Algorithms and Data Structures for External Memory

5
a b
a
3
c
a
0
b
a
4
b
6 6
a b a b
6 10 4 7 7 7 8
abaaba
abaabbabba
abac
bcbcaba
bcbcabb
14.2 String B-Trees 125
Fig. 14.1 Patricia trie representation of a single node of an SB-tree, with branching factor
B = 8. The seven strings used for partitioning are pictured at the leaves; in the actual data
structure, pointers to the strings, not the strings themselves, are stored at the leaves. The
pointers to the B children of the SB-tree node are also stored at the leaves.
label for each of its outgoing edges. Navigation from root to leaf in
the Patricia trie is done using the bit representation of the strings. For
example, suppose we want to search for the leaf “abac.” We start at
the root, which has index 0; the index indicates that we should examine
character 0 of the search string “abac” (namely, “a”), which leads us to
follow the branch labeled “a” (left branch). The next node we encounter
has index 3, and so we examine character 3 (namely, “c”), follow the
branch labeled “c” (right branch), and arrive at the leaf “abac.”
Searching for a text string that does not match one of the leaves
is more complicated and exploits the full power of the data structure,
using an amortization technique of Ajtai et al. [25]. Suppose we want
to search for “bcbabcba.” Starting at the root, with index 0, we examine
character 0 of “bcbabcba” (namely, “b”) and follow the branch
labeled “b” (right branch). We continue searching in this manner, which
leads along the rightmost path, examining indexes 4 and 6, and eventually
we arrive at the far-right leaf “bcbcbbba.” However, the search
string “bcbabcba” does not match the leaf string “bcbcbbba.” The
problem is that they differ at index 3, but only indexes 0, 4, and 6 were
examined in the traversal, and thus the difference was not detected.
In order to determine efficiently whether or not there is a match, we
go back and sequentially compare the characters of the search string
bcbcbba
bcbcbbba