Teaching Kids Programming – Maximum Sum of K Numbers from Front and Back of Array (Prefix/Suffix Sum Algorithm)

Teaching Kids Programming: Videos on Data Structures and Algorithms

There are several cards arranged in a row, and each card has an associated number of points. The points are given in the integer array cardPoints. In one step, you can take one card from the beginning or from the end of the row. You have to take exactly k cards. Your score is the sum of the points of the cards you have taken. Given the integer array cardPoints and the integer k, return the maximum score you can obtain.

Example 1:
Input: cardPoints = [1,2,3,4,5,6,1], k = 3
Output: 12
Explanation: After the first step, your score will always be 1. However, choosing the rightmost card first will maximize your total score. The optimal strategy is to take the three cards on the right, giving a final score of 1 + 6 + 5 = 12.

Example 2:
Input: cardPoints = [2,2,2], k = 2
Output: 4
Explanation: Regardless of which two cards you take, your score will always be 4.

Example 3:
Input: cardPoints = [9,7,7,9,7,7,9], k = 7
Output: 55
Explanation: You have to take all the cards. Your score is the sum of points of all cards.

Constraints:
1 <= cardPoints.length <= 10^5
1 <= cardPoints[i] <= 10^4
1 <= k <= cardPoints.length

Hints:
Let the sum of all points be total_pts. You need to remove a sub-array from cardPoints with length n – k.
Keep a window of size n – k over the array. The answer is max(answer, total_pts – sumOfCurrentWindow)

Maximum Points You Can Obtain from Cards (Prefix/Suffix Sum Algorithm)

To pick K numbers from front/rear of the array, we can pick i=[0, 1, 2, …K] numbers from the front and (K-i) numbers from the rear. We can compute the prefix sum via the accumulate function and with that we can compute the suffix sum. For example, S = sum of the entire array, and the suffix sum can be computed via S-p[i] where p[i] is the prefix sum. Alternatively, we can pre-calculate the suffix and store them in a array like the prefix sum.

We then can bruteforce the K+1 choices. The accumulate function in Python computes the prefix sum, but we can compute it via a running sum as well.

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class Solution:
    def maxScore(self, arr: List[int], k: int) -> int:
        s = sum(arr)
        a = [0] + list(accumulate(arr))
        n = len(arr)
        ans = 0
        for i in range(k + 1):
            c = a[i] + s - a[n - (k - i)]
            ans = max(ans, c)
        return ans

class Solution:
    def maxScore(self, arr: List[int], k: int) -> int:
        s = sum(arr)
        a = [0] + list(accumulate(arr))
        n = len(arr)
        ans = 0
        for i in range(k + 1):
            c = a[i] + s - a[n - (k - i)]
            ans = max(ans, c)
        return ans

The time complexity is O(N) and the space complexity is O(N).

See also: Teaching Kids Programming – Maximum Sum of K Numbers from Front and Back of Array (Sliding Window Algorithm)

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