3174. Clear Digits - Detailed Explanation

Problem Statement

You are given an integer n. In one move, you can subtract any nonzero digit of n from n itself. Return the minimum number of moves required to reduce n to 0.

Example 1

• Input:
  n = 27
• Output:
  5
• Explanation:
  One possible sequence of moves is:
  • 27 − 7 = 20
  • 20 − 2 = 18
  • 18 − 8 = 10
  • 10 − 1 = 9
  • 9 − 9 = 0
    Thus, a total of 5 moves are required.

Example 2

• Input:
  n = 10
• Output:
  2
• Explanation:
  The sequence of moves could be:
  • 10 − 1 = 9
  • 9 − 9 = 0
    Hence, 2 moves are needed.

Example 3

• Input:
  n = 1
• Output:
  1
• Explanation:
  Since 1 is the only digit (and it is nonzero), subtracting it gives:
  • 1 − 1 = 0
    So only 1 move is required.

Constraints

• n is a non-negative integer.
• n can be large enough that a brute-force solution without optimization might not run efficiently.

Hints and Intuition

• Hint 1:
  Think about how you can reduce the problem to smaller subproblems. If you subtract one of the digits from n, you obtain a new, smaller number. This naturally suggests a dynamic programming approach.

• Hint 2:
  Consider defining a state dp[i] as the minimum number of moves needed to reduce the number i to 0. For each number i, you can iterate over its digits (ignoring 0) and update dp[i] = 1 + min(dp[i - digit]).

Approaches

A. Top-Down Recursive Dynamic Programming (with Memoization)

• Idea:
  Write a recursive function that, for a given number, tries subtracting every nonzero digit. Use a cache (memoization) to avoid recalculating the minimum moves for the same number.
• Advantages:
  Easy to implement and understand when thinking recursively.
• Potential Pitfall:
  Without memoization, the same state may be computed repeatedly, leading to inefficiency.

B. Bottom-Up Dynamic Programming

• Idea:
  Create an array dp of size n + 1, where dp[0] = 0 (since no moves are needed to reduce 0). For each number i from 1 to n, iterate over the digits in i (ignoring 0) and update
  dp[i] = min(dp[i], dp[i - digit] + 1).
• Advantages:
  This iterative approach guarantees that each subproblem is solved only once.
• Potential Pitfall:
  Be sure to convert the number into its digits correctly and ignore any 0 digits.

Step-by-Step Walkthrough (Using Bottom-Up DP)

Let’s consider n = 27:

1. Initialization:
   Create an array dp with indices 0 through 27. Set dp[0] = 0 and initialize other entries with a large number (e.g., infinity).

2. Processing Each Number:
   For each i from 1 to 27, do the following:

   • Convert i to its string representation to extract each digit.
   • For every digit d in i (skip if d is 0):
     • Update dp[i] as:
       dp[i] = min(dp[i], dp[i - d] + 1)

3. Example for i = 27:

   • The digits of 27 are 2 and 7.
   • Consider subtracting 2: Look up dp[27 - 2] = dp[25].
   • Consider subtracting 7: Look up dp[27 - 7] = dp[20].
   • Choose the minimum among these two options and add 1 for the current move.
   • Through this iterative process, dp[27] will eventually be computed as 5.

4. Final Answer:
   The value dp[27] gives the minimum number of moves to clear the digits of 27, which is 5.

Code Implementation

Python Implementation