The.Algorithm.Design.Manual.Springer-Verlag.1998

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]:
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