a new approach to planning in STRIPS-like domains based on constructing and analyzing a compact structure we call a Planning Graph. We describe a new planner, Graphplan, that uses this paradigm. Graphplan always returns a shortest possible partial-order plan, or states that no valid plan exists. We provide empirical evidence in favor of this approach, showing that Graphplan outperforms the total-order planner, Prodigy, and the partial-order planner, UCPOP, on a variety of interesting natural and artificial planning problems. We also give empirical evidence that the plans produced by Graphplan are quite sensible. Since searches made by this approach are fundamentally different from the searches of other common planning methods, they provide a new perspective on the planning problem

Fast planning through planning graph analysis

We introduce a new approach to planning in STRIPS-like domains based on constructing and analyzing a compact structure we call a Planning Graph. We describe a new planner, Graphplan, that uses this paradigm. Graphplan always returns a shortestpossible partial-order plan, or states that no valid plan exists. We provide empirical evidence in favor of this approach, showing that Graphplan outperforms the total-order planner, Prodigy, and the partial-order planner, UCPOP, on a variety of interesting natural and artificial planning problems. We also give empirical evidence that the plans produced by Graphplan are quite sensible. Since searches made by this approach are fundamentally different from the searches of other common planning methods, they provide a new perspective on the planning problem

What To Do With Your Free Time: Algorithms for Infrequent Requests and Randomized Weighted Caching

We consider an extension of the standard on-line model to settings in which an on-line algorithm has free time between successive requests in an input sequence. During this free time, the algorithm may perform operations without charge before receiving the next request. For instance, in planning the motion of fire trucks, there may be time in between fires that one could use to reposition the trucks in anticipation of the next fire. We prove both upper and lower bounds on the power of deterministic and randomized algorithms in this model. As our main lemma, we show an O(log 2 k)-competitive algorithm and an\Omega\Gamma/44 k) lower bound on the competitive ratio for any weighted caching problem on (k + 1)-point spaces in the standard on-line model, thus making progress on an open problem of [MMS88b, You91]. These results also apply to any metrical task system on spaces corresponding to weighted star graphs. We also consider extensions to the standard on-line model in which both free t..

Finding a Maximum-Genius Graph Imbedding

The computational complexity of constructing the imbeddings of a given graph into surfaces of different genus is not well-understood. In this paper, topological methods and a reduction to linear matroid parity are used to develop a polynomial-time algorithm to find a maximum-genus cellular imbedding. This seems to be the first imbedding algorithm for which the running time is not exponential in the genus of the imbedding surface

Multi-party protocols

process communication have been examined from a complexity pbint of view [SP, Y]. We study a new model, in which a collection of processes eo, "'', e~:~