import sys
from collections import deque
import matplotlib.pyplot as plt
import networkx as nx
import numpy as np
plt.rcParams['font.sans-serif'] = ['SimHei'] # 解决中文显示问题
plt.rcParams['axes.unicode_minus'] = False # 解决负号显示问题
class DinicSourceSinkVisual:
def __init__(self, n, edges, source, sink, visualize=True):
"""
:param n: 节点数
:param edges: 边列表 [(u, v, lb, ub)]
:param source: 源点
:param sink: 汇点
:param visualize: 是否可视化
"""
self.n = n
self.source = source
self.sink = sink
self.original_edges = edges.copy() # 保存原始边
self.visualize = visualize
self.fig, self.ax = plt.subplots(figsize=(14, 10))
self.fig.suptitle("有源汇上下界可行流算法动态可视化", fontsize=16)
# 初始化超级源汇
self.super_source = n
self.super_sink = n + 1
self.total_nodes = n + 2
# 计算每个节点的流量差
self.A = [0] * (n + 2)
for u, v, lb, ub in self.original_edges:
self.A[u] -= lb
self.A[v] += lb
# 添加源汇之间无限容量的边 - 关键修改
# 标准的有源汇上下界处理方法：添加t->s的无限容量边
edges.append((sink, source, 0, float('inf')))
# 创建Dinic数据结构
self.graph = [[] for _ in range(self.total_nodes)]
self.level = [-1] * self.total_nodes
self.cur = [0] * self.total_nodes
self.edge_info = {} # 存储边信息
# 添加图中的边
self.edge_refs = []
for i, (u, v, lb, ub) in enumerate(edges):
cap = ub - lb
# 添加边并记录信息
self.add_edge(u, v, cap, (i, lb, ub, f"e{i}"))
# 仅原始边（不包括后添加的sink->source边）记录在edge_refs中
if i < len(edges) - 1: # 最后一条是后添加的sink->source边
self.edge_refs.append((u, v, len(self.graph[u]) - 1, lb))
# 添加超级源汇的边
self.total_flow = 0
for i in range(n + 2): # 包含所有节点
if self.A[i] > 0:
self.add_edge(self.super_source, i, self.A[i], (f"S→{i}", "super_source"))
self.total_flow += self.A[i]
elif self.A[i] < 0:
self.add_edge(i, self.super_sink, -self.A[i], (f"{i}→T", "super_sink"))
# 初始化可视化
if self.visualize:
self.initialize_visualization()
def add_edge(self, u, v, cap, info=None):
"""添加边并存储信息"""
forward = [v, cap, 0, info] # [目标, 容量, 流量, 信息]
reverse = [u, 0, 0, None] # 反向边
forward[2] = reverse
reverse[2] = forward
self.graph[u].append(forward)
self.graph[v].append(reverse)
# 存储边信息用于可视化
if info:
self.edge_info[(u, v)] = {
'capacity': cap,
'flow': 0,
'info': info
}
return forward
def bfs(self):
"""BFS分层并可视化"""
self.level = [-1] * self.total_nodes
queue = deque([self.super_source])
self.level[self.super_source] = 0
# 可视化：显示BFS搜索过程
if self.visualize:
self.visualize_step(f"BFS分层: 访问超级源点S (L0)")
plt.pause(1.0)
while queue:
u = queue.popleft()
for i, edge in enumerate(self.graph[u]):
v, cap, rev, info = edge
if cap > 0 and self.level[v] == -1:
self.level[v] = self.level[u] + 1
queue.append(v)
# 可视化：显示新访问的节点
if self.visualize:
if v < self.n:
node_label = f"节点{v}"
elif v == self.super_sink:
node_label = "超级汇点T"
else:
node_label = "超级源点S"
self.visualize_step(f"BFS分层: 访问{node_label} (L{self.level[v]})")
plt.pause(0.3)
return self.level[self.super_sink] != -1
def dfs(self, u, t, flow, path=None):
"""DFS查找增广路径并可视化"""
if path is None:
path = []
if u == t:
