# Energy Constrained Depth First Search

Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. This paper considers a variant of such search strategies for edge-weighted trees, where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes R_1,...,R_l, each of length at most B such that: R_i starts at the root, reaches directly the farthest point of R visited by R_i-1, then R_i continues along the path R as far as possible, and finally R_i returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum for any piecemeal exploration, again regardless of the choice of the original depth first search traversal R. The fact that R can be chosen arbitrarily makes our approach suitable for construction of distributed and online algorithms, i.e., when the input tree T is not known in advance, R_i's can be constructed locally without any knowledge of the yet unvisited part of T. Moreover, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for the exploration of arbitrary graphs.

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