Codeforces Round #103 (Div. 2) D题 SPFA

本文介绍如何使用SPFA算法求解给定源点到图中所有点的最短路径,并筛选出距离为L的点的数量。通过遍历图中的所有边和节点,实现高效的路径查找。

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题意就是求离源点距离为L的点的个数,点可以在节点上,也可以在路上,但是必须都是早源点的最短距离为L

用SPFA求一遍距离,然后扫描一遍点,再扫描一遍边

扫描边得时候注意了,有的边上有1个位置,有的边上有2个位置,并且要去重


/* ID: sdj22251 PROG: subset LANG: C++ */ #include <iostream> #include <vector> #include <list> #include <map> #include <set> #include <deque> #include <queue> #include <stack> #include <bitset> #include <algorithm> #include <functional> #include <numeric> #include <utility> #include <sstream> #include <iomanip> #include <cstdio> #include <cmath> #include <cstdlib> #include <cctype> #include <string> #include <cstring> #include <cmath> #include <ctime> #define LOCA #define MAXN 100005 #define INF 100000005 #define eps 1e-7 using namespace std; struct node { int v, w; node *next; } edge[MAXN], temp[3 * MAXN]; int d[MAXN], n, m, pos = 0; int q[MAXN * 4]; bool visited[MAXN]; int uu[MAXN], vv[MAXN], ww[MAXN]; void spfa(int src, int *ecost) //src是起点, ecost是边权 { int h, t, u, i; node *ptr; h = 0, t = 1; memset(visited, 0, sizeof(visited)); q[0] = src; ecost[src] = 0; visited[src] = true; while(h < t) { u = q[h++]; visited[u] = false; ptr = edge[u].next; while(ptr) { if(ecost[ptr -> v] > ecost[u] + ptr -> w) { ecost[ptr -> v] = ecost[u] + ptr -> w; if(!visited[ptr -> v]) { q[t++] = ptr -> v; visited[ptr -> v] = true; } } ptr = ptr -> next; } } } void insert(const int &x, const int &y, const int &w) { node *ptr = &temp[pos++]; ptr -> v = y; ptr -> w = w; ptr -> next = edge[x].next; edge[x].next = ptr; } void init() { for(int i = 0; i <= n; i++) { edge[i].next = NULL; d[i] = INF; } } int main() { int s, w, len; scanf("%d%d%d", &n, &m, &s); init(); for(int i = 0; i < m; i++) { scanf("%d%d%d", &uu[i], &vv[i], &ww[i]); insert(uu[i], vv[i], ww[i]); insert(vv[i], uu[i], ww[i]); } scanf("%d", &len); spfa(s, d); int ans = 0; for(int i = 1; i <= n; i++) { if(d[i] == len) ans++; } for(int i = 0; i < m; i++) { int mi = min(d[uu[i]], d[vv[i]]); int mx = max(d[uu[i]], d[vv[i]]); if(mi < len && mx < len) //如果两个点到源点的最短距离都小于L,就有可能出现边上有两个位置符合题意 { if(mi + mx + ww[i] < 2 * len) //由于题目要求是到源点的最短距离为L,那么两点分到源点的最短距离之和加上边权如果小于2*L,显然任何位置的最短距离都是小于L的 continue; if(mi + mx + ww[i] == 2 * len) //去重,当某个位置通过两个结点到到达的源点都是L的时候 ans++; else if(mi + mx + ww[i] > 2 * len) ans += 2; } else if(mi < len) { if(mi + ww[i] > len) ans++; } } printf("%d\n", ans); return 0; }

