I'm tackling a challenging problem and seeking assistance to find a solution with linear time complexity (Theta(n)). The objective is to compute the maximum strength a creature can harness from a variable number of preceding creatures in a line.
Problem Overview:
We have two lists, x and k, of equal length, representing attributes of creatures in a line. List x contains the strengths of these creatures, while list k indicates the count of creatures in front of the i-th creature (including itself) it can call upon for assistance.
Detailed Example:
Consider x = [5, 10, 7, 2, 20] and k = [1, 2, 1, 3, 2]. For each creature, we need to find the maximum strength available to it based on k. For instance, for the fourth creature (index 3):
- It can consider up to 3 creatures (itself and two before it), as
k[3] = 3. - The strengths to consider are
[10, 7, 2]. - Thus, the maximum strength it can use is
max([10, 7, 2]) = 10.
We aim to create a list where each element is the maximum strength accessible to each creature, as described.
Current Algorithm and Its Limitation:
def calculate_xj(x, k):
x_j = []
for i in range(len(x)):
start_index = max(0, i - k[i])
end_index = i + 1
x_j.append(max(x[start_index:end_index]))
return x_j
This algorithm doesn't achieve linear time complexity. The primary issue is the use of the max function within the loop, which iterates over each element in the sliced list. This results in a nested iteration, leading to O(n^2) complexity, as for each creature, we potentially iterate over a list subset, varying based on k.
Seeking Suggestions: I'm looking for algorithms or data structures that can achieve this calculation in linear time. Any guidance on optimizing my current approach or suggesting a new algorithm, particularly those utilizing sliding windows or other techniques suitable for variable-range calculations, would be greatly appreciated.