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z_WBCd$9|{8*R9;QXL_;j{TZjudCrmkS-Y(^yZT$mwMDl5n`)~+y;$#m_JL3AtcmNt zgJ57SN5%hZ?a4b=n=~7zX1VR_t+6vtTd_Fi^{#l7{59bKgVKTtQVzYTfyi486dJDq pi_fgyvJE0AnPXJ-U -## 01 背包 - -有N件物品和一个最多能被重量为W 的背包。第i件物品的重量是weight[i],得到的价值是value[i] 。**每件物品只能用一次**,求解将哪些物品装入背包里物品价值总和最大。 - -这是标准的背包问题,以至于很多同学看了这个自然就会想到背包,甚至都不知道暴力的解法应该怎么解。 - -这样其实就是没有从底向上去思考,而是习惯性的只知道背包了,那么暴力的解法应该是怎么样的呢? - -每一件物品其实只有两个状态,取或者不取,所以可以使用回溯法搜索出所有的情况,那么时间复杂度就是O(2^n),这里的n表示物品数量。 - -所以暴力的解法是指数级别的时间复杂度。进而才需要动态规划的解法来进行优化! - -目前leetcode上没有发现有纯01背包的题目,leetcode上相关01背包问题都是需要某种条件转化为01背包问题,所以 我举一个纯01背包的例子来给大家讲解。 - -把01背包理论和代码理解透彻了,我们再刷leetcode上的题目。 - -下面的讲解中,我举一个例子: - -背包最大重量为4。 - -物品为: - -| | 重量 | 价值 | -| --- | --- | --- | -| 书 | 1 | 15 | -| 台灯 | 3 | 20 | -| 乐高 | 4 | 30 | - -以下讲解和图示中出现的数字都是以这个例子为例。 - -* 确定dp数组以及下标的含义 - -对于背包问题,有一种写法, 是使用二维数组,即**dp[i][j] 表示从下标为[0-i]的物品里任意取,放进容量为j的背包,价值总和最大是多少**。 - -只看这个二维数组的定义,大家一定会有点懵,看下面这个图: - - - -要时刻记着这个dp数组的含义,下面的一些步骤都围绕这dp数组的含义进行的。 - -* dp数组如何初始化 - -**关于初始化,一定要和dp数组的定义吻合,否则到递推公式的时候就会越来越乱**。 - -首先从dp[i][j]的定义触发,如果背包容量j为0的话,无论是选取哪些物品,背包价值总和一定为0。如图: - - - - -那么其他下标应该初始化多少呢? - -dp[i][j]在推导的时候一定是取价值最大的数,如果题目给的价值都是正整数那么非0下标都初始化为0就可以了,因为0就是最小的了,不会影响去最大价值的结果。 - -如果题目给的价值有负数,那么非0下标就要初始化为负无穷了。例如:一个物品的价值是-2,但对应的位置依然初始化为0,那么去最大值的时候,就会取0而不是-2了,所以要初始化为负无穷。 - -这样才能让dp数组在递归公式的过程中取的最大的价值,而不是被初始值覆盖了。 - -而本题价值都是正整数,所以初始化为0就可以了。 - -如图: - - - -**很明显,红框的位置就是我们要求的结果** - -* 确定递推公式 - -再回顾一下dp[i][j]的含义:从下标为[0-i]的物品里任意取,放进容量为j的背包,价值总和最大是多少。 - -那么可以有两个方向推出来dp[i][j], - -* 又dp[i - 1][j]推出,即不放背包里不放物品i的最大价值,此时dp[i][j]就是dp[i - 1][j] -* 又dp[i - 1][j - weight[i]]推出,dp[i - 1][j - weight[i]] 为背包容量为j - weight[i]的时候不放物品i的最大价值,那么dp[i - 1][j - weight[i]] + value[i] (物品i的价值),就是背包放物品i得到的最大价值 - -那么 dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - weight[i]] + value[i]); - -* 确定遍历顺序 - -确定递归公式之后,还要确定遍历顺序。 - -在如下图中,可以看出,有两个遍历的维度:物品与背包重量 - - -那么问题来了,先遍历 物品还是先遍历背包重量呢? - -**其实都可以!! 但是先遍历物品更方便一些**。下面讲到具体原因的时候来在分析原因。 - -那么首先遍历物品,然后遍历背包重量。 - - -************************ 首先我们来看dp数组的推导过程: - - - -注意 状态转移方程 dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - weight[i]] + value[i]); 中有两个下表为负数的情况,即:i - 1 和 j - weight[i]。 - -所以代码如下: - -``` -for(int i = 1; i < weight.size(); i++) { // 遍历物品 - for(int j = 0; j <= bagWeight; j++) { // 遍历背包重量 - if (j < weight[i]) dp[i][j] = dp[i - 1][j]; - else dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - weight[i]] + value[i]); - - } -} -``` - -来看一下对应的dp数组的数值,如图: - - - -建议大家此时自己在纸上推导一遍,看看dp数组里每一个数值是不是这样的。 - -**做动态规划的题目,最好的过程就是自己在纸上举一个例子把对应的dp数组的数值推导一下,然后在动手写代码!** - -很多同学做dp题目,遇到各种问题,然后凭感觉东改改西改改,怎么改都不对,或者稀里糊涂就改过了。 - -主要就是自己没有动手推导一下dp数组的演变过程,如果推导明白了,代码写出来就算有问题,只要把dp数组打印出来,对比一下和自己推导的有什么差异,很快就可以发现问题了。 - - - - -因为 dp 每次用上一行的值进行计算的,没有重复利用本行的数值,所以不会重复使用同一个物品 -``` -for (int j = bagWeight; j >= weight[0]; j--) { - dp[0][j] = dp[0][j - weight[0]] + value[0]; -} - -// weight数组的大小 就是物品个数 -for(int i = 1; i < weight.size(); i++) { // 遍历物品 - for(int j = 0; j <= bagWeight; j++) { // 遍历背包重量 - if (j < weight[i]) dp[i][j] = dp[i - 1][j]; - else dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - weight[i]] + value[i]); - - } -} -``` - -``` -0 15 15 15 15 -0 0 0 20 35 -0 0 0 0 35 -``` - - -* 确定dp数组以及下标的含义 - -对于背包问题,有一种写法, 是使用二维数组,即dp[i][j] 表示从下标为0-i的物品里任意取,放进容量为j的背包,价值总和最大是多少。 - -对于这种写法,不仅空间多用了一个维度,而且思路比较绕。 - -我习惯直接使用一维数组(相对于二维数组,一维可以说是滚动数组)。在后面的讲解中,我也直接使用一维数组。 - -在一维dp数组中,dp[i]表示:容量为i的背包,所背的物品价值可以最大为dp[i] - - -``` -0 15 15 15 15 -0 15 15 20 35 -0 15 15 20 35 -``` - -* dp数组如何初始化 - -**关于初始化,一定要和dp数组的定义吻合,否则到递推公式的时候就会越来越乱**。 - -dp[i]表示:容量为i的背包,所背的物品价值可以最大为dp[i],那么dp[0]就应该是0,因为背包容量为0所背的物品的最大价值就是0。 - -那么dp数组除了下标0的位置,初始为0,其他下标应该初始化多少呢? - -在回顾一下dp数组的含义:容量为i的背包,所背的物品价值可以最大为dp[i]。 - -那么dp数组在推导的时候一定是去价值最大的数,如果题目给的价值都是正整数那么非0下标都初始化为0就可以了,如果题目给的价值有负数,那么非0下标就要初始化为负无穷。 - -这样才能让dp数组在递归公式的过程中取的最大的价值,而不是被初始值覆盖了。 - -那么我假设物品价值都是大于0的,所以dp数组初始化的时候,都初始为0就可以了。 - -代码如下: - -``` -vector dp(背包容量V, 0); -``` - -* 确定递推公式 - -有N件物品和一个容量为V 的背包。放入第i件物品耗费的空间是space[i],得到的价值是value[i] 。**每件物品只能用一次**,求解将哪些物品装入背包可使价值总和最大。 - -dp[j]为 容量为j的背包所背的最大价值,那么如何推导dp[j]呢? - -dp[j]可以通过dp[j - space[i]]推导出来,dp[j - space[i]]表示容量为j - space[i]的背包所背的最大价值。 - -dp[j - space[i]] + value[i] 表示 容量为 j - 物品i体积 的背包 加上 物品i的价值。 - -那么最大的dp[j]可能就是 dp[j - space[i]] + value[i]。 - -那么此时dp[j]有两个选择,一个是取子集dp[j],一个是取dp[j - space[i]] + value[i],指定是取最大的,毕竟是求最大价值, - -所以递归公式为: - -``` -dp[j] = max(dp[j], dp[j - space[i]] + value[i]); -``` - -* 确定遍历顺序 - -这里需要两层for循环,第一层遍历物品数量,第二层遍历背包的各个容量,来寻找最大值 - -****************** 讲讲for循环的顺序问题 - -代码如下: - -``` -for(int i = 0; i < 物品数量N; i++) { // 遍历物品 - for(int j = 背包容量V; j >= space[i]; j--) { // 遍历背包容量 - dp[j] = max(dp[j], dp[j - space[i]] + value[i]); - } -} -``` - -此时大家会发现为什么,第二层for循环要从大到小遍历。 - - - -注意这里第二个for循环是从大到小的,这样才能保证每件物品只使用一次。 - - -如果物品装不满背包,dp[V]也是返回最大价值。 - - -如果是第一种问法,要求恰好装满背包,那么在初始化时除了F [0]为0,其 它F [1..V ]均设为−∞,这样就可以保证最终得到的F [V ]是一种恰好装满背包的 最优解。 -如果并没有要求必须把背包装满,而是只希望价格尽量大,初始化时应该 将F [0..V ]全部设为0。 - -这是为什么呢?可以这样理解:初始化的F 数组事实上就是在没有任何物 品可以放入背包时的合法状态。如果要求背包恰好装满,那么此时只有容量 为0的背包可以在什么也不装且价值为0的情况下被“恰好装满”,其它容量的 背包均没有合法的解,属于未定义的状态,应该被赋值为-∞了。如果背包并非 必须被装满,那么任何容量的背包都有一个合法解“什么都不装”,这个解的 价值为0,所以初始时状态的值也就全部为0了。 # 完全背包 diff --git a/problems/背包理论基础01背包.md b/problems/背包理论基础01背包-1.md similarity index 66% rename from problems/背包理论基础01背包.md rename to problems/背包理论基础01背包-1.md index 39520694..e3528086 100644 --- a/problems/背包理论基础01背包.md +++ b/problems/背包理论基础01背包-1.md @@ -1,3 +1,5 @@ + +leetcode上没有纯01背包的问题,都是需要转化为01背包的题目,所以我先把通过纯01背包问题,把01背包原理讲清楚,后序讲解leetcode题目的时候,重点就是如何转化为01背包问题了。 ## 01 背包 有N件物品和一个最多能被重量为W 的背包。第i件物品的重量是weight[i],得到的价值是value[i] 。**每件物品只能用一次**,求解将哪些物品装入背包里物品价值总和最大。 @@ -226,133 +228,3 @@ dp[i-1][j]和dp[i - 1][j - weight[i]] 都在dp[i][j]的左上角方向(包括 这一点其实很多题解都没有说清楚,此时大家应该对01背包,两层for控制两个维度,究竟先遍历哪一个有一个深刻理解了。 -## 滚动数组 - -对于背包问题其实状态都是可以压缩的。 - -在使用二维数组的时候,递推公式:dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - weight[i]] + value[i]); - -这里其实可以发现如果把dp[i-1]那一层拷贝到dp[i]上,表达式完全可以是:dp[i][j] = max(dp[i][j], dp[i][j - weight[i]] + value[i]); - -读到这里估计大家都忘了 dp[i][j]里的i和j表达的是什么了,i是物品,j是背包容量,一定要时刻记住这里i和j的含义,要不然很容易看懵了。 - -**与其把dp[i-1]这一层拷贝到dp[i]上,不如只用一个一维数组了**。 - -只用dp[j]!(一维数组,也可以理解是一个滚动数组) - -在一维dp数组中,dp[j]表示:容量为j的背包,所背的物品价值可以最大为dp[j] - -* 一维dp数组如何初始化 - -**关于初始化,一定要和dp数组的定义吻合,否则到递推公式的时候就会越来越乱**。 - -dp[j]表示:容量为j的背包,所背的物品价值可以最大为dp[j],那么dp[0]就应该是0,因为背包容量为0所背的物品的最大价值就是0。 - -那么dp数组除了下标0的位置,初始为0,其他下标应该初始化多少呢? - -在回顾一下dp数组的含义:容量为j的背包,所背的物品价值可以最大为dp[j]。 - -那么dp数组在推导的时候一定是取价值最大的数,如果题目给的价值都是正整数那么非0下标都初始化为0就可以了,如果题目给的价值有负数,那么非0下标就要初始化为负无穷。 - -**这样才能让dp数组在递归公式的过程中取的最大的价值,而不是被初始值覆盖了**。 - -那么我假设物品价值都是大于0的,所以dp数组初始化的时候,都初始为0就可以了。 - - -* 一维dp数组的递推公式 - -dp[j]为 容量为j的背包所背的最大价值,那么如何推导dp[j]呢? - -dp[j]可以通过dp[j - weight[j]]推导出来,dp[j - weight[i]]表示容量为j - weight[i]的背包所背的最大价值。 - -dp[j - weight[i]] + value[i] 表示 容量为 j - 物品i体积 的背包 加上 物品i的价值。 - -那么最大的dp[j]可能就是 dp[j - weight[i]] + value[i]。 - -那么此时dp[j]有两个选择,一个是取自己dp[j],一个是取dp[j - weight[i]] + value[i],指定是取最大的,毕竟是求最大价值, - -所以递归公式为: - -``` -dp[j] = max(dp[j], dp[j - weight[i]] + value[i]); -``` - -可以看出相对于二维dp数组的写法,就是把dp[i][j]中i的维度去掉了。 - -* 一维dp数组遍历顺序 - -代码如下: - -``` -for(int i = 0; i < weight.size(); i++) { // 遍历物品 - for(int j = bagWeight; j >= weight[i]; j--) { // 遍历背包容量 - dp[j] = max(dp[j], dp[j - weight[i]] + value[i]); - - } -} -``` - -**这里大家发现和二维dp的写法中,遍历背包的顺序是不一样的!** - -二维dp遍历的时候,背包容量是从小到大,而一维dp遍历的时候,背包是从大到小。 - -为什么呢? - -**倒叙遍历是为了保证物品i只被放入一次!**,在上面讲解二维dp遍历第0行的时候已经提到过一次。 - -举一个例子:物品0的重量weight[0] = 1,价值value[0] = 15 - -如果正序遍历 - -dp[1] = dp[1 - weight[0]] + value[0] = 15 - -dp[2] = dp[2 - weight[0]] + value[0] = 30 - -此时dp[2]就已经是30了,意味着物品0,被放入了两次,所以不能正序遍历。 - -为什么倒叙遍历,就可以保证物品只放入一次呢? - -倒叙就是先算dp[2] - -dp[2] = dp[2 - weight[0]] + value[0] = 15 (dp数组已经都初始化为0) - -dp[1] = dp[1 - weight[0]] + value[0] = 15 - -通过这个例子大家应该理解了为什么倒叙遍历可以保证数组只放入一次! - -**那么问题又来了,为什么二维dp遍历的时候不用倒叙呢?** - -因为对数二维dp,dp[i][j]都是通过上一层即dp[i-1][j]计算而来,不用考虑重复计算dp[i][j]的问题! - - -再来看看两个嵌套for循环的顺序,代码中是先遍历物品嵌套遍历背包容量,那可不可以先背包容量嵌套遍历物品呢? - -不可以! - -因为一维dp的写法,背包容量一定是要倒序遍历(原因上面已经讲了),如果遍历背包容量放在上一层,dp[bagWeight]只会放入一个物品。 - -一维dp01背包完整测试代码如下: - -``` -void test_1_wei_bag_problem() { - int w[] = {1, 3, 4}; - int v[] = {15, 20, 30}; - int bagWeight = 4; - vector weight(w, w + sizeof(w)/sizeof(int)); - vector value(v, v + sizeof(v)/sizeof(int)); - int dp[6] = {0}; - for(int i = 0; i < weight.size(); i++) { // 遍历物品 - for(int j = bagWeight; j >= weight[i]; j--) { // 遍历背包容量 - dp[j] = max(dp[j], dp[j - weight[i]] + value[i]); - } - // 把dp数组打印出来,看看对不对 - for (int j = 0; j <= bagWeight; j++) { - cout << dp[j] << " "; - } - cout << endl; - } - // 输出结果 - cout << dp[bagWeight] << endl; -} - -``` diff --git a/problems/背包理论基础01背包-2.md b/problems/背包理论基础01背包-2.md new file mode 100644 index 00000000..f12a29e7 --- /dev/null +++ b/problems/背包理论基础01背包-2.md @@ -0,0 +1,131 @@ + +## 滚动数组 + +对于背包问题其实状态都是可以压缩的。 + +在使用二维数组的时候,递推公式:dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - weight[i]] + value[i]); + +这里其实可以发现如果把dp[i-1]那一层拷贝到dp[i]上,表达式完全可以是:dp[i][j] = max(dp[i][j], dp[i][j - weight[i]] + value[i]); + +读到这里估计大家都忘了 dp[i][j]里的i和j表达的是什么了,i是物品,j是背包容量,一定要时刻记住这里i和j的含义,要不然很容易看懵了。 + +**与其把dp[i-1]这一层拷贝到dp[i]上,不如只用一个一维数组了**。 + +只用dp[j]!(一维数组,也可以理解是一个滚动数组) + +在一维dp数组中,dp[j]表示:容量为j的背包,所背的物品价值可以最大为dp[j] + +* 一维dp数组如何初始化 + +**关于初始化,一定要和dp数组的定义吻合,否则到递推公式的时候就会越来越乱**。 + +dp[j]表示:容量为j的背包,所背的物品价值可以最大为dp[j],那么dp[0]就应该是0,因为背包容量为0所背的物品的最大价值就是0。 + +那么dp数组除了下标0的位置,初始为0,其他下标应该初始化多少呢? + +在回顾一下dp数组的含义:容量为j的背包,所背的物品价值可以最大为dp[j]。 + +那么dp数组在推导的时候一定是取价值最大的数,如果题目给的价值都是正整数那么非0下标都初始化为0就可以了,如果题目给的价值有负数,那么非0下标就要初始化为负无穷。 + +**这样才能让dp数组在递归公式的过程中取的最大的价值,而不是被初始值覆盖了**。 + +那么我假设物品价值都是大于0的,所以dp数组初始化的时候,都初始为0就可以了。 + + +* 一维dp数组的递推公式 + +dp[j]为 容量为j的背包所背的最大价值,那么如何推导dp[j]呢? + +dp[j]可以通过dp[j - weight[j]]推导出来,dp[j - weight[i]]表示容量为j - weight[i]的背包所背的最大价值。 + +dp[j - weight[i]] + value[i] 表示 容量为 j - 物品i体积 的背包 加上 物品i的价值。 + +那么最大的dp[j]可能就是 dp[j - weight[i]] + value[i]。 + +那么此时dp[j]有两个选择,一个是取自己dp[j],一个是取dp[j - weight[i]] + value[i],指定是取最大的,毕竟是求最大价值, + +所以递归公式为: + +``` +dp[j] = max(dp[j], dp[j - weight[i]] + value[i]); +``` + +可以看出相对于二维dp数组的写法,就是把dp[i][j]中i的维度去掉了。 + +* 一维dp数组遍历顺序 + +代码如下: + +``` +for(int i = 0; i < weight.size(); i++) { // 遍历物品 + for(int j = bagWeight; j >= weight[i]; j--) { // 遍历背包容量 + dp[j] = max(dp[j], dp[j - weight[i]] + value[i]); + + } +} +``` + +**这里大家发现和二维dp的写法中,遍历背包的顺序是不一样的!** + +二维dp遍历的时候,背包容量是从小到大,而一维dp遍历的时候,背包是从大到小。 + +为什么呢? + +**倒叙遍历是为了保证物品i只被放入一次!**,在上面讲解二维dp遍历第0行的时候已经提到过一次。 + +举一个例子:物品0的重量weight[0] = 1,价值value[0] = 15 + +如果正序遍历 + +dp[1] = dp[1 - weight[0]] + value[0] = 15 + +dp[2] = dp[2 - weight[0]] + value[0] = 30 + +此时dp[2]就已经是30了,意味着物品0,被放入了两次,所以不能正序遍历。 + +为什么倒叙遍历,就可以保证物品只放入一次呢? + +倒叙就是先算dp[2] + +dp[2] = dp[2 - weight[0]] + value[0] = 15 (dp数组已经都初始化为0) + +dp[1] = dp[1 - weight[0]] + value[0] = 15 + +通过这个例子大家应该理解了为什么倒叙遍历可以保证数组只放入一次! + +**那么问题又来了,为什么二维dp遍历的时候不用倒叙呢?** + +因为对数二维dp,dp[i][j]都是通过上一层即dp[i-1][j]计算而来,不用考虑重复计算dp[i][j]的问题! + + +再来看看两个嵌套for循环的顺序,代码中是先遍历物品嵌套遍历背包容量,那可不可以先背包容量嵌套遍历物品呢? + +不可以! + +因为一维dp的写法,背包容量一定是要倒序遍历(原因上面已经讲了),如果遍历背包容量放在上一层,dp[bagWeight]只会放入一个物品。 + +一维dp01背包完整测试代码如下: + +``` +void test_1_wei_bag_problem() { + int w[] = {1, 3, 4}; + int v[] = {15, 20, 30}; + int bagWeight = 4; + vector weight(w, w + sizeof(w)/sizeof(int)); + vector value(v, v + sizeof(v)/sizeof(int)); + int dp[6] = {0}; + for(int i = 0; i < weight.size(); i++) { // 遍历物品 + for(int j = bagWeight; j >= weight[i]; j--) { // 遍历背包容量 + dp[j] = max(dp[j], dp[j - weight[i]] + value[i]); + } + // 把dp数组打印出来,看看对不对 + for (int j = 0; j <= bagWeight; j++) { + cout << dp[j] << " "; + } + cout << endl; + } + // 输出结果 + cout << dp[bagWeight] << endl; +} + +```