13.3 Checkpoint: An Exercise
Some much needed practice.
Exercise: Apply techniques 2A and 2B to dup2
.
Calculate the counts of each operation for the following code with respect to N.
Predict the rough magnitudes of each one.
Solution:
Note: It's okay if you were slightly off—as mentioned earlier, you want rough estimates.
Operation | Symbolic Count | Count (for N=10000) |
---|---|---|
i = 0 | 1 | 1 |
j = i+1 | 0 to | 0 to 10,000 |
< | 0 to | 0 to 9,999 |
== | 1 to | 1 to 9,999 |
array accesses | 2 to | 2 to 19998 |
Let us compare the dup1
table with the dup2
table:
dup1
table:
Operation | Symbolic Count | Count (for N=10000) |
---|---|---|
i = 0 | 1 | 1 |
j = i+1 | 1 to | 1 (in the best case) to 10000 (in the worst case) |
< | 2 to | 2 to 50,015,001 |
+= 1 | 0 to | 0 to 50,005,000 |
== | 1 to | 1 to 49,995,000 |
array accesses | 2 to | 2 to 99,990,000 |
dup2
table:
Operation | Symbolic Count | Count (for N=10000) |
---|---|---|
i = 0 | 1 | 1 |
j = i+1 | 0 to | 0 to 10,000 |
< | 0 to | 0 to 9,999 |
== | 1 to | 1 to 9,999 |
array accesses | 2 to | 2 to 19998 |
We can see that dup2
performs significantly better than dup1
in the worst case!
One way to rationalize this is that it takes fewer operations for dup2
to accomplish the same goal as dup1
.
A better realization is that the algorithm for dup2
scales much better in the worst case (e.g. ( vs )
An even better realization is that parabolas () always grow faster than lines ().
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