1. Speaker: C. C. Lin Adviser: R. C. T. Lee
String Matching with k Mismatches by Using Kangaroo Method Efficient string with k mismatches, Landau, G.M., and Vishkin, U., Theoret. Comput Sci 43, 1986, pp
Speaker: C. C. Lin Adviser: R. C. T. Lee

2. Input: A text T with length n , a pattern P with
Problem definition:
Input: A text T with length n , a pattern P with
length m and a mismatching threshold k.
Output: All sub-strings of T with length m
matching P with k maximal number of
mismatches.
If k = 2 k: 1 4 3 2
T = A G C T G C D C A C G I A B...
P = A G C C
P = A G C C
P = A G C C
P = A G C C

3. The concept of the Kangaroo method can be
explained as the following figure.
Assume that it is known before hand there
t1t2…ta=p1p2…pa and ta+1 is not equal to pa+1.
Thus we do not have to examine t1t2…ta+1 with
p1p2…pa+1 and jump directly to match the suffixes
beginning from ta+2 and pa+2.
Text: t1 t2… ta ta+1 ta+2 ta+3…tk…………
Pattern: p1p2…pa pa+1 pa+2pa+3...pk…………
mismatch

4. Kangaroo method will process as follows.
P = ETBDBCCDFDC
T = ABCCABDADBDETADBAADFDAAEERDXTDADCT…
start

5. Kangaroo method will process as follows.
P = ETBDBCCDFDC
T = ABCCABDADBDETADBAADFDAAEERDXTDADCT…

6. Kangaroo method will process as follows.
P = ETBDBCCDFDC
T = ABCCABDADBDETADBAADFDAAEERDXTDADCT…

7. Kangaroo method will process as follows.
P = ETBDBCCDFDC
T = ABCCABDADBDETADBAADFDAAEERDXTDADCT…

8. Kangaroo method will process as follows.
P = ETBDBCCDFDC
T = ABCCABDADBDETADBAADFDAAEERDXTDADCT…

9. We continue the above process. Whenever we
come to the situation that it is known a
substring of T exactly matching with a substring
of P, we skip this substring. This process is
stopped when k+1 mismatches have been found.
Input: T=ABAABBCCDD, P=ACDCB and k=2.
T=ABAABCCDD
P=ACDCB
k=3, we stop and discard ABAAB, then we start to
compare “BAADB” and “ACDCB”.

10. Before we introduce the Kangaroo algorithm,
we shall first introduce the suffix tree and the
lowest common ancestor of two nodes.
The properties of suffix tree and the lowest
common ancestor of two nodes will be used in
Kangaroo algorithm.

11. S = ABCDEADDBE
Suffix tree of a string with length n can be constructed in O(n).
Weiner, 1973
McCreight, 1976
Ukkonen, 1995

12. The lowest common ancestor of two leaf nodes can be found in O(1) by O(n) preprocessing in constructing time.
Harel and Tarjan, 1984

13. The Kangaroo method constructs a suffix tree
for text T and pattern P. Let the leaf node
corresponding to the substring starting from the
location be denoted as X. Let the leaf
corresponding to the pattern be denoted as Y.
The Kangaroo Method finds the lowest common
ancestor of X and Y to verify a text location with
k mismatches in O(k).
Let us consider the next page to figure out the
Kangaroo method.

14. Two suffix strings: ANBECF$ ANCEC$
Then we can know that they have the same prefix “AN” and a mismatch “B” and “C”.
ANBECF$
ANCEC$
ANBECF$
ANCEC$
ANBECF$
ANCEC$
We now have to find whether there is any mismatches between ECF and EC.
mismatches=1

15. We get remaining suffix strings: ECF$ EC$
Then we can know that they have the same prefix “EC” and because we touch $, we finish the verification.
ECF$
EC$
Thus we could know that
the mismatches between
“ANBECF” and “ANCEC”
is 1.
ECF$
EC$
mismatches=1

16. We will not have to compare all characters by
using the finding of the lowest common ancestor of
two strings of text and pattern in the suffix tree.
This is useful if there are many equivalent
characters between the text and the pattern because
we will not have to compare those equivalent
characters.
Finding the lowest common ancestor between two
suffixes is to find the next mismatch between two
strings.

17. Input: T=ABCCBDCDBC, P=ABCD and k=2
The suffix tree of T and P is:

18. The lowest common ancestor of “ABCD” and
“ABCCBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=1, return “ABCC”.

19. The lowest common ancestor of “ABCD” and
“BCCBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=1.

20. The lowest common ancestor of “BCD” and
“CCBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=2.

21. The lowest common ancestor of “CD” and
“CBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=3, discard “BCCB”.

22. The lowest common ancestor of “ABCD” and
“CCBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=1.

23. The lowest common ancestor of “BCD” and
“CBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=2.

24. The lowest common ancestor of “CD” and
“BDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=3, discard “CCBD”.

25. The lowest common ancestor of “ABCD” and
“CBDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=1.

26. The lowest common ancestor of “BCD” and
“BDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=2.

27. The lowest common ancestor of “D” and
“CDBC”.
T=ABCCBDCDBC
P=ABCD
k=3, discard “CBDC”.

28. The lowest common ancestor of “ABCD” and
“BDCDBC”.
T=ABCCBDCDBC
P=ABCD
k=1.

29. The lowest common ancestor of “BCD” and
“DCDBC”.
T=ABCCBDCDBC
P=ABCD
k=2.

30. The lowest common ancestor of “CD” and
“CDBC”.
T=ABCCBDCDBC
P=ABCD
k=2, return “BDCD”.

31. The lowest common ancestor of “ABCD” and
“DCDBC”.
T=ABCCBDCDBC
P=ABCD
k=1.

32. The lowest common ancestor of “BCD” and
“CDBC”.
T=ABCCBDCDBC
P=ABCD
k=2.

33. The lowest common ancestor of “CD” and
“DBC”.
T=ABCCBDCDBC
P=ABCD
k=3, discard “DCDB”.

34. The lowest common ancestor of “ABCD” and
“CDBC”.
T=ABCCBDCDBC
P=ABCD
k=1.

35. The lowest common ancestor of “BCD” and
“DBC”.
T=ABCCBDCDBC
P=ABCD
k=2.

36. The lowest common ancestor of “CD” and
“BC”.
T=ABCCBDCDBC
P=ABCD
k=3, discard “CDBC”.

37. Input: T=ABCCBDCDBC, P=ABCD and k=2.
Output: “ABCC” and “BDCD”.

38. In order to use Kangaroo method, we construct
a suffix tree for the text T with the length n and
the pattern p with the length m in O(n+m).
By using Kangaroo method, we take O(1) time
to find one mismatch. We stop when there are
more than k mismatches. Therefore, we take
O(k) time to find at most k mismatches.

39. Thus, the time complexity of finding out all
locations of text T with k maximal mismatches
with the pattern P is O(nk).

40. References
For Construction of Suffix trees:
[M76] McCreight, E.M., A Space-Economical
Suffix Tree Construction Algorithm, J. ACM 23
(1976):
[U95] Ukkonen, E., On-line Construction of
Suffix Trees, Algorithmica 41 (1995):
For Finding Lowest Common Ancestor:
[HT84] Harel, D. and Tarjan, R.E., Fast
Algorithms for Finding Nearest Common Ancestor,
SIAM Journal on Computing 13 (1984):

41. References
For String Matching with k Mismatches:
[LV86] Landau, G.M., and Vishkin, U., Efficient
string with k mismatches, Theoret. Comput Sci 43
(1986):

42. Thank you