Transcript NP-Completeness

Algorithm Design Methodologies
• Divide & Conquer
• Dynamic Programming
• Backtracking
Optimization Problems
• Dynamic programming is typically applied
to optimization problems
• In such problems, there are many feasible
solutions
• We wish to find a solution with the optimal
(maximum or minimum value)
• Examples: Minimum spanning tree, shortest
paths
Matrix Chain Multiplication
• To multiply two matrices A (p by q) and B
(q by r) produces a matrix of dimensions (p
by r) and takes p * q * r “simple” scalar
multiplications
Matrix Chain Multiplication
Given a chain of matrices to multiply:
A1 * A2 * A3 * A4
we must decide how we will paranthesize
the matrix chain:
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(A1*A2)*(A3*A4)
A1 * (A2 * (A3*A4))
A1 * ((A2*A3) * A4)
(A1 * (A2*A3)) * A4
((A1*A2) * A3) * A4
Matrix Chain Multiplication
• We define m[i,j] as the minimum number of
scalar multiplications needed to compute
Ai..j
• Thus, the cheapest cost of multiplying the
entire chain of n matrices is A [1,n]
• If i <> j, we know
m[i,j] = m[i,k] + m[k+1,j] + p[i-1]*p[k]*p[j]
for some value of k  [i,j)
Elements of Dynamic
Programming
• Optimal Substructure
• Overlapping Subproblems
Optimal Substructure
• This means the optimal solution for a
problem contains within it optimal solutions
for subproblems.
• For example, if the optimal solution for the
chain A1*A2*…*A6 is
((A1*(A2*A3))*A4)*(A5*A6)
then this implies the optimal solution for the
subchain A1*A2*….*A4 is
((A1*(A2*A3))*A4)
Overlapping Subproblems
• Dynamic programming is appropriate when
a recursive solution would revisit the same
subproblems over and over
• In contrast, a divide and conquer solution is
appropriate when new subproblems are
produced at each recurrence