PROBLEM LINK:
Author: Sergey Kulik
Testers: Kevin Atienza and Vasya Antoniuk
Translators: Sergey Kulik (Russian), Team VNOI (Vietnamese) and Hu Zecong (Mandarin)
Editorialist: Kevin Atienza
DIFFICULTY:
Easy
PREREQUISITES:
Maximum Subarray Sum
PROBLEM:
Given an array A[1\ldots N], what is the maximum subarray sum you can get by removing at most one element?
QUICK EXPLANATION:
For every position i, compute the maximum subarray sum that ends in position i. Call it E[i].
For every position i, compute the maximum subarray sum that starts in position i. Call it S[i].
Both can be done in O(N) using Kadane’s algorithm/dynamic programming.
The answer is the largest among the following values:
- The maximum in E.
- The maximum E[i-1] + S[i+1] for every position 1 < i < N.
EXPLANATION:
No removals
Let’s first answer a simpler question: What is the maximum subarray sum of the array (without any removals)? This is actually a standard problem and you might already be familiar with it, but let’s describe it anyway.
Here’s a naïve solution (in pseudocode):
answer = -INFINITY
for j=1..N:
current = 0
for i=j..1 by -1:
current += A[i]
answer = max(answer, current)
It simply tries out all subarrays. Unfortunately, this is slow; it runs in O(N^2) time, and for N \approx 10^5 this will not pass the time limit.
To improve this solution, notice that we don’t have to check all the subarrays that end at j. We can classify all such subarrays into two types: whether it has length 1 or length greater than 1.
- If it has length 1, then the sum is simply A[j].
- If it has length greater than 1, then we can break this subarray into two parts. The first part is a subarray that ends in j-1, and the second is A[j] itself. Thus, in order to maximize the sum, we must choose the subarray that ends in j-1 with the maximum sum.
That last part is crucial; if you study the code above more closely, notice that what we’re computing is, for every possible rightmost index j, the maximum subarray sum that ends in j. And we’re doing this in increasing order of j. Thus, when we’re trying to compute it for j, we already have the answer for j-1, so we don’t need a loop any more!
The following code illustrates it better:
answer = -INFINITY
current = 0
for j=1..N:
current = max(A[j], current + A[j])
answer = max(answer, current)
Notice that before the line current = max(A[j], current + A[j]), the variable current contains the maximum sum that ends in j-1. Thus, current + A[j] is the maximum sum of any subarray that ends in j with length greater than 1.
This is now much faster: it runs in O(N) time! Also, if you’re curious, this algorithm is actually famous and has a name: Kadane’s algorithm.
Implementation notes:
- Note that \left|A[i]\right| can reach up to 10^9, so
currentandanswerwill exceed the bounds for 32-bit integers. So you should use 64-bit variables (long longin C/C++,longin Java). - For
INFINITY, you can use a very large number that will exceed all finite numbers under consideration. Since the maximum absolute sum is only 10^5\cdot 10^9 = 10^{14}, anINFINITYvalue of, say, 10^{18} is good enough for our purposes!
One removal
Now, let’s answer the original problem, where you are allowed to remove at most one element from the array. We already know the answer when you don’t remove any element, so all that remains it to compute the answer if we remove exactly one element. The maximum among these two is the answer. More specifically,
- Let M_0 be the answer if you don’t remove any element, and
- Let M_1 be the answer if you remove one element.
Then the answer is \max(M_0, M_1). We already know M_0 from above, so all that remains is computing M_1.
Obviously, we can try removing each element in turn, and computing the maximum subarray sum, and obtaining M_1 as the maximum among all these maximums. But this takes O(N^2) time which is too slow. We will need a better algorithm.
Suppose we remove the element A[i]. Then there are only three possible cases for the maximum subarray sum:
- The maximum subarray is completely to the left of A[i].
- The maximum subarray is completely to the right of A[i].
- The maximum subarray contains A[i-1] and A[i+1].
But here’s an important observation: in the first two cases, the subarrays are actually also subarrays of the original array, which means their sums cannot be more than M_0. And since we’re trying to maximize the answer and we alreaady know that the answer is \ge M_0, we can safely ignore these cases!
So all that remains is computing the maximum sum of any subarray that contains A[i-1] and A[i+1] (ignoring A[i] of course). We can decompose such a subarray into two parts:
- A subarray that ends in A[i-1], and
- A subarray that starts at A[i+1].
These two subarrays don’t overlap, so in order to maximize the sum, we want to maximize them individually. But we have already computed all maximum subarray sums that end in every position, from the previous algorithm! So if we just store the partial results that we got, then we can answer the first part quickly!
In more detail, let E[i] be the maximum subarray sum that ends in A[i]. We can compute E[1\ldots N] using the previous algorithm:
answer = -INFINITY
current = 0
for j=1..N:
current = max(A[j], current + A[j])
answer = max(answer, current)
E[j] = current
Similarly, we can define S[i] to be the maximum subarray sum that starts at A[i]. Computing S[1\ldots N] by performing the algorithm in reverse, as in the following:
current = 0
for j=N..1 by -1:
current = max(A[j], current + A[j])
answer = max(answer, current)
S[j] = current
With arrays S[1\ldots N] and E[1\ldots N], we can now compute the maximum sum of any subarray that contains A[i-1] and A[i+1], assuming A[i] is removed: It is simply E[i-1] + S[i+1]!
for i=2..N-1:
answer = max(answer, E[i-1] + S[i+1])
By combining all these snippets, we now have the answer to the problem!
Time Complexity:
O(N)