13.3 Checkpoint: An Exercise

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.

for (int i = 0; i < A.length - 1; i += 1){
  if (A[i] == A[i + 1]) { 
    return true; 
  }
}
return false;

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 $N$	0 to 10,000
<	0 to $N-1$	0 to 9,999
==	1 to $N-1$	1 to 9,999
array accesses	2 to $2N-2$	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 $N$	1 (in the best case) to 10000 (in the worst case)
<	2 to $(N² + 3N + 2)/2$	2 to 50,015,001
+= 1	0 to $(N² + N)/2$	0 to 50,005,000
==	1 to $(N² - N)/2$	1 to 49,995,000
array accesses	2 to $N²-N$	2 to 99,990,000

dup2 table:

Operation	Symbolic Count	Count (for N=10000)
i = 0	1	1
j = i+1	0 to $N$	0 to 10,000
<	0 to $N-1$	0 to 9,999
==	1 to $N-1$	1 to 9,999
array accesses	2 to $2N-2$	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. ($N² + 3N + 2)/2$ vs $N$)

An even better realization is that parabolas ($N²$) always grow faster than lines ($N$).