原生
python
运行速度很慢,只要数据量大于500
,求解就变得十分困难
五组测试数据
测试用例的第一行为图的节点数和边数,第二行为最大流算法的起始节点和中止节点,剩余所有行均为有向加权边,其中前两个数字代表边的两个端点,后一个数字代表边的权重。
·测试用例1(将txt
压缩为xz
后使用base16384转为文本)
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·测试用例2(将txt
压缩为xz
后使用base16384转为文本)
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·测试用例3、4、5
原生代码实现
思路是保持解的最优性,寻找解的可行性。
为实现此目的,为每个点赋予一个高度值,其中源点高度在初始时设置为节点数目。
源点具有无限容量(盈余),每个点只能向比自己高度低的点推流,且每次推流都尽可能大。
如果自己有盈余却没有任何点可以推流,则增加自己的高度(Relabel
)后再次尝试。
算法直到除了源点和宿点外,所有点的盈余都为0(找到可行解)为止。
#!/usr/bin/env python3
#preflow.py
#fumiama 20201225
from time import time
from heapq import heappop, heappush, heapify
class Node(object):
def __init__(self):
self.remain = 0
self.height = 0
def relabel(self): self.height += 1
def push(self, flow): self.remain += flow
def pour(self, flow): self.remain -= flow
#def __repr__(self): return "盈余: "+str(self.remain)+" 高度: "+str(self.height)
class Graph(object):
def __init__(self, nodeSize):
self.nodeSize = nodeSize
self.node2 = dict()
self.nodes = [Node() for i in range(nodeSize+1)]
def addEdge(self, a, b, w):
if a not in self.node2.keys(): self.node2[a] = []
heappush(self.node2[a], (b, w))
def addWeight(self, a, b, w):
#print("addWeight:", a, b, w)
if a not in self.node2.keys(): self.addEdge(a, b, w)
else:
notChange = 1
for node in self.node2[a]:
if node[0] == b:
index = self.node2[a].index(node)
#print("index:", index, "node:", node)
previousWeight = self.node2[a][index][1]
newWeight = previousWeight + w
if newWeight: self.node2[a][index] = (b, newWeight)
else: self.delEdge(a, b)
notChange = 0
break
if notChange: self.addEdge(a, b, w)
def delEdge(self, a, b):
for node in self.node2[a]:
if node[0] == b: self.node2[a].remove(node)
heapify(self.node2[a])
def getWeight(self, a, b):
if a in self.node2.keys():
for node in self.node2[a]:
if node[0] == b: return node[1]
return -1
def findMaxFlow(self, s, t):
stack = []
self.nodes[s].height = self.nodeSize
while self.node2[s]:
node = self.node2[s].pop() #删除正向边
self.nodes[node[0]].remain = node[1]
stack.append(node[0])
self.addWeight(node[0], s, node[1]) #反向加边
##print("初始点:", stack)
while stack:
#print("stack:", stack)
node = stack.pop()
nodeObj = self.nodes[node]
#print("弹出:", node, nodeObj)
if nodeObj.remain and node != t:
noPush = 1
#print("图:", self.node2)
for nextNode in self.node2[node]:
#print("(", node, ", ", nodeObj.remain, ")->", nextNode, sep='')
nextNodeObj = self.nodes[nextNode[0]]
if nodeObj.height > nextNodeObj.height:
noPush = 0
if nodeObj.remain <= nextNode[1]:
nextNodeObj.push(nodeObj.remain)
self.addWeight(node, nextNode[0], -nodeObj.remain)
#print("addWeight:", -nodeObj.remain)
self.addWeight(nextNode[0], node, nodeObj.remain)
#print("addBack:", nodeObj.remain)
nodeObj.remain = 0
#print("非饱和推送到:", nextNode, "该点", nextNodeObj)
else:
nextNodeObj.push(nextNode[1])
nodeObj.pour(nextNode[1])
self.delEdge(node, nextNode[0])
self.addWeight(nextNode[0], node, nextNode[1])
stack.append(node)
#print("饱和推送到:", nextNode, "该点", nextNodeObj)
if nextNode[0] != t and nextNode[0] != s and nextNodeObj.remain: stack.append(nextNode[0])
if noPush and self.node2[node]:
nodeObj.relabel()
stack.append(node)
#print("relabel:", node, "to", nodeObj.height)
#print('')
return self.nodes[t].remain
if __name__ == '__main__':
nodeSize, edgeSize = map(int, input().split())
s, t = map(int, input().split())
g = Graph(nodeSize)
while edgeSize:
a, b, w = map(int, input().split())
g.addEdge(a, b, w)
edgeSize -= 1
time_start = time()
print("最大流:", g.findMaxFlow(s, t))
print("用时:", time() - time_start)
原生实现的样例测试
$ ./preflow.py < 测试用例1.txt
最大流: 13
用时: 0.00015997886657714844
$ ./preflow.py < 测试用例2.txt
最大流: 38
用时: 0.001840829849243164
$ ./preflow.py < 测试用例3.txt
最大流: 869
用时: 0.011828899383544922
$ ./preflow.py < 测试用例4.txt
最大流: 17639
用时: 207.51951098442078
$ ./preflow.py < 测试用例5.txt
(十分钟内未计算完成)
可以看到,随着点的数量增加,求解所需时间迅速呈指数形式增加。但是相比Edmonds-Karp算法,速度已经有了不小的提升。
造成运行缓慢的一个主要原因是python
原生代码的缓慢,因此考虑使用cython
重写算法代码以加快求解速度。cython
会将我们的代码编译为以c语言
实现的python
库供我们调用。
为此,需要编写以下文件,并手动编译库文件:
pyx文件(代码主体)
#cython: language_level=3
#preflow_c.pyx
