A Relational Theory of Computing by John G. Sanderson (auth.)

By John G. Sanderson (auth.)

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Lemma 2. Let 1 ≤ bx , by , bz ≤ 2 and m = bx + by + bz − 3. Elements of a set S such that |S| = O(B 4/3 ) can be stored in a data structure that uses m+1 k O( |S| |S|) blocks of space and supports (bx , by , bz )-sided queries in O( B ) B log2 m I/O operations and updates in O(log2 (|S|)) I/O operations amortized. k In particular, we can support (2, 1, 2)-sided queries in O( B ) I/Os and updates in 2 4/3 O(log2 B) I/Os on a set S that contains Θ(B ) points using a data structure that needs O(B 1/3 log32 B) blocks of space.

C. -F. -C. Kuo A0 B 0 1 0 1 1 0 0 A 2 5 6 8 10 11 13 2 6 11 13 A1 5 8 10 Fig. 2. An illustration of bit-vector B Lemma 6. Let A0 and A1 be two sorted sequences and let A be the sorted sequence obtained by merging A0 and A1 . The bridges from A to A0 and A1 can be maintained using O(|A|) bits of space. Proof. Rather than storing a pointer from each element of A to its successor in A0 , we employ binary rank query to serve the function of bridges. The details are as follows. We create a bit-vector B of size |A| to indicate whether each element of A belongs to A0 or A1 .

Vk , we replace v and its children by a weight-balanced binary search tree with root v and leaves v1 , v2 , . . , vk , in which node vi has weight equal to its number of descendant leaves. The above transformation increases the tree height by O(log n) [18]. Thus, we can apply Chazelle’s data structures on BT to store the sorted sequences Av , where v is a node in ST with d(v) ≤ log n. For long patterns, Chazelle’s data structures can be directly applied on GT to store the sorted sequences Av .

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