### 关于网络流算法的模板目 网络流是一种经典的图论算法,主要用于解决最大流、最小割等问。以下是几个常见的网络流算法模板及其对应的经典目。 #### 1. 最大流问 最大流问是网络流中最基本的问之一,通常可以通过 **Edmonds-Karp 算法** 或者 **Dinic 算法** 解决。下面是一个简单的 Edmonds-Karp 算法实现: ```python from collections import deque class Edge: def __init__(self, v, flow, rev): self.v = v self.flow = flow self.rev = rev def add_edge(u, v, capacity, graph): edge_u_to_v = Edge(v, capacity, len(graph[v])) edge_v_to_u = Edge(u, 0, len(graph[u])) graph[u].append(edge_u_to_v) graph[v].append(edge_v_to_u) def bfs(s, t, parent, graph): visited = [False] * len(graph) queue = deque([s]) visited[s] = True while queue: u = queue.popleft() for idx, edge in enumerate(graph[u]): if not visited[edge.v] and edge.flow > 0: queue.append(edge.v) visited[edge.v] = True parent[edge.v] = u if edge.v == t: return True return False def edmonds_karp(n, s, t, graph): parent = [-1] * n max_flow = 0 while bfs(s, t, parent, graph): path_flow = float(&#39;Inf&#39;) v = t while v != s: u = parent[v] path_flow = min(path_flow, graph[u][next(i for i, e in enumerate(graph[u]) if e.v == v)].flow) v = u v = t while v != s: u = parent[v] index = next(i for i, e in enumerate(graph[u]) if e.v == v) graph[u][index].flow -= path_flow graph[v][graph[u][index].rev].flow += path_flow v = u max_flow += path_flow return max_flow ``` 上述代码实现了基于 BFS 的 Edmonds-Karp 算法来求解最大流问[^4]。 --- #### 2. 最小费用最大流问 如果需要考虑每条边的成本,则可以使用 **SPFA** 或 **Bellman-Ford** 结合最短路径的思想来计算最小费用的最大流。以下是一个 SPFA 实现的例子: ```python import heapq INF = int(1e9) def spfa_min_cost_max_flow(n, edges, start, end): adj_list = [[] for _ in range(n)] residual_graph = [[None]*n for _ in range(n)] for u, v, cap, cost in edges: adj_list[u].append((v, cap, cost)) adj_list[v].append((u, 0, -cost)) dist = [INF] * n potential = [0] * n prev_node = [-1] * n prev_edge = [-1] * n total_flow = 0 total_cost = 0 while True: pq = [] dist[start] = 0 heapq.heappush(pq, (dist[start], start)) while pq: d, node = heapq.heappop(pq) if d > dist[node]: continue for idx, (neighbor, cap, cost) in enumerate(adj_list[node]): if cap > 0 and dist[neighbor] > dist[node] + cost + potential[node] - potential[neighbor]: dist[neighbor] = dist[node] + cost + potential[node] - potential[neighbor] prev_node[neighbor] = node prev_edge[neighbor] = idx heapq.heappush(pq, (dist[neighbor], neighbor)) if dist[end] == INF: break for i in range(n): potential[i] += dist[i] flow = INF cur = end while cur != start: previous = prev_node[cur] edge_index = prev_edge[cur] flow = min(flow, adj_list[previous][edge_index][1]) cur = previous total_flow += flow total_cost += flow * potential[end] cur = end while cur != start: previous = prev_node[cur] edge_index = prev_edge[cur] adj_list[previous][edge_index] = ( adj_list[previous][edge_index][0], adj_list[previous][edge_index][1] - flow, adj_list[previous][edge_index][2] ) back_edge_index = None for j, (back_neighbor, _, _) in enumerate(adj_list[cur]): if back_neighbor == previous: back_edge_index = j break adj_list[cur][back_edge_index] = ( adj_list[cur][back_edge_index][0], adj_list[cur][back_edge_index][1] + flow, adj_list[cur][back_edge_index][2] ) cur = previous return total_flow, total_cost ``` 该代码通过调整势能函数优化了 SPFA,在处理负权边时更加高效[^5]。 --- #### 3. 经典模板推荐 以下是几道经典的网络流算法模板,适合初学者练习: 1. **POJ 1273 Drainage Ditches**: 这是一道典型的 Edmonds-Karp 算法入门2. **HDU 3549 Flow Problem**: 需要使用 Dinic 算法提高效率。 3. **Codeforces Round #XXX Div.2 C**: 涉及到最小费用最大流的应用场景。 4. **LeetCode 787 Cheapest Flights Within K Stops**: 虽然不是纯网络流问,但可以用类似思路建模。 ---
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