Bootstrap

代码随想录-动态规划

在这里插入图片描述
对于动态规划问题，我将拆解为如下五步曲，这五步都搞清楚了，才能说把动态规划真的掌握了！

  1. 确定dp数组（dp table）以及下标的含义
  2. 确定递推公式
  3. dp数组如何初始化
  4. 确定遍历顺序
  5. 举例推导dp数组

文章目录

509.斐波那契数

class Solution {
    public int fib(int n) {
        if (n <= 1) {
            return n;
        }
        int[] dp = new int[n + 1];
        dp[0] = 0;
        dp[1] = 1;
        for (int i = 2; i <= n; i++) {
            dp[i] = dp[i - 1] + dp[i - 2];
        }
        return dp[n];
    }
}

70. 爬楼梯

class Solution {
    public int climbStairs(int n) {
        int[] dp = new int[n+1];
        dp[0] = 1;
        dp[1] = 1;
        for(int i = 2;i <= n; i++){
            dp[i] = dp[i-1] + dp[i-2];
        }
        return dp[n];
    }
}

746.使用最小花费爬楼梯

class Solution {
    public int minCostClimbingStairs(int[] cost) {
        int len = cost.length;
        int[] dp = new int[len + 1];

        dp[0] = 0;
        dp[1] = 0;
        for (int i = 2; i <= len; i++) {
            dp[i] = Math.min((dp[i - 1] + cost[i - 1]), (dp[i - 2] + cost[i - 2]));
        }
        return dp[len];
    }
}

62. 不同路径

class Solution {
    public int uniquePaths(int m, int n) {
        int[][] dp = new int[m][n];
        for (int i = 0; i < m; i++) {
            dp[i][0] = 1;
        }
        for (int i = 0; i < n; i++) {
            dp[0][i] = 1;
        }
        for (int i = 1; i < m; i++) {
            for (int j = 1; j < n; j++) {
                dp[i][j] = dp[i - 1][j] + dp[i][j - 1];
            }
        }
        return dp[m - 1][n - 1];
    }
}

63. 不同路径 II

class Solution {
    public int uniquePathsWithObstacles(int[][] obstacleGrid) {
        int m = obstacleGrid.length;
        int n = obstacleGrid[0].length;
        int[][] dp = new int[m][n];

        if (obstacleGrid[m - 1][n - 1] == 1 || obstacleGrid[0][0] == 1) {
            return 0;
        }

        for (int i = 0; i < m && obstacleGrid[i][0] == 0; i++) {
            dp[i][0] = 1;
        }
        for (int i = 0; i < n && obstacleGrid[0][i] == 0; i++) {
            dp[0][i] = 1;
        }
        for (int i = 1; i < m; i++) {
            for (int j = 1; j < n; j++) {
                dp[i][j] = (obstacleGrid[i][j] == 0) ? dp[i - 1][j] + dp[i][j - 1] : 0;
            }
        }
        return dp[m - 1][n - 1];
    }
}

343. 整数拆分

class Solution {
    public int integerBreak(int n) {
        int[] dp = new int[n + 1];
        for (int i = 2; i <= n; i++) {
            int curMax = 0;
            for (int j = 1; j < i; j++) {
                curMax = Math.max(curMax, Math.max(j * (i - j), j * dp[i - j]));
            }
            dp[i] = curMax;
        }
        return dp[n];
    }
}

96. 不同的二叉搜索树

class Solution {
    public int numTrees(int n) {
        int[] dp = new int[n + 1];
        dp[0] = 1;
        dp[1] = 1;
        for (int i = 2; i <= n; i++) {
            for (int j = 1; j <= i; j++) {
                dp[i] += dp[j - 1] * dp[i - j];
            }
        }
        return dp[n];
    }
}

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01背包和完全背包就够用了。
而完全背包又是也是01背包稍作变化而来，即：完全背包的物品数量是无限的。

