Suffix Trees
CMSC 423
Preprocessing Strings
• Over the next few lectures, we’ll see several methods for
preprocessing string data into data structures that make many
questions (like searching) easy to answer:
•
•
•
•
•
Suffix Trees
Suffix Arrays
BorrowsWheeler transform
Typical setting: A long, known, and fixed text string (like a genome) and
many unknown, changing query strings.
•
•
Suffix Tries
Allowed to preprocess the text string once in anticipation of the
future unknown queries.
Data structures will be useful in other settings as well.
Suffix Tries
• A trie, pronounced “try”, is a tree that exploits some
structure in the keys

e.g. if the keys are strings, a binary search tree would
compare the entire strings, but a trie would look at their
individual characters

Suffix trie are a spaceefficient data structure to store a string
that allows many kinds of queries to be answered quickly.

Suffix trees are hugely important for searching large sequences
like genomes. The basis for a tool called “MUMMer” (developed
by UMD faculty).
Suffix Tries
s = abaaba$
a
b
$
SufTrie(s) = suffix trie representing string s.
b a $
a
a
Edges of the suffix trie are labeled with
letters from the alphabet ∑ (say {A,C,G,T}).
a
b
$
b
a
$
a
a
b
Every path from the root to a solid node
represents a suffix of s.
$
Every suffix of s is represented by some
path from the root to a solid node.
$
a
$
Why are all the solid nodes leaves?
How many leaves will there be?
Processing Strings Using Suffix Tries
Given a suffix trie T, and a string q, how can we:
•
•
•
•
•
determine whether q is a substring of T?
check whether q is a suffix of T?
count how many times q appears in T?
find the longest repeat in T?
find the longest common substring of T and q?
Main idea:
every substring of s is a prefix of some suffix of s.
Searching Suffix Tries
s = abaaba$
a
b a $
a
b
$
Is “baa” a substring of s?
a
Follow the path given by
the query string.
a
b
$
b
a
$
a
a
b
$
$
a
$
After we’ve built the suffix trees,
queries can be answered in time:
O(query)
regardless of the text size.
Searching Suffix Tries
s = abaaba$
a
b a $
a
b
$
Is “baa” a substring of s?
a
Follow the path given by
the query string.
a
b
$
b
a
$
a
a
b
$
$
a
$
After we’ve built the suffix trees,
queries can be answered in time:
O(query)
regardless of the text size.
Applications of Suffix Tries (1)
Check whether q is a substring of T:
Check whether q is a suffix of T:
Count # of occurrences of q in T:
Find the longest repeat in T:
Find the lexicographically (alphabetically) first suffix:
Applications of Suffix Tries (1)
Check whether q is a substring of T:
Follow the path for q starting from the root.
If you exhaust the query string, then q is in T.
Check whether q is a suffix of T:
Count # of occurrences of q in T:
Find the longest repeat in T:
Find the lexicographically (alphabetically) first suffix:
Applications of Suffix Tries (1)
Check whether q is a substring of T:
Follow the path for q starting from the root.
If you exhaust the query string, then q is in T.
Check whether q is a suffix of T:
Follow the path for q starting from the root.
If you end at a leaf at the end of q, then q is a suffix of T
Count # of occurrences of q in T:
Find the longest repeat in T:
Find the lexicographically (alphabetically) first suffix:
Applications of Suffix Tries (1)
Check whether q is a substring of T:
Follow the path for q starting from the root.
If you exhaust the query string, then q is in T.
Check whether q is a suffix of T:
Follow the path for q starting from the root.
If you end at a leaf at the end of q, then q is a suffix of T
Count # of occurrences of q in T:
Follow the path for q starting from the root.
The number of leaves under the node you end up in is the
number of occurrences of q.
Find the longest repeat in T:
Find the lexicographically (alphabetically) first suffix:
Applications of Suffix Tries (1)
Check whether q is a substring of T:
Follow the path for q starting from the root.
If you exhaust the query string, then q is in T.
Check whether q is a suffix of T:
Follow the path for q starting from the root.
If you end at a leaf at the end of q, then q is a suffix of T
Count # of occurrences of q in T:
Follow the path for q starting from the root.
The number of leaves under the node you end up in is the
number of occurrences of q.
Find the longest repeat in T:
Find the deepest node that has at least 2 leaves under it.
Find the lexicographically (alphabetically) first suffix:
Applications of Suffix Tries (1)
Check whether q is a substring of T:
Follow the path for q starting from the root.
If you exhaust the query string, then q is in T.
Check whether q is a suffix of T:
Follow the path for q starting from the root.
If you end at a leaf at the end of q, then q is a suffix of T
Count # of occurrences of q in T:
Follow the path for q starting from the root.
The number of leaves under the node you end up in is the
number of occurrences of q.
