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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.
1from collections import deque
2
3def constrainedSubsequenceSum(nums, k):
4 dp = [0] * len(nums)
5 deq = deque()
6 dp[0] = nums[0]
7 deq.append(0)
8 max_sum = nums[0]
9
10 for i in range(1, len(nums)):
11 if deq[0] < i - k:
12 deq.popleft()
13 dp[i] = nums[i] + (dp[deq[0]] if deq else 0)
14 max_sum = max(max_sum, dp[i])
15 while deq and dp[deq[-1]] <= dp[i]:
16 deq.pop()
17 deq.append(i)
18
19 return max_sum
20
21nums = [10, 2, -10, 5, 20]
22k = 2
23print(constrainedSubsequenceSum(nums, k))Here, Python uses a deque and list to find the maximum subsequence sum. The deque aids in maintaining a list of potential max sums 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.