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**Stata Programming Reference Manual Release 10**

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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 .

Of the 13th VLDB, pp. 444–455 (2004) 8. : The Rainbow Skip Graph: A FaultTolerant Constant-Degree Distributed Data Structure. In: Proc. of the 17th SODA, pp. 384–393 (2006) 9. : Deterministic SkipNet. In: Proc. of the 22nd PODC, pp. 152–153 (2003) 10. : BATON: a Balanced Tree Structure for Peerto-Peer Networks. In: Proc. of the 31st VLDB, pp. 661–672 (2005) 11. : Chord: A Scalable Peer-to-Peer Lookup Service for Internet Applications. In: Proc. of the SIGCOMM, pp. 149–160 (2001) 12. : Symphony: Distributed hashing in a small world.

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].