01背包

01背包

// 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++) { // 遍历背包容量
        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]);

    }
}

public class BagProblem {
    public static void main(String[] args) {
        int[] weight = {1,3,4};
        int[] value = {15,20,30};
        int bagSize = 4;
        testWeightBagProblem(weight,value,bagSize);
    }

    /**
     * 动态规划获得结果
     * @param weight  物品的重量
     * @param value   物品的价值
     * @param bagSize 背包的容量
     */
    public static void testWeightBagProblem(int[] weight, int[] value, int bagSize){

        // 创建dp数组
        int goods = weight.length;  // 获取物品的数量
        int[][] dp = new int[goods][bagSize + 1];

        // 初始化dp数组
        // 创建数组后，其中默认的值就是0
        for (int j = weight[0]; j <= bagSize; j++) {
            dp[0][j] = value[0];
        }

        // 填充dp数组
        for (int i = 1; i < weight.length; i++) {
            for (int j = 1; j <= bagSize; j++) {
                if (j < weight[i]) {
                    /**
                     * 当前背包的容量都没有当前物品i大的时候，是不放物品i的
                     * 那么前i-1个物品能放下的最大价值就是当前情况的最大价值
                     */
                    dp[i][j] = dp[i-1][j];
                } else {
                    /**
                     * 当前背包的容量可以放下物品i
                     * 那么此时分两种情况：
                     *    1、不放物品i
                     *    2、放物品i
                     * 比较这两种情况下，哪种背包中物品的最大价值最大
                     */
                    dp[i][j] = Math.max(dp[i-1][j] , dp[i-1][j-weight[i]] + value[i]);
                }
            }
        }

        // 打印dp数组
        for (int i = 0; i < goods; i++) {
            for (int j = 0; j <= bagSize; j++) {
                System.out.print(dp[i][j] + "\t");
            }
            System.out.println("\n");
        }
    }
}

01背包-滚动数组

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。

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]);
    }
}

public static void main(String[] args) {
    int[] weight = {1, 3, 4};
    int[] value = {15, 20, 30};
    int bagWight = 4;
    testWeightBagProblem(weight, value, bagWight);
}

public static void testWeightBagProblem(int[] weight, int[] value, int bagWeight){
    int wLen = weight.length;
    //定义dp数组：dp[j]表示背包容量为j时，能获得的最大价值
    int[] dp = new int[bagWeight + 1];
    //遍历顺序：先遍历物品，再遍历背包容量
    for (int i = 0; i < wLen; i++){
        for (int j = bagWeight; j >= weight[i]; j--){
            dp[j] = Math.max(dp[j], dp[j - weight[i]] + value[i]);
        }
    }
    //打印dp数组
    for (int j = 0; j <= bagWeight; j++){
        System.out.print(dp[j] + " ");
    }
 }

416. 分割等和子集

class Solution {
    public boolean canPartition(int[] nums) {
        if (nums == null || nums.length == 0) {
            return false;
        }
        int sum = 0;
        for (int num : nums) {
            sum += num;
        }
        if (sum % 2 != 0)
            return false;
        int target = sum / 2;
        int[] dp = new int[target + 1];
        for (int i = 0; i < nums.length; i++) {
            for (int j = target; j >= nums[i]; j--) {
                dp[j] = Math.max(dp[j], dp[j - nums[i]] + nums[i]);
            }
            if (dp[target] == target) {
                return true;
            }
        }
        return dp[target] == target;
    }
}
  1. 0/1背包最值问题
  2. 0/1背包存在问题
  3. 0/1背包组合问题
  4. 完全背包最值问题
  5. 完全背包存在问题
  6. 完全背包组合问题
  7. 分组背包最值问题
  8. 分组背包存在问题
  9. 分组背包组合问题

一篇文章吃透背包问题！（细致引入+解题模板+例题分析+代码呈现）
首先是背包分类的模板：

  1. 0/1背包：外循环nums,内循环target,target倒序且target>=nums[i];
  2. 完全背包：外循环nums,内循环target,target正序且target>=nums[i];