# 可视化：显示找到的增广路径
if self.visualize:
path_desc = "→".join(
[f"{'S' if p == self.super_source else 'T' if p == self.super_sink else p}"
for p in path + [t]]
)
self.visualize_step(f"找到增广路径: {path_desc}\n流量: {flow}")
plt.pause(1.5)
return flow
for i in range(self.cur[u], len(self.graph[u])):
self.cur[u] = i
edge = self.graph[u][i]
v, cap, rev, info = edge
if cap > 0 and self.level[v] == self.level[u] + 1:
# 可视化：显示当前探索的边
if self.visualize:
edge_desc = self.get_edge_description(u, v)
self.visualize_step(f"探索: {edge_desc} (剩余容量: {cap})")
plt.pause(0.5)
f = self.dfs(v, t, min(flow, cap), path + [u])
if f > 0:
# 更新边流量
edge[1] -= f
rev[1] += f
# 更新可视化信息
if (u, v) in self.edge_info:
self.edge_info[(u, v)]['flow'] += f
elif (v, u) in self.edge_info: # 处理反向边
self.edge_info[(v, u)]['flow'] -= f
# 可视化：显示流量更新
if self.visualize:
edge_desc = self.get_edge_description(u, v)
self.visualize_step(f"更新: {edge_desc}\n增加流量: {f}")
plt.pause(0.8)
return f
return 0
def max_flow(self):
"""计算最大流并动态可视化"""
total_flow = 0
iteration = 1
while self.bfs():
self.cur = [0] * self.total_nodes
if self.visualize:
self.visualize_step(f"开始阶段 {iteration} (分层完成)")
plt.pause(1.0)
while True:
flow = self.dfs(self.super_source, self.super_sink, float('inf'))
if flow == 0:
break
total_flow += flow
if self.visualize:
self.visualize_step(f"阶段 {iteration} 完成\n累计流量: {total_flow}/{self.total_flow}")
plt.pause(1.0)
iteration += 1
# 检查可行解
if total_flow != self.total_flow:
if self.visualize:
self.visualize_step(f"无可行解!\n需求流量: {self.total_flow}, 实际流量: {total_flow}")
plt.pause(3.0)
return None
# 计算原图中每条边的实际流量
flows = []
for u, v, idx, lb in self.edge_refs:
# 跳过最后添加的sink->source边
if u == self.sink and v == self.source:
continue
edge = self.graph[u][idx]
actual_flow = lb + edge[1] # 实际流量 = 下界 + 残余网络中的剩余容量
# 对于正常边，实际流量 = 下界 + 流量
# 但对于t->s边不计算
flows.append(actual_flow)
if self.visualize:
self.visualize_final_flow(flows)
plt.pause(5.0)
return flows
def get_edge_description(self, u, v):
"""获取边的描述信息"""
if u == self.super_source:
return f"S → {v}"
elif v == self.super_sink:
return f"{u} → T"
elif u == self.source and v == self.sink:
return f"{u}→{v} (源汇边)"
elif (u, v) in self.edge_info:
info = self.edge_info[(u, v)]['info']
if isinstance(info, tuple) and len(info) > 3:
return f"{u} → {v} ({info[3]})"
return f"{u} → {v}"
def initialize_visualization(self):
"""初始化可视化布局"""
self.G = nx.DiGraph()
# 添加节点
for i in range(self.n):
self.G.add_node(i, label=f"{i}")
self.G.add_node(self.super_source, label="S")
self.G.add_node(self.super_sink, label="T")
# 添加边
for u in range(self.total_nodes):
for edge in self.graph[u]:
v, cap, _, info = edge
if cap > 0: # 只添加正向边
self.G.add_edge(u, v, capacity=cap, flow=0)
# 创建环形布局
self.pos = {}
# 普通节点布置在圆上
angles = np.linspace(0, 2 * np.pi, self.n, endpoint=False)
for i in range(self.n):
angle = angles[i]
self.pos[i] = (np.cos(angle), np.sin(angle))
