1087 北大poj 解题报告,非常经典,超给力

1087A Plug For Unix
张法睿
2004-7-18
Part I 题目
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You are in charge of setting up the press room for the inaugural meeting of the United Nations Internet eXecutive(UNIX),which has an international mandate to make the freeflow of information and ideas on the Internet as cumbersome and bureaucratic as possible.Since the room was designed to accommodate reporters and journalists from around the world,it is equipped with electrical receptacles to suit the different shapes of plugs and voltages used by appliances in all of the countries that existed when the room was built.Unfortunately, the room was built many years ago when reporters used very few electric and electronic devices and is equipped with only one receptacle of each type.These days,like everyone else,reporters require many such devices to do their jobs: laptops,cell phones,tape recorders,pagers,coffee pots,microwave ovens,blow dryers,curling irons,tooth brushes,etc.Naturally,many of these devices can operate on batteries,but since the meeting is likely to be long and tedious,you want to be able to plug in as many as you can.Before the meeting begins,you gather up all the devices that the reporters would like to use,and attempt to set them up.You notice that some of the devices use plugs for which there is no receptacle.You wonder if these devices are from countries that didn’t exist when the room was built.For some receptacles,there are several devices that use the corresponding plug.For other receptacles,there are no devices that use the corresponding plug.In order to try to solve the problem you visit a nearby parts supply store.The store sells adapters that allow one type of plug to be used in a different type of outlet.Moreover,adapters are allowed to be plugged into other adapters.The store does not have adapters for all possible combinations of plugs and receptacles,but there is essentially an unlimited supply of the ones they do have.
中文简述有最多100个插座，100件电器，100种转接头，试求最多能有多少件电器可以插在插座上，注意：转接头可以串联
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Part II 分析
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这是网络最大流问题的一个经典题目，如果我们假设每件电器都消耗消耗单位功率的能量，那么能插上的最多的电器数，和电网中能量流的最大值是等价的。

同时，能量有统一的起点（电源）和统一的终点（耗散掉），可以视为一个单源单汇网络，这是网络最大流问题中一个比较简单的题型。

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Part III 算法
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朴素的网络最大流算法
将网络中的流量初始化为0
while（存在流量不满的路径）do
用尽可能大的流量填充这条路径。

返回网络中的流量
证明比较复杂，我们最后再涉及。

有兴趣的同学不妨阅读《算法导论》（INTRODUCTION TO ALGORITHMS2th,作者Thomas H.Cormen,Charles E.Leiserson,Ronald L.Rivest,Clifford Stein.MIT Press出版，高教出版社影印）的第26章，有比较详尽的论述。

数据结构两张无向图，分别描述网络中现有流量和最大流过能力。

因为题目的规模比较小，所以使用邻接矩阵描述。

空间复杂度O(n2)
流量不满路径的的查找算法我使用的是带标记的深度优先搜索，通过标记一个节点是否达到过，在哪种情况下达到过（探路过程中，或探路失败后，或探路成功后），来避免出现环路或者重复访问节点
因为每个节点至多被访问一次，寻找一条路经的时间复杂度为O(V)
在探路成功后返回时，通过一个队列记录所有经过的节点，即搜索到的流量不满的路径。

填充填充算法比较简单，根据找到的路径，找出该路径最“窄”处能容许的流量增量，时间复杂度比较粗略的估计O(V)
可能存在的问题如果数据不好的话，可能每次探路都只找到一条增量非常小的路径，以至最大流量增长的非常缓慢，但本题的流量属于一个很小的范围（0∼100），即便增长缓慢，时间的消耗也可以认为是一个常数。

6。