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Can dist tables be merged in linear time? an open problem

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1 Can dist tables be merged in linear time? an open problem
4/22/2019 Can dist tables be merged in linear time? an open problem Gad M. Landau Prague 2006 4/22/2019

2 Prague-Landau Yechezkel Landau — The Nodah Biyehudah (1713-1793 )
Rabbi Yechezkel Landau was born in 1713 in Opatow, Poland, and received his Rabbinical education in Galicia. From 1746 a Rabbi in Yampole on the upper Dniester, he was appointed Chief Rabbi of Prague and head of the Yeshiva (Talmudic school) in 1755. He is best known to us today through his monumental work of she'alot uteshuvot entitled Noda Biyehudah, one of the most authoritative sources in Halacha. He was a devoted Rosh Yeshiva, a compassionate Rabbi, a strong community leader and a skillful negotiator with governments during the numerous wars and threatened expulsions that plagued the Jewish community. The Nodah Biyehudah is buried in the Fibichova cemetery, near the television tower in Prague 2. Prague 2006 4/22/2019

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17 Common Subgraph = max( + ) B C A D O LCS[T ,Y] ( T stands for T ..T )
3 2 1 4 x+1 7 I 5 6 8 9 O = max( + LCS[T ,Y] ) x x = 0 Common Subgraph j ( T stands for T ..T ) Prague 2006 4/22/2019

18 DIST DIST (K,L) = The best path from IK to OL IK OL Prague 2006
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19 DIST DIST (K,L) = Edit (ak … aL ; B) ak . . . . . . . . aL B
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58 To DIST 1 2 3 4 5 6 7 8 From Prague 2006 4/22/2019

59 OUT To From At most O(1) per column and O(1) per row 2 3 1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 From 2 3 At most O(1) per column and O(1) per row Prague 2006 4/22/2019

60 Dist and OUT Linear Representation
Total Monotonicity – Column maxima in linear time Computing O0 …. On in linear time (LZ01) Prague 2006 4/22/2019

61 LCS A B D C 1 2 3 E Prague 2006 4/22/2019

62 DIST To  1 2  3  4  5  6 1 3  2 From DIST(i,j)=j-i+1- #points (Tiskin CPM 2006) Prague 2006 4/22/2019

63 Computing DIST Jeanette P. Schmidt SIAM J. Comput. (1998) O(n2) time
Prague 2006 4/22/2019

64 What Has been Shown: What is DIST The use of DIST
The properties of DIST How to compute DIST Prague 2006 4/22/2019

65 NEXT Can DIST tables be merged in linear time? an open problem
Prague 2006 4/22/2019

66 A=a1 …. An B= Uk=xyzxyz….xyz (U=xyz) |U|=p ; |B|= m = kp LCS (A,B) –
Prague 2006 4/22/2019

67 A=a1 …. An B= Uk=xyzxyz….xyz (U=xyz) |U|=p ; |B|= m = kp LCS (A,B) –
O(nm) Hirschberg O(nm/log n) CLZ O(n(p+k)) LZ O(n L_1 + k L_2) LSZ+LMS (L_2 = LCS(A,B) << n and L_1 = LCS(A,U) << p.) O(n(p+m0.5) Tiskin GOAL: O(n(p+log k) ) Prague 2006 4/22/2019

68 On the Complexity of Sparse Exon Assembly
4/22/2019 CPM2005 On the Complexity of Sparse Exon Assembly Carmel Kent Gad M. Landau Michal Ziv-Ukelson Good morning. I’m going to talk On the complexity of sparse exon assembly. This is a joint work with Gadi Landau and Michal Ziv Ukelson. In this paper we are addressing the problem of Gene Prediction. Prague 2006 4/22/2019

69 [Gelfand et al-96’]: Gene Prediction As a Network Graph
DP T DP DP DP 1 8 4 T 6 DP T T 2 DP DP 9 T 5 T DP DP 3 7 Prague 2006 4/22/2019

70 Replacing DP tables DIST S DIST DIST DIST DIST DIST DIST DIST DIST O O
1 8 4 O T 6 I O I DP DIST T T 2 I I DP DIST DP DIST 9 T 5 T I DP DIST DP DIST 3 7 O O O O Prague 2006 4/22/2019

71 Strings A and B of length m, T of length n
Input: DISTAT of strings A and T DISTBT of strings B and T Output: DISTABT of strings AB and T GOAL: Find the n points of DISTABT in O(n) time Prague 2006 4/22/2019

72 DISTABT (Tiskin06) At most n points Prague 2006 4/22/2019

73 DISTABT (Tiskin06) Prague 2006 4/22/2019

74 DISTABT (Tiskin06) ? O(x) time Total O(n1.5) time X X Prague 2006
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75 DISTABT (Tiskin06) DIST(i,j)=j-i+1- # points ? Prague 2006 4/22/2019

76 DISTABT (Tiskin06) = MAX [DISTAT (i,k)+ DISTAB T (i,j)=j-i+1- # points
DISTBT (k,j)] DISTAB T (i,j)=j-i+1- # points T i A B j Prague 2006 4/22/2019

77 DISTABT (Tiskin06) = MAX [DISTAT (i,k)+ DISTAB T (i,j)=j-i+1- # points
DISTBT (k,j)] DISTAB T (i,j)=j-i+1- # points #pointsABT (I,j) = MIN [#pointsAT (I,k) + #pointsBT(k,j)] Prague 2006 4/22/2019

78 #pointsABT (i,j) = MIN [#pointsAT (i,k) +#pointsBT(k,j)]
DISTAT DISTBT x j k k i x Prague 2006 4/22/2019

79 References Agarwal, Klawe, Moran, Shor, and Wilbur, 1987
Raffaele Giancarlo in Pattern matching algorithms,  Apostolico, Galil, 1997 Jeanette P. Schmidt, 1998 Crochemore, Landau, Schieber and Ziv-Ukelson, 2004 and Landau, Myers and ZU 2004 Tiskin, CPM06 Prague 2006 4/22/2019

80 NEXT Can DIST tables be merged in linear time? an open problem
Prague 2006 4/22/2019

81 THANKS Prague 2006 4/22/2019

82 #pointsABT (I,j) = MIN [#pointsAT (I,k) +#pointsBT(k,j)]
X points ? Prague 2006 4/22/2019

83 #pointsABT (I,j) = MIN [#pointsAT (I,k) + #pointsBT(k,j)]
Prague 2006 4/22/2019

84 Authors: Maxime Crochemore Carmel Kent Gene Myers Baruch Schieber
Jeanette Schmidt Michal Ziv Ukelson Prague 2006 4/22/2019

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