Find the longest repeat in T:
Find the deepest node that has at least 2 leaves under it.
Find the lexicographically (alphabetically) first suffix:
Start at the root, and follow the edge labeled with the
lexicographically (alphabetically) smallest letter.
Suffix Links
s = abaaba$
a
b
$
• Suffix links connect node
b a
a
$
representing “xα” to a node
representing “α”.
a
• Most important suffix links are
a
b
$
b
a
$
a
a
b
$
$
a
$
the ones connecting suffixes of
the full string (shown at right).
• But every node has a suffix link.
• Why?
• How do we know a node
representing α exists for
every node representing xα?
Suffix Tries
s = abaaba$
a
b a $
a
b
$
A node represents the prefix of some
suffix:
a
abaaba$
a
b
$
b
a
$
a
a
b
$
$
a
$
s
The node’s suffix link should link to the
prefix of the suffix s that is 1 character
shorter.
Since the suffix trie contains all
suffixes, it contains a path representing
s, and therefore contains a node
representing every prefix of s.
Suffix Tries
s = abaaba$
a
b a $
a
b
$
A node represents the prefix of some
suffix:
a
abaaba$
a
b
$
b
a
$
a
a
b
$
$
a
$
s
The node’s suffix link should link to the
prefix of the suffix s that is 1 character
shorter.
Since the suffix trie contains all
suffixes, it contains a path representing
s, and therefore contains a node
representing every prefix of s.
Applications of Suffix Tries (II)
abaaba$
Find the longest common substring of T and q:
$
a
b
$
a
b
$
a
a
a
b
b
a
$
T = abaaba$
q = bbaa
a
$
a
b
a
$
$
Applications of Suffix Tries (II)
Find the longest common substring of T and q:
Walk down the tree following q.
If you hit a dead end, save the current depth,
and follow the suffix link from the current
node.
When you exhaust q, return the longest
substring found.
abaaba$
$
a
b
$
a
b
$
a
a
a
b
b
a
$
T = abaaba$
q = bbaa
a
$
a
b
a
$
$
Constructing Suffix Tries
Suppose we want to build suffix trie for string:
s = abbacabaa
We will walk down the string from left to right:
abbacabaa
building suffix tries for s[0], s[0..1], s[0..2], ..., s[0..n]
To build suffix trie for s[0..i], we
will use the suffix trie for s[0..i1]
built in previous step
To convert SufTrie(S[0..i1]) → SufTrie(s[0..i]), add character s[i] to all the suffixes:
abbacabaa
i=4
Need to add nodes for
the suffixes:
abbac
bbac
bac
ac
c
Purple are suffixes that
will exist in
SufTrie(s[0..i1]) Why?
How can we find these
suffixes quickly?
Suppose we want to build suffix trie for string:
s = abbacabaa
We will walk down the string from left to right:
abbacabaa
building suffix tries for s[0], s[0..1], s[0..2], ..., s[0..n]
To build suffix trie for s[0..i], we
will use the suffix trie for s[0..i1]
built in previous step
To convert SufTrie(S[0..i1]) → SufTrie(s[0..i]), add character s[i] to all the suffixes:
abbacabaa
i=4
Need to add nodes for
the suffixes:
abbac
bbac
bac
ac
c
Purple are suffixes that
will exist in
SufTrie(s[0..i1]) Why?
How can we find these
suffixes quickly?
abbacabaa
Need to add nodes for
the suffixes:
i=4
b
a
a
a
Purple are suffixes that
will exist in
SufTrie(s[0..i1]) Why?
abbac
bbac
bac
ac
c
How can we find these
suffixes quickly?
b
c
a
b
c
b
b
b
c
a
a
b
b
c
a
a
Where is the new
deepest node? (aka
longest suffix)
c
SufTrie(abba)
SufTrie(abbac)
How do we add the
suffix links for the
new nodes?
abbacabaa
Need to add nodes for
the suffixes:
i=4
b
a
a
a
Purple are suffixes that
will exist in
SufTrie(s[0..i1]) Why?
abbac
bbac
bac
ac
c
How can we find these
suffixes quickly?
b
c
a
b
c
b
b
b
c
a
a
b
b
c
a
a
Where is the new
deepest node? (aka
longest suffix)
c
SufTrie(abba)
SufTrie(abbac)
How do we add the
suffix links for the
new nodes?
To build SufTrie(s[0..i]) from SufTrie(s[0..i1]):
CurrentSuffix = longest (aka deepest suffix)
until you reach the
root or the current
node already has an
edge labeled s[i]
leaving it.
Because if you
already have a node
for suffix αs[i]
then you have a
node for every
smaller suffix.
Repeat:
Add child labeled s[i] to CurrentSuffix.
