1 Example
2 Introduction
3.1 Trees
3.2 Nodes
3.3 Labels
string->label/ with-sentinel
vector->label/ with-sentinel
3.4 Other utilities
4 Caveats
5 References
Version: 5.2.1

Suffix trees with Ukkonen’s algorithm

Danny Yoo <dyoo@hashcollision.org>

This is an implementation of suffix trees and their linear-time construction with the Ukkonen algorithm. This implementation is based on notes from Gusfield, "Algorithms on Strings, Trees, and Sequences".

1 Example

Let’s rush into a minimal example:

> (require (planet dyoo/suffixtree))
> (define tree (make-tree))
> (tree-add! tree (string->label "00010010$"))
> (define root (tree-root tree))
> (node-children root)

'(#<node> #<node> #<node>)

> (label->string (node-up-label (car (node-children root))))


2 Introduction

Suffix trees encode all the nonempty suffixes of a string. For example, the string "0100101$" corresponds to the following suffix tree.




 +--- $


 +--- 1 --- $

 |    |

 |    +---- 0 --- 0101$

 |          |

 |          +---- 1$


 +--- 0 --- 0101$


      +---- 1 ---- $


            +----- 0 --- 1$


                   +---- 0101$

Every path from the root to any leaf spells out a suffix of the string "0100101$", and every suffix is accounted for. This in itself might not sound too sexy, but by preprocessing a string as a suffix tree, we can then do some amazing things.

For example, we can see if a substring is present in a suffix tree in time bounded by the length of the substring by following characters starting from the root. Suffix trees also allow us to find the longest common substring between strings in linear time. Dan Gusfield’s book "Algorithms on Strings, Trees, and Sequences" sings praises about suffix trees, and deservedly so.

Constructing a suffix tree can be done in linear-time; the algorithm used here is Ukkonen’s algorithm, since it’s one of the simplest to code. That being said, the algorithm is not quite simple; for more information on the construction algorithm, see the References section below.


 (require (planet dyoo/suffixtree:1:=2))

The API consists of the main suffix tree algorithm, and auxillary utilities and applications.

The main structures are trees, nodes, and labels.

3.1 Trees

A suffix tree consists of a root. This implementation allows multiple labels to be added to the tree.

(make-tree)  tree
Constructs an empty suffix tree with a single root node.

For example:
> (define a-tree (make-tree))
> a-tree


(tree? datum)  boolean
  datum : any
Returns #t if datum is a suffix tree.

For example:
> (define a-tree (make-tree))
> (tree? a-tree)


(tree-root a-tree)  node
  a-tree : tree
Selects the root node from a tree.

For example:
> (define a-tree (make-tree))
> (tree-root a-tree)


(tree-add! a-tree a-label)  void
  a-tree : tree
  a-label : label
Adds a label and all of its nonempty suffixes to the tree.

For example:
> (define a-tree (make-tree))
> (tree-add! a-tree (string->label "supercalifragilisticexpialidocious"))

(tree-walk a-tree a-label succeed-f fail-f)  (or/c A B)
  a-tree : tree
  a-label : label
  succeed-f : (node number -> A)
  fail-f : (node number number -> B)
Starting from the tree-root, walks along a path whose path label exactly matches the input label.

If the label matched completely, calls succeed-f with the position where the matching had succeeded.

If the label mismatched, calls fail-f with the tree position where the matching had failed.

The return value from tree-walk will either be A or B.

For example:
> (define a-tree (make-tree))
> (tree-add! a-tree (string->label "banana"))
> (define (success-f node up-label-offset)
    (list node up-label-offset))
> (define (fail-f node up-label-offset input-label-offset)
    (list node up-label-offset input-label-offset))
> (tree-walk a-tree (string->label "banana") success-f fail-f)

'(#<node> 6)

> (tree-walk a-tree (string->label "ban") success-f fail-f)

'(#<node> 3)

> (tree-walk a-tree (string->label "ana") success-f fail-f)

'(#<node> 3)

> (tree-walk a-tree (string->label "apple") success-f fail-f)

'(#<node> 1 1)

(tree-contains? a-tree a-label)  boolean
  a-tree : tree
  a-label : label
Returns #t if a path exists starting from the tree-root of the tree whose path-label exactly matches a-label.

For example:
> (define a-tree (make-tree))
> (tree-add! a-tree (string->label "0100101$"))
> (tree-contains? a-tree (string->label "01"))


> (tree-contains? a-tree (string->label "001"))


> (tree-contains? a-tree (string->label "011"))


> (tree-contains? a-tree (string->label "0101"))


tree-contains? is an application of tree-walk:

(define (tree-contains? tree label)
  (tree-walk tree label
             (lambda (node up-label-offset) #t)
             (lambda (node up-label-offset input-label-offset) #f)))

3.2 Nodes

Nodes form the structure of the suffix tree, and link up children nodes as well. Every internal node I of a suffix tree will also have a suffix-node whose path-label is the immediate suffix of node I.

