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前言：这是一段生成扑克牌的教学，用PYTHON进行实现 我们要生成一副扑克牌，目标结果如下(仅为示例)： 正式教程： 1.首先定义存储牌的可迭代容器，这里使用列表： 2.然后定义花色列表和数字列表： 3.接下来，推导式生成： 注意，这里每一副牌都是一个字典，例如♠ 5，就是{"KEY":"♠","TYPE":"5"} 4.添加大小王： 5.使用RANDOM库进行洗牌： 6.使用切片方法分出PLAYER1和PLAYER2两副牌： 7.格式化输出(重点): 如果你用的是VSCODE等强大一些的工具，那就请你照常格式化输出(高效版写法)： 但大部分编辑器都会出现问题，请你这样写(兼容版写法)： 全部代码： 想了解更多？点击这里！ 有问题请点击这里提问或在下方留言~

输入一个 8×88\times88×8 的 .Q.Q.Q 矩阵，试试结果

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前言 本次飞翔杯的基础组题解，总体难度属于入门～普及的难度 T1 偷吃 题目大意 给出包含nnn个整数的序列AAA，进行kkk次操作，每次删除其中的A1A_1A1 ，并将A2A_2A2 到AnA_nAn 中的所有元素前挪一位变为A1,A2...An−1A_1,A_2...A_{n-1}A1 ,A2 ...An−1 ，末尾AnA_nAn 变为0。 题解思路 根据题目意思模拟即可，方法很多，本处选择使用数组模拟，使用数组模拟可以参考以下思路。 1. 输入nnn个数字至AAA数组当中 2. 若满足 k<nk < nk<n 则输出a[k+1]a[k+1]a[k+1]至a[n]a[n]a[n]的数字，同时补上kkk个0在末尾 。 3. 若满足 k>=nk >= nk>=n 则输出nnn个0 。 参考代码 T2 INTERNETSHOP 题目大意 给出nnn个数字，找到第一个大于hih_ihi 的数字。否则输出-1. 题解思路 在题目给出的nnn个数字之中，针对于h1h_1h1 ，for循环枚举从左往右找到比他更大的第一个hih_ihi 即，否则输出−1-1−1。 参考代码 T3 YUILICE的盛大PARTY 题目大意 给出mmm个游戏，每场游戏共有kkk名参与者，同时给出他们编号，要求你记录下每个人之间是否有共同做过游戏，最后根据所有人是否都两两做过游戏输出Yes或No。 题解思路 我们可以使用二维数组来管理第iii人与第jjj人是否参与了一个游戏，对于每一场游戏我们都可以根据参与人数的编号来更新他们之间的关系，让二维数组arr[i][j]arr[i][j]arr[i][j]记录下第iii人与第jjj人是否参与了一个游戏。 最终需要时间复杂度O(n3)O(n^3)O(n3)。 参考代码 T4 RED AND GREEN 题目大意 给定两个水桶，体积为RRR与GGG，并且RRR于GGG，可以进行三种操作 1. RRR倒空水 2. GGG倒给RRR，满或GGG为空为止 3. GGG灌满水 求解kkk次操作后，有多少水还在两个水桶当中。 题解思路 我们设定x,yx,yx,y分别代表此时两个桶的水量，其中xxx代表红色水桶, yyy代表绿色水桶，来模拟其中的操作。 如果x=Rx = Rx=R,则设置x=0x = 0x=0，同理，如果y=0y = 0y=0,则设置y=Ry = Ry=R. 如果都不满足，则执行将xxx当中的水倒入yyy当中，需要移动的毫升为yyy毫升或RRR毫升，取最小值，所以只需要让倒入的水z=min(y,R−x)z = min(y , R - x)z=min(y,R−x)即可，那么结果为 xxx增加zzz, yyy减少zzz。 因为K≤100K \leq 100K≤100，因此暴力模拟即可。 参考代码

