Efficient STR-EC-LCS Computation Algorithm

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"Explore a new algorithm for solving the substring-excluding constrained LCS problem efficiently, outperforming existing methods. Learn how to find the longest common subsequence of two strings while excluding a specified substring."

  • Algorithm
  • Substring
  • LCS
  • Computation
  • Efficiency
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  1. Faster STR-EC-LCS Computation Kohei Yamada ,Yuto Nakashima , Shunsuke Inenaga , Hideo Bannai Masayuki Takeda Japan Science and Technology Agency, Kawaguchi( ), Japan Presenter : En-An Sung Date : Jan 5, 2021

  2. Abstract(1/2) The longest common subsequence (LCS) problem is a central problem in stringology that finds the longest common subsequence of given two strings A and B. More recently, a set of four constrained LCS problems (called generalized constrained LCS problem) were proposed by Chen and Chao [J. Comb. Optim, 2011]. In this paper, we consider the substring-excluding constrained LCS (STR-ECLCS) problem. A string Z is said to be an STR-EC-LCS of two given strings A and B excluding P if, Z is one of the longest common subsequences of A and B that does not contain P as a substring.

  3. Abstract(2/2) Wang et al. proposed a dynamic programming solution which computes an STR-EC-LCS in O (mnr) time and space where m = |A|, n = |B|, r = |P| [Inf. Process. Lett., 2013]. In this paper, we show a new solution for the STR-EC-LCS problem. Our algorithm computes an STR-EC-LCS in O ( n| | + ( L + 1)( m L + 1) r) time where | | min {m, n} denotes the set of distinct characters occurring in both A and B, and L is the length of the STR-EC-LCS. This algorithm is faster than the O (mnr)-time algorithm for short/long STR-EC-LCS (namely, L O(1) or m L O(1)), and is at least as efficient as the O (mnr)-time algorithm for all cases.

  4. STR-EC-LCS X = abcabac Y = acbcaacbaa P = abc LCS(X,Y) = abcaba STR-EC-LCS(X,Y,P) = bcaac

  5. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7

  6. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 2 3 4 5 6 7

  7. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 2 3 3 4 5 6 7

  8. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 2 3 3 4 4 5 6 7

  9. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 2 3 3 4 4 5 5 8 6 9 7 *

  10. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 4 4 5 5 8 6 9 7 *

  11. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 3 4 4 5 5 8 6 9 7 *

  12. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 3 4 4 4 5 5 5 8 6 6 9 7 *

  13. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 3 4 4 4 5 5 5 8 6 6 9 7 7 *

  14. LCS by Nakatsu X[i] 0 1 2 3 4 5 6 7 X = abcabac Y = acbcaacbaa LCS length 0 0 0 0 0 0 0 0 0 1 1 1 2 3 3 LCS(X,Y) = abcaac 3 4 4 4 5 5 5 8 6 6 9 7 7 *

  15. STR-EC-LCS X = abcabac Y = acbcaacbaa P = abc STR-EC-LCS(X,Y,P) = bcaac

  16. STR-EC-LCS P = abc k = 0 : suffix = (ex : bcababb) k = 1 : suffix = a (ex : bcaba) k = 2 : suffix = ab (ex : bcabab) b b bcaba(k=1,length=5) bcabab (k=2,length=6) bcababb(k=0,length=7)

  17. STR-EC-LCS X = abcabac Y = acbcaacbaa 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 2 3 4 5 6 7 suffix = a suffix = ab suffix =

  18. STR-EC-LCS X = abcabac Y = acbcaacbaa + a = a 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 2 3 4 5 6 7 suffix = a suffix = ab suffix =

  19. STR-EC-LCS X = abcabac Y = acbcaacbaa + a = a 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * 2 3 4 5 6 7 suffix = a suffix = ab suffix =

  20. STR-EC-LCS X = abcabac Y = acbcaacbaa a + b = ab 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * 2 3 4 5 6 7 suffix = a suffix = ab suffix =

  21. STR-EC-LCS X = abcabac Y = acbcaacbaa a + b = ab 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 2 * 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 2 * 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * 2 3 3 4 5 6 7 suffix = a suffix = ab suffix =

  22. STR-EC-LCS X = abcabac Y = acbcaacbaa ab + c = abc 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 2 * 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 2 * 3 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * 2 3 3 4 5 6 7 suffix = a suffix = ab suffix =

  23. STR-EC-LCS X = abcabac Y = acbcaacbaa ab + c = abc 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * 2 3 3 * 4 5 6 7 suffix = a suffix = ab suffix =

  24. STR-EC-LCS X = abcabac Y = acbcaacbaa + b = b 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * 2 3 3 * 4 5 6 7 suffix = a suffix = ab suffix =

  25. STR-EC-LCS X = abcabac Y = acbcaacbaa + b = b 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 1 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 * 4 5 6 7 suffix = a suffix = ab suffix =

  26. STR-EC-LCS X = abcabac Y = acbcaacbaa + c = c 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 * 4 5 6 7 suffix = a suffix = ab suffix =

  27. STR-EC-LCS X = abcabac Y = acbcaacbaa + c = c 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * 4 5 6 7 suffix = a suffix = ab suffix =

  28. STR-EC-LCS X = abcabac Y = acbcaacbaa + a = a ab + a = aba 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * 4 5 6 7 suffix = a suffix = ab suffix =

  29. STR-EC-LCS X = abcabac Y = acbcaacbaa + a = a ab + a = aba 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 suffix = a suffix = ab suffix =

  30. STR-EC-LCS X = abcabac Y = acbcaacbaa a + b = ab 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 suffix = a suffix = ab suffix =

  31. STR-EC-LCS X = abcabac Y = acbcaacbaa a + b = ab 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 * * 8 suffix = a suffix = ab suffix =

  32. STR-EC-LCS X = abcabac Y = acbcaacbaa ab + a = aba 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 * * 8 suffix = a suffix = ab suffix =

  33. STR-EC-LCS X = abcabac Y = acbcaacbaa ab + a = aba 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 * * 8 * 9 * suffix = a suffix = ab suffix =

  34. STR-EC-LCS X = abcabac Y = acbcaacbaa 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 * * 8 9 * * * * * suffix = a suffix = ab suffix =

  35. STR-EC-LCS X = abcabac Y = acbcaacbaa 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 * 3 2 * 4 3 * * 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 1 * 2 * * 3 * 5 4 5 6 7 0 1 2 3 4 5 6 7 0 * * * * * * * * 1 * * 2 3 3 3 * * 4 5 6 7 * * 8 9 * * * * * suffix = a suffix = ab suffix =

  36. Time complexity O((L + 1)*(m L + 1)*r) r : P s length

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