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-https://leetcode-cn.com/problems/reconstruct-itinerary/ +> 这也可以用回溯法? 其实深搜和回溯也是相辅相成的,毕竟都用递归。 -# 332. 重新安排行程 +# 332.重新安排行程 + +题目地址:https://leetcode-cn.com/problems/reconstruct-itinerary/ + +给定一个机票的字符串二维数组 [from, to],子数组中的两个成员分别表示飞机出发和降落的机场地点,对该行程进行重新规划排序。所有这些机票都属于一个从 JFK(肯尼迪国际机场)出发的先生,所以该行程必须从 JFK 开始。 + +提示: +* 如果存在多种有效的行程,请你按字符自然排序返回最小的行程组合。例如,行程 ["JFK", "LGA"] 与 ["JFK", "LGB"] 相比就更小,排序更靠前 +* 所有的机场都用三个大写字母表示(机场代码)。 +* 假定所有机票至少存在一种合理的行程。 +* 所有的机票必须都用一次 且 只能用一次。 +  + +示例 1: +输入:[["MUC", "LHR"], ["JFK", "MUC"], ["SFO", "SJC"], ["LHR", "SFO"]] +输出:["JFK", "MUC", "LHR", "SFO", "SJC"] + +示例 2: +输入:[["JFK","SFO"],["JFK","ATL"],["SFO","ATL"],["ATL","JFK"],["ATL","SFO"]] +输出:["JFK","ATL","JFK","SFO","ATL","SFO"] +解释:另一种有效的行程是 ["JFK","SFO","ATL","JFK","ATL","SFO"]。但是它自然排序更大更靠后。 # 思路 -举一个有重复机场的例子: +这道题目还是很难的,之前我们用回溯法解决了如下问题:[组合问题](https://mp.weixin.qq.com/s/OnBjbLzuipWz_u4QfmgcqQ),[分割问题](https://mp.weixin.qq.com/s/v--VmA8tp9vs4bXCqHhBuA),[子集问题](https://mp.weixin.qq.com/s/NNRzX-vJ_pjK4qxohd_LtA),[排列问题](https://mp.weixin.qq.com/s/SCOjeMX1t41wcvJq49GhMw)。 - +直觉上来看 这道题和回溯法没有什么关系,更像是图论中的深度优先搜索。 + +实际上确实是深搜,但这是深搜中使用了回溯的例子,在查找路径的时候,如果不回溯,怎么能查到目标路径呢。 + +所以我倾向于说本题应该使用回溯法,那么我也用回溯法的思路来讲解本题,其实深搜一般都使用了回溯法的思路,在图论系列中我会再详细讲解深搜。 + +**这里就是先给大家拓展一下,原来回溯法还可以这么玩!** + +**这道题目有几个难点:** + +1. 一个行程中,如果航班处理不好容易变成一个圈,成为死循环 +2. 有多种解法,字母序靠前排在前面,让很多同学望而退步,如何该记录映射关系呢 ? +3. 使用回溯法(也可以说深搜) 的话,那么终止条件是什么呢? +4. 搜索的过程中,如何遍历一个机场所对应的所有机场。 + +针对以上问题我来逐一解答! + +## 如何理解死循环 + +对于死循环,我来举一个有重复机场的例子: + +![332.重新安排行程](https://img-blog.csdnimg.cn/20201115180537865.png) 为什么要举这个例子呢,就是告诉大家,出发机场和到达机场也会重复的,**如果在解题的过程中没有对集合元素处理好,就会死循环。** -这道题目有几个难点: +## 该记录映射关系 -1. 有多种解法,字母序靠前排在前面,让很多同学望而退步,如何该记录映射关系呢 ? -2. 这是一个图不是一棵树,使用深搜/回溯 终止条件是什么呢? -3. 回溯的过程中,如何遍历一个城市所对应的所有城市。 +有多种解法,字母序靠前排在前面,让很多同学望而退步,如何该记录映射关系呢 ? -首先这道题目是使用回溯法(也可以说深搜),那么按照我总结的回溯模板来。 - -``` -backtracking() { - if (终止条件) { - 存放结果; - } - - for (枚举同一个位置的所有可能性,可以想成节点孩子的数量) { - 递归,处理节点; - backtracking(); - 回溯,撤销处理结果 - } -} -``` - -## 1. 有多种解法,字母序靠前排在前面,让很多同学望而退步,如何该记录映射关系呢 ? - -一个城市映射多个城市,城市之间要靠字母序排列,一个城市映射多个城市,可以使用std::unordered_map,如果让多个城市之间再有顺序的话,就是用std::map 或者 或者std::multimap 或者 std::multiset。 +一个机场映射多个机场,机场之间要靠字母序排列,一个机场映射多个机场,可以使用std::unordered_map,如果让多个机场之间再有顺序的话,就是用std::map 或者std::multimap 或者 std::multiset。 如果对map 和 set 的实现机制不太了解,也不清楚为什么 map、multimap就是有序的同学,可以看这篇文章[关于哈希表,你该了解这些!](https://mp.weixin.qq.com/s/g8N6WmoQmsCUw3_BaWxHZA)。 @@ -44,34 +64,112 @@ backtracking() { 含义如下: -`unordered_map> targets`:`unordered_map<出发城市, 到达城市的集合> targets` -`unordered_map> targets`:`unordered_map<出发城市, map<到达城市, 航班次数>> targets` +`unordered_map> targets`:`unordered_map<出发机场, 到达机场的集合> targets` +`unordered_map> targets`:`unordered_map<出发机场, map<到达机场, 航班次数>> targets` -这两个结构,我们选择了后者,因为如果使用`unordered_map> targets` 遍历multiset的时候,不能删除元素,一旦删除元素,迭代器就失效了。而本地在回溯的过程中就是要不断的增删 multiset里的元素,所以 我们使用`unordered_map> targets`。 +这两个结构,我选择了后者,因为如果使用`unordered_map> targets` 遍历multiset的时候,不能删除元素,一旦删除元素,迭代器就失效了。 -在遍历 `unordered_map<出发城市, map<到达城市, 航班次数>> targets`的过程中,可以使用航班次数这个字段的数字 --或者++,来标记到达城市是否使用过了,而不用对集合做删除元素或者增加元素的操作。 +**再说一下为什么一定要增删元素呢,正如开篇我给出的图中所示,出发机场和到达机场是会重复的,搜索的过程没及时删除目的机场就会死循环。** -## 2. 这是一个图不是一棵树,使用深搜/回溯 终止条件是什么呢? +所以搜索的过程中就是要不断的删multiset里的元素,那么推荐使用`unordered_map> targets`。 -你看有多少个航班,那题目中的示例为例,输入: [["MUC", "LHR"], ["JFK", "MUC"], ["SFO", "SJC"], ["LHR", "SFO"]] ,这是有4个航班,那么只要找出一种行程,行程里的机场个数是5就可以了。 +在遍历 `unordered_map<出发机场, map<到达机场, 航班次数>> targets`的过程中,**可以使用"航班次数"这个字段的数字做相应的增减,来标记到达机场是否使用过了。** -所以终止条件 我们回溯遍历的过程中,遇到的机场个数,如果达到了(航班数量+1),那么我们就找到了一个行程,把所有航班串在一起了。 -## 3. 回溯的过程中,如何遍历一个城市所对应的所有城市。 +如果“航班次数”大于零,说明目的地还可以飞,如果如果“航班次数”等于零说明目的地不能飞了,而不用对集合做删除元素或者增加元素的操作。 -这里刚刚说过,在选择映射函数的时候,不能选择`unordered_map> targets`, 因为一旦有元素增删multiset的迭代器就会失效,有一些题解使用了 例如list的迭代器,使用splice来保证 迭代器不失效。 +**相当于说我不删,我就做一个标记!** -可以说既要找到一个对数据经行排序的容器,而且这个容易增删元素,迭代器还不能失效。 +## 回溯法 -**再说一下为什么一定要增删元素呢,正如开篇我给出的图中所示,出发机场和到达机场是会重复的,搜索的过程没及时删除元素就会死循环。** +这道题目我使用回溯法,那么下面按照我总结的回溯模板来: -所以我选择了`unordered_map> targets` 来基于映射条件。 +``` +void backtracking(参数) { + if (终止条件) { + 存放结果; + return; + } + + for (选择:本层集合中元素(树中节点孩子的数量就是集合的大小)) { + 处理节点; + backtracking(路径,选择列表); // 递归 + 回溯,撤销处理结果 + } +} +``` + +本题以输入:[["JFK", "KUL"], ["JFK", "NRT"], ["NRT", "JFK"]为例,抽象为树形结构如下: + + + +开始回溯三部曲讲解: + +* 递归函数参数 + +在讲解映射关系的时候,已经讲过了,使用`unordered_map> targets;` 来记录航班的映射关系,我定义为全局变量。 + +当然把参数放进函数里传进去也是可以的,我是尽量控制函数里参数的长度。 + +参数里还需要ticketNum,表示有多少个航班(终止条件会用上)。 + +代码如下: + +``` +// unordered_map<出发机场, map<到达机场, 航班次数>> targets +unordered_map> targets; +bool backtracking(int ticketNum, vector& result) { +``` + +**注意函数返回值我用的是bool!** + +我们之前讲解回溯算法的时候,一般函数返回值都是void,这次为什么是bool呢? + +因为我们只需要找到一个行程,就是在树形结构中唯一的一条通向叶子节点的路线,如图: + + + +所以找到了这个叶子节点了直接返回,这个递归函数的返回值问题我们在讲解二叉树的系列的时候,在这篇[二叉树:递归函数究竟什么时候需要返回值,什么时候不要返回值?](https://mp.weixin.qq.com/s/6TWAVjxQ34kVqROWgcRFOg)详细介绍过。 + +当然本题的targets和result都需要初始化,代码如下: +``` +for (const vector& vec : tickets) { + targets[vec[0]][vec[1]]++; // 记录映射关系 +} +result.push_back("JFK"); // 起始机场 +``` + +* 递归终止条件 + +拿题目中的示例为例,输入: [["MUC", "LHR"], ["JFK", "MUC"], ["SFO", "SJC"], ["LHR", "SFO"]] ,这是有4个航班,那么只要找出一种行程,行程里的机场个数是5就可以了。 + +所以终止条件是:我们回溯遍历的过程中,遇到的机场个数,如果达到了(航班数量+1),那么我们就找到了一个行程,把所有航班串在一起了。 + +代码如下: + +``` +if (result.size() == ticketNum + 1) { + return true; +} +``` + +已经看习惯回溯法代码的同学,到叶子节点了习惯性的想要收集结果,但发现并不需要,本题的result相当于 [回溯算法:求组合总和!](https://mp.weixin.qq.com/s/HX7WW6ixbFZJASkRnCTC3w)中的path,也就是本题的result就是记录路径的(就一条),在如下单层搜索的逻辑中result就添加元素了。 + +* 单层搜索的逻辑 + +回溯的过程中,如何遍历一个机场所对应的所有机场呢? + +这里刚刚说过,在选择映射函数的时候,不能选择`unordered_map> targets`, 因为一旦有元素增删multiset的迭代器就会失效,当然可能有牛逼的容器删除元素迭代器不会失效,这里就不在讨论了。 + +**可以说本题既要找到一个对数据进行排序的容器,而且还要容易增删元素,迭代器还不能失效**。 + +所以我选择了`unordered_map> targets` 来做机场之间的映射。 遍历过程如下: ``` for (pair& target : targets[result[result.size() - 1]]) { - if (target.second > 0 ) { + if (target.second > 0 ) { // 记录到达机场是否飞过了 result.push_back(target.first); target.second--; if (backtracking(ticketNum, index + 1, result)) return true; @@ -81,24 +179,23 @@ backtracking() { } ``` -可以看出 通过`unordered_map> targets`里的int字段来判断 这个集合使用使用完了,这样避免了 增删元素。 +可以看出 通过`unordered_map> targets`里的int字段来判断 这个集合里的机场是否使用过,这样避免了直接去删元素。 -此时完整代码如下: +分析完毕,此时完整C++代码如下: # C++代码 ``` - class Solution { private: -// unordered_map<出发城市, map<到达城市, 航班次数>> targets +// unordered_map<出发机场, map<到达机场, 航班次数>> targets unordered_map> targets; bool backtracking(int ticketNum, vector& result) { if (result.size() == ticketNum + 1) { return true; } for (pair& target : targets[result[result.size() - 1]]) { - if (target.second > 0 ) { // 使用int字段来记录到达城市是否使用过了 + if (target.second > 0 ) { // 记录到达机场是否飞过了 result.push_back(target.first); target.second--; if (backtracking(ticketNum, result)) return true; @@ -110,16 +207,31 @@ bool backtracking(int ticketNum, vector& result) { } public: vector findItinerary(vector>& tickets) { + targets.clear(); vector result; for (const vector& vec : tickets) { targets[vec[0]][vec[1]]++; // 记录映射关系 } - result.push_back("JFK"); + result.push_back("JFK"); // 起始机场 backtracking(tickets.size(), result); return result; } }; - ``` +一波分析之后,可以看出我就是按照回溯算法的模板来的。 + +# 总结 + +本题其实可以算是一道hard的题目了,关于本题的难点我在文中已经列出了。 + +**如果单纯的回溯搜索(深搜)并不难,难还难在容器的选择和使用上**。 + +本题其实是一道深度优先搜索的题目,但是我完全使用回溯法的思路来讲解这道题题目,**算是给大家拓展一下思维方式,其实深搜和回溯也是分不开的,毕竟最终都是用递归**。 + +如果最终代码,发现照着回溯法模板画的话好像也能画出来,但难就难如何知道可以使用回溯,以及如果套进去,所以我再写了这么长的一篇来详细讲解。 + +就酱,很多录友表示和「代码随想录」相见恨晚,那么帮Carl宣传一波吧,让更多同学知道这里! + + > 更多算法干货文章持续更新,可以微信搜索「代码随想录」第一时间围观,关注后,回复「Java」「C++」 「python」「简历模板」「数据结构与算法」等等,就可以获得我多年整理的学习资料。