Fast Algorithms for Computing Tree LCS

The LCS of two rooted, ordered, and labeled trees Fand Gis the largest forest that can be obtained from both trees by deleting nodes. We present algorithms for computing tree LCS which exploit the sparsityinherent to the tree LCS problem. Assuming Gis smaller than F, our first algorithm runs in time $O(r\cdot {\rm height}(F) \cdot {\rm height}(G)\cdot \lg\lg |G|)$, where ris the number of pairs (vi?? F, wi?? G) such that vand whave the same label. Our second algorithm runs in time $O(L r \lg r \cdot \lg\lg|G|)$, where Lis the size of the LCS of Fand G. For this algorithm we present a novel three dimensional alignment graph. Our third algorithm is intended for the constrained variant of the problem in which only nodes with zero or one children can be deleted. For this case we obtain an $O(r h \lg \lg|G|)$ time algorithm, where h= height(F) + height(G).

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