#fumiama 20201226
from time import time
from libc.string cimport memcpy, memset
from libc.stdlib cimport malloc, free
cdef int nodeSize
cdef int** arr
cdef set(int x, int y, int item):
global arr
#print("set", x, y, item)
arr[x][y] = item
cdef int get(int x, int y):
global arr
#print("get", x, y)
return arr[x][y]
cpdef createArray(int x, int y):
global arr
cdef int i
arr = <int**>malloc(x * sizeof(int*))
for i in range(x):
arr[i] = <int*>malloc(y * sizeof(int))
memset(arr[i], 0, y * sizeof(int))
cdef int* stack
cdef int sp = 0
cdef int* touch
cdef initStack(int length):
global stack, sp, touch
stack = <int*>malloc(length)
touch = <int*>malloc(length)
memset(touch, 0, length)
sp = 0
cdef push(int item):
global stack, sp, touch
stack[sp] = item
touch[item] = 1
sp += 1
cdef int pop():
global stack, sp, touch
cdef int re
sp -= 1
re = stack[sp]
touch[re] = 0
return re
cdef int inStack(int item):
global touch
return touch[item]
#cdef printStack():
# global stack, sp
# cdef int i
# print("stack:[", end=' ')
# for i in range(sp): print(stack[i], end=" ")
# print(']')
cdef int* remain
cdef int* height
cdef initGraph(int nSize):
global nodeSize, arr, remain, height
nodeSize = nSize
length = nSize * sizeof(int)
createArray(nSize, nSize)
remain = <int*>malloc(length)
height = <int*>malloc(length)
memset(remain, 0, length)
memset(height, 0, length)
cdef addEdge(int a, int b, int w): set(a-1, b-1, w)
cdef addWeight(int a, int b, int w): set(a, b, w + get(a, b))
cdef delEdge(int a, int b): set(a, b, 0)
cdef int findMaxFlow(int s, int t):
global nodeSize, height, sp
cdef int length = nodeSize * sizeof(int)
cdef int sM1 = s-1
cdef int tmp, noPush, nextNode, notEmpty
initStack(length)
#print("Stack initialized.")
height[sM1] = nodeSize
for nextNode in range(nodeSize):
tmp = get(sM1, nextNode)
if tmp:
#print("设置出边:", s, nextNode+1, tmp)
set(sM1, nextNode, 0)
remain[nextNode] = tmp
push(nextNode)
addWeight(nextNode, sM1, tmp)
#print("Pushed from souce")
while sp:
#printStack()
node = pop()
#print(sp, end=' ')
if remain[node] and node != t-1:
noPush = 1
for nextNode in range(nodeSize):
tmp = get(node, nextNode)
#print("inStack", nextNode, inStack(nextNode))
if tmp and height[node] > height[nextNode]:
noPush = 0
#print("找到边:", node, nextNode, tmp)
if remain[node] <= tmp:
if nextNode != s-1: remain[nextNode] += remain[node] #nextNodeObj.push(nodeObj.remain)
addWeight(node, nextNode, -remain[node])
addWeight(nextNode, node, remain[node])
remain[node] = 0
else:
if nextNode != s-1: remain[nextNode] += tmp
remain[node] -= tmp
delEdge(node, nextNode)
addWeight(nextNode, node, tmp)
if not inStack(node) and node != t-1: push(node)
#print(remain[node], end=' ')
if nextNode+1 != t and nextNode+1 != s and remain[nextNode] and not inStack(nextNode): push(nextNode)
notEmpty = 0
for nextNode in range(nodeSize):
tmp = get(node, nextNode)
if tmp and nextNode+1 != s:
notEmpty = 1
break
if noPush and notEmpty and remain[node]:
height[node] += 1
push(node)
return remain[t-1]
def main():
cdef int nodeSize, edgeSize, s, t, a, b, w
cdef double time_start
nodeSize, edgeSize = map(int, input().split())
s, t = map(int, input().split())
initGraph(nodeSize)
#print("Graph initialized.")
while edgeSize:
a, b, w = map(int, input().split())
#print("Add edge:", a, b, w)
addEdge(a, b, w)
edgeSize -= 1
time_start = time()
print("最大流:", findMaxFlow(s, t))
print("用时:", time() - time_start)
setup文件(构建库)
# cython: language_level=3
from distutils.core import setup
from Cython.Build import cythonize
setup(
ext_modules = cythonize("preflow_c.pyx")
)
编译
$ python3 ./preflow_c_setup.py build_ext --inplace
调用库
#edmonds_c_run.py
import preflow_c
preflow_c.main()
进行测试
$ ./preflow_c_run.py < 测试用例1.txt
最大流: 13
用时: 3.0994415283203125e-05
$ ./preflow_c_run.py < 测试用例2.txt
最大流: 38
用时: 6.29425048828125e-05
$ ./preflow_c_run.py < 测试用例3.txt
最大流: 869
用时: 0.0002541542053222656
$ ./preflow_c_run.py < 测试用例4.txt
最大流: 17639
用时: 0.7053031921386719
$ ./preflow_c_run.py < 测试用例5.txt
最大流: 62401
用时: 4.692018032073975
可见在使用cython
优化之后,运行速度有了极大提升,基本可以胜任大部分场景下的计算需求。