# 特殊节点位置
self.pos[self.source] = (0, 1.2) # 源点在上方
self.pos[self.sink] = (0, -1.2) # 汇点在下方
self.pos[self.super_source] = (-1.5, 0) # 超级源点在左侧
self.pos[self.super_sink] = (1.5, 0) # 超级汇点在右侧
# 初始绘图
self.ax.clear()
# 节点颜色：普通节点-浅蓝，源汇点-浅绿，超级源汇-浅红
node_colors = []
for node in self.G.nodes():
if node == self.source or node == self.sink:
node_colors.append('lightgreen')
elif node == self.super_source or node == self.super_sink:
node_colors.append('salmon')
else:
node_colors.append('lightblue')
nx.draw_networkx_nodes(self.G, self.pos, node_size=800, node_color=node_colors)
nx.draw_networkx_labels(self.G, self.pos,
labels={n: d['label'] for n, d in self.G.nodes(data=True)})
# 绘制边
self.edge_collection = nx.draw_networkx_edges(
self.G, self.pos, arrowstyle='->', arrowsize=20,
edge_color='gray', width=1, ax=self.ax
)
# 初始化边标签
self.edge_labels = {}
for u, v in self.G.edges():
self.edge_labels[(u, v)] = self.ax.text(0, 0, "",
fontsize=8, ha='center', va='center')
self.ax.set_title("初始化网络", fontsize=14)
self.ax.set_axis_off()
plt.tight_layout()
plt.pause(2.0)
def visualize_step(self, message):
"""可视化当前步骤"""
self.ax.clear()
# 节点颜色
node_colors = []
for node in self.G.nodes():
if node == self.source or node == self.sink:
node_colors.append('lightgreen')
elif node == self.super_source or node == self.super_sink:
node_colors.append('salmon')
else:
node_colors.append('lightblue')
# 绘制节点
nx.draw_networkx_nodes(self.G, self.pos, node_size=800, node_color=node_colors)
nx.draw_networkx_labels(self.G, self.pos,
labels={n: d['label'] for n, d in self.G.nodes(data=True)})
# 绘制边并设置颜色和宽度
edge_colors = []
edge_widths = []
for u, v in self.G.edges():
# 获取当前边的状态
cap = self.G[u][v]['capacity']
flow = self.edge_info.get((u, v), {}).get('flow', 0)
# 计算饱和度
saturation = flow / cap if cap > 0 else 0
# 使用颜色表示饱和度
edge_colors.append(plt.cm.RdYlGn(saturation))
# 使用宽度表示流量
edge_widths.append(1 + 3 * saturation)
# 绘制边
nx.draw_networkx_edges(
self.G, self.pos, arrowstyle='->', arrowsize=20,
edge_color=edge_colors, width=edge_widths, ax=self.ax
)
# 更新边标签
for (u, v), text in self.edge_labels.items():
# 获取边信息
cap = self.G[u][v]['capacity']
flow = self.edge_info.get((u, v), {}).get('flow', 0)
# 特殊边处理
if u == self.super_source or v == self.super_sink:
label = f"{flow}/{cap}"
else:
# 获取原始边信息
info = self.edge_info.get((u, v), {}).get('info', None)
if info and isinstance(info, tuple):
_, lb, ub, name = info
actual_flow = lb + flow
label = f"{name}: {actual_flow}/{ub}\n[{lb},{ub}]"
else:
label = f"{flow}/{cap}"
# 计算边的中点位置
x = (self.pos[u][0] + self.pos[v][0]) / 2
y = (self.pos[u][1] + self.pos[v][1]) / 2
# 更新文本位置和内容
text.set_position((x, y))
text.set_text(label)
self.ax.add_artist(text)
# 显示当前信息
self.ax.set_title(message, fontsize=14)
self.ax.set_axis_off()
plt.tight_layout()
plt.draw()
def visualize_final_flow(self, flows):
"""可视化最终可行流分配（仅显示原图边）"""
