Time Needed to Rearrange a Binary String

Time Needed to Rearrange a Binary String

What is this problem about?

Imagine a binary string of '0's and '1's. In each second, any occurrence of the substring "01" is replaced with "10". This effectively means the '1's are moving to the left and the '0's are moving to the right. The process continues until no more "01" substrings exist (i.e., all '1's are to the left of all '0's). You need to find the total time in seconds until the string is fully rearranged.

Why is this asked in interviews?

This time needed to rearrange a binary string interview question is a medium-level problem used by ServiceNow, PayPal, and Amazon. It tests your ability to either simulate a string manipulation process or identify a dynamic programming pattern to calculate the result more efficiently. It evaluates your understanding of how "bottlenecks" occur in moving data and whether you can model those delays mathematically.

Algorithmic pattern used

The problem utilizes the String, Simulation, Dynamic Programming interview pattern.

1. Simulation approach: Repeatedly replace "01" with "10" using a loop until the string stops changing. This is O(N^2) and might be slow for very long strings.
2. DP approach (O(N)):
   • Iterate through the string from left to right.
   • Keep track of the number of zeros encountered so far.
   • If you see a '1', it needs to move past all the zeros to its left.
   • The time taken for this '1' to reach its final position is at least the number of zeros.
   • However, if the '1' is blocked by another '1' moving ahead of it, it might take 1 extra second.
   • time[i] = max(time[i-1] + 1, count_zeros) if the current char is '1' and there's at least one zero.

Example explanation

String: "011"

1. Time 0: "011"
2. Time 1: "101" (The first "01" becomes "10")
3. Time 2: "110" (The new "01" becomes "10") Total time = 2. Using DP:

• i=0: '0'. zeros=1, time=0.
• i=1: '1'. time = max(0+1, 1) = 1.
• i=2: '1'. time = max(1+1, 1) = 2. Result: 2.

Common mistakes candidates make

In "Time Needed to Rearrange a Binary String coding problem," a common mistake is only counting the number of zeros, which fails when '1's are clustered together and cause a delay. Another error is not correctly handling strings that are already sorted, where the result should be 0. Finally, simulation can be tricky to implement without creating new string objects in every iteration, which is memory-intensive.

Interview preparation tip

Many string movement problems have a "bottleneck" property. Practice identifying when one element's progress is limited by the element in front of it. This often leads to a simple linear DP solution that is far more efficient than a full simulation.