给定的参考引用中未提及sdutoj平台上关于最小生成树的题目及解题方法相关内容。
最小生成树问题是图论中的经典问题,在很多OJ平台上都有相关题目。常见的求解最小生成树的算法有Prim算法和Kruskal算法。
Prim算法的基本思想是从一个顶点开始,每次选择与当前生成树相连的边中权值最小的边,将对应的顶点加入到生成树中,直到所有顶点都被加入。以下是Prim算法的Python示例代码:
```python
import heapq
def prim(graph):
num_vertices = len(graph)
mst = []
visited = [False] * num_vertices
start_vertex = 0
visited[start_vertex] = True
edges = [(weight, start_vertex, to) for to, weight in graph[start_vertex]]
heapq.heapify(edges)
while edges:
weight, frm, to = heapq.heappop(edges)
if not visited[to]:
visited[to] = True
mst.append((frm, to, weight))
for next_to, next_weight in graph[to]:
if not visited[next_to]:
heapq.heappush(edges, (next_weight, to, next_to))
return mst
# 示例图的邻接表表示
graph = [
[(1, 2), (2, 3)],
[(0, 2), (2, 1), (3, 4)],
[(0, 3), (1, 1), (3, 5)],
[(1, 4), (2, 5)]
]
mst = prim(graph)
print(mst)
```
Kruskal算法的基本思想是将所有边按照权值从小到大排序,然后依次选择边,如果这条边的两个端点不在同一个连通分量中,则将这条边加入到生成树中,直到生成树包含所有顶点。以下是Kruskal算法的Python示例代码:
```python
class UnionFind:
def __init__(self, n):
self.parent = list(range(n))
def find(self, x):
if self.parent[x] != x:
self.parent[x] = self.find(self.parent[x])
return self.parent[x]
def union(self, x, y):
root_x = self.find(x)
root_y = self.find(y)
if root_x != root_y:
self.parent[root_x] = root_y
return True
return False
def kruskal(edges, num_vertices):
edges.sort(key=lambda x: x[2])
uf = UnionFind(num_vertices)
mst = []
for u, v, weight in edges:
if uf.union(u, v):
mst.append((u, v, weight))
return mst
# 示例边的列表
edges = [
(0, 1, 2),
(0, 2, 3),
(1, 2, 1),
(1, 3, 4),
(2, 3, 5)
]
num_vertices = 4
mst = kruskal(edges, num_vertices)
print(mst)
```