然后是问题分类的模板：

  1. 最值问题: dp[j] = max/min(dp[j], dp[i-nums[j] + 1)或dp[j] = max/min(dp[j], dp[j-nums[i]] + nums[i])
  2. 存在问题(bool)：dp[j] = dp[j] || dp[j - nums[i];
  3. 组合问题：dp[j] += dp[j - nums[i]]

这样遇到问题将两个模板往上一套大部分问题就可以迎刃而解

1049.最后一块石头的重量 II

class Solution {
    public int lastStoneWeightII(int[] stones) {
        int sum = 0;
        for (int stone : stones) {
            sum += stone;
        }
        int target = sum / 2;
        int[] dp = new int[target + 1];
        for (int i = 0; i < stones.length; i++) {
            for (int j = target; j >= stones[i]; j--) {
                dp[j] = Math.max(dp[j], dp[j - stones[i]] + stones[i]);
            }
        }
        return sum - dp[target] - dp[target];
    }
}

494.目标和

// 01背包-组合问题
class Solution {
    public int findTargetSumWays(int[] nums, int target) {
        int sum = 0;
        for (int num : nums) {
            sum += num;
        }
        if (Math.abs(target) > sum)
            return 0;
        if ((target + sum) % 2 == 1)
            return 0;
        int bagSize = (target + sum) / 2;
        int[] dp = new int[bagSize + 1];
        dp[0] = 1;
        for (int i = 0; i < nums.length; i++) {
            for (int j = bagSize; j >= nums[i]; j--) {
                dp[j] += dp[j - nums[i]];
            }
        }
        return dp[bagSize];
    }
}

474.一和零

class Solution {
    public int findMaxForm(String[] strs, int m, int n) {
        int[][] dp = new int[m+1][n+1];
        int oneNum, zeroNum;
        for (String str : strs) {
            oneNum = 0;
            zeroNum = 0;
            for (char ch : str.toCharArray()) {
                if (ch == '0') {
                    zeroNum++;
                } else {
                    oneNum++;
                }
            }
            for (int i = m; i >= zeroNum; i--) {
                for (int j = n; j >= oneNum; j--) {
                    dp[i][j] = Math.max(dp[i][j], dp[i - zeroNum][j - oneNum] + 1);
                }
            }
        }
        return dp[m][n];
    }
}

完全背包

// 先遍历物品，再遍历背包
for(int i = 0; i < weight.size(); i++) { // 遍历物品
    for(int j = weight[i]; j <= bagWeight ; j++) { // 遍历背包容量
        dp[j] = max(dp[j], dp[j - weight[i]] + value[i]);

    }
}

如果求组合数就是 外层for循环遍历物品，内层for遍历背包。
如果求排列数就是 外层for遍历背包，内层for循环遍历物品。

518.零钱兑换 II

class Solution {
    public int change(int amount, int[] coins) {
        int[] dp = new int[amount + 1];
        dp[0] = 1;
        for (int i = 0; i < coins.length; i++) {
            for (int j = coins[i]; j <= amount; j++) {
                dp[j] += dp[j - coins[i]];
            }
        }
        return dp[amount];
    }
}

377.组合总和 Ⅳ

// 先遍历背包时，最里面需要添加判断条件（当前的容量要大于选择物品的体积）
class Solution {
    public int combinationSum4(int[] nums, int target) {
        int[] dp = new int[target + 1];
        dp[0] = 1;

        for (int i = 0; i <= target; i++) {
            for (int j = 0; j < nums.length; j++) {
                if (i >= nums[j]) {
                    dp[i] += dp[i - nums[j]];
                }
            }
        }
        return dp[target];
    }
}

