CSE 780: Design and Analysis of Algorithms

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CSE 780 Design and Analysis of Algorithms

• Lecture 11
• DP-algorithm
• Matrix multiplication
• Min-weight triangulation

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Matrix Multiplication

• Given matrix A (size p ? q) and B (size q ? r)
• Compute C A B
• A and B must be compatible
• Time to compute C (number of scalar
  multiplications) pqr

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Matrix-chain Multiplication

• Different ways to compute C
• Matrix multiplication is associative
• So output will be the same
• However, time cost can be very different
• Example

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Problem Definition

• fully parenthesize the product in a way that
  minimizes the number of scalar multiplications
• ( ( ) ( ) ) ( ( ) ( ( ) ( ( ) ( ) ) )
  )

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Number of Parenthesizations

• T(n) number of parenthesizations for n matrices
• Recurrences

Number of parenthesizations exponential
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Extract the Problem

• First try
• m(i)
• smallest number of scalar multiplications from A1
  to Ai.

Doesnt work
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Extract the Problem

• Second try
• Define the problem MM(i, j)
• let m(i,j) smallest number of scalar
  multiplications
• Goal MM(1, n) (with m(1,n) )

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Structure of Optimal Parenthesization

• Given MM(i, j)
• ( ( ) ( ) ) ( ( ) ( ( ) ( ( ) ( ) ) )
  )
• Imagine we take the leftmost left-parenthesis and
  its pairing right-parenthesis
• Break MM(i,j) into two parts

• Suppose break at kth position
• Subproblems MM(i,k), MM(k1, j)
• Optimal substructure property
• Where to break

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A Recursive Solution

• Base case for mi,j
• i j, m i , i 0
• Otherwise, if its broken at i k lt j

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Compute Optimal Cost

• How to do bottom-up?

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Example

• A1 (30 ? 35)
• A2 (35 ? 15)
• A3 (15 ? 5)
• A4 (5 ? 10)
• A5 (10 ? 20)

1000
750
4375
2625
5775
15750
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Construct Opt-solution

• Remember the breaking position chosen for each
  table entry.
• Total time complexity
• Each table entry
• Number of choices O(j-i)
• Each choice O(1)
• O(n3) running time

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Remarks

• More than constant choices
• The order to build table

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Simple Polygon Triangulation
diagonal, triangulation
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Minimum-weight Triangulation

• Weight of triangle perimeter of triangle
• Weight of triangulation sum of perimeters
• gt sum of length of diagonals
• Hence now define weight sum of length of
  diagonals!
• Goal find the triangulation with min-weight
• for a convex polygon

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Observation
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Extract Problem

• MT(i, j) minimum weight of sub-polygon from
  vertex i to j
• wi,j weight of MT(i,j)
• Goal MT(1,n)

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Recursion

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DP Table

• 2D table
• Size O(n2)
• Total time complexity
• O(n3)