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C++递推：从基础递推关系构建高效算法解决方案 在C编程领域，递推是一种极为重要且基础的算法思想与编程技术。它通过依据已知的初始条件或边界情况，按照特定的递推关系逐步推导出后续的结果，从而有效地解决一系列具有规律性和重复性的问题。递推算法以其简洁直观的逻辑、相对较低的时间与空间复杂度以及广泛的应用场景，成为C程序员解决实际问题的得力工具。本文将深入全面地介绍C中的递推，涵盖其基本概念、核心原理、与递归的区别与联系、常见的递推类型、在经典算法中的应用、递推算法的设计与实现要点、性能优化策略以及实际应用中的注意事项与拓展等多方面内容，旨在为读者清晰呈现递推算法的全貌，助力其在C编程实践中熟练运用递推思想，高效构建优质的程序解决方案。 一、递推的基本概念 递推，从本质上讲，是一种基于已知信息逐步推导出未知信息的过程。在数学与计算机科学中，它通常表现为通过给定的初始值或边界条件，依据某种固定的递推公式或规则，依次计算出后续的一系列值。 例如，考虑一个简单的斐波那契数列问题。斐波那契数列的定义为：F(0)=0F(0)=0F(0)=0，F(1)=1F(1)=1F(1)=1，且对于n≥2n\geq2n≥2，有F(n)=F(n−1)+F(n−2)F(n)=F(n - 1)+F(n - 2)F(n)=F(n−1)+F(n−2)。这里，F(0)F(0)F(0)和F(1)F(1)F(1)就是初始条件，而F(n)=F(n−1)+F(n−2)F(n)=F(n - 1)+F(n - 2)F(n)=F(n−1)+F(n−2)则是递推关系。通过这个递推关系，从已知的F(0)F(0)F(0)和F(1)F(1)F(1)开始，我们可以逐步计算出F(2)=F(1)+F(0)=1F(2)=F(1)+F(0)=1F(2)=F(1)+F(0)=1，F(3)=F(2)+F(1)=2F(3)=F(2)+F(1)=2F(3)=F(2)+F(1)=2，以此类推，计算出整个斐波那契数列的各项值。 二、递推的核心原理 递推的核心原理在于利用问题本身所具有的重复性和规律性结构，建立起前后项之间的明确关联，即递推关系。这种递推关系能够将复杂的问题逐步分解为一系列简单的、可重复执行的计算步骤，从初始状态出发，按照既定的规则逐步推进，直至得到所需的最终结果。 在实际应用中，递推通常需要明确两个关键要素：一是初始条件或边界情况，这些是递推的起点，为后续的推导提供了基础数据；二是递推公式或规则，它描述了如何从已知的前序项推导出后续项，是递推过程的核心逻辑依据。 以计算阶乘为例，阶乘的递推定义为：n!=n×(n−1)!n! = n\times(n - 1)!n!=n×(n−1)!，其中初始条件为0!=10!=10!=1。从这个初始条件出发，根据递推公式，我们可以依次计算出1!=1×0!=11!=1\times0!=11!=1×0!=1，2!=2×1!=22!=2\times1!=22!=2×1!=2，3!=3×2!=63!=3\times2!=63!=3×2!=6，等等，逐步推导出任意正整数nnn的阶乘值。 三、递推与递归的区别与联系 （一）区别 1. 实现方式： * 递推是一种迭代的过程，通常通过循环结构（如for循环或while循环）来实现。它从初始条件开始，按照递推关系逐步计算出后续结果，在计算过程中只保留当前和必要的前序结果，不需要像递归那样进行大量的函数调用和栈操作。 * 递归则是函数在其自身内部调用自身，通过不断地深入递归调用，直到满足终止条件后再逐步回溯返回结果。递归在每次调用时都会创建新的函数调用帧，将当前的局部变量、参数等信息压入栈中，这会导致较大的空间开销，尤其是在递归深度较大时。 2. 