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Find Shortest Square-Sum Path

PinduoduoSoftware Engineer · Technical Screen
HardSoftware EngineerTechnical ScreenCoding & Algorithms

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This problem tests your ability to model a shortest-path graph search problem where states are integers and transitions involve adding perfect squares.

Problem Statement

Given a target integer n (1 <= n <= 10^5), find the minimum number of perfect square numbers that sum up to n. While many numbers can be represented as a sum of squares, you must determine the smallest count required to reach exactly n.

Implement the following function:

def min_square_sum(n: int) -> int:
    pass

Examples

Example 1:

  • Input: n = 12
  • Output: 3
  • Explanation: 12 = 4 + 4 + 4 (3 squares).

Example 2:

  • Input: n = 13
  • Output: 2
  • Explanation: 13 = 4 + 9 (2 squares).

Constraints

  • 1 <= n <= 10^5
  • Time complexity should be better than O(n * sqrt(n)).

Considerations

  • Consider the edge cases where n is already a perfect square.
  • Lagrange's Four-Square Theorem states that every natural number can be represented as the sum of four integer squares; use this to bound your search space.

Hints

Hint 1

This problem can be modeled as a shortest-path problem on a graph where nodes are numbers 0 to n and edges exist between `i` and `i + k^2`.

Hint 2

A Dynamic Programming approach where `dp[i]` represents the minimum squares for `i` works by iterating through all possible perfect squares `k^2 <= i`.

Hint 3

Legendre's three-square theorem provides a faster way to check if a number can be represented as a sum of three squares, which can optimize the search for the result.

Follow-up questions

  • How would you reconstruct the actual squares used to form the sum?
  • If we were restricted to using each perfect square at most once, how would the problem complexity change?
  • What is the theoretical upper bound on the number of squares required for any integer, and how does that affect your algorithm?