Leetcode #2787: Ways to Express an Integer as Sum of Powers
In this guide, we solve Leetcode #2787 Ways to Express an Integer as Sum of Powers in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Given two positive integers n and x. Return the number of ways n can be expressed as the sum of the xth power of unique positive integers, in other words, the number of sets of unique integers [n1, n2, ..., nk] where n = n1x + n2x + ...
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Dynamic Programming
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: n = 10, x = 2
Output: 1
Explanation: We can express n as the following: n = 32 + 12 = 10.
It can be shown that it is the only way to express 10 as the sum of the 2nd power of unique integers.
Python Solution
class Solution:
def numberOfWays(self, n: int, x: int) -> int:
mod = 10**9 + 7
f = [[0] * (n + 1) for _ in range(n + 1)]
f[0][0] = 1
for i in range(1, n + 1):
k = pow(i, x)
for j in range(n + 1):
f[i][j] = f[i - 1][j]
if k <= j:
f[i][j] = (f[i][j] + f[i - 1][j - k]) % mod
return f[n][n]
Complexity
The time complexity is , and the space complexity is , where is the given integer in the. The space complexity is , where is the given integer in the.
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.