空间复杂度： * 由于递推主要使用循环和有限的变量来存储中间结果，其空间复杂度相对较低，通常只与问题的规模相关的常数级或线性级。例如，在计算斐波那契数列的递推实现中，只需要几个变量来存储当前和前两个数，空间复杂度为O(1)O(1)O(1)（如果忽略结果存储所需的空间）。 * 递归的空间复杂度较高，因为它依赖于函数调用栈来保存每次调用的信息。在最坏情况下，递归深度可能与问题规模相等，导致空间复杂度为O(n)O(n)O(n)，其中nnn为问题规模。例如，在计算斐波那契数列的递归实现中，如果计算较大的斐波那契数，递归调用栈会迅速增长，可能导致栈溢出错误。 3. 执行效率： * 递推在执行效率上通常较高，尤其是对于大规模数据的处理。因为它避免了递归调用的开销，能够快速地通过循环迭代计算出结果。例如，在计算较大的斐波那契数时，递推实现的速度明显快于递归实现。 * 递归由于函数调用和返回的开销，以及可能存在的重复计算（如在斐波那契数列递归实现中，会多次计算相同的子问题），其执行效率相对较低，特别是对于深度较大的递归情况。 （二）联系 1. 问题解决思路：递推和递归在某些情况下可以用于解决相同类型的问题，它们都基于问题的重复性和规律性。例如，斐波那契数列既可以用递推算法实现，也可以用递归算法实现，只是实现方式和性能特点有所不同。 2. 相互转换：在一些情况下，可以将递归算法转换为递推算法，反之亦然。例如，通过分析递归算法的执行过程，提取出其中的递推关系，就可以将其改写为递推实现，以提高性能。同样，对于一些复杂的递推问题，如果难以直接用循环实现，也可以考虑用递归的思路来解决，然后再进行优化。 四、常见的递推类型 （一）线性递推 线性递推是最常见的递推类型之一，其递推关系表现为当前项只与前一项或前几项成线性关系。例如，斐波那契数列就是一种线性递推关系，其中F(n)F(n)F(n)只与F(n−1)F(n - 1)F(n−1)和F(n−2)F(n - 2)F(n−2)相关。再如，等差数列也是线性递推的典型例子，其递推公式为an=an−1+da_{n}=a_{n - 1}+dan =an−1 +d，其中ddd为公差，初始条件为a1a_{1}a1 的值。 （二）非线性递推 非线性递推关系则相对复杂，当前项与前一项或前几项之间的关系不是简单的线性组合。例如，汉诺塔问题的递推关系就属于非线性递推。设将nnn个圆盘从起始柱移动到目标柱所需的最少步数为T(n)T(n)T(n)，则有T(n)=2T(n−1)+1T(n)=2T(n - 1)+1T(n)=2T(n−1)+1，初始条件为T(1)=1T(1)=1T(1)=1。这种递推关系中，T(n)T(n)T(n)与T(n−1)T(n - 1)T(n−1)之间不是简单的线性相加关系，而是涉及到倍数和常数项的复杂组合。 （三）多维递推 多维递推是指递推关系涉及到多个维度的变量。例如，在计算二维数组中从左上角到右下角的路径数量时，可以使用二维递推关系。设dp[i][j]dp[i][j]dp[i][j]表示从(0,0)(0,0)(0,0)到(i,j)(i,j)(i,j)的路径数量，对于一个m×nm\times nm×n的二维数组，有递推关系dp[i][j]=dp[i−1][j]+dp[i][j−1]dp[i][j]=dp[i - 1][j]+dp[i][j - 1]dp[i][j]=dp[i−1][j]+dp[i][j−1]（当i>0i>0i>0且j>0j>0j>0时），边界条件为dp[0][j]=1dp[0][j]=1dp[0][j]=1（当j≥0j\geq0j≥0时）和dp[i][0]=1dp[i][0]=1dp[i][0]=1（当i≥0i\geq0i≥0时）。这里的递推关系涉及到二维数组中的两个维度变量iii和jjj。 五、递推在经典算法中的应用 （一）斐波那契数列计算 如前所述，斐波那契数列可以通过递推高效地计算。