In a given array nums of positive integers, find three non-overlapping subarrays with maximum sum.
Each subarray will be of size k, and we want to maximize the sum of all 3*k entries.
Return the result as a list of indices representing the starting position of each interval (0-indexed). If there are multiple answers, return the lexicographically smallest one.
Example:
Input: [1,2,1,2,6,7,5,1], 2 Output: [0, 3, 5] Explanation: Subarrays [1, 2], [2, 6], [7, 5] correspond to the starting indices [0, 3, 5]. We could have also taken [2, 1], but an answer of [1, 3, 5] would be lexicographically larger.
Note:
-
nums.lengthwill be between 1 and 20000. -
nums[i]will be between 1 and 65535. -
kwill be between 1 and floor(nums.length / 3).
给一个由正数组成的数组，找三个长度为k的不重叠的子数组，使得三个子数组的数字之和最大。
解法： DP，思路类似于123. Best Time to Buy and Sell Stock III，先分别从左和右两个方向求出每一个位置i之前的长度为k的元素和最大值，这样做的好处是之后想要得到某一位置的最大和时能马上知道。然后在用一个循环找出三段的最大和。
Java:
class Solution {
public int[] maxSumOfThreeSubarrays(int[] nums, int k) {
int n = nums.length, maxsum = 0;
int[] sum = new int[n+1], posLeft = new int[n], posRight = new int[n], ans = new int[3];
for (int i = 0; i < n; i++) sum[i+1] = sum[i]+nums[i];
// DP for starting index of the left max sum interval
for (int i = k, tot = sum[k]-sum[0]; i < n; i++) {
if (sum[i+1]-sum[i+1-k] > tot) {
posLeft[i] = i+1-k;
tot = sum[i+1]-sum[i+1-k];
}
else
posLeft[i] = posLeft[i-1];
}
// DP for starting index of the right max sum interval
// caution: the condition is ">= tot" for right interval, and "> tot" for left interval
posRight[n-k] = n-k;
for (int i = n-k-1, tot = sum[n]-sum[n-k]; i >= 0; i--) {
if (sum[i+k]-sum[i] >= tot) {
posRight[i] = i;
tot = sum[i+k]-sum[i];
}
else
posRight[i] = posRight[i+1];
}
// test all possible middle interval
for (int i = k; i <= n-2*k; i++) {
int l = posLeft[i-1], r = posRight[i+k];
int tot = (sum[i+k]-sum[i]) + (sum[l+k]-sum[l]) + (sum[r+k]-sum[r]);
if (tot > maxsum) {
maxsum = tot;
ans[0] = l; ans[1] = i; ans[2] = r;
}
}
return ans;
}
}
Python:
class Solution(object):
def maxSumOfThreeSubarrays(self, nums, k):
"""
:type nums: List[int]
:type k: int
:rtype: List[int]
"""
n = len(nums)
accu = [0]
for num in nums:
accu.append(accu[-1]+num)
left_pos = [0] * n
total = accu[k]-accu[0]
for i in xrange(k, n):
if accu[i+1]-accu[i+1-k] > total:
left_pos[i] = i+1-k
total = accu[i+1]-accu[i+1-k]
else:
left_pos[i] = left_pos[i-1]
right_pos = [n-k] * n
total = accu[n]-accu[n-k]
for i in reversed(xrange(n-k)):
if accu[i+k]-accu[i] > total:
right_pos[i] = i;
total = accu[i+k]-accu[i]
else:
right_pos[i] = right_pos[i+1]
result, max_sum = [], 0
for i in xrange(k, n-2*k+1):
left, right = left_pos[i-1], right_pos[i+k]
total = (accu[i+k]-accu[i]) + (accu[left+k]-accu[left]) + (accu[right+k]-accu[right])
if total > max_sum:
max_sum = total
result = [left, i, right]
return result
C++:
class Solution {
public:
vector<int> maxSumOfThreeSubarrays(vector<int>& nums, int k) {
int n = nums.size(), maxsum = 0;
vector<int> sum = {0}, posLeft(n, 0), posRight(n, n-k), ans(3, 0);
for (int i:nums) sum.push_back(sum.back()+i);
// DP for starting index of the left max sum interval
for (int i = k, tot = sum[k]-sum[0]; i < n; i++) {
if (sum[i+1]-sum[i+1-k] > tot) {
posLeft[i] = i+1-k;
tot = sum[i+1]-sum[i+1-k];
}
else
posLeft[i] = posLeft[i-1];
}
// DP for starting index of the right max sum interval
// caution: the condition is ">= tot" for right interval, and "> tot" for left interval
for (int i = n-k-1, tot = sum[n]-sum[n-k]; i >= 0; i--) {
if (sum[i+k]-sum[i] >= tot) {
posRight[i] = i;
tot = sum[i+k]-sum[i];
}
else
posRight[i] = posRight[i+1];
}
// test all possible middle interval
for (int i = k; i <= n-2*k; i++) {
int l = posLeft[i-1], r = posRight[i+k];
int tot = (sum[i+k]-sum[i]) + (sum[l+k]-sum[l]) + (sum[r+k]-sum[r]);
if (tot > maxsum) {
maxsum = tot;
ans = {l, i, r};
}
}
return ans;
}
};
类似题目：
[LeetCode] 123. Best Time to Buy and Sell Stock III 买卖股票的最佳时间 III
[LeetCode] 689. Maximum Sum of 3 Non-Overlapping Subarrays 三个非重叠子数组的最大和