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**Sample text**

3 shows an example. Then, we see that i2 s(i, j) = i=i1 wj (v) s(i, j) = v∈Si1 ,i2 i∈L(v) v∈Si1 ,i2 cj (u), u∈Anc(v) where the last equality follows from Lemma 3. Note that given a consecutive sequence of leaves {i1 , . . , i2 } of T , we can find its representing ancestors Si1 ,i2 in O(log n) time in a bottom-up fashion. This also shows that the number of representing ancestors is O(log n). Now we are ready to describe our algorithm to compute s(1, j), . . , s(n + 1, j) from s(1, j − 1), . .

S(n + 1, j) in an ordered binary tree T . The tree T has height O(log n), with a root r, and with n + 1 leaves corresponding to the intervals I1 , . . , In+1 : We name the leaves 1, . . , n + 1 and the leaf i corresponds to the interval Ii . The structure of T does not change through computation, but only the values carried by nodes change. To represent s(i, j), every node v ∈ T carries two values wj (v) and cj (v) satisfying the following three conditions; wj (v) = cj (v) · (wj (u1 ) + wj (u2 )) for every inner node v, (1) s(i, j) = w (i) · (2) c (u) for every leaf i ∈ {1, .

However, we must take care of the disjointness of the paths. Let s = v0 , . . , v = t be the sequence of vertices on the path from s to t in T . For each of these vertices vi = s, t, we deﬁne Ws (v) as the set of vertices in Tvi that are reachable from s in T without crossing the path R. Each of the vertices w ∈ Ws (vi ) deﬁnes a path Ps (w). Analogously, we deﬁne the set of vertices Wt (vi ) in Tvi that are reachable from t in T without crossing R. Each of those vertices w ∈ Wt (vi ) deﬁnes a path Pt (w) from t to w.