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使用贪心算法解决集合覆盖问题

原创 作者:jeanron100 时间:2018-04-06 16:22:44 0 删除 编辑

使用贪心算法解决集合覆盖问题

在《算法图解》里面有一个蛮有意思的小案例,背景是一个广播节目,要让全美的50个周的听众都能够听到,但是每个电台可能覆盖多个州,每在一个电台播出就需要一笔费用,所以就是从成本的角度来看,怎么尽可能在所有的州都播出,这是一个典型的集合覆盖的问题,而且在我们的生活中算是比较典型。

比如我们先缩小范围,指定5个州,那么50个州也是同样的算法。

states_need = set(["mt", "wa", "or", "id", "nv", "ut", "ca", "az"]) # 传入一个数组, 它被转换为集合

有的同学可能对这些州没概念,这个简称就跟京代表北京,鲁代表山东,甘代表甘肃一样,细细一看,都是西部的一些州。

使用贪心算法解决集合覆盖问题

如何使用贪心算法呢,就是选择覆盖尽可能多的州的电台,然后逐步缩小范围。那么覆盖面广的州所对应的电台就优先被选中,依次类推。

程序的实现是指定了一个集合states_need,里面包含所有的州,每个电台对应的州是通过初始化的数组元素来实现的,按照一二三四五的顺序来命名,当然实际上这种元素的排列set不是按照数组名的顺序,在这个场景里是kfive,ktwo,kthree,kone,kfour

然后逐步缩小范围来收敛,里面比较特别的一点就是集合的运算,使用了 & ,得到的是交集,如果是并集是 |,差集是 -,

程序代码如下:

#!/usr/bin/env
 python# coding:utf-8states_need = set(["mt", "wa", "or", "id", "nv", 
"ut", "ca", "az"]) # 传入一个数组, 它被转换为集合# 可供选择的广播台清单stations = 
{}stations["kone"] = set(["id", "nv", "ut"])stations["ktwo"] = 
set(["wa", "id", "mt"])stations["kthree"] = set(["or", "nv", 
"ca"])stations["kfour"] = set(["nv", "ut"])stations["kfive"] = 
set(["ca", "az"])print(stations)# 最终选择的广播台集合final_stations = set()while 
states_need:best_station = Nonestates_covered = set()for station, 
states_for_station in stations.items():covered = states_need & 
states_for_station # 
求交集print("states_need:",states_need,"states_for_station:",states_for_station,"covered:",covered)if
 len(covered) > len(states_covered):best_station = 
stationstates_covered = coveredstates_need -= 
states_coveredfinal_stations.add(best_station)print("states_needed:",states_need,"best_station:",best_station,"final_stations:",final_stations)print("---")print("Final_stations:",final_stations)

