Solution Explanation: Finding the Most Similar Path in a Graph

This problem asks to find a path in a graph that is most similar to a given target path, minimizing the edit distance. The solution employs dynamic programming to efficiently explore all possible paths and find the optimal one.

1. Graph Representation:

The input roads defines the graph's edges. We first convert this into an adjacency list (g) for efficient neighbor lookups. g[i] contains a list of all cities directly connected to city i.

2. Dynamic Programming State:

The core of the solution is a 2D DP array f[i][j]. f[i][j] stores the minimum edit distance between:

• The first i cities in the targetPath.
• A path in the graph ending at city j.

3. Recurrence Relation:

The DP state is computed recursively:

f[i][j] = min_{k∈g[j]} { f[i-1][k] + cost(i, j) }

Where:

• g[j] is the set of neighbors of city j.
• cost(i,j) is the cost of adding city j to the path at position i. This cost is 1 if targetPath[i] and names[j] are different (an edit is needed), and 0 otherwise.

The base case is f[0][j], which is the edit distance between the first city of targetPath and city j. This is 0 if they match and 1 otherwise.

4. Path Reconstruction:

The algorithm simultaneously tracks the predecessor city (pre[i][j]) for each state. After computing the DP array, we backtrack from the state with the minimum edit distance (the last row of f) to reconstruct the optimal path.

5. Time and Space Complexity:

• Time Complexity: O(m * n^2), where m is the length of targetPath and n is the number of cities. The nested loops iterate through all possible path extensions.
• Space Complexity: O(m * n) to store the f and pre arrays.

Code Explanation (Python as Example):

class Solution:
    def mostSimilar(
        self, n: int, roads: List[List[int]], names: List[str], targetPath: List[str]
    ) -> List[int]:
        #Adjacency List Creation
        g = [[] for _ in range(n)]
        for a, b in roads:
            g[a].append(b)
            g[b].append(a)
 
        m = len(targetPath)
        
        #DP array initialization
        f = [[float('inf')] * n for _ in range(m)]
        pre = [[-1] * n for _ in range(m)]
 
        #Base Case: First city of targetPath
        for j, s in enumerate(names):
            f[0][j] = int(targetPath[0] != s)
 
        #DP Calculation
        for i in range(1, m):
            for j in range(n):
                for k in g[j]:
                    cost = 1 if targetPath[i] != names[j] else 0
                    if f[i-1][k] + cost < f[i][j]:
                        f[i][j] = f[i-1][k] + cost
                        pre[i][j] = k
 
        #Optimal path reconstruction
        k = 0
        mi = float('inf')
        for j in range(n):
            if f[-1][j] < mi:
                mi = f[-1][j]
                k = j
 
        ans = [0] * m
        for i in range(m - 1, -1, -1):
            ans[i] = k
            k = pre[i][k]
        return ans

The code in other languages follows a very similar structure, adapting syntax and data structures appropriately. The core algorithm remains consistent across implementations.