Apply Operations to Make Two Strings Equal

Apply Operations to Make Two Strings Equal

What is this problem about?

The Apply Operations to Make Two Strings Equal interview question involves two binary strings of the same length. You are allowed to flip bits in the first string to make it identical to the second string using two types of operations with different costs. One operation flips two adjacent bits (usually for a small fixed cost), and the other flips any two bits at any distance (usually for a higher cost x). The goal is to find the minimum cost to make the strings match.

Why is this asked in interviews?

Google and Zeta ask this Apply Operations to Make Two Strings Equal coding problem to assess a candidate's proficiency with Dynamic Programming (DP). It is a complex variation of the edit distance or shortest path problem, requiring you to carefully consider how "unmatched" indices can be paired up to minimize cost.

Algorithmic pattern used

This follows the String, Dynamic Programming interview pattern. Because each flip affects two bits, the total number of differences between the two strings must be even for a solution to exist. You identify the indices where the strings differ and then use DP to decide whether to pair the current difference with the previous one (adjacent flip cost) or pair it with some other difference (half of the x cost).

Example explanation

Let s1 = "1011", s2 = "0001", and cost x = 3. Differences are at indices 0 and 2.

• Choice 1: Use the general flip for two indices (0 and 2). Cost = x = 3.
• Choice 2: Use adjacent flips. Flip (0,1) then (1,2). If adjacent flip cost is 1, total cost is 1+1=2. The DP state dp[i] represents the minimum cost to fix the first i differences.

Common mistakes candidates make

• Ignoring Parity: Failing to realize that if the number of differing bits is odd, it's impossible to make the strings equal (since every operation flips exactly two bits).
• Greedy approach: Trying to always use the cheapest operation locally, which might lead to a higher total cost later.
• State Definition: Struggling to define a DP state that accounts for an "unpaired" bit that is waiting for a future bit to be flipped.

Interview preparation tip

When you see a problem where you need to pair up items (like mismatched indices) for minimum cost, and the cost depends on the distance between items, think about Dynamic Programming. Practice problems involving "matching" or "pairing" in a linear sequence.