python --- 二分图匈牙利算法和KM算法

基础概念

关于匈牙利算法的基础概念就不作具体描述了，不清楚的可以自己搜索相关知识
主要需要了解的知识点

• 二分图
• 匹配：最大匹配，完美匹配
• 路径：交错路径，增广路径

算法核心：通过不断寻找增广路径找到最大匹配的道路

算法实现

1. 使用线性规划库scipy

默认取最小组合，设置maximize为True时取最大组合

import numpy as np
from scipy.optimize import linear_sum_assignment

a = np.array([[84, 65, 3, 34], [65, 56, 23, 35], [63, 18, 35, 12]])
row, col = linear_sum_assignment(a)
print("行坐标:", row, "列坐标:", col, "最小组合:", a[row, col])
row, col = linear_sum_assignment(a, True)	
print("行坐标:", row, "列坐标:", col, "最大组合:", a[row, col])

输出

行坐标: [0 1 2] 列坐标: [2 3 1] 最小组合: [ 3 35 18]
行坐标: [0 1 2] 列坐标: [0 1 2] 最大组合: [84 56 35]

2. 使用munkres库

源码：https://github.com/bmc/munkres
文档：http://software.clapper.org/munkres/

目前该库已经可以使用pip install munkres安装

默认是取最小组合，需要取最大组合则使用make_cost_matrix转换数据矩阵

import numpy as np
from munkres import Munkres, make_cost_matrix, DISALLOWED

a = np.array([[84, 65, 3, 34], [65, 56, 23, 35], [63, 18, 35, 12]])
b = make_cost_matrix(a, lambda cost: (a.max() - cost) if (cost != DISALLOWED) else DISALLOWED)

mk = Munkres()
# 最小组合
indexes = mk.compute(a.copy()) # 会改变输入的源数据
print("最小组合:",indexes, a[[i[0] for i in indexes], [i[1] for i in indexes]])
# 最大组合
indexes = mk.compute(b)
print("最大组合：", indexes, a[[i[0] for i in indexes], [i[1] for i in indexes]])

输出

最小组合：[(0, 2), (1, 3), (2, 1)] [ 3 35 18]
最大组合：[(0, 0), (1, 1), (2, 2)] [84 56 35]

注意使用np.array输入，mk.compute会改变输入的源数据

3. KM算法python实现

基本思想：通过引入顶标，将最优权值匹配转化为最大匹配问题
参考：
https://blog.youkuaiyun.com/u010510549/article/details/91350549
https://www.cnblogs.com/fzl194/p/8848061.html

实现了矩阵的自动补0和最大最小组合计算

import numpy as np

class KM:
    def __init__(self):
        self.matrix = None
        self.max_weight = 0
        self.row, self.col = 0, 0  # 源数据行列
        self.size = 0   # 方阵大小
        self.lx = None  # 左侧权值
        self.ly = None  # 右侧权值
        self.match = None   # 匹配结果
        self.slack = None   # 边权和顶标最小的差值
        self.visx = None    # 左侧是否加入增广路
        self.visy = None    # 右侧是否加入增广路

    # 调整数据
    def pad_matrix(self, min):
        if min:
            max = self.matrix.max() + 1
            self.matrix = max-self.matrix

        if self.row > self.col:   # 行大于列，添加列
            self.matrix = np.c_[self.matrix, np.array([[0] * (self.row - self.col)] * self.row)]
        elif self.col > self.row:  # 列大于行，添加行
            self.matrix = np.r_[self.matrix, np.array([[0] * self.col] * (self.col - self.row))]

    def reset_slack(self):
        self.slack.fill(self.max_weight + 1)

    def reset_vis(self):
        self.visx.fill(False)
        self.visy.fill(False)

    def find_path(self, x):
        self.visx[x] = True
        for y in range(self.size):
            if self.visy[y]:
                continue
            tmp_delta = self.lx[x] + self.ly[y] - self.matrix[x][y]
            if tmp_delta == 0:
                self.visy[y] = True
                if self.match[y] == -1 or self.find_path(self.match[y]):
                    self.match[y] = x
                    return True
            elif self.slack[y] > tmp_delta:
                self.slack[y] = tmp_delta

        return False

    def km_cal(self):
        for x in range(self.size):
            self.reset_slack()
            while True:
                self.reset_vis()
                if self.find_path(x):
                    break
                else:  # update slack
                    delta = self.slack[~self.visy].min()
                    self.lx[self.visx] -= delta
                    self.ly[self.visy] += delta
                    self.slack[~self.visy] -= delta

    def compute(self, datas, min=False):
        """
        :param datas: 权值矩阵
        :param min: 是否取最小组合，默认最大组合
        :return: 输出行对应的结果位置
        """
        self.matrix = np.array(datas) if not isinstance(datas, np.ndarray) else datas
        self.max_weight = self.matrix.sum()
        self.row, self.col = self.matrix.shape  # 源数据行列
        self.size = max(self.row, self.col)
        self.pad_matrix(min)
        print(self.matrix)
        self.lx = self.matrix.max(1)
        self.ly = np.array([0] * self.size, dtype=int)
        self.match = np.array([-1] * self.size, dtype=int)
        self.slack = np.array([0] * self.size, dtype=int)
        self.visx = np.array([False] * self.size, dtype=bool)
        self.visy = np.array([False] * self.size, dtype=bool)

        self.km_cal()

        match = [i[0] for i in sorted(enumerate(self.match), key=lambda x: x[1])]
        result = []
        for i in range(self.row):
            result.append((i, match[i] if match[i] < self.col else -1))  # 没有对应的值给-1
        return result


if __name__ == '__main__':
    a = np.array([[84, 65, 3, 34], [65, 56, 23, 35], [63, 18, 35, 12]])
    # a = np.array([[84, 65], [3, 34], [63, 18], [35, 12]])
   
    km = KM()
    min_ = km.compute(a.copy(), True)
    print("最小组合:", min_, a[[i[0] for i in min_], [i[1] for i in min_]])
    max_ = km.compute(a.copy())
    print("最大组合:", max_, a[[i[0] for i in max_], [i[1] for i in max_]])

输出：

最小组合: [(0, 2), (1, 3), (2, 1)] [ 3 35 18]
最大组合: [(0, 0), (1, 1), (2, 2)] [84 56 35]