For example, we can inspect the node-up-label of each child of the root:
> (define a-tree (make-tree))
> (tree-add! a-tree
            (string->label "peter piper"))
> (for ([i (in-naturals)]
        [c (node-children (tree-root a-tree))])
    (printf "~a: ~a\n" i (label->string (node-up-label c))))

0: iper

1: p

2:  piper

3: r piper

4: e

5: ter piper

> (define p-node (node-find-child (tree-root a-tree) #\p))
> (for ([i (in-naturals)]
        [c (node-children p-node)])
    (printf "~a: ~a\n" i (label->string (node-up-label c))))

0: e

1: iper

(node-up-label a-node)  label
  a-node : node
Selects the label of the edge that connects this node to its parent. The up-label of the root node is empty.

(node-parent a-node)  (or/c node #f)
  a-node : node
Selects the parent of this node. The root of a suffix tree has no parent, so (node-parent (tree-root tree)) returns #f.

(node-suffix-link a-node)  node
  a-node : node
Selects the suffix node of this node. If the suffix-link is not set, returns #f.

(node-find-child a-node a-label-element)  (or/c node #f)
  a-node : node
  a-label-element : label-element
Selects the child whose up-label starts with the label-element. If no such child can be found, returns #f.

(node-children a-node)  (listof node)
  a-node : node
Selects the list of children nodes to this node. If the node is a leaf, returns '().

3.3 Labels

Labels represent an immutable sequence of label-elements. Label-elements can be anything that compare with equal?, but the most common label-elements will be characters. Labels can be sublabeled with efficiency.

(string->label a-string)  label
  a-string : string
Constructs a label from a string. Each of the label-elements of this label will be a character.

(string->label/with-sentinel a-str)  label
  a-str : string
Constructs a label from a string with a trailing sentinel character to guarantee that all suffixes can be explicitely represented in a suffix tree. (See the Caveats section below for details.)

Note that label->string can’t be directly used on a label with a sentinel.

(label->string a-label)  string
  a-label : label
Constructs a string from a label, assuming that all label-elements of the string are characters.

(vector->label a-vec)  label
  a-vec : vector
Constructs a label from a vector.

(vector->label/with-sentinel a-vec)  label
  a-vec : vector
Constructs a label from a vector with a trailing sentinel character.

(label->vector a-label)  vector
  a-label : label
Selects a vector of the label-elements that represent the label. This vector is immutable.

(sublabel a-label left-offset [right-offset])  label
  a-label : label
  left-offset : number
  right-offset : number = (label-length label)
Derives a new sliced label from the parent label, along the half-open interval [left-offset, right-offset).

If right-offset is omitted, it defaults to (label-length label).

(<= left-offset right-offset) should be #t.

(label-ref a-label a-offset)  label-element
  a-label : label
  a-offset : number
Returns the label’s label-element at that offset.

(label-length a-label)  number
  a-label : label
Returns the length of a label.

(label-equal? label-1 label-2)  boolean
  label-1 : label
  label-2 : label
Produce true if the two labels have equal content.

Warning: two labels may have equal content, but come from different sources.

(label-source-eq? label-1 label-2)  boolean
  label-1 : label
  label-2 : label
Returns #t if both labels share a common derivation from sublabeling.

(label-source-id a-label)  number
  a-label : label
Returns an numeric identifier for this label.

(label-source-eq? label-1 label-2)

logically implies:

(= (label-source-id label-1) (label-source-id label-2))

3.4 Other utilities

This module provides some example applications of suffix trees.

(longest-common-substring string-1    
  string-2)  string
  string-1 : string
  string-2 : string
Returns the longest common substring between the two strings.

For example:
> (longest-common-substring
   "Lambda: the Ultimate Imperative"
   "Procedure Call Implementations Considered Harmful, or, Lambda: the Ultimate GOTO")

"Lambda: the Ultimate "

(longest-common-sublabel label-1 label-2)  label
  label-1 : label
  label-2 : label
Returns the longest sublabel that’s shared between label-1 and label-2.

(path-label a-node)  label
  a-node : node
Returns a new label that represents the path from the root to this node.

4 Caveats

The code in tree-add! assumes that the construction of the full suffix tree on its input string is possible. Certain strings don’t have an full explicit suffix tree, such as "foo".

+-- "foo"


+-- "oo"

In this case, when we try to construct a suffix tree out of "foo", we have an implicit suffix tree, where not every leaf corresponds to a suffix of the input string. The suffix "o" is implicit in this tree.

In order to guarantee that all suffixes will have a place in the suffix tree, we’ll often add a sentinel character at the end the string to make sure all suffixes have a unique path in the suffix tree. For example, assuming that we use "$" as our sentinel:

+-- "foo$"


+-- "o" -- "o$"

|    |

|    +---- "$"


+-- "$"

The API has the function string->label/with-sentinel to automatically add a unique sentinel character at the end of a string.

(let ([tree (make-tree)]
      [label (string->label/with-sentinel "foo")])
  (tree-add! label))

so be sure to use this if you need to ensure the representation of all suffixes in the suffix tree.

5 References

Dan Gusfield. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, New York, NY, 1997.

Lloyd Allison. Suffix Trees. http://www.allisons.org/ll/AlgDS/Tree/Suffix/

Mark Nelson. Fast String Searching With Suffix Trees. Dr. Dobb’s Journal, August, 1996. http://www.dogma.net/markn/articles/suffixt/suffixt.htm

Mummer: Ultra-fast alignment of large-scale DNA and protein sequences. http://mummer.sourceforge.net/