Download presentation

Presentation is loading. Please wait.

Published bySavion Parcel Modified over 3 years ago

1
Approximate On-line Palindrome Recognition, and Applications Amihood Amir Benny Porat

2
Moskva River

3
Confluence of 4 Streams Palindrome Recognition Approximate Matching Interchange Matching Online Algorithms CPM 2014

4
Palindrome Recognition - Voz'mi-ka slovo ropot, - govoril Cincinnatu ego shurin, ostriak, -- I prochti obratno. A? Smeshno poluchaetsia? Vladimir Nabokov, Invitation to a Beheading (1) "Take the word ropot [murmur]," Cincinnatus' brother-in-law, the wit, was saying to him, "and read it backwards. Eh? Comes out funny, doesn't it?" [--› topor: the axe] A palindrome is a string that is the same whether read from right to left or from left to right: Examples: доход доход A man, a plan, a cat, a ham, a yak, a yam, a hat, a canal-Panama!

5
Palindrome Example Ibn Ezra: Medieval Jewish philosopher, poet, Biblical commentator, and mathematician. Was asked: " אבי אל חי שמך למה מלך משיח לא יבא " [ My Father, the Living God, why does the king messiah not arrive?] His response: "דעו מאביכם כי לא בוש אבוש, שוב אשוב אליכם כי בא מועד" [ Know you from your Father that I will not be delayed. I will return to you when the time will come ]

6
Palindromes in Computer Science Great programming exercise in CS 101. Example of a problem that can be solved by a RAM in linear time, but not by a 1-tape Turing machine. (Can be done in linear time by a 2-tape TM)

7
Palindrome Concatenation We may be interested in finding out whether a string is a concatenation of palindromes of length > 1. Example: ABCCBABBCCBCAACB Why would we be interested in such a funny problem? – we’ll soon see Exercise: Do this in linear time… ABCCBABBCCBCAACB

8
Stream 2 - Approximations As in exact matching, there may be errors. Find the minimum number of errors that, if fixed, will give a string that is a concatenation of palindromes of length > 1 Example: ABCCBCBBCCBCABCB For Hamming distance: A-Porat [ISAAC 13]: Algorithm of time O(n 2 ) ABCCBABBCCBCAACB

9
Stream 3 - Reversals Why is this funny problem interesting? Sorting by reversals: In the evolutionary process a substring may “detach” and “reconnect” in reverse: ABCABCDAABCBADCBAADCB ABCABCDAABCBAD

10
Sorting by Reversals What is the minimum number of reversals that, when applied to string A, result in string B? History: Introduced: Bafna & Pevzner [95] NP-hard: Carpara [97] Approximations: Christie [98] Berman, Hannenhalli, Karpinski [02] Hartman [03]

11
Sorting by Reversals – Polynomial time Relaxations Signed reversals: Hannenhalli & Pevzner [99] Kaplan, Shamir, Tarjan [00] Tannier & Sagot [04]... Disjointness: Swap Matching Muthu [96] Two constraints: 1. The length of the reversed substring is limited to 2. 2. All swaps are disjoint.

12
Reversal Distance (RD): – The RD between s 1 and s 2 is the minimum number k, such that there exist s 2 ’, where HAM(s 1,s 2 ’) =k, and s 1 reversal match s 2. ABDEABCDA ECBABAADA S1:S1: S2:S2: RD(S 1,S 2 ) = 2 Pattern Matching with Disjoint Reversals

13
Interleave Strings: ABDEABCDA EDBABAADC S1:S1: S2:S2: Connection between Reversal Matching and Palindrome Matching A C D D C A B A A B E A D B B D A E

14
On-line Input Suppose that we get the input a byte at a time: For the palindrome problem: ACDACABBAAEBB A A A EADDD

15
On-line Input Suppose that we get the input a byte at a time: For the reversal problem: ACAC CACA BABA ABAB EAEA BDBD A A A AEAED DBDB

16
Main Idea – Palindrome Fingerprint s 0,s 1,s 2,…s m-1 Φ R (S)=r -1 s 0 + r -2 s 1 +… r -m s m-1 mod (p) Φ (S)=r 1 s 0 + r 2 s 1 +… r m s m-1 mod (p) The Rabin Karp Fingerprint If r m+1 Φ R (S) = Φ(S) => S is a palindrome. w.h.p. The Reversal Fingerprint

17
Palindrome Fingerprint If r m+1 Φ R (S) = Φ(S) => S is a palindrome. Example: S = A B C B A r 6 Φ R (S)= r 6 (1/r A + 1/r 2 B + 1/r 3 C + 1/r 4 B + 1/r 5 A) = r 5 A + r 4 B + r 3 C + r 2 B + r A = Φ(S) Φ R (S)=r -1 s 0 + r -2 s 1 +… r -m s m-1 mod (p) Φ (S)=r 1 s 0 + r 2 s 1 +… r m s m-1 mod (p)

18
Simple Online Algorithm for Finding a Palindrome in a Text t 1,t 2,t 3, … t i,t i+1,t i+2, … t i+m, t i+m+1, … t n Φ R =r -1 t i + r -2 t i+1 +… r -m t i+m mod (p) Φ =r 1 t i + r 2 t i+1 +… r m t i+m mod (p) If not, then for the next position: If r m+1 Φ R = Φ = > there is a palindrome starting in the i-th position. Φ = Φ + r m+1 t i+m+1 mod (p) Φ R = Φ R + r -(m+1) t i+m+1 mod (p) Note: This algorithm finds online whether the prefix of a text is a permutation. For finding online whether the text is a concatenation of permutations, assume even-length permutations, otherwise, every text is a concatenation of length-1 permutations.

