Longest common subsequence

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Longest common subsequence
INPUT two strings OUTPUT longest common
subsequence
ACTGAACTCTGTGCACT TGACTCAGCACAAAAAC
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Longest common subsequence
INPUT two strings OUTPUT longest common
subsequence
ACTGAACTCTGTGCACT TGACTCAGCACAAAAAC
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Longest common subsequence
If the sequences end with the same symbol s, then
LCS ends with s.
s
s
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Longest common subsequence
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
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Longest common subsequence
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
if xi yj then
LCS(i,j)
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Longest common subsequence
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
if xi yj then
LCS(i,j) 1 LCS(i-1,j-1)
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Longest common subsequence
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
if xi ? yj then
LCS(i,j) max (LCS(i-1,j),LCS(i,j-1))
xi and yj cannot both be in LCS
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Longest common subsequence
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
if xi yj then
LCS(i,j) 1 LCS(i-1,j-1)
if xi ? yj then
LCS(i,j) max (LCS(i-1,j),LCS(i,j-1))
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Longest common subsequence
Running time ?
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
if xi yj then
LCS(i,j) 1 LCS(i-1,j-1)
if xi ? yj then
LCS(i,j) max (LCS(i-1,j),LCS(i,j-1))
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Longest common subsequence
Running time O(mn)
Sequences x1,,xn, and y1,,ym
LCS(i,j) length of a longest common
subsequence of x1,,xi and y1,,yj
if xi yj then
LCS(i,j) 1 LCS(i-1,j-1)
if xi ? yj then
LCS(i,j) max (LCS(i-1,j),LCS(i,j-1))
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Optimal matrix multiplication
c
b
b

a
A a x b matrix B b x c matrix How many
operations to compute AB ?
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Optimal matrix multiplication
c
b
b

a
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Optimal matrix multiplication
c
b
b

a
each entry of A B takes ?(b) time
need to compute ac entries ? ?(abc) time

total
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Optimal matrix multiplication
N x N matrix
N x N matrix
B
A
Compute AB, time ?
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Optimal matrix multiplication
N x N matrix
N x N matrix
B
A
Compute AB, time ?(N3)
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Optimal matrix multiplication
N x N matrix
N x N matrix
N x 1 matrix
B
C
A
Compute AB, time ?(N3) Compute ABC, time ?
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Optimal matrix multiplication
N x N matrix
N x N matrix
N x 1 matrix
B
C
A
Compute DBC, time ? Compute AD, time
? Compute ABC, time ?
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Optimal matrix multiplication
N x N matrix
N x N matrix
N x 1 matrix
B
C
A
Compute DBC, time ?(N2) Compute AD, time
? Compute ABC, time ?
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Optimal matrix multiplication
N x N matrix
N x 1 matrix
D
A
Compute DBC, time ?(N2) Compute AD, time
? Compute ABC, time ?
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Optimal matrix multiplication
N x N matrix
N x 1 matrix
D
A
Compute DBC, time ?(N2) Compute AD, time
?(N2) Compute ABC, time ?(N2)
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Optimal matrix multiplication
N x N matrix
N x N matrix
N x 1 matrix
B
C
A
(AB)C ABC A(BC)
The order of evaluation does not
change the result can change the
amount of work needed
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Optimal matrix multiplication
a1,a2,a3,.,an n-1 matrices of sizes a1 x a2
B1 a2 x a3
B2 a3 x a4 B3 .
an-1 x an Bn-1
What order should we multiply them in?
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Optimal matrix multiplication
B1 B2 B3 B4 Bn-1
B1 (B2 B3 B4 Bn-1 ) (B1 B2) (B3 B4 Bn-1 ) (B1
B2 B3) (B4 Bn-1 ) (B1 B2 B3 B4 ) Bn-1
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Optimal matrix multiplication
B1 B2 B3 B4 Bn-1
Ki,j the minimal number of operations
needed to multiply Bi Bj
Bi (Bi1 Bi2 Bj ) (Bi Bi1) (Bi2 Bj ) (Bi
Bi1 Bi2) ( Bj ) (B1 B2
B3 ) Bj
Ki,i Ki1,j aiai1aj1
Ki,i1 Ki2,j aiai2aj1
Ki,i2 Ki3,j aiai3aj1
Ki,j-1 Kj,j aiajaj1
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Optimal matrix multiplication
B1 B2 B3 B4 Bn-1
Ki,j the minimal number of operations
needed to multiply Bi Bj
Ki,j min Ki,w Kw1,j ai aw1
aj
Ki,i0
i?wlt j
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Travelling Salesman Problem
INPUT N cities, NxN symmetric matrix D,
D(i,j) distance between city i and j OUTPUT
the shortest tour visiting all the cities
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Travelling Salesman Problem
Algorithm 1 try all possibilities for each
permutation ? of 1,...,n visit the cities in
the order ?, compute distance travelled, pick
the best solution running time ?
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Travelling Salesman Problem
Algorithm 1 try all possibilities for each
permutation ? of 1,...,n visit the cities in
the order ?, compute distance travelled, pick
the best solution running time ? n!
is (n1)! O(n!) ?
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Travelling Salesman Problem
for each subset S of the cities with S ? 2 and
each u,v? S KS,u,v the length of the shortest
path that starts at u
ends at v visits all cities
in S
How large is K ?
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Travelling Salesman Problem
for each subset S of the cities with S ? 2 and
each u,v? S KS,u,v the length of the shortest
path that starts at u
ends at v visits all cities
in S
How large is K ?
? 2n n2
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Travelling Salesman Problem
KS,u,v
some vertex w ? S u,v must be visited first
d(u,w) we get to w KS-u,w,v
we need to get from w to v
and visit all vertices in S-u
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Travelling Salesman Problem
KS,u,v the length of the shortest path that
starts at u ends at
v visits all cities in S
if Su,v then KS,u,vd(u,v) if Sgt2
then KS,u,v min KS-u,w,v d(u,w)
w?S-u,v
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Travelling Salesman Problem
if Su,v then KS,u,vd(u,v) if Sgt2
then KS,u,v min KS-u,w,v d(u,w)
w?S-u,v
Running time ?
K ? 2n n2
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Travelling Salesman Problem
if Su,v then KS,u,vd(u,v) if Sgt2
then KS,u,v min KS-u,w,v d(u,w)
w?S-u,v
Running time O(n3 2n)
K ? 2n n2
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Travelling Salesman Problem
dynamic programming O(n3 2n) brute force O(n!)
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Longest increasing subsequence
INPUT numbers a1, a2, ... , an OUTPUT longest
increasing subsequence
1,9,2,4,7,5,6
1,9,2,4,7,5,6
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Longest increasing subsequence
INPUT numbers a1, a2, ... , an OUTPUT longest
increasing subsequence
reduce to a problem that we saw today
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Longest increasing subsequence
INPUT numbers a1, a2, ... , an OUTPUT longest
increasing subsequence
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Longest increasing subsequence
INPUT numbers a1, a2, ... , an OUTPUT longest
increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
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Longest increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
true/false Ki,j ? Ki,j1 ?
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Longest increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
K0,j ? for j ? 1 K0,0 ?
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Longest increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
K0,j ? for j ? 1 K0,0
- ?
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Longest increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
Ki,j ?
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Longest increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
Ki,j ai if ai lt
Ki-1,j
and
ai gt Ki-1,j-1

