|
Data Structures and Algorithms
|
| Matrix Chain Multiplication
|
Problem
We are given a sequence of matrices to multiply:
A1
A2
A3
...
An
Matrix multiplication is associative, so
A1 ( A2 A3 ) =
( A1 A2 ) A3
that is, we can can generate the product in two ways.
The cost of multiplying an
nxm
by an
mxp
one is O(nmp) (or
O(n3) for two
nxn
ones).
A poor choice of parenthesisation can be expensive:
eg if we have
| Matrix | Rows | Columns |
|---|
| A1 | 10 | 100 |
| A2 | 100 | 5 |
| A3 | 5 | 50 |
the cost for
( A1 A2 ) A3 is
| A1A2 | 10x100x5 | = 5000 |
=> A1 A2 (10x5) |
| (A1A2) A3 | 10x5x50 | = 2500 |
=> A1A2A3 (10x50) |
| Total | | 7500 | |
but for
A1 ( A2 A3 )
| A2A3 | 100x5x50 | = 25000 |
=> A2A3 (100x5) |
| A1(A2A3) | 10x100x50 | = 50000 |
=> A1A2A3 (10x50) |
| Total | | 75000 | |
Clearly demonstrating the benefit of calculating the optimum order
before commencing the product calculation!
Optimal Sub-structure
As with the optimal binary search tree, we can observe that
if we divide a chain of matrices to be multiplied into two optimal
sub-chains:
(A1
A2
A3
...
Aj)
(Aj+1
...
An
)
then the optimal parenthesisations of the sub-chains must be
composed of optimal chains.
If they were not, then we could replace them with cheaper parenthesisations.
This property, known as optimal sub-structure
is a hallmark of dynamic algorithms:
it enables us to solve the small problems (the sub-structure) and use those
solutions to generate solutions to larger problems.
For matrix chain multiplication, the procedure is now almost identical to
that used for constructing an optimal binary search tree.
We gradually fill in two matrices,
- one containing the costs of multiplying all the sub-chains.
The diagonal below the main diagonal contains the costs of all
pair-wise multiplications:
cost[1,2] contains the cost of generating product
A1A2, etc.
The diagonal below that contains the costs of triple products:
eg cost[1,3] contains the cost of generating product
A1A2A3, which we derived
from comparing cost[1,2] and cost[2,3], etc.
- one containing the index of last array in the left parenthesisation
(similar to the root of the optimal sub-tree in the optimal binary
search tree, but there's no root here - the chain is divided into
left and right sub-products), so that
best[1,3] might contain 2 to indicate that the
left sub-chain contains
A1A2 and the right one is
A3 in the optimal parenthesisation of
A1A2A3.
As before, if we have n matrices to multiply,
it will take O(n) time to generate each of the
O(n2) costs and entries in the
best matrix for an overall complexity of
O(n3) time
at a cost of
O(n2) space.
Animation
Matrix Chain Multiplication Animation
This animation was written by Woi Ang.
If you don't have a high resolution display,
the bottom of the screen will be clipped!
|
|
Please email comments to:
morris@ee.uwa.edu.au
|
| Key terms |
- optimal sub-structure
- a property of optimisation problems in which the sub-problems
which constitute the solution to the problem itself are themselves
optimal solutions to those sub-problems.
This property permits the construction of dynamic algorithms to
solve the problem.
|
© John Morris, 1998