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Since Ar [i] = i for i = 1, 2, . . , n, the value of Aμ(P ) [k] can be found in O(d(v)) time. In summary, given a PPM query (P, s), we ﬁrst locate μ(P ) in ST . By following the bridges the value k = succ−1(Aμ(P ) , s) is also determined. Then, tracing from μ(P ) back to the tree root r, we compute k as the position of Aμ(P ) [k] in Ar . Finally, we report k as the answer. We have the following. Theorem 1. We can construct an O(n)-word index for a string T over a ﬁnite alphabet, so that a positional pattern matching query can be answered in O(p) time for any short pattern P .

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Pinter [17] had a linear-time algorithm for determining whether P occurs in T . His algorithm is based upon the following observation: if P1 does not occur in T , then neither does P ; otherwise, P occurs in T if and only if P2 ∗ . . ∗Pm occurs in T [k1 + |P1 |, n], where k1 is the ﬁrst occurrence of P1 in T . Consider the example in Fig 3. The ﬁrst occurrence of P1 is at position 3. Thus, our problem reduces to determining whether P2 ∗P3 occurs in T [8, n]. Similarly, since the ﬁrst occurrence of P2 in T [8, n] is at position 11, the problem further reduces to determining whether P3 occurs in T [15, n].