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882. Reachable Nodes In Subdivided Graph

  • Time: $O((|V| + |E|)\log |V|)$
  • Space: $O(|E| + |V|)$
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class Solution {
 public:
  int reachableNodes(vector<vector<int>>& edges, int maxMoves, int n) {
    vector<vector<pair<int, int>>> graph(n);
    vector<int> dist(graph.size(), maxMoves + 1);

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

    const int reachableNodes = dijkstra(graph, 0, maxMoves, dist);
    int reachableSubnodes = 0;

    for (const vector<int>& edge : edges) {
      const int u = edge[0];
      const int v = edge[1];
      const int cnt = edge[2];
      // the number of reachable nodes of `edge` from `u`
      const int a = dist[u] > maxMoves ? 0 : min(maxMoves - dist[u], cnt);
      // the number of reachable nodes of `edge` from `v`
      const int b = dist[v] > maxMoves ? 0 : min(maxMoves - dist[v], cnt);
      reachableSubnodes += min(a + b, cnt);
    }

    return reachableNodes + reachableSubnodes;
  }

 private:
  int dijkstra(const vector<vector<pair<int, int>>>& graph, int src,
               int maxMoves, vector<int>& dist) {
    dist[src] = 0;
    using P = pair<int, int>;  // (d, u)
    priority_queue<P, vector<P>, greater<>> minHeap;
    minHeap.emplace(dist[src], src);

    while (!minHeap.empty()) {
      const auto [d, u] = minHeap.top();
      minHeap.pop();
      // Already took `maxMoves` to reach `u`, so can't explore anymore.
      if (d >= maxMoves)
        break;
      if (d > dist[u])
        continue;
      for (const auto& [v, w] : graph[u])
        if (d + w + 1 < dist[v]) {
          dist[v] = d + w + 1;
          minHeap.emplace(dist[v], v);
        }
    }

    return ranges::count_if(dist, [&](int d) { return d <= maxMoves; });
  }
};

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class Solution {
  public int reachableNodes(int[][] edges, int maxMoves, int n) {
    List<Pair<Integer, Integer>>[] graph = new List[n];
    int[] dist = new int[n];
    Arrays.fill(dist, maxMoves + 1);

    for (int i = 0; i < n; ++i)
      graph[i] = new ArrayList<>();

    for (int[] edge : edges) {
      final int u = edge[0];
      final int v = edge[1];
      final int cnt = edge[2];
      graph[u].add(new Pair<>(v, cnt));
      graph[v].add(new Pair<>(u, cnt));
    }

    final int reachableNodes = dijkstra(graph, 0, maxMoves, dist);
    int reachableSubnodes = 0;

    for (int[] edge : edges) {
      final int u = edge[0];
      final int v = edge[1];
      final int cnt = edge[2];
      // the number of reachable nodes of `edge` from `u`
      final int a = dist[u] > maxMoves ? 0 : Math.min(maxMoves - dist[u], cnt);
      // the number of reachable nodes of `edge` from `v`
      final int b = dist[v] > maxMoves ? 0 : Math.min(maxMoves - dist[v], cnt);
      reachableSubnodes += Math.min(a + b, cnt);
    }

    return reachableNodes + reachableSubnodes;
  }

  private int dijkstra(List<Pair<Integer, Integer>>[] graph, int src, int maxMoves, int[] dist) {
    dist[src] = 0;
    Queue<Pair<Integer, Integer>> minHeap =
        new PriorityQueue<>(Comparator.comparing(Pair::getKey)) {
          { offer(new Pair<>(dist[src], src)); } // (d, u)
        };

    while (!minHeap.isEmpty()) {
      final int d = minHeap.peek().getKey();
      final int u = minHeap.poll().getValue();
      // Already took `maxMoves` to reach `u`, so can't explore anymore.
      if (d >= maxMoves)
        break;
      if (d > dist[u])
        continue;
      for (Pair<Integer, Integer> pair : graph[u]) {
        final int v = pair.getKey();
        final int w = pair.getValue();
        if (d + w + 1 < dist[v]) {
          dist[v] = d + w + 1;
          minHeap.offer(new Pair<>(dist[v], v));
        }
      }
    }

    return (int) Arrays.stream(dist).filter(d -> d <= maxMoves).count();
  }
}

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class Solution:
  def reachableNodes(
      self,
      edges: list[list[int]],
      maxMoves: int,
      n: int,
  ) -> int:
    graph = [[] for _ in range(n)]
    dist = [maxMoves + 1] * n

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

    reachableNodes = self._dijkstra(graph, 0, maxMoves, dist)
    reachableSubnodes = 0

    for u, v, cnt in edges:
      # the number of reachable nodes of (u, v) from `u`
      a = 0 if dist[u] > maxMoves else min(maxMoves - dist[u], cnt)
      # the number of reachable nodes of (u, v) from `v`
      b = 0 if dist[v] > maxMoves else min(maxMoves - dist[v], cnt)
      reachableSubnodes += min(a + b, cnt)

    return reachableNodes + reachableSubnodes

  def _dijkstra(
      self,
      graph: list[list[tuple[int, int]]],
      src: int,
      maxMoves: int,
      dist: list[int],
  ) -> int:
    dist[src] = 0
    minHeap = [(dist[src], src)]  # (d, u)

    while minHeap:
      d, u = heapq.heappop(minHeap)
      # Already took `maxMoves` to reach `u`, so can't explore anymore.
      if dist[u] >= maxMoves:
        break
      if d > dist[u]:
        continue
      for v, w in graph[u]:
        newDist = d + w + 1
        if newDist < dist[v]:
          dist[v] = newDist
          heapq.heappush(minHeap, (newDist, v))

    return sum(d <= maxMoves for d in dist)