帮我整理一下图论有关的竞赛模板
时间: 2023-05-28 09:02:33 浏览: 67
以下是图论有关的竞赛模板:
1. 搜索算法(DFS/BFS):在图上搜索能够帮助解决一些问题,比如求连通块、寻找最短路径等。
2. 最短路算法(Dijkstra、Floyd、SPFA等):求解从一个节点到另一个节点的最短路径问题。
3. 最小生成树算法(Prim、Kruskal等):用于解决无向图的最小生成树问题。
4. 强连通分量算法(Tarjan、Kosaraju等):用于解决有向图的强连通分量问题。
5. 拓扑排序算法:用于解决有向无环图的拓扑序问题。
6. 二分图算法(匈牙利算法、染色法等):用于解决二分图匹配问题。
7. 最大流算法(Dinic、EK等):求解网络流问题,如最大流、最小割等。
以上是图论有关的一些竞赛模板,可能还有部分相关概念没有列举出来。
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帮我整理一下图论有关的竞赛代码
这里整理一下图论有关的竞赛代码:
1. 最短路算法(Dijkstra算法、Bellman-Ford算法、Floyd算法)
```python
# Dijkstra算法
def dijkstra(graph, start):
dist = {node: float('inf') for node in graph}
dist[start] = 0
pq = [(0, start)]
while pq:
(d, node) = heapq.heappop(pq)
if d > dist[node]:
continue
for neighbor, weight in graph[node].items():
new_dist = dist[node] + weight
if new_dist < dist[neighbor]:
dist[neighbor] = new_dist
heapq.heappush(pq, (new_dist, neighbor))
return dist
# Bellman-Ford算法
def bellman_ford(graph, start):
dist = {node: float('inf') for node in graph}
dist[start] = 0
for i in range(len(graph) - 1):
for u in graph:
for v, w in graph[u].items():
if dist[u] + w < dist[v]:
dist[v] = dist[u] + w
return dist
# Floyd算法
def floyd(graph):
dist = {i: {j: float('inf') for j in graph} for i in graph}
nxt = {i: {j: None for j in graph} for i in graph}
for i in graph:
dist[i][i] = 0
for j in graph[i]:
dist[i][j] = graph[i][j]
nxt[i][j] = j
for k in graph:
for i in graph:
for j in graph:
if dist[i][k] + dist[k][j] < dist[i][j]:
dist[i][j] = dist[i][k] + dist[k][j]
nxt[i][j] = nxt[i][k]
return dist, nxt
```
2. 最小生成树算法(Prim算法、Kruskal算法)
```python
# Prim算法
def prim(graph):
mst = set()
visited = set()
start_node = next(iter(graph))
visited.add(start_node)
edges = list(graph[start_node].items())
heapq.heapify(edges)
while edges:
weight, (node1, node2) = heapq.heappop(edges)
if node1 in visited and node2 in visited:
continue
mst.add((node1, node2, weight))
visited.add(node1)
visited.add(node2)
for neighbor, weight in graph[node2].items():
if neighbor not in visited:
heapq.heappush(edges, (weight, (node2, neighbor)))
return mst
# Kruskal算法
def kruskal(graph):
mst = set()
parent = {node: node for node in graph}
def find(parent, node):
if parent[node] == node:
return node
return find(parent, parent[node])
edges = list()
for node in graph:
for neighbor, weight in graph[node].items():
edges.append((weight, node, neighbor))
edges.sort()
for weight, node1, node2 in edges:
root1 = find(parent, node1)
root2 = find(parent, node2)
if root1 != root2:
mst.add((node1, node2, weight))
parent[root1] = root2
return mst
```
3. 拓扑排序算法(Kahn算法)
```python
def kahn(graph):
in_degree = {node: 0 for node in graph}
for node in graph:
for neighbor in graph[node]:
in_degree[neighbor] += 1
queue = [node for node in graph if in_degree[node] == 0]
result = []
while queue:
node = queue.pop(0)
result.append(node)
for neighbor in graph[node]:
in_degree[neighbor] -= 1
if in_degree[neighbor] == 0:
queue.append(neighbor)
if len(result) == len(graph):
return result
return None
```
4. 最大流算法(Ford-Fulkerson算法、Edmonds-Karp算法、Dinic算法)
```python
# Ford-Fulkerson算法
def dfs(graph, start, end, path, visited):
if start == end:
return path
visited.add(start)
for neighbor, capacity in graph[start].items():
if neighbor not in visited and capacity > 0:
result = dfs(graph, neighbor, end, path + [(start, neighbor)], visited)