322.零钱兑换

class Solution {
    public int coinChange(int[] coins, int amount) {
        int[] dp = new int[amount + 1];
        Arrays.fill(dp, Integer.MAX_VALUE);
        dp[0] = 0;
        for (int i = 0; i < coins.length; i++) {
            for (int j = coins[i]; j <= amount; j++) {
                if (dp[j - coins[i]] != Integer.MAX_VALUE) {
                    dp[j] = Math.min(dp[j], dp[j - coins[i]] + 1);
                }
            }
        }
        return dp[amount] == Integer.MAX_VALUE ? -1 : dp[amount];
    }
}

279.完全平方数

class Solution {
    public int numSquares(int n) {
        int[] dp = new int[n + 1];
        Arrays.fill(dp, Integer.MAX_VALUE);
        dp[0] = 0;
        for (int i = 1; i <= n; i++) {
            for (int j = 1; j * j <= i; j++) {
                dp[i] = Math.min(dp[i], dp[i - j * j] + 1);
            }
        }
        return dp[n];
    }
}

139.单词拆分

class Solution {
    public boolean wordBreak(String s, List<String> wordDict) {
        Set<String> set = new HashSet<>(wordDict);
        boolean[] dp = new boolean[s.length() + 1];
        dp[0] = true;

        for (int i = 1; i <= s.length(); i++) {
            for (int j = 0; j < i; j++) {
                if (set.contains(s.substring(j, i)) && dp[j]) {
                    dp[i] = true;
                }
            }
        }

        return dp[s.length()];
    }
}

打家劫舍

198. 打家劫舍

class Solution {
    public int rob(int[] nums) {
        if (nums == null || nums.length == 0)
            return 0;
        if (nums.length == 1)
            return nums[0];

        int[] dp = new int[nums.length];
        dp[0] = nums[0];
        dp[1] = Math.max(dp[0], nums[1]);
        for (int i = 2; i < nums.length; i++) {
            dp[i] = Math.max(dp[i - 1], dp[i - 2] + nums[i]);
        }

        return dp[nums.length - 1];
    }
}

213. 打家劫舍 II

class Solution {
    public int rob(int[] nums) {
        if (nums.length == 0)
            return 0;
        if (nums.length == 1)
            return nums[0];

        return Math.max(robAction(nums, 0, nums.length - 2), robAction(nums, 1, nums.length - 1));
    }

    public int robAction(int[] nums, int start, int end) {
        if (start == end) {
            return nums[start];
        }
        int[] dp = new int[nums.length];
        dp[start] = nums[start];
        dp[start + 1] = Math.max(nums[start], nums[start + 1]);
        for (int i = start + 2; i <= end; i++) {
            dp[i] = Math.max(dp[i - 1], dp[i - 2] + nums[i]);
        }
        return dp[end];
    }
}

337. 打家劫舍 III

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    public int rob(TreeNode root) {
        int[] dp = robAction(root);
        return Math.max(dp[0], dp[1]);
    }

    public int[] robAction(TreeNode root) {
        if (root == null)
            return new int[2];
        // 0 代表不偷，1 代表偷
        int[] dp = new int[2];

        int[] left = robAction(root.left);
        int[] right = robAction(root.right);

        dp[0] = Math.max(left[0], left[1]) + Math.max(right[0], right[1]);
        dp[1] = left[0] + right[0] + root.val;

        return dp;
    }
}

股票问题

121. 买卖股票的最佳时机

class Solution {
    public int maxProfit(int[] prices) {
        int low = Integer.MAX_VALUE;
        int res = 0;
        for (int i = 0; i < prices.length; i++) {
            low = Math.min(low, prices[i]);
            res = Math.max(res, prices[i] - low);
        }
        return res;
    }
}

public class Solution {
    public int maxProfit(int[] prices) {
        int maxMoney = 0; // 最大利润
        int slow = 0; // 慢指针
        int fast = 1; // 快指针
        while (fast < prices.length) {
            int money = prices[fast] - prices[slow];
            if (money > maxMoney) {
                maxMoney = money;
            }
            if (money < 0) {
                slow = fast;
            }
            fast++;
        }
        return maxMoney;
    }
}

