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zNW45Yl4#{=K26V0a-FkupUjs}(370#64de5h^z{c6gkSR!%qlM1_kjIibCInS8ME4 z-4qyE5rpNiB~}PBF)@McICvr{*!eVA8XFr|T^hr&v&(~!(s|cx2^wC6k;eQYd-@Js ztLSlP|OWCJ?l&_W!W=Hy#(Gm|l&BZep_-{>4-Zn{i?*y5tq-Av! z0Oz0}NKf~gYa^^$10KFFDDDrWt6d1Y{06hQU{n{3AnXad+|!iY(fmns)oqg=meF~Z zj+sB%@dc^d0_`Bv?4dKtbmgY#pPx%OWO_Q!C|^FgLHuON0+c94+JhNn1c;n!&h$kJ zibnaJf8V(L`Ak8Te4*dVS_x_(?jxEB% z!ps^+7k;_4i3 看懂很容易,彻底掌握需要下功夫 + # 第39题. 组合总和 +题目链接:https://leetcode-cn.com/problems/combination-sum/ + 给定一个无重复元素的数组 candidates 和一个目标数 target ,找出 candidates 中所有可以使数字和为 target 的组合。 candidates 中的数字可以无限制重复被选取。 -**说明:** +说明: -所有数字(包括 target)都是正整数。 -解集不能包含重复的组合。  +* 所有数字(包括 target)都是正整数。 +* 解集不能包含重复的组合。  -示例 1: +示例 1: +输入:candidates = [2,3,6,7], target = 7, +所求解集为: +[ + [7], + [2,2,3] +] -输入:candidates = [2,3,6,7], target = 7, -所求解集为: -[ - [7], - [2,2,3] -] - -示例 2: - -输入:candidates = [2,3,5], target = 8, -所求解集为: -[ -  [2,2,2,2], -  [2,3,3], -  [3,5] -] +示例 2: +输入:candidates = [2,3,5], target = 8, +所求解集为: +[ +  [2,2,2,2], +  [2,3,3], +  [3,5] +] # 思路 题目中的**无限制重复被选取,吓得我赶紧想想 出现0 可咋办**,然后看到下面提示:1 <= candidates[i] <= 200,我就放心了。 +本题和[回溯算法:求组合问题!](https://mp.weixin.qq.com/s/OnBjbLzuipWz_u4QfmgcqQ),[回溯算法:求组合总和!](https://mp.weixin.qq.com/s/HX7WW6ixbFZJASkRnCTC3w)和区别是:本题没有数量要求,可以无限重复,但是有总和的限制,所以间接的也是有个数的限制。 -这道题上来可以这么想,看看一个数能不能构成target,一个for循环遍历一遍,再看看两个数能不能构成target,两个for循环遍历,在看看三个数能不能构成target,三个for循环遍历,直到candidates.size()个for循环遍历一遍。 - -遇到这种问题,就要想到递归的层级嵌套关系就可以解决这种多层for循环的问题,而回溯则帮我们选择每一个合适的集合! - -那么使用回溯的时候,要知道求的是排列,还是组合,排列和组合是不一样的。 - -一些同学可能海分不清,我大概说一下: - -**组合是不强调元素顺序的,排列是强调元素顺序的。** - -例如 集合 1,2 和 集合 2,1 在组合上,就是一个集合,因为不强调顺序,而要是排列的话,1,2 和 2,1 就是两个集合了。 - -**求组合,和求排列的回溯写法是不一样的,代码上有小小细节上的改变。** - -本题选组过程如下: +本题搜索的过程抽象成树形结构如下: +注意图中叶子节点的返回条件,因为本题没有组合数量要求,仅仅是总和的限制,所以递归没有层数的限制,只要选取的元素总和超过target,就返回! + +而在[回溯算法:求组合问题!](https://mp.weixin.qq.com/s/OnBjbLzuipWz_u4QfmgcqQ)和[回溯算法:求组合总和!](https://mp.weixin.qq.com/s/HX7WW6ixbFZJASkRnCTC3w) 中都可以知道要递归K层,因为要取k个元素的组合。 + +## 回溯三部曲 + +* 递归函数参数 + +这里依然是定义两个全局变量,二维数组result存放结果集,数组path存放符合条件的结果。(这两个变量可以作为函数参数传入) + +首先是题目中给出的参数,集合candidates, 和目标值target。 + +此外我还定义了int型的sum变量来统计单一结果path里的总和,其实这个sum也可以不用,用target做相应的减法就可以了,最后如何target==0就说明找到符合的结果了,但为了代码逻辑清晰,我依然用了sum。 + +**本题还需要startIndex来控制for循环的起始位置,对于组合问题,什么时候需要startIndex呢?** + +我举过例子,如果是一个集合来求组合的话,就需要startIndex,例如:[回溯算法:求组合问题!](https://mp.weixin.qq.com/s/OnBjbLzuipWz_u4QfmgcqQ),[回溯算法:求组合总和!](https://mp.weixin.qq.com/s/HX7WW6ixbFZJASkRnCTC3w)。 + +如果是多个集合取组合,各个集合之间相互不影响,那么就不用startIndex,例如:[回溯算法:电话号码的字母组合](https://mp.weixin.qq.com/s/e2ua2cmkE_vpYjM3j6HY0A) + +**注意以上我只是说求组合的情况,如果是排列问题,又是另一套分析的套路,后面我再讲解排列的时候就重点介绍**。 + +代码如下: -分析完过程,回溯算法的模板框架如下: ``` -backtracking() { - if (终止条件) { - 存放结果; - } +vector> result; +vector path; +void backtracking(vector& candidates, int target, int sum, int startIndex) +``` - for (选择:选择列表(可以想成树中节点孩子的数量)) { - 递归,处理节点; - backtracking(); - 回溯,撤销处理结果 - } +* 递归终止条件 + +在如下树形结构中: + + + +从叶子节点可以清晰看到,终止只有两种情况,sum大于target和sum等于target。 + +sum等于target的时候,需要收集结果,代码如下: + +``` +if (sum > target) { + return; +} +if (sum == target) { + result.push_back(path); + return; } ``` -按照模板不难写出如下代码,但很一些细节,我在注释中标记出来了。 +* 单层搜索的逻辑 -# C++代码 +单层for循环依然是从startIndex开始,搜索candidates集合。 + +**注意本题和[回溯算法:求组合问题!](https://mp.weixin.qq.com/s/OnBjbLzuipWz_u4QfmgcqQ)、[回溯算法:求组合总和!](https://mp.weixin.qq.com/s/HX7WW6ixbFZJASkRnCTC3w)的一个区别是:本题元素为可重复选取的**。 + +如何重复选取呢,看代码,注释部分: + +``` +for (int i = startIndex; i < candidates.size(); i++) { + sum += candidates[i]; + path.push_back(candidates[i]); + backtracking(candidates, target, sum, i); // 关键点:不用i+1了,表示可以重复读取当前的数 + sum -= candidates[i]; // 回溯 + path.pop_back(); // 回溯 +} +``` + +按照[关于回溯算法,你该了解这些!](https://mp.weixin.qq.com/s/gjSgJbNbd1eAA5WkA-HeWw)中给出的模板,不难写出如下C++完整代码: + +``` +// 版本一 +class Solution { +private: + vector> result; + vector path; + void backtracking(vector& candidates, int target, int sum, int startIndex) { + if (sum > target) { + return; + } + if (sum == target) { + result.push_back(path); + return; + } + + for (int i = startIndex; i < candidates.size(); i++) { + sum += candidates[i]; + path.push_back(candidates[i]); + backtracking(candidates, target, sum, i); // 不用i+1了,表示可以重复读取当前的数 + sum -= candidates[i]; + path.pop_back(); + } + } +public: + vector> combinationSum(vector& candidates, int target) { + result.clear(); + path.clear(); + backtracking(candidates, target, 0, 0); + return result; + } +}; +``` + +## 剪枝优化 + +在这个树形结构中: + + + +以及上面的版本一的代码大家可以看到,对于sum已经大于target的情况,其实是依然进入了下一层递归,只是下一层递归结束判断的时候,会判断sum > target的话就返回。 + +其实如果已经知道下一层的sum会大于target,就没有必要进入下一层递归了。 + +那么可以在for循环的搜索范围上做做文章了。 + +**对总集合排序之后,如果下一层的sum(就是本层的 sum + candidates[i])已经大于target,就可以结束本轮for循环的遍历**。 + +如图: + + + + +for循环剪枝代码如下: + +``` +for (int i = startIndex; i < candidates.size() && sum + candidates[i] <= target; i++) +``` + +整体代码如下:(注意注释的部分) ``` class Solution { @@ -85,11 +185,11 @@ private: return; } - // 这里i 依然从 startIndex开始,因为求的是组合,如果求的是排列,那么i每次都从0开始 - for (int i = startIndex; i < candidates.size(); i++) { + // 如果 sum + candidates[i] > target 就终止遍历 + for (int i = startIndex; i < candidates.size() && sum + candidates[i] <= target; i++) { sum += candidates[i]; path.push_back(candidates[i]); - backtracking(candidates, target, sum, i); // 关键点在这里,不用i+1了,表示可以重复读取当前的数 + backtracking(candidates, target, sum, i); sum -= candidates[i]; path.pop_back(); @@ -97,8 +197,31 @@ private: } public: vector> combinationSum(vector& candidates, int target) { + result.clear(); + path.clear(); + sort(candidates.begin(), candidates.end()); // 需要排序 backtracking(candidates, target, 0, 0); return result; } }; ``` + +# 总结 + +本题和我们之前讲过的[回溯算法:求组合问题!](https://mp.weixin.qq.com/s/OnBjbLzuipWz_u4QfmgcqQ)、[回溯算法:求组合总和!](https://mp.weixin.qq.com/s/HX7WW6ixbFZJASkRnCTC3w)有两点不同: + +* 组合没有数量要求 +* 元素可无限重复选取 + +针对这两个问题,我都做了详细的分析。 + +并且给出了对于组合问题,什么时候用startIndex,什么时候不用,并用[回溯算法:电话号码的字母组合](https://mp.weixin.qq.com/s/e2ua2cmkE_vpYjM3j6HY0A)做了对比。 + +最后还给出了本题的剪枝优化,这个优化如果是初学者的话并不容易想到。 + +**在求和问题中,排序之后加剪枝是常见的套路!** + +可以看出我写的文章都会大量引用之前的文章,就是要不断作对比,分析其差异,然后给出代码解决的方法,这样才能彻底理解题目的本质与难点。 + +**就酱,如果感觉很给力,就帮Carl宣传一波吧,哈哈**。 +

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