Follow suffix link to set CurrentSuffix to next
shortest suffix.
Add suffix links connecting nodes you just added in
the order in which you added them.
In practice, you add these links as you go
along, rather than at the end.
Python Code to Build a Suffix Trie
class SuffixNode:
def __init__(self, suffix_link = None):
self.children = {}
if suffix_link is not None:
self.suffix_link = suffix_link
else:
self.suffix_link = self
def add_link(self, c, v):
"""link this node to node v via string c"""
self.children[c] = v
def build_suffix_trie(s):
"""Construct a suffix trie."""
assert len(s) > 0
# explicitly build the twonode suffix tree
Root = SuffixNode()
# the root node
s[0]
Longest = SuffixNode(suffix_link = Root)
Root.add_link(s[0], Longest)
# for every character left in the string
for c in s[1:]:
Current = Longest; Previous = None
while c not in Current.children:
# create new node r1 with transition Current c>r1
r1 = SuffixNode()
Current.add_link(c, r1)
# if we came from some previous node, make that
# node's suffix link point here
if Previous is not None:
Previous.suffix_link = r1
# walk down the suffix links
Previous = r1
Current = Current.suffix_link
# make the last suffix link
if Current is Root:
Previous.suffix_link = Root
else:
Previous.suffix_link = Current.children[c]
# move to the newly added child of the longest path
# (which is the new longest path)
Longest = Longest.children[c]
return Root
current
current
s[i]
s[i]
longest
s[i]
s[i]
longest
s[i]
s[i]
u
s[i]
u
s[i]
s[i]
Prev
Prev
current
s[i]
boundary path
s[i]
s[i]
s[i]
s[i]
longest
Prev
a
a
ab
a
a
a
b
b
aba
ab
a
a
a
b
b
a
b
a
b
a
Note: there's already a path for
suffix "a", so we don't change it (we
just add a suffix link to it)
ab
a
a
a
b
abaa
aba
b
a
b
a
a
b
a
Note: there's already a path for
suffix "a", so we don't change it (we
just add a suffix link to it)
b
a
b
a
a
a
a
ab
a
a
a
b
abaa
aba
b
a
b
a
a
b
a
Note: there's already a path for
suffix "a", so we don't change it (we
just add a suffix link to it)
abaab
b
a
a
b
b
a
b
a
a
a
a
a
a
a
b
b
a
b
ab
a
b
a
b
a
abaa
aba
a
a
b
abaaba
b
a
a
a
a
b
a
a
b
a
a
b
a
b
a
b
a
a
a
Note: there's already a path for
suffix "a", so we don't change it (we
just add a suffix link to it)
b
a
a
a
a
b
b
a
b
abaab
a
a
a
b
b
a
b
ab
a
b
a
b
a
abaa
aba
a
b
b
a
a
a
b
a
a
a
a
a
a
a
b
a
Note: there's already a path for
suffix "a", so we don't change it (we
just add a suffix link to it)
b
a
b
abaaba$
abaaba
a
b
a
b
a
b
abaab
$
a
b
b
$
a
a
a
$
a
a
b
a
b
a
b
a
a
b
b
a
b
a
$
a
b
a
a
$
a
b
a
$
$
How many nodes can a suffix trie
have?
a
s = aaabbb
a
b
• 1 root node
• n nodes in a path of “b”s
• n paths of n+1 “b” nodes
b
a
b
b
• s = anbn will have
b
b
b
b
• Total = n(n+1)+n+1 = O(n2)
nodes.
• This is not very efficient.
b
b
b
b
• How could you make it
smaller?
So... we have to “trie” again...
SpaceEfficient Suffix Trees
A More Compact Representation
s = abaaba$
1234567
$
a
s = abaaba$
1234567
7:7
6:6
ba
5:6
$
7:7
ba
5:6
aba$
4:7
aba$
aba$
4:7
$
4:7
$
• Compress paths where
there are no choices.
7:7
7:7
• Represent sequence
along the path using a
range [i,j] that refers to
the input string s.
Space usage:
• In the compressed representation:

# leaves = O(n) [one leaf for each position in the string]
Every internal node is at least a binary split.
Each edge uses O(1) space.
• Therefore, # number of internal nodes is about equal
to the number of leaves.
• And # of edges ≈ number of leaves, and space per
edge is O(1).
• Hence, linear space.
Constructing Suffix Trees Ukkonen’s Algorithm
• The same idea as with the suffix trie
algorithm.
• Main difference: not every trie node is
s = abab
u
explicitly represented in the tree.
bab
• Solution: represent trie nodes as pairs (u,
abab
α), where u is a real node in the tree and
α is some string leaving it.
v
suffix_link[v] = (u, ab)
• Some additional tricks to get to O(n)
time.
