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the best way to understand it is form a simple example.
say there are 4 of us in a line: 1. me 2. you 3. bob 4. sue
and we all start out with zero points.
Thcan be represented as (where my points are in the first index, your in the second, bob's in the third, sue's in fourth):
0 0 0 0
Furthermore, we go through rounds, where in each round a contiguous block of us can receive some set amount of points.
So in round one, say 2 points are awarded to anything in the range of start index = 1, and end index = 2. This means that you and bob, who are in the range, get 2 points.
But rather than write the current score as:
0 2 2 0
We instead want to write the score as:
0 2 0 -2
Because we want each value to represent how much greater/less than it is from the previous-index value. Doing this allows us to only ever need to change two elements in the list for a given round.
Now say we play one more round, and 1 point is awarded to all people in range of index = 0 to index = 1. This gives you
1 2 -1 -2
All I did was add the point value to the start index, and subtract it from the "end index + 1".
Then calculating the max is simply a matter of going through that list, adding each element to an accumulator variable (as this summing process reveals the actual value of an element at any given point - e.g., you would have 3 points at the end because your score is the result of 1 + 2), and having a variable which keeps track of the running max.