The quantity m[i - 1] is undefined when i=0. The ternary operator ensures that this undefined value is not used by checking if i is nonzero.

I think the TLE is occurring because you are using memset(mem, -1, sizeof(mem)). Using memset will set the byte of each one to -1, so you will not end up with -1 in each cell, but something like 10000000100000001000000010000000 instead. That is why you are getting TLE, because it's recalculating each time.

UPD: that, and I think that your algorithm is N^3 = TLE.

This is pretty neat. The code without the green parts computes m[-1] = nonsense, m[0] = nonsense + a[0], m[1] = nonsense + a[0] + a[1], etc., and then solve uses this for prefix sums -- a[i] + ... + a[j] = m[j] - m[i-1] (the random value coming from m[-1] cancels out). Essentially it implements prefix sums without branching and with an array of only size n.

It's wildly undefined behavior, of course, and it would not be surprising if a compiler noticed that and e.g. optimized away certain loop iterations. (It would also crash if you compiled your program with ASan and UBSan, which I highly recommend.) Simplest fix would be to make m be of size n+1 and move each entry up one step, presumably setting m[0] = 0.