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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 <iostream>
2#include <vector>
3#include <deque>
4using namespace std;
5
6int constrainedSubsequenceSum(vector<int>& nums, int k) {
7 vector<int> dp(nums.size(), 0);
8 deque<int> deq;
9 int maxSum = nums[0];
10 dp[0] = nums[0];
11 deq.push_back(0);
12
13 for (int i = 1; i < nums.size(); ++i) {
14 if (!deq.empty() && deq.front() < i - k) {
15 deq.pop_front();
16 }
17 dp[i] = nums[i] + (deq.empty() ? 0 : dp[deq.front()]);
18 maxSum = max(maxSum, dp[i]);
19 while (!deq.empty() && dp[deq.back()] <= dp[i]) {
20 deq.pop_back();
21 }
22 deq.push_back(i);
23 }
24
25 return maxSum;
26}
27
28int main() {
29 vector<int> nums = {10, 2, -10, 5, 20};
30 int k = 2;
31 cout << constrainedSubsequenceSum(nums, k) << endl;
32 return 0;
33}This solution implements a C++ version using vectors and deques. The deque keeps track of indices with potential maximum sums, helping compute the current index's maximum sum efficiently.
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.