Tries and Suffix Tries

8/10/2019 Tries and Suffix Tries

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Tries

A trie (pronounced try) is a tree representing a collection of strings withone node per common prefix

Each key is spelled out along some path starting at the root

Each edge is labeled with a character c!

A node has at most one outgoing edge labeled c, for c!

Smallest tree such that:

Natural way to represent either a setor a mapwhere keys are strings
http://xlinux.nist.gov/dads/HTML/prefix.htmlhttp://xlinux.nist.gov/dads/HTML/prefix.htmlhttp://xlinux.nist.gov/dads/HTML/node.htmlhttp://xlinux.nist.gov/dads/HTML/node.html


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Tries: example

Checking for presence of a key P,where n= | P |, is ??? time

If total length of all keys is N, trie

has ??? nodes

O(n)

O

(N

)

What about | | ?

Depends how we represent outgoing

edges. If we dont assume |

| is asmall constant, it shows up in one orboth bounds.

i

n

s

t

a

n

t

t

e

r

n

a

l

e

t1

2 3


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Tries: another example

We can index T with a trie. The trie mapssubstrings to offsets where they occur

ac 4

ag 8

at 14

cc 12

cc 2

ct 6

gt 18

gt 0

ta 10tt 16

ac

g

t

cg

t

c

t

t

a

t

4

8

14

12, 2

6

18, 0

10

16

root:


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Tries: implementation

!"#$$&'()*#+,-./)!012

333 &'() (4+")4)50#0(-5 -6 # 4#+7 8$$-!(#0(59 :);$ ,$0'(59$ -' -0) +#(' 333

!>' E$)"67'--0 6-'! (5:2 H 6-' )#!< !'L!M EFG H (6 5-0 0'L!M

!>'LN@#">)NM E@ H #0 0'2

')0>'5P-5)H :); ?#$5N0 (5 0'L!M

H 9)0 @#">)D -' P-5) (6 0'5!>'79)0,N@#">)N1

Python example:

http://nbviewer.ipython.org/6603619
http://nbviewer.ipython.org/6603619http://nbviewer.ipython.org/6603619http://nbviewer.ipython.org/6603619http://nbviewer.ipython.org/6603619


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Tries: alternatives

Tries arent the only tree structure that can encode sets or maps with string

keys. E.g. binary or ternary search trees.

i

Example from: Bentley, Jon L., and Robert Sedgewick. "Fast algorithms for sorting and searching

strings." Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms. Society for

Industrial and Applied Mathematics, 1997

b s o

a e h n t n t

Ternary search tree for as, at, be, by, he, in, is, it, of, on, or, to

s y e f r o

t


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Indexing with suffixes

Until now, our indexes have been based on extracting substrings from T

A very different approach is to extract suffixesfrom T. This will lead us tosome interesting and practical index data structures:

6

5

3

1

0

4

2

$

A$

ANA$

ANANA$

BANANA$

NA$

NANA$

$B A N A N A

A$ B A N A N

A N A $ B A N

A N A N A $B

BA N A N A$

N A $ B A N A

N A N A $ BA

Suffix Tree Suffix Array FM IndexSuffix Trie


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Suffix trie

Build a triecontaining all suffixesof a text T

G T T A T A G C T G A T C G C G G C G T A G C G G $G T T A T A G C T G A T C G C G G C G T A G C G G $ T T A T A G C T G A T C G C G G C G T A G C G G $ T A T A G C T G A T C G C G G C G T A G C G G $ A T A G C T G A T C G C G G C G T A G C G G $ T A G C T G A T C G C G G C G T A G C G G $ A G C T G A T C G C G G C G T A G C G G $ G C T G A T C G C G G C G T A G C G G $ C T G A T C G C G G C G T A G C G G $ T G A T C G C G G C G T A G C G G $ G A T C G C G G C G T A G C G G $ A T C G C G G C G T A G C G G $ T C G C G G C G T A G C G G $ C G C G G C G T A G C G G $ G C G G C G T A G C G G $ C G G C G T A G C G G $ G G C G T A G C G G $ G C G T A G C G G $ C G T A G C G G $ G T A G C G G $ T A G C G G $ A G C G G $ G C G G $ C G G $ G G $ G $ $

T:

m(m+1)/2

chars


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Suffix trie

Each path from root to leaf represents asuffix; each suffix is represented by somepath from root to leaf

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Shortest

(non-empty)suffix

Longest suffix

T: abaaba abaaba$T$:

Would this still be the case if we hadntadded $?


