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3108. Minimum Cost Walk in Weighted Graph 👍

  • Time: $O(n\log^* n)$
  • Space: $O(n)$
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class UnionFind {
 public:
  // 2^17 - 1 is the minimum number in the form 2^x - 1 > 10^5.
  UnionFind(int n) : id(n), rank(n), weight(n, (1 << 17) - 1) {
    iota(id.begin(), id.end(), 0);
  }

  void unionByRank(int u, int v, int w) {
    const int i = find(u);
    const int j = find(v);
    const int newWeight = weight[i] & weight[j] & w;
    weight[i] = newWeight;
    weight[j] = newWeight;
    if (i == j)
      return;
    if (rank[i] < rank[j]) {
      id[i] = j;
    } else if (rank[i] > rank[j]) {
      id[j] = i;
    } else {
      id[i] = j;
      ++rank[j];
    }
  }

  int getMinCost(int u, int v) {
    if (u == v)
      return 0;
    const int i = find(u);
    const int j = find(v);
    return i == j ? weight[i] : -1;
  }

 private:
  vector<int> id;
  vector<int> rank;
  vector<int> weight;

  int find(int u) {
    return id[u] == u ? u : id[u] = find(id[u]);
  }
};

class Solution {
 public:
  vector<int> minimumCost(int n, vector<vector<int>>& edges,
                          vector<vector<int>>& query) {
    vector<int> ans;
    UnionFind uf(n);

    for (const vector<int>& edge : edges) {
      const int u = edge[0];
      const int v = edge[1];
      const int w = edge[2];
      uf.unionByRank(u, v, w);
    }

    for (const vector<int>& q : query) {
      const int u = q[0];
      const int v = q[1];
      ans.push_back(uf.getMinCost(u, v));
    }

    return ans;
  }
};

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class UnionFind {
  public UnionFind(int n) {
    id = new int[n];
    rank = new int[n];
    weight = new int[n];
    for (int i = 0; i < n; ++i)
      id[i] = i;
    // 2^17 - 1 is the minimum number in the form 2^x - 1 > 10^5.
    Arrays.fill(weight, (1 << 17) - 1);
  }

  public void unionByRank(int u, int v, int w) {
    final int i = find(u);
    final int j = find(v);
    final int newWeight = weight[i] & weight[j] & w;
    weight[i] = newWeight;
    weight[j] = newWeight;
    if (i == j)
      return;
    if (rank[i] < rank[j]) {
      id[i] = j;
    } else if (rank[i] > rank[j]) {
      id[j] = i;
    } else {
      id[i] = j;
      ++rank[j];
    }
  }

  public int getMinCost(int u, int v) {
    if (u == v)
      return 0;
    final int i = find(u);
    final int j = find(v);
    return i == j ? weight[i] : -1;
  }

  private int[] id;
  private int[] rank;
  private int[] weight;

  private int find(int u) {
    return id[u] == u ? u : (id[u] = find(id[u]));
  }
}

class Solution {
  public int[] minimumCost(int n, int[][] edges, int[][] query) {
    int[] ans = new int[query.length];
    UnionFind uf = new UnionFind(n);

    for (int[] edge : edges) {
      final int u = edge[0];
      final int v = edge[1];
      final int w = edge[2];
      uf.unionByRank(u, v, w);
    }

    for (int i = 0; i < query.length; ++i) {
      final int u = query[i][0];
      final int v = query[i][1];
      ans[i] = uf.getMinCost(u, v);
    }

    return ans;
  }
}

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class UnionFind:
  def __init__(self, n: int):
    self.id = list(range(n))
    self.rank = [0] * n
    # 2^17 - 1 is the minimum number in the form 2^x - 1 > 10^5.
    self.weight = [(1 << 17) - 1] * n

  def unionByRank(self, u: int, v: int, w: int) -> None:
    i = self._find(u)
    j = self._find(v)
    newWeight = self.weight[i] & self.weight[j] & w
    self.weight[i] = newWeight
    self.weight[j] = newWeight
    if i == j:
      return
    if self.rank[i] < self.rank[j]:
      self.id[i] = j
    elif self.rank[i] > self.rank[j]:
      self.id[j] = i
    else:
      self.id[i] = j
      self.rank[j] += 1

  def getMinCost(self, u: int, v: int) -> int:
    if u == v:
      return 0
    i = self._find(u)
    j = self._find(v)
    return self.weight[i] if i == j else -1

  def _find(self, u: int) -> int:
    if self.id[u] != u:
      self.id[u] = self._find(self.id[u])
    return self.id[u]


class Solution:
  def minimumCost(
      self,
      n: int,
      edges: list[list[int]],
      query: list[list[int]],
  ) -> list[int]:
    uf = UnionFind(n)

    for u, v, w in edges:
      uf.unionByRank(u, v, w)

    return [uf.getMinCost(u, v) for u, v in query]