122. 买卖股票的最佳时机 II

class Solution {
    public int maxProfit(int[] prices) {
        int result = 0;
        for (int i = 1; i < prices.length; i++) {
            result += Math.max(prices[i] - prices[i - 1], 0);
        }
        return result;
    }
}

子序列问题

子序列（不连续）

300. 最长递增子序列

class Solution {
    public int lengthOfLIS(int[] nums) {
        if (nums.length <= 1) {
            return nums.length;
        }
        int[] dp = new int[nums.length];
        Arrays.fill(dp, 1);
        int res = 0;
        for (int i = 1; i < nums.length; i++) {
            for (int j = 0; j < i; j++) {
                if (nums[j] < nums[i]) {
                    dp[i] = Math.max(dp[i], dp[j] + 1);
                }
            }
            res = Math.max(dp[i], res);
        }
        return res;
    }
}

1143. 最长公共子序列

class Solution {
    public int longestCommonSubsequence(String text1, String text2) {
        int[][] dp = new int[text1.length() + 1][text2.length() + 1];

        for (int i = 1; i <= text1.length(); i++) {
            for (int j = 1; j <= text2.length(); j++) {
                if (text1.charAt(i - 1) == text2.charAt(j - 1)) {
                    dp[i][j] = dp[i - 1][j - 1] + 1;
                } else {
                    dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]);
                }
            }
        }
        return dp[text1.length()][text2.length()];

    }
}

子序列（连续）

674. 最长连续递增序列

class Solution {
    public int findLengthOfLCIS(int[] nums) {
        if (nums.length <= 1) {
            return nums.length;
        }
        int[] dp = new int[nums.length];
        Arrays.fill(dp, 1);
        int res = 0;
        for (int i = 1; i < nums.length; i++) {
            if (nums[i - 1] < nums[i]) {
                dp[i] = dp[i - 1] + 1;
            }
            res = Math.max(res, dp[i]);
        }
        return res;
    }
}

718. 最长重复子数组

// 0 ~ i-1
// 0 ~ j-1
class Solution {
    public int findLength(int[] nums1, int[] nums2) {
        int res = 0;
        int[][] dp = new int[nums1.length + 1][nums2.length + 1];

        for (int i = 1; i <= nums1.length; i++) {
            for (int j = 1; j <= nums2.length; j++) {
                if (nums1[i - 1] == nums2[j - 1]) {
                    dp[i][j] = dp[i - 1][j - 2] + 1;
                    res = Math.max(res, dp[i][j]);
                }
            }
        }
        return res;
    }
}

53. 最大子数组和

class Solution {
    public int maxSubArray(int[] nums) {
        if (nums.length == 1) {
            return nums[0];
        }
        int[] dp = new int[nums.length];

        dp[0] = nums[0];
        int res = nums[0];
        for (int i = 1; i < nums.length; i++) {
            dp[i] = Math.max(dp[i - 1] + nums[i], nums[i]);
            res = Math.max(res, dp[i]);
        }
        return res;
    }
};

编辑距离

392. 判断子序列

class Solution {
    public boolean isSubsequence(String s, String t) {
        int lenS = s.length();
        int lenT = t.length();

        int[][] dp = new int[lenS + 1][lenT + 1];
        for (int i = 1; i <= lenS; i++) {
            for (int j = 1; j <= lenT; j++) {
                if (s.charAt(i - 1) == t.charAt(j - 1)) {
                    dp[i][j] = dp[i - 1][j - 1] + 1;
                } else {
                    dp[i][j] = dp[i][j - 1];
                }
            }
        }
        return dp[lenS][lenT] == lenS;
    }
}