为了方便调试和得到一个迭代的结果,我加了几处输出日志,工参考。

{'kfive':
 set(['ca', 'az']), 'ktwo': set(['mt', 'id', 'wa']), 'kthree': 
set(['ca', 'or', 'nv']), 'kone': set(['ut', 'id', 'nv']), 'kfour': 
set(['ut', 'nv'])}('states_need:', set(['wa', 'ut', 'ca', 'id', 'mt', 
'az', 'or', 'nv']), 'states_for_station:', set(['ca', 'az']), 
'covered:', set(['ca', 'az']))('states_needed:', set(['wa', 'ut', 'id', 
'mt', 'or', 'nv']), 'best_station:', 'kfive', 'final_stations:', 
set(['kfive']))---('states_need:', set(['wa', 'ut', 'id', 'mt', 'or', 
'nv']), 'states_for_station:', set(['mt', 'id', 'wa']), 'covered:', 
set(['mt', 'id', 'wa']))('states_needed:', set(['ut', 'or', 'nv']), 
'best_station:', 'ktwo', 'final_stations:', set(['ktwo', 
'kfive']))---('states_need:', set(['ut', 'or', 'nv']), 
'states_for_station:', set(['ca', 'or', 'nv']), 'covered:', set(['or', 
'nv']))('states_needed:', set(['ut', 'or', 'nv']), 'best_station:', 
'ktwo', 'final_stations:', set(['ktwo', 'kfive']))---('states_need:', 
set(['ut', 'or', 'nv']), 'states_for_station:', set(['ut', 'id', 'nv']),
 'covered:', set(['ut', 'nv']))('states_needed:', set(['ut', 'or', 
'nv']), 'best_station:', 'ktwo', 'final_stations:', set(['ktwo', 
'kfive']))---('states_need:', set(['ut', 'or', 'nv']), 
'states_for_station:', set(['ut', 'nv']), 'covered:', set(['ut', 
'nv']))('states_needed:', set(['ut', 'or', 'nv']), 'best_station:', 
'ktwo', 'final_stations:', set(['ktwo', 'kfive']))---('states_need:', 
set(['ut', 'or', 'nv']), 'states_for_station:', set(['ca', 'az']), 
'covered:', set([]))('states_needed:', set(['ut', 'or', 'nv']), 
'best_station:', None, 'final_stations:', set(['ktwo', None, 
'kfive']))---('states_need:', set(['ut', 'or', 'nv']), 
'states_for_station:', set(['mt', 'id', 'wa']), 'covered:', 
set([]))('states_needed:', set(['ut', 'or', 'nv']), 'best_station:', 
None, 'final_stations:', set(['ktwo', None, 
'kfive']))---('states_need:', set(['ut', 'or', 'nv']), 
'states_for_station:', set(['ca', 'or', 'nv']), 'covered:', set(['or', 
'nv']))('states_needed:', set(['ut']), 'best_station:', 'kthree', 
'final_stations:', set(['ktwo', 'kthree', None, 
'kfive']))---('states_need:', set(['ut']), 'states_for_station:', 
set(['ut', 'id', 'nv']), 'covered:', set(['ut']))('states_needed:', 
set(['ut']), 'best_station:', 'kthree', 'final_stations:', set(['ktwo', 
'kthree', None, 'kfive']))---('states_need:', set(['ut']), 
'states_for_station:', set(['ut', 'nv']), 'covered:', 
set(['ut']))('states_needed:', set(['ut']), 'best_station:', 'kthree', 
'final_stations:', set(['ktwo', 'kthree', None, 
'kfive']))---('states_need:', set(['ut']), 'states_for_station:', 
set(['ca', 'az']), 'covered:', set([]))('states_needed:', set(['ut']), 
'best_station:', None, 'final_stations:', set(['ktwo', 'kthree', None, 
'kfive']))---('states_need:', set(['ut']), 'states_for_station:', 
set(['mt', 'id', 'wa']), 'covered:', set([]))('states_needed:', 
set(['ut']), 'best_station:', None, 'final_stations:', set(['ktwo', 
'kthree', None, 'kfive']))---('states_need:', set(['ut']), 
'states_for_station:', set(['ca', 'or', 'nv']), 'covered:', 
set([]))('states_needed:', set(['ut']), 'best_station:', None, 
'final_stations:', set(['ktwo', 'kthree', None, 
'kfive']))---('states_need:', set(['ut']), 'states_for_station:', 
set(['ut', 'id', 'nv']), 'covered:', set(['ut']))('states_needed:', 
set([]), 'best_station:', 'kone', 'final_stations:', set(['ktwo', 
'kthree', None, 'kfive', 'kone']))---('states_need:', set([]), 
'states_for_station:', set(['ut', 'nv']), 'covered:', 
set([]))('states_needed:', set([]), 'best_station:', 'kone', 
'final_stations:', set(['ktwo', 'kthree', None, 'kfive', 
'kone']))---('Final_stations:', set(['ktwo', 'kthree', None, 'kfive', 
'kone']))

最后的结果是:ktwo,kthree,kfive,kone这四个电台。

当然贪心算法得到的不是精确的结果,即可能不是最优解,算是一种近似算法,能够基本得到的最优解,而且效率很高。

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