This document presents a method for creating smaller, fully-functional bidirectional indexes that support efficient searching in both directions of a text. The method constructs an index structure called a maximal repeat graph that encodes the repeats and frequencies in a text. Operations like locating suffixes, computing suffix links and reverse intervals can be performed on this structure in linear or near-linear time and space. This allows supporting bidirectional search while using less space than a traditional suffix tree or Burrows-Wheeler transform.

1. Smaller fully-functional
bi irectional in e e
Djamal Belazzougui1
Fabio Cunial2,3
1 DTISI, CERIST, Algiers.
2 Max Plan Ins i u e or Mole ular Cell Biolog an ene i s, Dres en.
3 Cen er or S s ems Biolog Dres en

2. Constant-space descriptor of a string W:
Bidirectional index (synchrono s
pp ication: aria e-order de r n grap encoding a orders
and a fre encies si taneo s

3. Time PracticalSpace
[1]
[1] and [2]
[1] constant
wo ds
ts
wo ds
wo ds
wo ds
[1] Belazzougui, Cunial. Fully-functional bidirectional Burrows-Wheeler indexes and infinite-order de Bru n gra hs. C 2 1 .
[2] agie, a arro, rezza. Fully functional suffix trees and o ti al text searching in BW -runs bounded s ace. C 2 2 .
Ne
Ne
Bidirectional inde nc rono

4. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Maximal re ea
= 8

5. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Right-extensions of maximal re eats
≥
≥

6. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Depth of the m m epe t t ee
= 3

7. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
String depth of the maximal repeat tree
= 6

8. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Frontier m im re e t

9. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Rightmost m im ts

10. Backgrou

11. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Left-contraction from right-maximal

12. C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Suffix link

13. Suffix link
CG CGCG
G GG
C
C
C
C
C
C
C
C
C
(a) (b) ( )

14. Left-contraction from non-right-maximal
C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Belazzougui, Cunial. Fully-functional bidirectional Burrows-Wheeler indexes and infinite-order de Bruijn graphs. C .

15. Left-contraction from non-right-maximal
C
A
A•# #
#
#
#
#
C
C•
C•A
A
A A C•
A C
ST
A
A
• •
•
C•
•
Belazzougui, Cunial. Fully-functional bidirectional Burrows-Wheeler indexes and infinite-order de Bruijn graphs. C .

16. Left-contraction from non-right-maximal
Belazzougui, Cunial. Fully-functional bidirectional Burrows-Wheeler indexes and infinite-order de Bruijn graphs. CPM .
Suffix link
Checking if a no e i a axi al e ea
eng h of a axi al e ea
eigh e le el ance o ue ie
on axi al e ea u g a h
ee t

17. word

18. Pruning the ST topology
ST
...

19. Size of the pruned topology
Run-length encoded takes wo ds
he e a e at ost ed nodes
CCCCCCCCCCCCCCCCCCCCCCCAAAAAAAAAACCCAAAAA

20. Suffix link with the pruned topolo
CG CGCG
G GG
C
C
C
C
C
C
C
C
C
(a) (b) ( )
n h n ed

21. Suffix link with the pruned topolo
ST

22. Suffix link with the pruned topolo
ST

23. Suffix link with the pruned topolo
ST

24. Left-contraction from non-right-maximal
parent of a red node
Supported by
Suffix l n
e n f a node a ax al repeat
en t of a ax al repeat
e ted le el an e tor uer e
on ax al repeat ub rap
o e t ax al repeat an e tor
ee t

25. Computing the reverse interval
Reduces to implementing
on the reverse RLBWT (from a maximal repeat W),
and exploiting the isomorphism of subtrees of ST.
in RL T can be implemented via
from a maximal repeat can also be implemented
in RL T.
Nontrivial only when W is left maximal but not right maximal
by traversing the for r maximal repeat tree top down
time, words.

26. word

27. Compressing the pruned topology

28. CCCAAACCCCGTTTCAAAAACCCAAACCCCC
Size of the compressed topolog
might still contain unary paths of black nodes.
Every node in such a path can be charged to a
distinct run boundary.
The deepest nodes in (black) are rightmost
maximal repeats, so they can be charged to distinct
run boundaries.
Thus, there are black nodes with more than
one maximal repeat child.
Every blue node can be charged to its black
descendant.
Red nodes can still be charged to a constant
number of runs or of blue/black nodes.
Run-length encoded takes words

29. Suffix link with the comp e e topolo
After the LCA, we might end u in ue n de
ut we w nt the inter f the highe t n de
in ST with tring de th t e t
Let den te thi r em with tu e

30. words
In t s tr
ST

31. words
In t s tr
STST Weine in

32. words
In t s tr
STST Weine in

33. words
Hig s od d
ST

34. words
Hig s od d
STST Weine in

35. words
Hig s od d
STST Weine in

36. words
Weine in
Hig s od d
STST

37. words
Several other cases are possible
not left-maximal
is blue
is a subpath of a blue path
straddles multiple blue paths
straddles itself
It can be shown that all cases can be handled with
time
After at most H Weiner links, every maximal repeat loses
its right-maximality permanently.
So the length of the ST path between the first interval of an instance
and the solution interval becomes zero.

38. Suffix link with the comp e e topolo
After the LCA from interval , e mi ht en in a
re no e of that i the hil of a l e no e
t e ant the interval of the re no e of that
ontain , an of it l e arent
Can e olve ith a imilar re r ive ro e re

39. Computing the reverse interval
Nontrivial only when W is left- a i al t not ri ht- a i al
an the lo s of W is a l e no e of
i le sol tion iss e a nown n er of
eries on the reverse in e
ti e
till nee s on the reverse in e
fro a a i al re eat W.
ti e

40. Unidirectional operations
TimeSpace ordsor ard direction
(from maximal repeat)
(needs ID of longest
left-maximal suffix)
amortized
Application: variable-order de Bruijn graph
that supports just one direction

41. Smaller fully-functional
bi irectional in e e
Djamal Belazzougui1
Fabio Cunial2,3
1 DTISI, CERIST, Algiers.
2 Max Plan Ins i u e or Mole ular Cell Biolog an ene i s, Dres en.
3 Cen er or S s ems Biolog Dres en