The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Suffix Trees and Arrays Next: Graph Data Structures Up: Data Structures Previous: Priority Queues Suffix Trees and Arrays Input description: A reference string S. Problem description: Build a data structure for quickly finding all places where an arbitrary query string q is a substring of S. Discussion: Suffix trees and arrays are phenomenally useful data structures for solving string problems efficiently and with elegance. If you need to speed up a string processing algorithm from to linear time, proper use of suffix trees is quite likely the answer. Indeed, suffix trees are the hero of the war story reported in Section . In its simplest instantiation, a suffix tree is simply a trie of the n strings that are suffixes of an n-character string S. A trie is a tree structure, where each node represents one character, and the root represents the null string. Thus each path from the root represents a string, described by the characters labeling the nodes traversed. Any finite set of words defines a trie, and two words with common prefixes will branch off from each other at the first distinguishing character. Each leaf represents the end of a string. Figure illustrates a simple trie. file:///E|/BOOK/BOOK3/NODE131.HTM (1 of 4) [19/1/2003 1:30:07]
Suffix Trees and Arrays Figure: A trie on strings the, their, there, was, and when Tries are useful for testing whether a given query string q is in the set. Starting with the first character of q, we traverse the trie along the branch defined by the next character of q. If this branch does not exist in the trie, then q cannot be one of the set of strings. Otherwise we find q in |q| character comparisons regardless of how many strings are in the trie. Tries are very simple to build (repeatedly insert new strings) and very fast to search, although they can be expensive in terms of memory. A suffix tree is simply a trie of all the proper suffixes of S. The suffix tree enables you to quickly test whether q is a substring of S, because any substring of S is the prefix of some suffix (got it?). The search time is again linear in the length of q. The catch is that constructing a full suffix tree in this manner can require time and, even worse, space, since the average length of the n suffices is n/2 and there is likely to be relatively little overlap representing shared prefixes. However, linear space suffices to represent a full suffix tree by being clever. Observe that most of the nodes in a trie-based suffix tree occur on simple paths between branch nodes in the tree. Each of these simple paths corresponds to a substring of the original string. By storing the original string in an array and collapsing each such path into a single node described by the starting and ending array indices representing the substring, we have all the information of the full suffix tree in only O(n) space. The output figure for this section displays a collapsed suffix tree in all its glory. Even better, there exist linear-time algorithms to construct this collapsed tree that make clever use of pointers to minimize construction time. The additional pointers used to facilitate construction can also be used to speed up many applications of suffix trees. But what can you do with suffix trees? Consider the following applications. For more details see the books by Gusfield [Gus97] or Crochemore and Rytter [CR94]: file:///E|/BOOK/BOOK3/NODE131.HTM (2 of 4) [19/1/2003 1:30:07]
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Suffix Trees and Arrays<br />
Next: Graph Data Structures Up: Data Structures Previous: Priority Queues<br />
Suffix Trees and Arrays<br />
Input description: A reference string S.<br />
Problem description: Build a data structure for quickly finding all places where an arbitrary query string<br />
q is a substring of S.<br />
Discussion: Suffix trees and arrays are phenomenally useful data structures for solving string problems<br />
efficiently and with elegance. If you need to speed up a string processing algorithm from to linear<br />
time, proper use of suffix trees is quite likely the answer. Indeed, suffix trees are the hero of the war story<br />
reported in Section . In its simplest instantiation, a suffix tree is simply a trie of the n strings that<br />
are suffixes of an n-character string S. A trie is a tree structure, where each node represents one<br />
character, and the root represents the null string. Thus each path from the root represents a string,<br />
described by the characters labeling the nodes traversed. Any finite set of words defines a trie, and two<br />
words with common prefixes will branch off from each other at the first distinguishing character. Each<br />
leaf represents the end of a string. Figure illustrates a simple trie.<br />
file:///E|/BOOK/BOOK3/NODE131.HTM (1 of 4) [19/1/2003 1:30:07]
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