algo(maximumSubarray): add algo + docs + ojAns + plot

Author: domfarolino

src/algorithm_practice/Dynamic_Programming/maximumSubarray/README.md

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+# Maximum (Sum) Subarray
+
+## Naïve solution
+
+This is quite an interesting problem, and lends itself to a rather straightforward
+naive solution. Basically if we were to try and find the subarray with the largest sum
+(the maximum sum subarray), we could do so pretty easily by just seeing what the maximum
+possible subarray starting at a given element would be, and just do that for each element,
+keeping track of the max. Consider the following array:
+
+[1, 2, -4, 5, -2, 9]
+
+All of the subarrays starting with `1` are:
+
+| Subarray      | Sum |
+|---------------|-----|
+| [1]           | 1   |
+| [1, 2]        | 3   |
+| [1, 2, -4]    | -1  |
+| [1, 2, -4, 5] | 4   |
+| [...so on]    | x   |
+
+The max of which is finally 11. We can keep track of this max, and compare it against the max of
+all of the other `n - 1` tables. We can see that the number of subarrays at a given index is bounded
+by `n`, thus making this trivial solution quite slow. Ultimately in this example, we'll see that the
+max of the max subarrays has a sum of 12, and is the array [5, -2, 9].
+
+### Complexity analysis:
+
+ - Time complexity: O(n<sup>2</sup>)
+ - Space complexity: O(1)
+
+## Optimal Solution (dynamic programming)
+
+In the above example, we're doing a ton of work. It'd be nice to make the realization that the reason
+[5, -2, 9] is the maximum subarray, as opposed to [1, 2, -4, 5, -2, 9], is because the subarray
+[1, 2, -4] has a negative sum, so any subarray concatenated onto it is damaged by having it as a prefix.
+In other words, when we get to the point where we are to see what the result of adding `5` to [1, 2, -4]
+will be, we can optimize our approach by saying "Wait, that's less than just starting over at `5`. No need
+to tack `5` onto the negative prefix if I'd be in a better place just starting over at `5`".
+
+It feels like we just skipped over all subarrays that could start with `2` here, or for that matter,
+any value in between the original start, and the new start that we jumped to right? We did! And the reason
+that's ok is because by the time we got to those values, it was still beneficial to have the prefix subarray
+we started before getting there. There's no reason to drop the `1`, from [1, 2, ...], because it'll just lead
+to a lower value. Any subarray starting with `2` can _only be enhanced_ by adding `1` to the beginning. In other
+words, by the time we got to `2`, we didn't have a reason to drop the prefix, because it was helping. Finally, when
+we dipped into the negatives, we realized it'd make more sense to just start over. This solution is nice, and is
+basically a sliding window kinda solution rooted in dynamic programming principles.
+
+This is exactly the logic we use in the approach based off of Kadane's maximum sum subarray algorithm. It
+is a dynamic programming solution, where we essentially keep track of what the maximum sum subarray could be
+if it were to _end_ at any given element. If the sum is ever negative, we know that we can be in a better place
+by starting over at the next not-so-bad solo value. To read more about Kadane's algorithm approach, check out my
+[blog post](https://blog.domfarolino.com/Maximum-Subarray-Study/) on this topic, which goes into much more detail.
+
+### Complexity analysis:
+
+ - Time complexity: O(n)
+ - Space complexity: O(1)
+
+## Performance analysis
+
+Of course the difference between O(n<sup>2</sup>) and O(n) is probably obvious to most of the
+readers here, but since it's fun creating and analyzing data, here's a nifty chart I made with gnuplot:
+
+![alt text](plot.png)
+
+For more reading, check out my [blog post](https://blog.domfarolino.com/Maximum-Subarray-Study/) post which
+goes into a little more detail.