以下是使用递推实现计算斐波那契数列第nnn项的C++代码示例： 在这个代码中，首先处理了初始条件n=0n=0n=0和n=1n=1n=1的情况，然后通过循环利用递推关系F(n)=F(n−1)+F(n−2)F(n)=F(n - 1)+F(n - 2)F(n)=F(n−1)+F(n−2)计算出后续的斐波那契数，只需要三个变量a、b、c来存储当前和前两个数，空间复杂度低且执行效率较高。 （二）动态规划中的递推应用 动态规划算法本质上是一种基于递推思想的优化算法策略。例如，在求解最长上升子序列问题时，可以使用递推来构建动态规划解决方案。设dp[i]表示以第i个元素结尾的最长上升子序列的长度，则有递推关系dp[i]=max(dp[j])+1（其中j<i且nums[j]<nums[i]），初始条件为dp[i]=1（当没有比nums[i]更小的前序元素时）。通过两层循环，外层循环遍历数组元素，内层循环用于寻找满足条件的前序元素并更新dp[i]，最终可以得到整个数组的最长上升子序列长度。 （三）数塔问题 数塔问题是一个经典的递推应用场景。给定一个数塔，从顶部出发，在每一步可以选择向左下或右下走，要求找到一条从塔顶到塔底的路径，使得路径上的数字之和最大。设dp[i][j]表示从塔顶到第i行第j列的最大路径和，则有递推关系dp[i][j]=max(dp[i - 1][j - 1], dp[i - 1][j])+nums[i][j]（当i>0且j>0且j<i时），边界条件为dp[0][0]=nums[0][0]。通过从塔顶开始，按照递推关系自顶向下或自底向上地计算dp数组，最终可以得到从塔顶到塔底的最大路径和。 以下是自底向上计算的C++代码示例： 六、递推算法的设计与实现要点 （一）确定递推关系 这是递推算法设计的核心步骤。需要深入分析问题的本质，找出前后项之间的内在联系，确定准确的递推公式或规则。在分析过程中，可以通过列举前几项的结果，观察其规律，尝试推导递推关系。例如，在解决上述数塔问题时，通过分析从塔顶到下一层的路径选择与数字和的关系，得出了dp[i][j]=max(dp[i - 1][j - 1], dp[i - 1][j])+nums[i][j]的递推关系。 （二）确定初始条件或边界情况 明确递推的起点，即初始条件或边界情况。这些条件是递推的基础，没有正确的初始条件，递推过程将无法正确展开。在不同的问题中，初始条件的形式和数量可能不同。例如，在斐波那契数列中，有两个初始条件F(0)=0和F(1)=1；在数塔问题中，边界条件为dp[0][0]=nums[0][0]。在确定初始条件时，要确保其完整性和正确性，覆盖所有可能的起始情况。 （三）选择合适的实现方式 根据递推关系和问题的特点，选择合适的实现方式，通常是使用循环结构。对于简单的一维递推，可以使用单个for循环；对于多维递推，则可能需要嵌套的循环结构。在循环中，要按照递推关系正确地更新变量或数组元素的值。例如，在计算斐波那契数列的递推代码中，使用一个for循环来迭代计算后续的斐波那契数；在数塔问题的自底向上计算代码中，使用两层嵌套的for循环来更新dp数组。 七、递推算法的性能优化策略 （一）空间优化 在一些递推算法中，可以通过分析递推过程中对前序结果的依赖情况，进行空间优化。例如，在计算斐波那契数列时，只需要存储当前和前两个数，不需要使用数组来存储整个数列的中间结果，从而减少了空间的占用。在动态规划的递推应用中，如最长上升子序列问题，如果只需要求最大长度而不需要具体的子序列，可以使用一个变量来记录当前的最大长度，而不需要完整的dp数组，将空间复杂度从O(n)O(n)O(n)降低到O(1)O(1)O(1)。 （二）避免重复计算 有些递推问题可能存在大量的重复计算。例如，在递归实现斐波那契数列时，会多次计算相同的子问题。