#3192

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Operations to Make Binary Array Elements Equal to One II

Solve the Minimum Operations to Make Binary Array Elements Equal to One II using state transition dynamic programming efficiently.

array dynamic programming greedy

Problem Statement

You are given a binary array nums consisting of 0s and 1s. Your goal is to transform all elements into 1s using a defined operation. Each operation allows you to flip a chosen element, changing 0 to 1 or 1 to 0, any number of times.

Determine the minimum number of operations needed to make all elements equal to 1. The challenge requires careful tracking of state transitions, as flipping one element may impact future operations due to dependencies on previous flips.

Examples

Example 1

Input: nums = [0,1,1,0,1]

Output: 4

We can do the following operations:

Example 2

Input: nums = [1,0,0,0]

Output: 1

We can do the following operation:

Constraints

1 <= nums.length <= 105
0 <= nums[i] <= 1

Solution Approach

State Transition Dynamic Programming

Define a DP array where dp[i] represents the minimum flips needed for the prefix ending at index i. Transition states by considering flipping nums[i] or leaving it, then updating future states with the minimum cost.

Greedy Optimization

Use a greedy approach to decide when flipping a 0 is unavoidable. Track contiguous zeros and apply flips at earliest possible indexes to reduce cumulative operations while maintaining correctness.

Index-Based Tracking

Maintain additional arrays or counters to track the effect of prior operations on current elements. This prevents redundant flips and ensures dp transitions correctly reflect minimal operation counts.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the chosen DP implementation, typically O(n) for a linear scan with index tracking. Space complexity varies with auxiliary arrays but is usually O(n) for storing DP states and flip effects.

What Interviewers Usually Probe

Asks if you considered state transitions for every index.
Questions how to minimize redundant flips over consecutive zeros.
Probes understanding of combining greedy steps with dynamic programming.

Common Pitfalls or Variants

Common pitfalls

Ignoring the impact of a flip on subsequent positions.
Double-counting operations when tracking state transitions.
Assuming all 0s can be flipped independently without affecting DP state.

Follow-up variants

Minimum flips required to make all elements 0 instead of 1.
Binary array with weighted flips, where some indices cost more.
Circular binary array where flips wrap around the end.

FAQ

What pattern is most important for solving this problem?

State transition dynamic programming is the core pattern, tracking how flips affect the prefix and subsequent operations.

Can I use a purely greedy approach?

Purely greedy may fail because early flips can influence future decisions; combining greedy with DP ensures correctness.

What is the time complexity for large arrays?

With linear DP and index tracking, the time complexity is O(n), which is efficient for arrays up to 10^5 elements.

Are there constraints on the input array?

Yes, array length is 1 to 105, and elements are either 0 or 1.

How does GhostInterview guide solving this exact problem?

It breaks down state transitions, suggests minimal greedy flips, and highlights index dependencies to achieve the minimal operation count.

terminal

Solution

Solution 1: Bit Manipulation

We notice that whenever we change an element at a certain position to 1, all the elements to its right are flipped. Therefore, we can use a variable $v$ to record whether the current position and all elements to its right have been flipped. If flipped, the value of $v$ is 1, otherwise, it is 0.

1

2

3

4

5

6

7

8

9

class Solution:
    def minOperations(self, nums: List[int]) -> int:
        ans = v = 0
        for x in nums:
            x ^= v
            if x == 0:
                ans += 1
                v ^= 1
        return ans

Continue Practicing

#3218 minimum cost for cutting cake i #3229 minimum operations to make array equal to target #3068 find the maximum sum of node values