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1548. The Most Similar Path in a Graph

  • Time: $O(n^2 \cdot |\texttt{targetPath}|)$
  • Space: $O(n^2 + n \cdot |\texttt{targetPath}|)$
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class Solution {
 public:
  vector<int> mostSimilar(int n, vector<vector<int>>& roads,
                          vector<string>& names, vector<string>& targetPath) {
    this->names = names;
    this->targetPath = targetPath;
    // cost[i][j] := the minimum cost to start from names[i] in path[j]
    this->cost.resize(names.size(), vector<int>(targetPath.size(), -1));
    // next[i][j] := the best next of names[i] in path[j]
    this->next.resize(names.size(), vector<int>(targetPath.size()));
    this->graph.resize(n);

    for (const vector<int>& road : roads) {
      graph[road[0]].push_back(road[1]);
      graph[road[1]].push_back(road[0]);
    }

    int minDist = INT_MAX;
    int start = 0;

    for (int i = 0; i < n; ++i) {
      const int dist = dfs(i, 0);
      if (dist < minDist) {
        minDist = dist;
        start = i;
      }
    }

    vector<int> ans;

    while (ans.size() < targetPath.size()) {
      ans.push_back(start);
      start = next[start][ans.size() - 1];
    }

    return ans;
  }

 private:
  vector<string> names;
  vector<string> targetPath;
  vector<vector<int>> cost;
  vector<vector<int>> next;
  vector<vector<int>> graph;

  int dfs(int nameIndex, int pathIndex) {
    if (cost[nameIndex][pathIndex] != -1)
      return cost[nameIndex][pathIndex];

    const int editDist = names[nameIndex] != targetPath[pathIndex];
    if (pathIndex == targetPath.size() - 1)
      return editDist;

    int minDist = INT_MAX;

    for (const int v : graph[nameIndex]) {
      const int dist = dfs(v, pathIndex + 1);
      if (dist < minDist) {
        minDist = dist;
        next[nameIndex][pathIndex] = v;
      }
    }

    return cost[nameIndex][pathIndex] = editDist + minDist;
  }
};

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class Solution {
  public List<Integer> mostSimilar(int n, int[][] roads, String[] names, String[] targetPath) {
    this.names = names;
    this.targetPath = targetPath;
    // cost[i][j] := the minimum cost to start from names[i] in path[j]
    this.cost = new int[names.length][targetPath.length];
    // next[i][j] := the best next of names[i] in path[j]
    this.next = new int[names.length][targetPath.length];
    this.graph = new List[n];

    Arrays.stream(cost).forEach(a -> Arrays.fill(a, -1));
    Arrays.setAll(graph, i -> new ArrayList<>());

    for (int[] road : roads) {
      graph[road[0]].add(road[1]);
      graph[road[1]].add(road[0]);
    }

    int minDist = Integer.MAX_VALUE;
    int start = 0;

    // Start from different names
    for (int i = 0; i < names.length; ++i) {
      final int dist = dfs(i, 0);
      if (dist < minDist) {
        minDist = dist;
        start = i;
      }
    }

    List<Integer> ans = new ArrayList<>();

    while (ans.size() < targetPath.length) {
      ans.add(start);
      start = next[start][ans.size() - 1];
    }

    return ans;
  }

  private String[] names;
  private String[] targetPath;
  private int[][] cost;
  private int[][] next;
  private List<Integer>[] graph;

  private int dfs(int nameIndex, int pathIndex) {
    if (cost[nameIndex][pathIndex] != -1)
      return cost[nameIndex][pathIndex];

    final int editDist = names[nameIndex].equals(targetPath[pathIndex]) ? 0 : 1;
    if (pathIndex == targetPath.length - 1)
      return editDist;

    int minDist = Integer.MAX_VALUE;

    for (final int v : graph[nameIndex]) {
      final int dist = dfs(v, pathIndex + 1);
      if (dist < minDist) {
        minDist = dist;
        next[nameIndex][pathIndex] = v;
      }
    }

    return cost[nameIndex][pathIndex] = editDist + minDist;
  }
}

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class Solution:
  def mostSimilar(self, n: int, roads: list[list[int]], names: list[str],
                  targetPath: list[str]) -> list[int]:
    # cost[i][j] := the minimum cost to start from names[i] in path[j]
    cost = [[-1] * len(targetPath) for _ in range(len(names))]
    # next[i][j] := the best next of names[i] in path[j]
    next = [[0] * len(targetPath) for _ in range(len(names))]
    graph = [[] for _ in range(n)]

    for u, v in roads:
      graph[u].append(v)
      graph[v].append(u)

    minDist = math.inf
    start = 0

    def dfs(nameIndex: int, pathIndex: int) -> int:
      if cost[nameIndex][pathIndex] != -1:
        return cost[nameIndex][pathIndex]

      editDist = names[nameIndex] != targetPath[pathIndex]
      if pathIndex == len(targetPath) - 1:
        return editDist

      minDist = math.inf

      for v in graph[nameIndex]:
        dist = dfs(v, pathIndex + 1)
        if dist < minDist:
          minDist = dist
          next[nameIndex][pathIndex] = v

      cost[nameIndex][pathIndex] = editDist + minDist
      return editDist + minDist

    for i in range(n):
      dist = dfs(i, 0)
      if dist < minDist:
        minDist = dist
        start = i

    ans = []

    while len(ans) < len(targetPath):
      ans.append(start)
      start = next[start][len(ans) - 1]

    return ans