if result is not None:
return result
return None
def ford_fulkerson(graph, source, sink):
residual = {node: {neighbor: capacity for neighbor, capacity in graph[node].items()} for node in graph}
max_flow = 0
path = dfs(residual, source, sink, [], set())
while path is not None:
flow = min(residual[node1][node2] for node1, node2 in path)
for node1, node2 in path:
residual[node1][node2] -= flow
residual[node2][node1] += flow
max_flow += flow
path = dfs(residual, source, sink, [], set())
return max_flow
# Edmonds-Karp算法
def edmonds_karp(graph, source, sink):
flow = 0
while True:
queue = [(source, [source], float('inf'))]
while queue:
(node, path, capacity) = queue.pop(0)
for neighbor, weight in graph[node].items():
residual_capacity = weight - capacity
if residual_capacity > 0 and neighbor not in path:
new_flow = min(capacity, residual_capacity)
if neighbor == sink:
return new_flow, path + [neighbor]
else:
queue.append((neighbor, path + [neighbor], new_flow))
if not queue:
return flow
path_flow, path = queue.pop()
flow += path_flow
for i in range(len(path) - 1):
node1 = path[i]
node2 = path[i + 1]
graph[node1][node2] -= path_flow
graph[node2][node1] += path_flow
return flow
# Dinic算法
def bfs(graph, start, end):
queue = [start]
visited = set()
visited.add(start)
parent = {node: None for node in graph}
while queue:
current_node = queue.pop(0)
if current_node == end:
break
for neighbor in graph[current_node]:
if neighbor not in visited and graph[current_node][neighbor] > 0:
visited.add(neighbor)
parent[neighbor] = current_node
queue.append(neighbor)
path = []
node = end
while node is not None:
path.append(node)
node = parent[node]
path.reverse()
return path
def dfs(graph, start, end, flow, parent):
if start == end:
return flow
for neighbor in graph[start]:
if neighbor not in parent and graph[start][neighbor] > 0:
f = dfs(graph, neighbor, end, min(flow, graph[start][neighbor]), parent)
if f > 0:
graph[start][neighbor] -= f
graph[neighbor][start] += f
return f
return 0
def dinic(graph, source, sink):
max_flow = 0
while True:
path = bfs(graph, source, sink)
if not path:
break
flow = float('inf')
for i in range(len(path) - 1):
u = path[i]
v = path[i + 1]
flow = min(flow, graph[u][v])
parent = {}
while flow > 0:
f = dfs(graph, source, sink, flow, parent)
if f == 0:
break
max_flow += f
flow -= f
return max_flow
```
希望这些代码可以对你有所帮助!
图论bfs算法模板实现c++
当然,下面是一个基本的图论 BFS 算法的 C++ 实现模板:
```cpp
#include <iostream>
#include <queue>
#include <vector>
using namespace std;
// 图的邻接表表示
vector<vector<int>> graph;
// BFS 函数
void bfs(int start) {
// 记录节点是否被访问过的数组
vector<bool> visited(graph.size(), false);
// 创建一个队列用于 BFS
queue<int> q;
// 将起始节点加入队列并标记为已访问
q.push(start);
visited[start] = true;
while (!q.empty()) {
// 取出队列的头节点
int node = q.front();
q.pop();
// 处理当前节点
cout << node << " ";
// 遍历当前节点的邻接节点
for (int neighbor : graph[node]) {
// 如果邻接节点未被访问过,则加入队列并标记为已访问
if (!visited[neighbor]) {
q.push(neighbor);
visited[neighbor] = true;
}
}
}
}
int main() {
// 假设图有 6 个节点,节点编号从 0 到 5
int n = 6;
graph.resize(n);
// 添加图的边
graph[0].push_back(1);
graph[0].push_back(2);
graph[1].push_back(3);
graph[1].push_back(4);
graph[2].push_back(4);
graph[3].push_back(5);
graph[4].push_back(5);
// 从节点 0 开始进行 BFS
bfs(0);
return 0;
}
```
这个模板实现了一个基本的图论 BFS 算法。你可以根据自己的需求修改和扩展这个模板来适应不同的场景。希望对你有帮助!