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ENGR 325 HOMEWORK #7 KEY FALL 2015
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ENGR 325 HOMEWORK #7 KEY FALL 2015Page |1
ENGR 325
HOMEWORK #7 KEY
FALL 2015
1. (5) In this exercise we look at memory locality properties of matrix computation. The following code
is written in C, where elements within the same row are stored contiguously. Assume each word is
a 32-bit integer. (P&H 5.1, §5.1)
for (I=0; I<8; I++)
for (J=0; J<8000; J++)
A[I][J]=B[I][0]+A[J][I];
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a. How many 32-bit integers can be stored in a 16-byte cache block?
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b. References to which variables exhibit temporal locality?
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c. References to which variables exhibit spatial locality?
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Locality is affected by both the reference order and data layout. The same computation can also be
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written below in Matlab, which differs from C by storing matrix elements within the same column
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contiguously in memory.
for I=1:8
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for J=1:8000
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A(I,J)=B(I,0)+A(J,I);
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end
end
d. How many 16-byte cache blocks are needed to store all 32-bit matrix elements being
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referenced?
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e. References to which variables exhibit temporal locality?
f. References to which variables exhibit spatial locality?
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SOLUTION:
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a. Each 32-bit integer requires 4 bytes to store, therefore FOUR 32-bit integers can be stored in the
16-byte block.
b. Consider the access sequence:
A[0][0] = B[0][0] + A[0][0]
A[1][0] = B[0][0] + A[0][1]
A[2][0] = B[0][0] + A[0][2]
…
A[7999][0] = B[0][0] + A[0][7999]
A[0][1] = B[1][0] + A[1][0]
…
⇒ Variables I, J, and B[I][0] exhibit temporal locality.
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c. Variables A[I][J] and B[I][0] exhibit spatial locality.
d. Consider the access sequence:
A(1,1) = B(1,0) + A(1,1)
A(1,2) = B(1,0) + A(2,1)
A(1,3) = B(1,0) + A(3,1)
…
A(1,8000) = B(1,0) + A(8000,1)
A(2,1) = B(2,0) + A(1,2)
A(2,2) = B(2,0) + A(2,2)
…
A(2,8000) = B(2,0) + B(8000,2)
…
A(8,1) = B(8,0) + A(1,8)
A(8,2) = B(8,0) + A(2,8)
…
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A(8,8000) = B(8,0) + A(8000,8)
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Matrix A begins as an 8-row by 8000-column matrix and is transformed into an 8000-row by 8-
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column matrix. So for the A matrices:
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Elements = 8 x 8000 = 64,000
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Total elements in the beginning and ending matrix: 64000 x 2 = 128,000
Matrix elements per cache block: 4, as per 5.1.1.
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Total cache blocks: 128,000 ÷ 4 = 32,000
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Note, however, that the “upper left” eight by eight elements overlap, so cache is not needed
for those. Therefore we can subtract 8 x 8 ÷ 4 = 16 cache blocks.
For the B matrix:
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Elements = 8 x 1 = 8
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Matrix Elements per cache block: 4, as above
Total cache blocks: 8 ÷ 4 = 2
Total 16-byte cache blocks = 32000 – 16 + 2 ⇒ 31,986 cache blocks are required.
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e. As in 5.1.2, variables I, J, and B(I,0) exhibit temporal locality.
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f. Unlike in C, in MATLAB matrix elements in the same column are stored contiguously. Therefore
the variables A(J,I) and B(I,0) exhibit spatial locality.
2. (10) Caches are important to providing a high-performance memory hierarchy to processors. Below
is a list of 32-bit memory address references, given as word addresses:
3, 180, 43, 2, 191, 88, 190, 14, 181, 44, 186, 253 (P&H 5.2, §5.3)
a. For each of these references, identify the binary address, the tag, and the index given a direct-
mapped cache with 16 one-word blocks. Also list if each reference is a hit or a miss, assuming
the cache is initially empty.
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