Abstract
We investigate the relation between the time complexity and the space complexity for the rendezvous problem with k agents in asynchronous tree networks. The rendezvous problem requires that all the agents in the system have to meet at a single node within finite time. First, we consider asymptotically time-optimal algorithms and investigate the minimum memory requirement per agent for asymptotically time-optimal algorithms. We show that there exists a tree with n nodes in which Ω(n) bits of memory per agent is required to solve the rendezvous problem in O(n) time (asymptotically time-optimal). Then, we present an asymptotically time-optimal rendezvous algorithm. This algorithm can be executed if each agent has O(n) bits of memory. From this lower/upper bound, this algorithm is asymptotically space-optimal on the condition that the time complexity is asymptotically optimal. Finally, we consider asymptotically space-optimal algorithms while allowing slowdown in time required to achieve rendezvous. We present an asymptotically space-optimal algorithm that each agent uses only O(logn) bits of memory. This algorithm terminates in O(Δn 8) time where Δ is the maximum degree of the tree.
This work is supported in part by Global COE Program of MEXT, Grant-in-Aid for Scientific Research ((B)19300017, (B)20300012) of JSPS, and the Kayamori Foundation of Informational Science Advancement.
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Baba, D., Izumi, T., Ooshita, F., Kakugawa, H., Masuzawa, T. (2010). Space-Optimal Rendezvous of Mobile Agents in Asynchronous Trees. In: Patt-Shamir, B., Ekim, T. (eds) Structural Information and Communication Complexity. SIROCCO 2010. Lecture Notes in Computer Science, vol 6058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13284-1_8
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DOI: https://doi.org/10.1007/978-3-642-13284-1_8
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