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Algorithmic Theory of Random graphs
, 1997
"... The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this pap ..."
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The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...
An Experimental Study on Transitive Closure Representations
"... We present two new compact transitive closure representations. The first uses intervals and the second chains to store the closure. Both representations are based on previous methods designed for acyclic graphs. The new representations are applicable to all kinds of graphs, and can be efficiently co ..."
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Cited by 1 (0 self)
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We present two new compact transitive closure representations. The first uses intervals and the second chains to store the closure. Both representations are based on previous methods designed for acyclic graphs. The new representations are applicable to all kinds of graphs, and can be efficiently constructed during a single traversal of the input graph. We compared experimentally the average size of these representations and traditional list based representations. The inputs were random graphs. The interval representation outperformed the other representations: it typically required a space at most linear to the number of vertices of the input graph. The chain representation did not save much space compared to a list representation. We also studied the complexity of constructing the interval representation. Our results indicate that in the models of random graphs that we used, the transitive closure can typically be computed in a time linear to the size of the input graph when the interv...
Average Costs of a Graph Exploration: Upper and Lower Bounds
"... We consider the exploration of random graphs. We give upper and lower bounds for the expected number of edges traversed during an exploration. This result implies a lower bound for the expected running time of a wide class of algorithms  e.g. BreadthFirstSearch, DepthFirstSearch, and algorithm ..."
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We consider the exploration of random graphs. We give upper and lower bounds for the expected number of edges traversed during an exploration. This result implies a lower bound for the expected running time of a wide class of algorithms  e.g. BreadthFirstSearch, DepthFirstSearch, and algorithms to determine a minimum spanning tree or to solve the single source shortest paths problem in a weighted graph. Furthermore, we investigate the connectedness of nonhomogeneous random graphs and we point out the relationship with the exploration algorithms. Key words: Average Case Analysis, Lower Bounds, Random Graphs. 1 Introduction Traversing a (directed) graph is a fundamental problem in graph theory since it is the abstraction of a variety of processes. Hence, search algorithms are widely used to solve lots of subproblems. Furthermore, Karp [5] pointed out that the analysis of some searching algorithms (like BreadthFirst search) can be used to analyze structural properties of random ...