Weighted Nearest Neighbor Algorithms for the Graph Exploration Problem on Cycles

Authors:
Yuichi Asahiro;Eiji Miyano;Shuichi Miyazaki;Takuro Yoshimuta
Affiliations:
Department of Social Information Systems, Kyushu Sangyo University, Fukuoka 813-8503, Japan;Department of Systems Innovation and Informatics, Kyushu Institute of Technology, Fukuoka 820-8502, Japan;Academic Center for Computing and Media Studies, Kyoto University, Kyoto 606-8501, Japan;Department of Systems Innovation and Informatics, Kyushu Institute of Technology, Fukuoka 820-8502, Japan
Venue:
SOFSEM '07 Proceedings of the 33rd conference on Current Trends in Theory and Practice of Computer Science
Year:
2007

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Abstract

In the graph exploration problem, a searcher explores the whole set of nodes of an unknown graph. The searcher is not aware of the existence of an edge until he/she visits one of its endpoints. The searcher's task is to visit all the nodes and go back to the starting node by traveling as a short tour as possible. One of the simplest strategies is the nearest neighbor algorithm (NN), which always chooses the unvisited node nearest to the searcher's current position. The weighted NN(WNN) is an extension of NN, which chooses the next node to visit by using the weighted distance. It is known that WNNwith weight 3 is 16-competitive for planar graphs. In this paper we prove that NNachieves the competitive ratio of 1.5 for cycles. In addition, we show that the analysis for the competitive ratio of NNis tight by providing an instance for which the bound of 1.5 is attained, and NNis the best for cycles among WNNwith all possible weights. Furthermore, we prove that no online algorithm is better than 1.25-competitive.