Abstract
In this paper, we investigate the q -gram distance for ordered unlabeled trees (trees, for short). First, we formulate a q-gram as simply a tree with q nodes isomorphic to a line graph, and the q-gram distance between two trees as similar as one between two strings. Then, by using the depth sequence based on postorder, we design the algorithm EnumGram to enumerate all q-grams in a tree T with n nodes which runs in O(n 2) time and in O(q) space. Furthermore, we improve it to the algorithm LinearEnumGram which runs in O(qn) time and in O(qd) space, where d is the depth of T. Hence, we can evaluate the q-gram distance D q (T 1,T 2) between T 1 and T 2 in O(q maxn 1, n 2) time and in O(q maxd 1, d 2) space, where n i and d i are the number of nodes in T i and the depth of T i , respectively. Finally, we show the relationship between the q-gram distance D q (T 1,T 2) and the edit distanceE(T 1,T 2) that D q (T 1,T 2)≤ (gl+1)E(T 1,T 2), where g = max{g 1, g 2}, l = max{l 1, l 2}, g i is the degree of T i and l i is the number of leaves in T i . In particular, for the top-down tree edit distanceF(T 1,T 2), this result implies that \(D_{q}(T_{1}, T_{2}) \leq {\rm min}\{g^{q-2}, l - 1\}\{F(T_{1}, T_{2})\}\).
This work is partially supported by Grand-in-Aid for Scientific Research 15700137, 16016275 and 17700138 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Ohkura, N., Hirata, K., Kuboyama, T., Harao, M. (2005). The q-Gram Distance for Ordered Unlabeled Trees. In: Hoffmann, A., Motoda, H., Scheffer, T. (eds) Discovery Science. DS 2005. Lecture Notes in Computer Science(), vol 3735. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11563983_17
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DOI: https://doi.org/10.1007/11563983_17
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