Sponsored
Sponsored
This approach involves using dynamic programming with a sliding window to maintain the maximum sum at each position. For each position i in the array, calculate the maximum sum that can be achieved till that position, considering the constraint j - i <= k. Use a deque to track the indices which offer the maximum sum within the range of k.
Time Complexity: O(n), where n is the number of elements in the array nums. The space complexity is O(n) due to the storage of the dp array and deque.
1#include <stdio.h>
2#include <stdlib.h>
3#include <limits.h>
4
5int constrainedSubsequenceSum(int* nums, int numsSize, int k) {
6 int* dp = malloc(numsSize * sizeof(int));
7 int maxSum = INT_MIN;
8 int deque[numsSize];
9 int front = 0, rear = 0;
10
11 for (int i = 0; i < numsSize; i++) {
12 if (front <= rear && deque[front] < i - k) {
13 front++;
14 }
15 dp[i] = nums[i];
16 if (front <= rear) {
17 dp[i] += dp[deque[front]];
18 }
19 if (dp[i] > maxSum) {
20 maxSum = dp[i];
21 }
22 while (front <= rear && dp[deque[rear]] <= dp[i]) {
23 rear--;
24 }
25 deque[++rear] = i;
26 }
27
28 free(dp);
29 return maxSum;
30}
31
32int main() {
33 int nums[] = {10, 2, -10, 5, 20};
34 int k = 2;
35 printf("%d\n", constrainedSubsequenceSum(nums, 5, k));
36 return 0;
37}The solution uses a dynamic programming array to store the maximum sum possible up to each index. It utilizes a deque to maintain useful indices that help in calculating the needed maximum sum over the moving window of size k without having to recompute every time.
By using a priority queue (or heap), we manage the maximum possible sum within the constraint more efficiently. We employ dynamic programming to calculate the possible maximal sum at each index while maintaining a priority queue to keep track of the relevant maximum sums.
Time Complexity: O(n log k) primarily due to heap operations. Space Complexity: O(n) is utilized by the dp array and the heap.
1using System.Collections.Generic;
class Program {
public static int ConstrainedSubsequenceSum(int[] nums, int k) {
SortedDictionary<int, List<int>> map = new SortedDictionary<int, List<int>>(Comparer<int>.Create((x, y) => y.CompareTo(x)));
int[] dp = new int[nums.Length];
dp[0] = nums[0];
if (!map.ContainsKey(nums[0])) map[nums[0]] = new List<int>();
map[nums[0]].Add(0);
int maxSum = nums[0];
for (int i = 1; i < nums.Length; i++) {
if (map.TryGetValue(map.Keys.GetEnumerator().Current, out List<int> indices) && indices[0] < i - k) {
int v = indices[0];
if (map[v].Count == 1) map.Remove(v);
else map[v].RemoveAt(0);
}
int maxPrior = map.Keys.GetEnumerator().Current;
dp[i] = nums[i] + maxPrior;
if (!map.ContainsKey(dp[i])) map[dp[i]] = new List<int>();
map[dp[i]].Add(i);
maxSum = Math.Max(maxSum, dp[i]);
}
return maxSum;
}
static void Main(string[] args) {
int[] nums = { 10, 2, -10, 5, 20 };
int k = 2;
Console.WriteLine(ConstrainedSubsequenceSum(nums, k));
}
}This C# utilizes a dictionary to maintain the highest attainable sequence sums in their respective windows, enhancing optimization over dynamic windows and constraints.