Ki,j Ki-1,j otherwise
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Longest increasing subsequence
K0..n,0..n
Ki,j the minimum last element of an
increasing seqence in a1, ... ,ai
of length j (if no sequence ? ?)
Ki,0 -? Ki,1 Ki,2 ... Ki,j
Ki,j1 ?
ai lt Ki-1,j and ai gt Ki-1,j-1

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Longest increasing subsequence
K0,0 -? K0,1 ? K0,2 ? K0,3
? K0,4 ? K0,5 ? K0,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K1,0 -? K1,1 1 K1,2 ? K1,3
? K1,4 ? K1,5 ? K1,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K1,0 -? K1,1 1 K1,2 ? K1,3
? K1,4 ? K1,5 ? K1,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K2,0 -? K2,1 1 K2,2 9 K2,3
? K2,4 ? K2,5 ? K2,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K2,0 -? K2,1 1 K2,2 9 K2,3
? K2,4 ? K2,5 ? K2,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K3,0 -? K3,1 1 K3,2 2 K3,3
? K3,4 ? K3,5 ? K3,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K3,0 -? K3,1 1 K3,2 2 K3,3
? K3,4 ? K3,5 ? K3,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K4,0 -? K4,1 1 K4,2 2 K4,3
4 K4,4 ? K4,5 ? K4,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K4,0 -? K4,1 1 K4,2 2 K4,3
4 K4,4 ? K4,5 ? K4,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K5,0 -? K5,1 1 K5,2 2 K5,3
4 K5,4 7 K5,5 ? K5,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K5,0 -? K5,1 1 K5,2 2 K5,3
4 K5,4 7 K5,5 ? K5,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K6,0 -? K6,1 1 K6,2 2 K6,3
4 K6,4 5 K6,5 ? K6,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K6,0 -? K6,1 1 K6,2 2 K6,3
4 K6,4 5 K6,5 ? K6,6 ?
1,9,2,4,7,5,6
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Longest increasing subsequence
K7,0 -? K7,1 1 K7,2 2 K7,3
4 K7,4 5 K7,5 6 K7,6 ?
answer 5
1,9,2,4,7,5,6