19
Palindrome with mismatches Start with 1 mismatch case.

20
1-Mismatch s 0,s 1,s 2, … s m-1 S= Choose l prime numbers q 1,…,q l < m such that

21
1-Mismatch s 0,s 1,s 2, … s m-1 s 0,s 2,s 4 … s 1 s 3,s 5 … s 0,s 3,s 6 … s 1,s 4,s 7 … s 2,s 5,s 8 … mod 2 mod 3 S= S 2,0 = S 2,1 = S 3,0 = S 3,1 = S 3,2 = For each q i construct q i subsequences of S as follows: subsequence S q i,j is all elements of S whose index is j mod q i. Examples: q 1 =2, q 2 =3

22
Example s 0,s 1,s 2, s 3,s 4,s 5 s 0,s 2,s 4 s 1 s 3,s 5 s 0,s 3 s 1,s 4 s 2,s 5 mod 2 mod 3 S= S 2,0 = S 2,1 = S 3,0 = S 3,1 = S 3,2 =

23
1-Mismatch We need to compare: We prove that in the partitions strings: s 0, s 1, s 2, … s m-2,s m-1 s m-1, s m-2, s m-3 … s 1, s 0 S q,j = S R q,(m-1-j)mod q

24
Example s 0,s 1,s 2,s 3,s 4,s 5 s 0,s 2,s 4 s 0,s 3 s 1,s 4 S= s 5,s 4,s 3,s 2,s 1,s 0 SR=SR= s 5 s 3,s 1 s 5,s 2 s 4,s 1 s 0,s 2,s 4 s 1 s 3,s 5 s 0,s 3 s 1,s 4 s 2,s 5 S 2,0 = S 2,1 = S 3,0 = S 3,1 = S 3,2 = S 2,0 = S R 2,1 = S 3,0 = S R 3,2 = S 3,1 = S R 3,1 =

25
Exact Matching Lemma: S=S R S q,j = S R q,(m-1-j) mod q for all q and all 0 ≤ j ≤ q.

26
1-Mismatch Lemma: S is a palindrome with 1-mismatch for each q, there is exactly one j such that: Φ (S q,j ) ≠ r |Sq,j| Φ R (S R q,(m-1-j)mod q )

27
1-Mismatch Lemma: There is exactly one mismatch There is exactly one subpattern in each group that does not match. C.R.T

28
Chinese Remainder Theorem Let n and m two positive integers. In our case: if two different indices, i and j, have an error, and only one subsequence is erroneous, since the product of all q’s > m, it means that i=j.

29
Complexity There exists a constant c such that, for any x

30
Complexity For each q i we compute 2q i different fingerprints: Overall space: Each character participates in exactly two fingerprints (the regular and the reverse). Overall time:

31
Online All fingerprint calculations can be done online We know the m at every input character, to compute the comparisons. Conclude: Our algorithm is online.

32
k-Mismatches Use Group testing…

33
k-Mismatches Group Testing Given n items with some positive ones, identify all positive ones by a small number of tests. Each test is on a subset of items. Test outcome is positive iff there is a positive item in the subset.

34
k-Mismatch Group: partition of the text. Test: distinguish between: (using the 1-mismatch algorithm) – match – 1-mismatch – more then 1-mismatch

35
k-Mismatches s 0,s 1,s 2, … s m-1 s 0,s 2,s 4 … s 1 s 3,s 5 … s 0,s 3,s 6 … s 1,s 4,s 7 … s 2,s 5,s 8 … mod 2 mod 3 S= S 2,0 = S 2,1 = S 3,0 = S 3,1 = S 3,2 = Similar to the 1-mismatch algorithm just with more prime numbers… Each S q,j is a group in our group testing

36
Our tests We define The reversal pair of S q,j to be S R q,(m-1-j)mod q Each partition is “tested against” its reversal pair.

37
Correctness s 0,s 1,s 2, … s j …. s m-1 For any group of k character i 1,i 2,..i k There exists a partition where s j appears alone i2i2 i5i5 i7i7 i9i9 i C.R.T

38
Correctness s 0,s 1,s 2, … s j …. s m-1 If s j invokes a mismatch we will catch it. i2i2 i5i5 i7i7 i9i9 i

39
Complexity Overall space: Overall time:

40
Approximate Reversal Distance Using the palindrome up to k-mismatches algorithm, can be solved in time, and space.

41
спасибо

Similar presentations

OK

Incremental Linear Programming Linear programming involves finding a solution to the constraints, one that maximizes the given linear function of variables.

Incremental Linear Programming Linear programming involves finding a solution to the constraints, one that maximizes the given linear function of variables.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on wireless mesh networks Ppt on fiscal policy 2012 Ppt on credit policy of a bank Ppt on electricity for class 10th date Ppt on printed circuit board Ppt on intelligent manufacturing system Ppt on standing order act packets Ppt on schottky diode application Ppt on solar eclipse and lunar eclipse Heat energy for kids ppt on batteries