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Tree Exploration with an Oracle

  • Pierre Fraigniaud
  • David Ilcinkas
  • Andrzej Pelc
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

We study the amount of knowledge about the network that is required in order to efficiently solve a task concerning this network. The impact of available information on the efficiency of solving network problems, such as communication or exploration, has been investigated before but assumptions concerned availability of particular items of information about the network, such as the size, the diameter, or a map of the network. In contrast, our approach is quantitative: we investigate the minimum number of bits of information (minimum oracle size) that has to be given to an algorithm in order to perform a task with given efficiency.

We illustrate this quantitative approach to available knowledge by the task of tree exploration. A mobile entity (robot) has to traverse all edges of an unknown tree, using as few edge traversals as possible. The quality of an exploration algorithm \({\cal A}\) is measured by its competitive ratio, i.e., by comparing its cost (number of edge traversals) to the length of the shortest path containing all edges of the tree. Depth-First-Search has competitive ratio 2 and, in the absence of any information about the tree, no algorithm can beat this value.

We determine the minimum number of bits of information that has to be given to an exploration algorithm in order to achieve competitive ratio strictly smaller than 2. Our main result establishes an exact threshold oracle size that turns out to be roughly loglogD, where D is the diameter of the tree. More precisely, for any constant c, we construct an exploration algorithm with competitive ratio smaller than 2, using an oracle of size at most loglogDc, and we show that every algorithm using an oracle of size loglogDg(D), for any function g unbounded from above, has competitive ratio at least 2.

Keywords

Competitive Ratio Binary String Global Knowledge Exploration Scheme Graph Exploration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pierre Fraigniaud
    • 1
  • David Ilcinkas
    • 2
  • Andrzej Pelc
    • 3
  1. 1.CNRS, Laboratoire de Recherche en Informatique (LRI)Université Paris-SudOrsayFrance
  2. 2.Laboratoire de Recherche en Informatique (LRI)Université Paris-SudOrsayFrance
  3. 3.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada

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