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Suffix trie

T: abaaba

Would this still be the case if we hadntadded $? No

! "

! "

"

!

!

!

"

!

!

!

"

!

Each path from root to leaf represents asuffix; each suffix is represented by somepath from root to leaf


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Suffix trie

We can think of nodes as having labels,where the label spells out characters on thepath from the root to the node

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

baa


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Suffix trie

How do we check whether a string Sis asubstring of T?

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Note: Each of Tssubstrings is spelled outalong a path from the root. I.e., everysubstringis aprefix of some suffixof T.

Start at the root and follow the edgeslabeled with the characters of S

If we fall off the trie -- i.e. there is no

outgoing edge for next character of S, thenSis not a substring of T

If we exhaust Swithout falling off, Sis asubstring of T

S = baaYes, its a substring


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Suffix trie

How do we check whether a string Sis asubstring of T?

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Note: Each of Tssubstrings is spelled outalong a path from the root. I.e., everysubstringis aprefix of some suffixof T.

Start at the root and follow the edgeslabeled with the characters of S

If we fall off the trie -- i.e. there is no

outgoing edge for next character of S, thenSis not a substring of T

If we exhaust Swithout falling off, Sis asubstring of T


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Suffix trie

How do we check whether a string Sis asuffixof T?

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Same procedure as for substring, butadditionally check whether the final node inthe walk has an outgoing edge labeled $

S = baaNota suffix


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Suffix trie

How do we check whether a string Sis asuffixof T?

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Same procedure as for substring, butadditionally check whether the final node inthe walk has an outgoing edge labeled $


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Suffix trie

How do we count the number of timesa string Soccurs as a substring of T?

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Follow path corresponding to S.Either we fall off, in which case

answer is 0, or we end up at node nand the answer = # of leaf nodes inthe subtree rooted at n.

Leaves can be counted with depth-firsttraversal.

n


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Suffix trie

How do we find the longest repeatedsubstringof T?

! " #

! " #

"

!

#

!

! #

"

!

#

!

! #

"

!

#

Find the deepest node with morethan one child


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Suffix trie: implementation!"#$$Q>66(R&'(),-./)!012

B)6CC(5(0CC,$)"6D 012

333 *#:) $>66(R 0'() 6'-4 0 333 0 SENTNH $+)!(#" 0)'4(5#0-' $;4.-" $)"67'--0 EFG 6-'( (5R'#59),")5,0112 H 6-' )#!< $>66(R !>' E$)"67'--0 6-'! (50L(2M2 H 6-' )#!< !66(R (6! 5-0(5!>'2 !>'L!M EFG H #BB ->09-(59 )B9) (6 5)!)$$#'; !>' E!>'L!M

B)66-""-?U#0'2 ')0>'5P-5) !>' E!>'L!M

')0>'5!>'

B)6.$0'(59,$)"6D $12 333 V)0>'5 0'>) (66 $ #++)#'$ #$ # $>.$0'(59 -6 0 333 ')0>'5$)"676-""-?U#0


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Suffix trie

How many nodes does the suffix trie have?

Is there a class of string where the numberof suffix trie nodes grows linearly with m?

Yes: e.g. a string of mas in a row (am)

! "

! "

! "

! "

"

T= aaaa

1 Root

mnodes withincoming aedge

m + 1 nodes withincoming $edge

2m+ 2 nodes


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Suffix trie

Is there a class of string where the numberof suffix trie nodes grows with m2?

Yes: anbn

1 root

nnodes along b chain, right nnodes along a chain, middle

nchains of nb nodes hanging offeacha chain node

2n + 1 $leaves (not shown)

n2+ 4n+ 2 nodes, where m= 2n

Figure & exampleby Carl Kingsford


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Suffix trie: upper bound on size

Suffix trie

Root

Deepest leaf

Max # nodes from top to bottom= length of longest suffix + 1= m+ 1

Max # nodes from left to right= max # distinct substrings of any length

m

O(m2) is worst case

Could worst-case # nodes be worse than O(m2)?


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Suffix trie: actual growth

0 100 200 300 400 500

0

50

000

100000

150000

200000

250000

Length prefix over which suffix trie was built

#s

uffix

trien

odes

m^2actual

mBuilt suffix tries for the first500 prefixes of the lambdaphage virus genome

Black curve shows how #nodes increases with prefixlength

Tries and Suffix Tries

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