Optimal Array Multiplication Sequence

Time Limit: 2 Seconds
Memory Limit: 65536 KB
Special Judge

Given two arrays A and B, we can determine the array C = A B using the standard
definition of matrix multiplication:

The number of columns in the A array must be the same as the number of rows
in the B array. Notationally, let's say that rows(A) and columns(A) are the
number of rows and columns, respectively, in the A array. The number of individual
multiplications required to compute the entire C array (which will have the
same number of rows as A and the same number of columns as B) is then rows(A)
columns(B) columns(A). For example, if A is a 10 x 20 array, and B is a 20 x
15 array, it will take 10 x 15 x 20, or 3000 multiplications to compute the
C array.

To perform multiplication of more than two arrays we have a choice of how to
proceed. For example, if X, Y, and Z are arrays, then to compute X Y Z we could
either compute (X Y) Z or X (Y Z). Suppose X is a 5 x 10 array, Y is a 10 x
20 array, and Z is a 20 x 35 array. Let's look at the number of multiplications
required to compute the product using the two different sequences:

(X Y) Z

5 x 20 x 10 = 1000 multiplications to determine the product (X Y), a 5 x 20
array.

Then 5 x 35 x 20 = 3500 multiplications to determine the final result.

Total multiplications: 4500.

X (Y Z)

10 x 35 x 20 = 7000 multiplications to determine the product (Y Z), a 10 x
35 array.

Then 5 x 35 x 10 = 1750 multiplications to determine the final result.

Total multiplications: 8750.

Clearly we'll be able to compute (X Y) Z using fewer individual multiplications.

Given the size of each array in a sequence of arrays to be multiplied, you
are to determine an optimal computational sequence. Optimality, for this problem,
is relative to the number of individual multiplcations required.

**Input**

For each array in the multiple sequences of arrays to be multiplied you will
be given only the dimensions of the array. Each sequence will consist of an
integer N which indicates the number of arrays to be multiplied, and then N
pairs of integers, each pair giving the number of rows and columns in an array;
the order in which the dimensions are given is the same as the order in which
the arrays are to be multiplied. A value of zero for N indicates the end of
the input. N will be no larger than 10.

**Output**

Assume the arrays are named A1, A2, ..., AN. Your output for each input case
is to be a line containing a parenthesized expression clearly indicating the
order in which the arrays are to be multiplied. Prefix the output for each case
with the case number (they are sequentially numbered, starting with 1). Your
output should strongly resemble that shown in the samples shown below. If, by
chance, there are multiple correct sequences, any of these will be accepted
as a valid answer.

**Sample Input**

3

1 5

5 20

20 1

3

5 10

10 20

20 35

6

30 35

35 15

15 5

5 10

10 20

20 25

0

**Sample Output**

Case 1: (A1 x (A2 x A3))

Case 2: ((A1 x A2) x A3)

Case 3: ((A1 x (A2 x A3)) x ((A4 x A5) x A6))

Source:

**North Central North America 1996**
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