Area : Computer Science

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• Area Computer Science
• Subject Algorithms
• Matrix Chain Multiplication
• Report submitted by- Akash Singhal (07005016)

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1. Dynamic Programming is a method of solving
complex problems by breaking them down into
simpler steps. It is applicable to problems that
exhibit the properties of overlapping
sub-problems and optimal substructure. 2. A
problem is said to have optimal substructure if
an optimal solution can be constructed
efficiently from optimal solutions to its
sub-problems.3. Matrix Chain Multiplication is
an optimization problem which finds the most
efficient way of multiplying a given set of
matrices.
Fig. showing the minimum number of steps required
for multiplying each subset of matrices
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AIM PROCEDUREThe aim of this animation is to
clarify the concepts of dynamic programming by
illustrating the problem of Matrix Chain
Multiplication.

• This animation will use a sample set of matrices
  however user can also give a set of matrices as
  input.
• The animation will display a pyramid showing the
  no. of steps required for multiplication of
  suitable subsets of matrices.
• The animation will finally display the minimum
  no. of steps required and the optimum way of
  multiplying the matrices.

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Problem Statement

• Given a set of matrices which can be multiplied,
  find the best possible way of multiplying the
  matrices such that the whole multiplication takes
  least number of possible steps.
• The problem of matrix chain multiplication can be
  broken down into sub-chains to find the number of
  steps required. This is a top-down approach and
  can be solved using recursion.
• Another solution to this problem uses dynamic
  programming which is a bottom-up approach and
  uses iteration instead of recursion.

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1575
2625
1000
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A B C
D E
1800
4375
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1575
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A B C
D E
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1800
4375
2500
1575
2625
1000
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0
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A B C
D E
2500
1950
2175
1800
4375
2500
1575
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1000
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A B C
D E
Construction of a matrix chain multiplication
pyramid
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Review Questions

• Matrix Chain Multiplication is an example of
  _____.( Ans- Dynamic Programming).
• Matrix Chain Multiplication using Dynamic
  Programming is a _______approach.
• a) Right-Left
• b) Left-Right
• c) Top-down
• d) Bottom-up (Ans- Bottom-up)
• Dynamic Programming uses recursion. True or
  False. (Ans- False)
• The running time of the algorithm is_______.
• a) O(n)
• b) O(n log n)
• c) O(n3)
• d) O(n2)
• ______________ means that the solution to a given
  optimization problem can be obtained by the
  combination of optimal solutions to its
  subproblems.
• (Ans- Optimal substructure)

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Further Interactivity

• The user can himself give a set of matrices as
  input. The animation will then construct a
  pyramid for that particular set of matrices and
  tell the user, the minimum number of steps
  required for the multiplication.
• Apart from just showing the minimum no. of steps,
  the optimal break-up of the set of matrices can
  be shown to give the user a better idea of matrix
  chain multiplication problem.

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Further Reading Links

• http//en.wikipedia.org/wiki/Dynamic_Programming
• http//en.wikipedia.org/wiki/Chain_matrix_multipli
  cation
• http//www.brian-borowski.com/Software/Matrix/
• http//www.columbia.edu/cs2035/courses/csor4231.F
  07/matrix-chain.pdf

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CREDITS