Storing more than one string with
Generalized Suffix Trees
Constructing Generalized Suffix
Trees
Goal. Represent a set of strings P = {s1, s2, s3, ..., sm}.
Example. att, tag, gat
Simple solution:
(1) build suffix tree for string aat#1tag#2gat#3
#1tag#2gat#3
#3
g
#2gat#3
a
t
#2gat#3
#3
#3
at#3
ag#2gat#3
#1tag#2gat#3
g#2gat#3
t
#1tag#2gat#3
at#1tag#2gat#3
#3
Constructing Generalized Suffix
Trees
Goal. Represent a set of strings P = {s1, s2, s3, ..., sm}.
Example. att, tag, gat
Simple solution:
(1) build suffix tree for string aat#1tag#2gat#3
(2) For every leaf node, remove
any text after the first # symbol.
#1tag#2gat#3
#3
g
#3
#2gat#3
#2
g
a
t
#1
a
t
#2gat#3
#3
#3
at#3
ag#2gat#3
#1tag#2gat#3
#2
#3
#3
at#3
ag#2
g#2gat#3
t
#1tag#2gat#3
#1
g#2
t
at#1tag#2gat#3
at#1
#3
#1
#3
Applications of Generalized Suffix
Trees
Longest common substring of S and T:
Determine the strings in a database {S1, S2, S3, ..., Sm} that contain
query string q:
Applications of Generalized Suffix
Trees
Longest common substring of S and T:
Build generalized suffix tree for {S, T}
Find the deepest node that has has descendants from both
strings (containing both #1 and #2)
Determine the strings in a database {S1, S2, S3, ..., Sm} that contain
query string q:
Applications of Generalized Suffix
Trees
Longest common substring of S and T:
Build generalized suffix tree for {S, T}
Find the deepest node that has has descendants from both
strings (containing both #1 and #2)
Determine the strings in a database {S1, S2, S3, ..., Sm} that contain
query string q:
Build generalized suffix tree for {S1, S2, S3, ..., Sm}
Follow the path for q in the suffix tree.
Suppose you end at node u: traverse the tree below u, and
output i if you find a string containing #i.
Longest Common Extension
Longest common extension: We are given strings S and T. In the future, many pairs (i,j) will be
provided as queries, and we want to quickly find:
the longest substring of S starting at i that matches a substring of T starting at j.
S
LCE(i,j)
LCE(i,j)
T
i
j
Build generalized suffix tree for S and T.
Preprocess tree so that lowest common
ancestors (LCA) can be found in constant time.
LCA(i,j)
Create an array mapping suffix numbers to leaf
nodes.
Given query (i,j):
Find the leaf nodes for i and j
Return string of LCA for i and j
j
i
i
j
Longest Common Extension
Longest common extension: We are given strings S and T. In the future, many pairs (i,j) will be
provided as queries, and we want to quickly find:
the longest substring of S starting at i that matches a substring of T starting at j.
S
LCE(i,j)
LCE(i,j)
T
i
j
Build generalized suffix tree for S and T.
O(S + T)
Preprocess tree so that lowest common O(S + T)
ancestors (LCA) can be found in constant time.
LCA(i,j)
Create an array mapping suffix numbers to leaf
O(S + T)
nodes.
Given query (i,j):
Find the leaf nodes for i and j
Return string of LCA for i and j
O(1)
O(1)
j
i
i
j
Using LCE to Find Palindromes
Maximal even palindrome at position i: the longest string to the left and right so that the left
half is equal to the reverse of the right half.
x≠y
y
x
S
i
= the reverse of
Goal: find all maximal palindromes in S.
Sr
Construct Sr, the reverse of S.
y
x
x≠y
ni
Preprocess S and Sr so that LCE queries can be solved in constant time (previous slide).
LCE(i, ni) is the length of the longest palindrome centered at i.
For every position i:
Compute LCE(i, ni)
Using LCE to Find Palindromes
Maximal even palindrome at position i: the longest string to the left and right so that the left
half is equal to the reverse of the right half.
x≠y
y
x
S
i
= the reverse of
Goal: find all maximal palindromes in S.
y
Sr
Construct Sr, the reverse of S. O(S)
x
x≠y
ni
Preprocess S and Sr so that LCE queries can be solved in constant time (previous slide). O(S)
LCE(i, ni) is the length of the longest palindrome centered at i.
For every position i:
Compute LCE(i, ni)
O(S)
O(1)
Total time = O(S)
Recap
• Suffix tries natural way to store a string  search, count
occurrences, and many other queries answerable easily.
• But they are not space efficient: O(n2) space.
• Suffix trees are space optimal: O(n), but require a little more
subtle algorithm to construct.
• Suffix trees can be constructed in O(n) time using Ukkonen’s
algorithm.
• Similar ideas can be used to store sets of strings.