The Most Similar Path in a Graph

The Most Similar Path in a Graph

What is this problem about?

Imagine you are trying to match a sequence of observed locations (like GPS pings) to a known map represented as a graph. However, the pings might be slightly inaccurate. "The Most Similar Path in a Graph" asks you to find a path in the graph of a specific length 'm' such that the sequence of city names in the path is as close as possible to a given "target" sequence. Similarity is measured by the edit distance: for each position in the sequence, you get a penalty of 1 if the city in your path doesn't match the city in the target sequence at that same position. You want to minimize the total penalty.

Why is this asked in interviews?

This the Most Similar Path in a Graph interview question is a sophisticated challenge used by Google. It tests your ability to apply Dynamic Programming (DP) to graph problems. It's essentially a variation of the Viterbi algorithm or the Hidden Markov Model (HMM) decoding problem. Interviewers look for your ability to define a state that captures all necessary information and to transition between states efficiently.

Algorithmic pattern used

The problem falls under the Graph, Dynamic Programming interview pattern.

1. Define dp[i][v] as the minimum penalty for a path of length i ending at vertex v.
2. Base case: dp[0][v] = (1 if city[v] != target[0] else 0).
3. Transition: dp[i][v] = min(dp[i-1][u] for u in neighbors[v]) + (1 if city[v] != target[i] else 0).
4. To reconstruct the path, you also need to store which vertex u gave you the minimum value for each dp[i][v].
5. The final answer is the path ending at the vertex v that has the minimum dp[m-1][v].

Example explanation

Target: ["NY", "LA"]. Graph: A(NY) -- B(CHI) -- C(LA).

1. i=0: dp[0][A]=0, dp[0][B]=1, dp[0][C]=1.
2. i=1:
   • For A: neighbors are [B]. dp[1][A] = dp[0][B] + (1 if A != LA) = 1 + 1 = 2.
   • For B: neighbors are [A, C]. dp[1][B] = min(dp[0][A], dp[0][C]) + (1 if B != LA) = min(0, 1) + 1 = 1.
   • For C: neighbors are [B]. dp[1][C] = dp[0][B] + (1 if C != LA) = 1 + 0 = 1. The minimum penalty is 1, and the path could be [NY, CHI] or [CHI, LA]. Wait, let's re-check. [NY, CHI] matches NY (penalty 0) but CHI != LA (penalty 1). Total 1. Correct.

Common mistakes candidates make

In "The Most Similar Path in a Graph coding problem," a common mistake is trying to use a greedy approach (picking the best neighbor at each step), which doesn't work because a bad choice now might allow for a much better match later. Another error is not correctly storing the "backpointers" needed to reconstruct the actual path of city names, returning only the minimum penalty instead.

Interview preparation tip

When you see a problem asking for an optimal sequence of decisions where each decision depends on the previous one, think Dynamic Programming. For graph paths, dp[length][last_vertex] is a very common state definition. Practice reconstructing the "actual path" from DP tables, as this is often the hardest part of the implementation.