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Formatted question description: https://leetcode.ca/all/882.html
882. Reachable Nodes In Subdivided Graph (Hard)
Starting with an undirected graph (the "original graph") with nodes from 0 to N-1, subdivisions are made to some of the edges.
The graph is given as follows: edges[k] is a list of integer pairs (i, j, n) such that (i, j) is an edge of the original graph,
and n is the total number of new nodes on that edge.
Then, the edge (i, j) is deleted from the original graph, n new nodes (x_1, x_2, ..., x_n) are added to the original graph,
and n+1 new edges (i, x_1), (x_1, x_2), (x_2, x_3), ..., (x_{n-1}, x_n), (x_n, j) are added to the original graph.
Now, you start at node 0 from the original graph, and in each move, you travel along one edge.
Return how many nodes you can reach in at most M moves.
Example 1:
Input:edges= [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3 Output: 13 Explanation: The nodes that are reachable in the final graph after M = 6 moves are indicated below.
Example 2:
Input: edges = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4
Output: 23
Note:
0 <= edges.length <= 100000 <= edges[i][0] < edges[i][1] < N- There does not exist any
i != jfor whichedges[i][0] == edges[j][0]andedges[i][1] == edges[j][1]. - The original graph has no parallel edges.
0 <= edges[i][2] <= 100000 <= M <= 10^91 <= N <= 3000- A reachable node is a node that can be travelled to using at most M moves starting from node 0.
Related Topics:
Heap
Solution 1. Dijkstra
// OJ: https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/
// Time: O(ElogN)
// Space: O(N)
// Ref: https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/solution/
class Solution {
typedef pair<int, int> pii;
public:
int reachableNodes(vector<vector<int>>& edges, int M, int N) {
vector<vector<pii>> G(N);
for (auto &e : edges) {
int u = e[0], v = e[1], w = e[2];
G[u].emplace_back(v, w);
G[v].emplace_back(u, w);
}
map<int, int> dist;
dist[0] = 0;
for (int i = 1; i < N; ++i) dist[i] = M + 1; // M is the maximum steps we can take, so M + 1 is the smallest invalid value.
map<pii, int> used;
int ans = 0;
priority_queue<pii, vector<pii>, greater<pii>> q; // weight, node
q.emplace(0, 0);
while (q.size()) {
auto top = q.top();
q.pop();
int d = top.first, u = top.second;
if (d > dist[u]) continue;
++ans;
for (auto [v, w] : G[u]) {
used[{ u, v }] = min(w, M - d); // M - d is how much further we can walk from this `u` node
int d2 = d + w + 1; // d2 is the minimal distance from source node to this `v` node.
if (d2 < min(dist[v], M + 1)) {
q.emplace(d2, v);
dist[v] = d2;
}
}
}
for (auto &e : edges) {
int u = e[0], v = e[1], w = e[2];
ans += min(w, used[{ u, v }] + used[{ v, u }]);
}
return ans;
}
};
-
class Solution { public int reachableNodes(int[][] edges, int M, int N) { Map<Integer, Map<Integer, Integer>> graphMap = new HashMap<Integer, Map<Integer, Integer>>(); for (int[] edge : edges) { int node0 = edge[0], node1 = edge[1], insertions = edge[2]; Map<Integer, Integer> map0 = graphMap.getOrDefault(node0, new HashMap<Integer, Integer>()); Map<Integer, Integer> map1 = graphMap.getOrDefault(node1, new HashMap<Integer, Integer>()); map0.put(node1, insertions); map1.put(node0, insertions); graphMap.put(node0, map0); graphMap.put(node1, map1); } int[] distances = new int[N]; Arrays.fill(distances, M + 1); distances[0] = 0; int[][] used = new int[N][N]; int reachable = 0; PriorityQueue<int[]> priorityQueue = new PriorityQueue<int[]>(new Comparator<int[]>() { public int compare(int[] nodeDistance1, int[] nodeDistance2) { return nodeDistance1[1] - nodeDistance2[1]; } }); priorityQueue.offer(new int[]{0, 0}); while (!priorityQueue.isEmpty()) { int[] nodeDistance = priorityQueue.poll(); int node = nodeDistance[0], distance = nodeDistance[1]; if (distance > distances[node]) continue; reachable++; if (graphMap.containsKey(node)) { Map<Integer, Integer> edgesMap = graphMap.get(node); Set<Integer> keySet = edgesMap.keySet(); for (int adjacent : keySet) { int insertions = edgesMap.get(adjacent); int currReachable = Math.min(insertions, M - distance); used[node][adjacent] = currReachable; int newDistance = distance + insertions + 1; if (newDistance < distances[adjacent]) { distances[adjacent] = newDistance; priorityQueue.offer(new int[]{adjacent, newDistance}); } } } } for (int[] edge : edges) { int node0 = edge[0], node1 = edge[1], insertions = edge[2]; reachable += Math.min(insertions, used[node0][node1] + used[node1][node0]); } return reachable; } } ############ class Solution { public int reachableNodes(int[][] edges, int maxMoves, int n) { List<int[]>[] g = new List[n]; Arrays.setAll(g, e -> new ArrayList<>()); for (var e : edges) { int u = e[0], v = e[1], cnt = e[2] + 1; g[u].add(new int[] {v, cnt}); g[v].add(new int[] {u, cnt}); } int[] dist = new int[n]; Arrays.fill(dist, 1 << 30); PriorityQueue<int[]> q = new PriorityQueue<>((a, b) -> a[0] - b[0]); q.offer(new int[] {0, 0}); dist[0] = 0; while (!q.isEmpty()) { var p = q.poll(); int d = p[0], u = p[1]; for (var nxt : g[u]) { int v = nxt[0], cnt = nxt[1]; if (d + cnt < dist[v]) { dist[v] = d + cnt; q.offer(new int[] {dist[v], v}); } } } int ans = 0; for (int d : dist) { if (d <= maxMoves) { ++ans; } } for (var e : edges) { int u = e[0], v = e[1], cnt = e[2]; int a = Math.min(cnt, Math.max(0, maxMoves - dist[u])); int b = Math.min(cnt, Math.max(0, maxMoves - dist[v])); ans += Math.min(cnt, a + b); } return ans; } } -
// OJ: https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/ // Time: O(ElogE) // Space: O(E) // Ref: https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/solution/ class Solution { public: int reachableNodes(vector<vector<int>>& E, int maxMoves, int n) { vector<vector<pair<int, int>>> G(n); for (auto &e : E) { int u = e[0], v = e[1], w = e[2]; G[u].emplace_back(v, w); G[v].emplace_back(u, w); } vector<int> step(n, -1); step[0] = maxMoves; priority_queue<pair<int, int>> pq; // Max heap. Each entry is (stepsLeft, nodeIndex) pq.emplace(maxMoves, 0); int ans = 0; while (pq.size()) { auto [stepsLeft, u] = pq.top(); pq.pop(); if (stepsLeft < step[u]) continue; ++ans; if (stepsLeft == 0) continue; for (auto &[v, w] : G[u]) { if (step[v] < stepsLeft - w - 1) { step[v] = stepsLeft - w - 1; pq.emplace(step[v], v); } } } for (auto &e : E) ans += min(e[2], max(0, step[e[0]]) + max(0, step[e[1]])); return ans; } }; -
class Solution: def reachableNodes(self, edges: List[List[int]], maxMoves: int, n: int) -> int: g = defaultdict(list) for u, v, cnt in edges: g[u].append((v, cnt + 1)) g[v].append((u, cnt + 1)) q = [(0, 0)] dist = [0] + [inf] * n while q: d, u = heappop(q) for v, cnt in g[u]: if (t := d + cnt) < dist[v]: dist[v] = t q.append((t, v)) ans = sum(d <= maxMoves for d in dist) for u, v, cnt in edges: a = min(cnt, max(0, maxMoves - dist[u])) b = min(cnt, max(0, maxMoves - dist[v])) ans += min(cnt, a + b) return ans -
func reachableNodes(edges [][]int, maxMoves int, n int) (ans int) { g := make([][]pair, n) for _, e := range edges { u, v, cnt := e[0], e[1], e[2]+1 g[u] = append(g[u], pair{cnt, v}) g[v] = append(g[v], pair{cnt, u}) } dist := make([]int, n) for i := range dist { dist[i] = 1 << 30 } dist[0] = 0 q := hp{ {0, 0} } for len(q) > 0 { p := heap.Pop(&q).(pair) d, u := p.v, p.i for _, nxt := range g[u] { v, cnt := nxt.i, nxt.v if t := d + cnt; t < dist[v] { dist[v] = t heap.Push(&q, pair{t, v}) } } } for _, d := range dist { if d <= maxMoves { ans++ } } for _, e := range edges { u, v, cnt := e[0], e[1], e[2] a := min(cnt, max(0, maxMoves-dist[u])) b := min(cnt, max(0, maxMoves-dist[v])) ans += min(cnt, a+b) } return } type pair struct{ v, i int } type hp []pair func (h hp) Len() int { return len(h) } func (h hp) Less(i, j int) bool { return h[i].v < h[j].v } func (h hp) Swap(i, j int) { h[i], h[j] = h[j], h[i] } func (h *hp) Push(v interface{}) { *h = append(*h, v.(pair)) } func (h *hp) Pop() interface{} { a := *h; v := a[len(a)-1]; *h = a[:len(a)-1]; return v } func max(a, b int) int { if a > b { return a } return b } func min(a, b int) int { if a < b { return a } return b }