Suffix Trees

1
Suffix Trees

• Lecture 11 October 6, 2005
• Algorithms in Biosequence Analysis
• Nathan Edwards - Fall, 2005

2
Suffix Trees

• Powerful preprocessing for the text T
• Indexes all substrings of T
• Build in O(m) time
• Answer substring queries in O(n) time
• independent of m!
• no preprocessing of pattern
• Can amortize cost of text preprocessing over many
  patterns

3
O(m) build algorithms

• Weiner 1973
• McCreight 1976
• Better space efficiency
• Ukkonen 1995
• Simpler exposition
• Many bioinformatics algorithms start with First
  build a suffix tree on S

4
Definition

• Suffix tree T of a string S, S m
• Rooted, directed tree
• Exactly m leaves numbered 1 m
• Each edge is labeled with a substring of S
• The edge-labels of root-to-leaf-i path spell out
  Sim
• Internal nodes have at least two children
• The edge-labels of a node have distinct first
  characters

5
Example

• Suffix tree of S xabxac

x
b
c
a
a
x
6
b
c
a
b
x
5
c
x
c
a
4
a
c
c
3
2
1
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Example

• Non-Suffix tree of S xabxa

x
b
a
a
x
b
a
b
x
x
a
a
3
1
2
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Example

• Suffix tree of S xabxa

x
b

a
a
x
6
b

a
b
x
5

x

a
4
a


3
2
1
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Definitions

• Path-label of node v is the concatenation of the
  edge-labels of root-to-v path.
• String-depth of node v is the length of its
  node-label
• A path from the root that ends inside an edge out
  of node u has label formed from the concatenation
  of us node-label and the partial edge-label

9
Naïve construction

• Looks just like a keyword tree for P S1m,
  S2m, , S(m)
• Build in O(m2) time
• Merge non-branching nodes into single edge

10
Exact string matching

• Find all occurrences of P in T in O(nm) time
• Build suffix tree T in O(m) time
• Match characters of P along path from the root of
  T until either P is exhausted or there is a
  mismatch
• If mismatch, no occurrence of P in T
• If match, every leaf below end of P is a start
  position of P in T

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Exact string matching

• O(n) time to find end of P
• P occurs in T at j iff P is a prefix of Tjm
• Use depth-first search from end of P to collect
  start positions
• edges 2 leaves
• O(k) time to enumerate occurrences
• Total time O(m n k)

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Naïve construction 2

• Insert suffixes one at a time, ala keyword tree
  construction O(m2)
• Initial tree
• single edge leaf 1 with label S.
• Insert each suffix S2m to in turn
• Follow path from root until mismatch
• If at node, add leaf edge for remainder
• If inside edge, split edge with new node and add
  leaf edge for remainder.

13
Implicit Suffix Tree

• Non-suffix tree built on S
• Some suffixes might end inside tree
• Implicit suffix tree of S is a suffix tree iff
  last character of S is nowhere else

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Example

• Suffix tree of S xabxa

x
b

a
a
x
6
b

a
b
x
5

x

a
4
a


3
2
1
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Example

• Implicit suffix tree of S xabxa

x
b
a
a
x
b
a
b
x
x
a
a
3
1
2
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Ukkonens O(m) construction

• Construct implicit suffix tree Ti of prefix
  S1i, from T1 to Tm.
• Phase i1 constructs Ti1 from Ti
• Phase i1 consists of i1 extensions, which add
  S(i1) to suffixes S1i, S2i, in Ti to
  make Ti1

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Ukkonens O(m) construction high-level

• Construct T1
• For i 1 to m 1 Phase i1 make Ti1 from
  Ti For j 1 to i 1 Extension j Find
  path from root labeled Sji Extend path with
  S(i1)

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Suffix extension rules

• Extension of Sji ß with S(i1)
• ß ends at a leaf ? add S(i1) to leaf edge label
• No path at end of ß starts with S(i1)? add new
  leaf edge at end of ß with label
  S(i1). create new node at end of ß if
  necessary
• A path at end of ß starts with S(i1)? do nothing

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Example

• Implicit suffix tree of S axabx

b
a
x
x
x
a
a
4
b
b
x
b
x
1
x
3
2
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Example

• Implicit suffix tree of S axabxbRule 1

b
a
x
x
x
a
a
b
b
b
4
x
b
x
b
x
b
1
b
3
2
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Example

• Implicit suffix tree of S axabxbRule 2

b
a
x
x
x
b
a
a
b
b
b
5
4
x
b
x
b
x
b
1
b
3
2
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Example

• Implicit suffix tree of S axabxbRule 3

b
a
x
x
x
b
a
a
b
b
b
5
4
x
b
x
b
x
b
1
b
3
2
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Extention j

• Constant time, once suffix ßSji of S1i is
  found.
• Extension j of phase i1 takes O(i 1 j)
  O(ß) time to walk from the root
• O(m3) construction algorithm!
• Suffix links find ß Sji of extension j of
  phase i1 in constant time.
• O(m2) algorithm.

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Suffix links

• Given
• String xa, character x, string a
• Internal node v with node-label xa
• If there is a node s(v) with node-label a, then v
  ? s(v) is called a suffix link.
• If a is empty, s(v) is the root.

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Suffix links

• Lemma (In extension j of phase i1) If a new
  internal node v with path-label xa is created,
  then either
• the path labeled a ends at an internal node, or
• or a new internal node will be created at the end
  of the path labeled a, or
• a is empty and s(v) is the root

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Suffix links

• Corollary (In extension j of phase i1) If a
  new internal node v with path-label xa is
  created, then it has a suffix link by the end of
  extension j1.
• Corollary If implicit suffix tree Ti has an
  internal node v with path label xa,Ti must have a
  node s(v) with path-label a.

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Using suffix links

• S1i always ends at a leaf
• longest string of Ti
• Finding path ß Sji given ß0 Sj-1iv
  node at or above end of ß0, ? (partial) label
• If v is root, find ß path from root
• Otherwise, jump to s(v) find ? below s(v)
• If Rule 2 created a new internal node w at end of
  ß0, then after ß is found and extension j
  completed, s(w) must exist. Create suffix link w
  ? s(w).

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Using suffix links
node s(v)
node v
a
a
b
b
?
c
c
?
suffix link
d
end of ß0
d
j-1
end of ß
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Skip/Count Edge Traversal

• No need to compare all characters of ? with
  edge-labels below s(v), we know it is there!
• Avoid extension in O( chars below s(v)) time
• Just check first character of each edge label
• skip the other comparisons
• need length of each edge label
• extension in O( nodes below s(v)) time

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Skip/Count Edge Traversal
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Skip/Count

• node-depth of v is number of nodes on root-to-v
  path.
• LemmaWhen any suffix link v ? s(v) is used,
  node-depth(v) node-depth(s(v))1

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Node-depth lemma
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O(m2) Running Time

• LemmaAll phases take O(m) time.
• CorollaryUkkonens algorithm with suffix links
  runs in O(m2) time.