Problem Description

In this problem, you're given an integer array nums and another integer k. The nums array represents a series of spots that you can stand on, and the value at each spot represents the score you can receive for standing there. The objective is to get from the first index (0) of the array to the last index (n - 1, n being the length of the array), accumulating the maximum possible score.

You can jump from your current index i to any index in the range [i + 1, min(n - 1, i + k)] inclusive. This means that at any point, you may take a leap forward by up to k steps, but no further, and you cannot jump outside the array boundaries. Each time you land on an index, you add up the values (nums[j]) from each of the indices j that you've visited. Your "score" is the sum of these values.

The challenge is to figure out which set of jumps will yield the highest possible score by the time you reach the end of the array.

Intuition

To solve this problem and find the path with the maximum score, one might consider using Dynamic Programming (DP). The DP approach works here because at each index, the best score only depends on the previous decisions, and we are looking for an overall maximum.

The intuition behind the solution can be likened to a 'sliding window' of possibilities for each jump, constrained by k. In DP terms, this window represents subproblems you need to solve on your way to the overall problem's solution. For each index i, you try to find the maximum score you could have if you landed on that index considering the best you could do to get to any permissible previous index.

Key Steps:

1. Initialization: Create an array f of the same length as nums to store the maximum score at each index.

2. Deque to Store Candidates: Use a deque q to keep indices of the array that are potential candidates for jumping to the next index. The deque is maintained in decreasing order of scores; the front always holds the maximum score attainable.

3. Iterate Each Index:

   • If the first element in the deque is more than k distance from the current index, remove it from the deque because it's no longer a valid jump option.
   • Calculate the current index's max score by adding the value at nums[i] to the score at the front of the deque (which represents the best previous score within a k-sized window).
   • Remove elements from the end of the deque as long as the score corresponding to the indices is less than or equal to the score at current index i because they would never contribute to a higher score in the future (due to the k constraint).

4. Append Current Index:

   • After adjusting the deque to maintain the decreasing order of scores, append the current index to it.

5. Return Score at Last Index: The last element of f (f[-1]) will hold the maximum score you could get upon reaching the end of the array.

Using this approach, the solution achieves an optimal balance between computing maximum scores and maintaining an efficient way to discard non-competitive options. The deque ensures that removal and addition of candidates are done in constant time, thus keeping the overall time complexity manageable.

Learn more about Queue, Dynamic Programming, Monotonic Queue and Heap (Priority Queue) patterns.

Solution Approach

The problem's solution employs the Dynamic Programming (DP) technique and a deque data structure to keep track of the potential indices that could give us the maximum score upon reaching each position.

1. Dynamic Initialization:

f = [0] * n
q = deque([0])

Here, f is initialized with zeros for all indices in nums. This array f will store the maximum score at each index as we traverse the array. The deque q is initialized with the first index (0) since our starting position is index 0.

2. Iterate Through nums:

For each index i in the range [0, n):

• Bound Check: If the current index i is out of the k-range from the first element in the deque, it means we can no longer jump from that element's index to i, and thus we remove it from the deque:

if i - q[0] > k:
    q.popleft()

• Calculate Score: The score at f[i] is computed as the sum of nums[i] and the score at f[q[0]] (as q[0] holds the index of the maximum score within k distance):

f[i] = nums[i] + f[q[0]]

• Maintain Decreasing Order in Deque: Before adding the current index to the deque, we pop from the deque's end while the score at the end is less than or equal to the current index's score. This is done because if f[current] is greater than or equal to f[deque's last element], then the last element will never be used as it's not going to contribute to a maximum score:

while q and f[q[-1]] <= f[i]:
    q.pop()

• Add Current Index to Deque: Finally, append the current index i to the deque q because it could potentially be the index from which a future position jumps to get a maximum score:

q.append(i)

3. Return Maximum Score: At the end of the loop, since we have computed the score for each index, the final score at the last index n-1 is the maximum score that can be obtained by any jump sequence, so we return it:

return f[-1]

The algorithm uses DP to efficiently calculate the maximum score at each index based on the best possible previous indices and uses a deque to maintain a sliding window of indices that are potential candidates to jump from, for each index i. The use of a doubly-ended queue allows for efficient addition and removal of indices from both ends, which is crucial for ensuring the algorithm's time efficiency.