115. 不同的子序列

在这里插入代码片

583. 两个字符串的删除操作

class Solution {
    public int minDistance(String word1, String word2) {
        int len1 = word1.length();
        int len2 = word2.length();
        int[][] dp = new int[len1 + 1][len2 + 1];

        for (int i = 1; i <= len1; i++) {
            for (int j = 1; j <= len2; j++) {
                if (word1.charAt(i - 1) == word2.charAt(j - 1)) {
                    dp[i][j] = dp[i - 1][j - 1] + 1;
                } else {
                    dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]);
                }
            }
        }
        return len1 + len2 - 2 * dp[len1][len2];
    }
}

class Solution {
    public int minDistance(String word1, String word2) {
        int len1 = word1.length();
        int len2 = word2.length();
        int[][] dp = new int[len1 + 1][len2 + 1];

        for (int i = 0; i <= len1; i++)
            dp[i][0] = i;
        for (int j = 0; j <= len2; j++)
            dp[0][j] = j;

        for (int i = 1; i <= len1; i++) {
            for (int j = 1; j <= len2; j++) {
                if (word1.charAt(i - 1) == word2.charAt(j - 1)) {
                    dp[i][j] = dp[i - 1][j - 1];
                } else {
                    dp[i][j] = Math.min(dp[i - 1][j - 1] + 2,
                            Math.min(dp[i - 1][j] + 1, dp[i][j - 1] + 1));
                }
            }
        }
        return dp[len1][len2];
    }
}

72. 编辑距离

class Solution {
    public int minDistance(String word1, String word2) {
        int len1 = word1.length();
        int len2 = word2.length();
        int[][] dp = new int[len1 + 1][len2 + 1];
        // 初始化
        for (int i = 1; i <= len1; i++) {
            dp[i][0] = i;
        }
        for (int j = 1; j <= len2; j++) {
            dp[0][j] = j;
        }
        for (int i = 1; i <= len1; i++) {
            for (int j = 1; j <= len2; j++) {
                // 因为dp数组有效位从1开始
                // 所以当前遍历到的字符串的位置为i-1 | j-1
                if (word1.charAt(i - 1) == word2.charAt(j - 1)) {
                    dp[i][j] = dp[i - 1][j - 1];
                } else {
                    dp[i][j] = Math.min(Math.min(dp[i - 1][j - 1], dp[i][j - 1]), dp[i - 1][j]) + 1;
                }
            }
        }
        return dp[len1][len2];
    }
}

回文

647. 回文子串

class Solution {
    public int countSubstrings(String s) {
        char[] chars = s.toCharArray();
        int len = chars.length;
        boolean[][] dp = new boolean[len][len];
        int res = 0;
        for (int i = len - 1; i >= 0; i--) {
            for (int j = i; j < len; j++) {
                if (chars[i] == chars[j]) {
                    if (j - i <= 1) { // 情况一 和 情况二
                        res++;
                        dp[i][j] = true;
                    } else if (dp[i + 1][j - 1]) { // 情况三
                        res++;
                        dp[i][j] = true;
                    }
                }
            }
        }
        return res;
    }
}

516. 最长回文子序列

class Solution {
    public int longestPalindromeSubseq(String s) {
        int len = s.length();
        int[][] dp = new int[len + 1][len + 1];
        for (int i = len - 1; i >= 0; i--) { // 从后往前遍历 保证情况不漏
            dp[i][i] = 1; // 初始化
            for (int j = i + 1; j < len; j++) {
                if (s.charAt(i) == s.charAt(j)) {
                    dp[i][j] = dp[i + 1][j - 1] + 2;
                } else {
                    dp[i][j] = Math.max(dp[i + 1][j], Math.max(dp[i][j], dp[i][j - 1]));
                }
            }
        }
        return dp[0][len - 1];
    }
}

悦读

道可道，非常道；名可名，非常名。 无名，天地之始，有名，万物之母。 故常无欲，以观其妙，常有欲，以观其徼。 此两者，同出而异名，同谓之玄，玄之又玄，众妙之门。

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