Dynamic Programming (16.0/15)

1
Dynamic Programming (16.0/15)

• The 3-d Paradigm
• 1st Divide and Conquer
• 2nd Greedy Algorithm
• Dynamic Programming metatechnique (not a
  particular algorithm)
• Programming tableau method, not
  writing a code

2
Longest Common Subsequence (16.3/15.4)

• Problem Given x1..m and y1..n, find LCS
• x A B C B D A B
• 
  ? B C B A
• y B D C A B A
• Brute-force algorithm
• for every subsequence of x check if it is in y
• O(n2m ) time
• 2m subsequences of x (each element either in
  or out
• O(n) for scanning y with x-subsequence (m ? n)

3
Recurrent Formula for LCS (16.3/15.4)

• Let ci,j length of LCS of Xix1..i,Yj
  y1..j
• Then cm,n length of LCS of x and y
• Theorem
• 
  if xi yj
• 
  otherwise
• Proof. xi yj ?
• LCS(Xi,Yj) LCS(Xi-1,Yj-1) xi

4
DP properties(16.2/15.3)

• 1st Observation
• any part of the optimal answer is also
  optimal
• The subsequence of LCS(X,Y) is LCS for some
  subsequences of X and Y.
• 2nd Observation
• subproblems overlap
• LCS(Xm,Yn-1) and LCS(Xm-1,Yn) has common
  subproblem LCS(Xm-1,Yn-1)
• There are few problems in total mn for LCS
• Unlike divide and conquer

5
DP (16.2/15.3)

• After computing solution of a subproblem, store
  in table
• Time O(mn)
• When computing ci,j we need O(1) time if we
  have
• xi, yj
• ci,j-1
• ci-1,j
• ci-1,j-1

6
DP Table for LCS
y B D C A B A
x A B C B D A B
7
DP Table for LCS
y B D C A B A
x A B C B D A B
0 0 0 0 0 0 0 0 0
0 0 1 1 1 0 1 1 1
1 2 2 0 1 1 2 2 2
2 0 1 1 2 2 3 3 0
1 2 2 2 3 3 0 1 2
2 3 3 4 0 1 2 2 3
4 4
8
Optimal Polygon Triangulation (16.4/?)

• Polygon has sides and vertices
• Polygon is simple not self-intersecting.
• Polygon P is convex if any segment with ends in P
  belong to P
• Triangulation of P is partition of P with chords
  into triangles.
• Problem Given a convex polygon and weight
  function defined on triangles (e.g. the
  perimeter). Find triangulation of minimum weight
  (of minimum total length).
• of triangles n - 2 triangles with n-3 chords

9
Optimal Polygon Triangulation (16.4/?)

• Optimal sub-triangulation of optimal
  triangulation
• Recurrent formula
• ti,j the weight of the optimal triangulation
  of the polygon
• ltvi-1,vi,...,vjgt If ij, then ti,j0
  else
• Runtime O(n3) and space is O(n2)

vi
vi
vi-1
vi-1
vj
vj
vk
vk