By utilizing DP combined with a sliding-window approach, the solution can efficiently tackle the challenge posed by the problem, keeping a running tally of the best scores for each index, and always having a quick access to the best score from which to "jump" to the next potential index in the window defined by k.

Example Walkthrough

To illustrate the solution approach, let's use a small example. Consider the array nums = [10, -5, 2, 4, 0, 3] and k = 2.

We want to find the maximum score we can get by the time we reach the end of the array following the jumping rules defined by k.

Initialization

We initialize an array f to store the maximum scores and a deque q to keep track of potential jumping indices.

f = [0, 0, 0, 0, 0, 0]  # Based on `nums` length
q = deque([0])

Iteration #1

Starting at index 0, nums[0] is 10, so f[0] will also be 10 (since it's the start point).

Iteration #2

Moving to index 1:

• No elements are removed from q since i - q[0] is 1 - 0 which is not greater than k (2).
• We calculate f[1] to be nums[1] + f[q[0]], which is -5 + 10 = 5.
• Since f[1] is less than f[0], we do not remove anything from q.
• We append 1 to q.

The arrays become: f = [10, 5, 0, 0, 0, 0], q = deque([0, 1]).

Iteration #3

At index 2:

• No elements from q are removed.
• f[2] is nums[2] + f[q[0]] which is 2 + 10 = 12.
• Now, f[2] is greater than f[1], so we pop 1 from q.
• We append 2 to q.

Now f = [10, 5, 12, 0, 0, 0], q = deque([0, 2]).

Iteration #4

At index 3:

• We remove 0 from q since 3 - q[0] (0) is greater than k.
• f[3] is nums[3] + f[q[0]] which is 4 + 12 = 16.
• f[3] is greater than f[2], so we pop 2 from q.
• We append 3 to q.

Arrays update to: f = [10, 5, 12, 16, 0, 0], q = deque([3]).

Iteration #5

At index 4:

• No elements from q are removed.
• Calculate f[4] as nums[4] + f[q[0]] which is 0 + 16 = 16.
• No changes to q, as f[4] is equal to f[3].
• We append 4 to q.

We have: f = [10, 5, 12, 16, 16, 0], q = deque([3, 4]).

Iteration #6

Finally, at index 5:

• No elements from q are removed.
• Calculate f[5] as nums[5] + f[q[0]] which is 3 + 16 = 19.
• Since f[5] is greater than f[4], we pop 4 from q.
• Append 5 to q.

The complete f array is f = [10, 5, 12, 16, 16, 19]. The q is deque([5]).

Return Maximum Score

The last element in f, f[-1], is the answer, which is 19. This is the maximum score achievable.

Summary

In this example walkthrough, the maximum score calculation has been demonstrated step by step using an example array and the process of maintaining a deque q for optimal substructure discovery, as well as dynamic programming array f for storing intermediate results. The process ensures that at every step, the most optimal choice for the next jump is made while also pruning choices that will no longer be beneficial for future jumps.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code is O(n), where n is the length of the input array nums. The main loop runs once for each element of the array. In that loop, the following operations occur:

• Checking if i - q[0] > k and popping from the left of the deque: In total, each index will be added and then removed exactly once throughout the entire loop. Therefore, these deque operations will not exceed O(n) over the course of the entire loop.

• Appending to the deque happens once for every element, which provides a time contribution of O(n) across all iterations.

• The while loop pops elements from the deque if f[q[-1]] <= f[i]. Each element will at most be popped once since an element is removed only if a new optimum is found, which is an O(1) operation per element. Therefore, this while loop will, in aggregate, not exceed O(n) over all the iterations.

Hence, the algorithm has a linear time complexity due to the deque operations being amortized over all the iterations.

Space Complexity

The space complexity of the code consists of the space required for:

• The list f, which takes up O(n) space.

• The deque q, which in the worst case, can contain all the elements of the list if the elements of nums are non-increasing, also leading to O(n) space.

Thus, taking into account the space required for the input and auxiliary data structures, the total space complexity would be O(n) as well.

Learn more about how to find time and space complexity quickly using problem constraints.