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Resource discovery is a fundamental service of the grid to find out resources suitable for executing user jobs. Traditional resource discovery services of the grid have static structure and centralized or hierarchized, such as Globus MDS. However, with the number of resources composing the grid increased, this approach has two problems, namely, on dynamic changes of the structure in the grid and obtaining the latest resource information. To solve these problems, we have been seeking for the possibility of a resource discovery mechanism adopting peer-to-peer approach, which is characterized by real-time retrieval of information using an overlay network spanned on physical computational resources. In this paper, we present the preliminary results of our research activities so far.