self.ax.clear()
# 创建仅包含原图节点和边的子图
H = nx.DiGraph()
for i in range(self.n):
H.add_node(i, label=f"{i}")
# 添加原图边（排除最后添加的sink->source边）
for i, (u, v, lb, ub) in enumerate(self.original_edges):
if i >= len(flows):
continue
H.add_edge(u, v, flow=flows[i], lb=lb, ub=ub, name=f"e{i}")
# 使用原布局，但只保留原图节点的位置
pos = {k: v for k, v in self.pos.items() if k in H.nodes()}
# 绘制节点
node_colors = ['lightgreen' if node == self.source or node == self.sink
else 'lightblue' for node in H.nodes()]
nx.draw_networkx_nodes(H, pos, node_size=800, node_color=node_colors)
nx.draw_networkx_labels(H, pos)
# 绘制边并设置颜色和宽度
edge_colors = []
edge_widths = []
for u, v in H.edges():
flow = H[u][v]['flow']
ub = H[u][v]['ub']
saturation = flow / ub
edge_colors.append(plt.cm.RdYlGn(saturation))
edge_widths.append(1 + 3 * saturation)
nx.draw_networkx_edges(
H, pos, arrowstyle='->', arrowsize=20,
edge_color=edge_colors, width=edge_widths, ax=self.ax
)
# 添加边标签
edge_labels = {}
for u, v in H.edges():
flow = H[u][v]['flow']
lb = H[u][v]['lb']
ub = H[u][v]['ub']
name = H[u][v]['name']
edge_labels[(u, v)] = f"{name}: {flow}\n[{lb},{ub}]"
nx.draw_networkx_edge_labels(H, pos, edge_labels=edge_labels, font_size=8)
self.ax.set_title("可行流分配结果（仅显示原图边）", fontsize=14)
self.ax.set_axis_off()
plt.tight_layout()
plt.draw()
def circulation_flow_visual(n, edges, source, sink):
"""有源汇上下界可行流求解与可视化"""
# 创建可视化实例
dinic_visual = DinicSourceSinkVisual(n, edges, source, sink, visualize=True)
# 计算可行流
flows = dinic_visual.max_flow()
if flows is None:
print("无可行流解")
return None
print("\n各边实际流量分配:")
for i, (u, v, lb, ub) in enumerate(edges[:-1]): # 排除最后添加的sink->source边
print(f"边 {u}→{v} ({lb},{ub}): {flows[i]}")
# 计算源点到汇点的总流量
source_flow = sum(flows[i] for i, (u, v, _, _) in enumerate(edges) if u == source)
sink_flow = sum(flows[i] for i, (u, v, _, _) in enumerate(edges) if v == sink)
print(f"\n源点({source})总输出流量: {source_flow}")
print(f"汇点({sink})总输入流量: {sink_flow}")
plt.show() # 保持窗口打开
return flows
if __name__ == "__main__":
# 15节点有可行解的网络示例 - 简化版
print("=" * 50)
print("15节点网络的有源汇上下界可行流计算 (保证有可行解)")
# 简化设计：确保网络平衡
n = 15
edges = [
# 源点→核心节点
(0, 1, 5, 10),
(0, 2, 5, 10),
# 核心环状结构
(1, 2, 0, 5),
(2, 3, 2, 8),
(3, 4, 2, 8),
(4, 1, 0, 5),
# 核心→中间节点
(1, 5, 1, 4),
(2, 6, 1, 4),
(3, 7, 1, 4),
(4, 8, 1, 4),
# 中间层平衡结构
(5, 6, 0, 5),
(6, 7, 0, 5),
(7, 8, 0, 5),
(8, 5, 0, 5),
# 中间→汇点
(5, 14, 3, 6),
(6, 14, 3, 6),
(7, 14, 2, 5),
(8, 14, 2, 5),
# 连接外围节点
(1, 9, 0, 3),
(2, 10, 0, 3),
(3, 11, 0, 3),
(4, 12, 0, 3),
(9, 13, 0, 3),
(10, 13, 0, 3),
(11, 13, 0, 3),
(12, 13, 0, 3),
(13, 14, 0, 5) # 汇点入口
]
# 设置源点和汇点
source = 0 # 节点0作为源点
sink = 14 # 节点14作为汇点
# 计算并可视化可行流
flows = circulation_flow_visual(n, edges, source, sink)
修改代码，使得其能够解决有源汇上下界最小流问题，
给出修改后的完整代码
最新发布
本文探讨了在图论中应用流量网络优化算法的实现方式，具体包括使用广度优先搜索（BFS）和深度优先搜索（DFS）算法进行最大流问题求解。通过实例展示了算法的应用步骤，并提供了代码实现细节。
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