The "Better Enumeration" maximum sub-array algorithm is described by the following pseudo-code:
BETTER-ENUMERATION-MAX-SUBARRAY(A[1, ..., N])
maximum sum = -Infinity
for i from 1 to N
current sum = 0
for j from i to N
current sum = current sum + A[j]
if current sum > maximum sum
maximum sum = current sum
start index = i
end index = j
return maximum sum, A[start index, ..., end index]The outer $i$ loop runs from $1$ to $N$, and the inner $j$ loop runs from $i$ to $N$. Inside the inner loop are constant time operations. We can compute the number of iterations of these constant time operations as:
\begin{equation} \begin{split} \sum_{i=1}^N \sum_{j=i}^N \Theta(1) & = \sum_{i=1}^N (N + 1 - i)\cdot \Theta(1) =N(N+1)\cdot \Theta(1) -\frac{1}{2}N(N+1)\cdot \Theta(1) \\ & = \frac{1}{2}N(N+1)\cdot \Theta(1) = \Theta(N^2) \end{split} \end{equation}Thus, the theoritical run-time is $\Theta(N^2)$.
For a series of array sizes $N$, 10 random arrays were generated and run through the "Better Enumeration" algorithm. The CPU clock time was recorded for each of the 10 random array inputs, and an average run time was computed. Below is the plot of average run time versus $N$ for the "Better Enumeration" algorithm.
A curve fit was applied to the average run time data, which resulted in the following fit equation as a function of $x$ standing in for $N$:
The fit curve for the plotted data has the same degree as the theoretical runtime of $\theta(N^{2})$ so the experimental appears to match the theoretical runtime.
Based on the average run time data curve fit, we would expect the "Better Enumeration" to be able to process the following number of elements in the given amount of time:
| Time | Max Input Size |
|---|---|
| 10 seconds | 12775 |
| 30 seconds | 22419 |
| 60 seconds | 32006 |