dijkstra 算法改进
时间: 2023-11-25 16:50:17 浏览: 51
Dijkstra算法是解决单源最短路径问题的经典算法,但是随着图问题规模的增大,算法的时间复杂度也会呈指数级增长,因此需要对其进行优化和改进。以下是两种Dijkstra算法的改进思路及对应的Python代码实现:
1.堆优化Dijkstra算法
在Dijkstra算法中,每次需要从未确定最短路径的节点中选取一个距离源点最近的节点进行松弛操作。如果使用线性查找的方式,时间复杂度为O(n^2),如果使用优先队列的方式,时间复杂度为O(mlogn),其中m为边数,n为节点数。但是,如果使用堆优化的方式,可以将时间复杂度降为O(mlogn)。
```python
import heapq
def dijkstra_heap(graph, start):
# 初始化距离字典
dist = {node: float('inf') for node in graph}
dist[start] = 0
# 初始化堆
heap = [(0, start)]
while heap:
# 弹出堆顶元素
(distance, node) = heapq.heappop(heap)
# 如果当前节点已经处理过,跳过
if distance > dist[node]:
continue
# 遍历当前节点的邻居节点
for neighbor, weight in graph[node].items():
# 计算新的距离
new_distance = dist[node] + weight
# 如果新的距离更短,更新距离字典和堆
if new_distance < dist[neighbor]:
dist[neighbor] = new_distance
heapq.heappush(heap, (new_distance, neighbor))
return dist
```
2.双向Dijkstra算法
双向Dijkstra算法是一种优化Dijkstra算法的方式,它从源点和终点同时开始搜索,直到两个搜索路径相遇。这种算法的时间复杂度为O(m^(2/3)),比普通Dijkstra算法的时间复杂度O(mlogn)更快。
```python
def bidirectional_dijkstra(graph, start, end):
# 初始化距离字典
forward_dist = {node: float('inf') for node in graph}
backward_dist = {node: float('inf') for node in graph}
forward_dist[start] = 0
backward_dist[end] = 0
# 初始化前向堆和后向堆
forward_heap = [(0, start)]
backward_heap = [(0, end)]
# 初始化前向节点集合和后向节点集合
forward_visited = set()
backward_visited = set()
while forward_heap and backward_heap:
# 处理前向堆
(forward_distance, forward_node) = heapq.heappop(forward_heap)
forward_visited.add(forward_node)
# 如果当前节点已经在后向节点集合中,返回结果
if forward_node in backward_visited:
return forward_dist[forward_node] + backward_dist[forward_node]
# 遍历当前节点的邻居节点
for neighbor, weight in graph[forward_node].items():
# 计算新的距离
new_distance = forward_dist[forward_node] + weight
# 如果新的距离更短,更新距离字典和堆
if new_distance < forward_dist[neighbor]:
forward_dist[neighbor] = new_distance
heapq.heappush(forward_heap, (new_distance, neighbor))
# 处理后向堆
(backward_distance, backward_node) = heapq.heappop(backward_heap)
backward_visited.add(backward_node)
# 如果当前节点已经在前向节点集合中,返回结果
if backward_node in forward_visited:
return forward_dist[backward_node] + backward_dist[backward_node]
# 遍历当前节点的邻居节点
for neighbor, weight in graph[backward_node].items():
# 计算新的距离
new_distance = backward_dist[backward_node] + weight
# 如果新的距离更短,更新距离字典和堆
if new_distance < backward_dist[neighbor]:
backward_dist[neighbor] = new_distance
heapq.heappush(backward_heap, (new_distance, neighbor))
return float('inf')
```
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