在递推实现中，可以通过记录已经计算过的结果来避免重复计算。一种常见的方法是使用数组或哈希表来存储中间结果，在计算之前先检查是否已经计算过，如果是，则直接使用存储的结果。这种技术称为记忆化，它可以显著提高递推算法的效率，尤其是对于具有重叠子问题的递推关系。 八、递推在实际应用中的注意事项与拓展 （一）注意事项 1. 递推关系的准确性：确保所确定的递推关系准确无误，否则会导致计算结果错误。在推导递推关系时，要进行充分的分析和验证，可以通过手工计算前几项结果与实际情况进行对比，或者使用数学归纳法等方法进行证明。 2. 边界情况处理：正确处理初始条件和边界情况，避免遗漏或错误设置。边界情况往往是递推算法的关键部分，如果处理不当，可能会导致程序运行错误或得到不正确的结果。在编写代码时，要对边界情况进行单独的测试和验证，确保其正确性。 3. 数据范围考虑：对于一些数值较大的递推问题，要考虑数据类型的选择，避免数据溢出。例如，在计算斐波那契数列的较大项时，如果使用普通的int类型，可能会因为数值过大而导致溢出错误。此时，可以考虑使用long long类型或高精度计算库来处理大数值。 （二）拓展 递推思想不仅仅局限于简单的数值计算，还可以拓展到其他领域。例如，在图形学中，可以使用递推算法来生成分形图形，如科赫雪花、谢尔宾斯基三角形等，通过不断地重复特定的几何变换规则来构建复杂而美丽的图形。在人工智能领域，一些搜索算法和决策算法也可以基于递推思想进行设计和优化，如马尔可夫决策过程中的值迭代算法，通过递推地计算状态值函数来确定最优策略。此外，递推还可以与其他算法思想相结合，如与贪心算法结合用于解决一些组合优化问题，与分治算法结合处理大规模数据的递推计算等，进一步拓展其应用范围和解决问题的能力。 九、总结 C递推作为一种基础且强大的算法思想与编程技术，以其简洁高效的特点在众多领域中发挥着重要作用。通过深入理解递推的基本概念、核心原理、与递归的区别与联系、常见递推类型、在经典算法中的应用、设计与实现要点、性能优化策略以及实际应用中的注意事项与拓展等多方面知识，程序员能够在C编程实践中熟练运用递推算法解决各种具有规律性和重复性的问题。无论是在数值计算、动态规划、图形学还是人工智能等领域，递推都为构建高效、优质的程序解决方案提供了有力的支持。在不断的实践和探索中，进一步挖掘递推算法的潜力，结合实际问题的特点进行灵活应用和创新拓展，将有助于提升C++编程水平，为解决复杂多变的现实世界问题提供更多有效的手段和方法。

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1.PE（Pattern Error）输出格式错误 可能是没加空格或者换行 2.RE（Runtime Error）运行错误 60%概率是递归、递推死循环了 tips：数组太大也会RE哦！ 3.TLE（Time Limit Exceed）超出时间限制/超时 时间复杂度太大了 tips：TLE一般是循环嵌套得太多层了，或者循环没出口！ 4.OLE（Output Limit Exceed）输出超出限制/输出超限 在此感谢用户ID“2927399”提供的 OLE 代码 5.AC（Accepted）答案正确/通过 6.WA（Wrong Answer）答案错误 以上就是ACGO中常见的错误！ 找图不易，点个爱心+关注吧 -_-" 有兴趣的可以看看用户“4398808”提供的题目 链接直达

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

一、复合运算符：+=、-=、*=、/=、%= 自增、自减符号：++、-- a+=1;------->a=a+1; a++;-------->a=a+1; 二、for循环 语法： for(①循环初值;②循环条件;④条件改变量){ ③循环体 }