#1611

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum One Bit Operations to Make Integers Zero

Compute the minimum number of one-bit operations to convert a given integer to zero using state transition dynamic programming.

dynamic programming bit manipulation memoization

Problem Statement

Given a non-negative integer n, you must transform it into 0 by repeatedly applying a specific one-bit operation: choose a bit position i and flip it only if all lower bits (0 through i-1) satisfy a defined condition. Compute the minimum number of such operations needed to reach zero.

For example, with n = 3 (binary "11"), you must determine the sequence of flips starting from the leftmost set bit to the rightmost, counting each operation. Return the minimal total steps required for any given integer n within the constraint 0 <= n <= 10^9.

Examples

Example 1

Input: n = 3

Output: 2

The binary representation of 3 is "11". "11" -> "01" with the 2nd operation since the 0th bit is 1. "01" -> "00" with the 1st operation.

Example 2

Input: n = 6

Output: 4

The binary representation of 6 is "110". "110" -> "010" with the 2nd operation since the 1st bit is 1 and 0th through 0th bits are 0. "010" -> "011" with the 1st operation. "011" -> "001" with the 2nd operation since the 0th bit is 1. "001" -> "000" with the 1st operation.

Constraints

0 <= n <= 109

Solution Approach

Binary Observation and State Definition

Represent n in binary and observe that the optimal sequence requires flipping bits starting from the highest set bit. Define a state f(x) representing the minimal operations to reduce x to 0, where transitions depend on the bit pattern and previously computed states.

Recursive Memoization

Use recursion with memoization to avoid recomputing states. For each bit i, compute the minimal flips considering whether lower bits are zero or one. The recurrence captures the left-to-right dependency inherent in this problem, ensuring O(log n) operations.

Iterative DP Implementation

Alternatively, implement the DP iteratively by constructing an array or formula where each state depends on the previous lower state. Update the count of operations according to the rule that each bit flip may require toggling lower bits first, achieving minimal total flips.

Complexity Analysis

Metric	Value
Time	O(\log{}n)
Space	O(1)

Time complexity is O(log n) because each bit in n contributes to a state computation. Space complexity is O(1) for the iterative solution or O(log n) for recursive memoization, as only one value per bit needs storage, avoiding full DP tables.

What Interviewers Usually Probe

Can you model the problem as a recursive state transition?
Are you recognizing how higher-order bits affect operation counts?
Have you optimized using memoization to prevent redundant bit calculations?

Common Pitfalls or Variants

Common pitfalls

Ignoring the dependency of a bit on all lower bits causes overcounting operations.
Attempting a greedy left-to-right flip without tracking state may fail on examples like n = 6.
Failing to memoize recursive computations leads to exponential time and timeouts.

Follow-up variants

Compute minimum operations for multiple integers in a batch.
Modify the rule to allow flipping only contiguous sequences of bits and find minimal flips.
Find the total sequence of bit positions flipped, not just the count.

FAQ

What is the pattern behind minimum one-bit operations to make integers zero?

The key is state transition dynamic programming: each bit flip depends on all lower bits, starting from the leftmost set bit to zero.

How does GhostInterview solve Minimum One Bit Operations to Make Integers Zero efficiently?

It applies recursive memoization or iterative DP to model each state transition, computing the minimal number of flips for every bit efficiently.

What is the time complexity for this problem?

Time complexity is O(log n) since each bit of n contributes a constant number of operations and states.

Can I implement this without recursion?

Yes, an iterative DP approach can track states from lowest to highest bit, maintaining minimal flip counts without recursion.

Why do naive greedy approaches fail for n = 6 or similar numbers?

Greedy flipping without considering lower bit dependencies can overcount operations or miss the optimal sequence, leading to incorrect results.

terminal

Solution

Solution 1: Gray Code Inverse Transform (Gray Code to Binary Code)

This problem essentially asks for the inverse transformation of Gray code at position $n$, i.e., constructing the original number from the Gray code.

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class Solution:
    def minimumOneBitOperations(self, n: int) -> int:
        ans = 0
        while n:
            ans ^= n
            n >>= 1
        return ans

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