回溯——3、5升杯倒4升水

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原创 2024-10-31 17:22:38 477 阅读 AI 写同款· GEO 优化 ›
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回溯应用

接前面

递归的改造——间隔挑硬币打印所挑选的硬币需要用到回溯。但书上的回溯没说一定要用递归调用。

书上说

回溯的本质是深度优先，还需要建立栈。
在这里插入图片描述

数学浅谈

一般说用扩展欧几里得除法的方法。

最大公约数

小学数学也要学复习。这里理解就是3升水和5升水都可以有1升水组成。4升和6升最大有两升组成。

g c d ( a , b ) = x ∗ a + y ∗ b gcd(a,b)=x*a+y*b gcd(a,b)=x∗a+y∗b

拙见最大公约数1升的，可以倒出所有1的倍数的水，最大公约数2升的，可以倒出所有2的倍数的水。
[基础定理]任意给定二个整数a,b, 总是存在二个自然数x, y, 使得 g c d ( a , b ) = x ∗ a + y ∗ b gcd(a,b)=x*a+y*b gcd(a,b)=x∗a+y∗b.这里要给数学家来点难度，只给一杯8升水怎么倒？

P 3 2 = 6 P_{3}^{2}=6 P32​=6

三杯里选两杯有6种情况。C是组合，这里是排列，搞错了。排列的英文是 Permutation 或者 Arrangement，因此在数学符号中，用 P 或者 A 表示都可以，二者意思完全一样。

只要一杯8升水

有6种情形可以倒，但要i杯有水且j杯有空间，才能i倒j，可以i倒满j，或者i倒空j未满出，不能洒；但这个倒法不能重复之前的状态,否则无限循环；直到最后有一杯出现了4升水。

代码

def next_step(num):
    for i in range(num):
        for j in range(num):
            if i != j:
                yield i, j
def isin(a, b):
    for i in b:
        if a == i:return True
    return False   
                 
def f(s, trace, X=[], d=4):
    #准备初值：s原始杯子状态二维列表,trace倒的步骤三维数组第一步默认状态[0,5], [8,8], [0,3],X解正确的步骤,d目标水升数
    global traceback
    traceback += 1
    for k in s:
        if d == k[0]:#最后的成功,倒出d了。
            X.append(copy.deepcopy(trace))
            return True
    #准备初值：P(3,2)三杯里选两杯有6种情况    
    a = next_step(len(s))
    for i, j in a:#(范围未超界并且工作未完成)
        #分析条件：i杯有水且j杯有空间，才能i倒j
        if 0 != s[i][0] and 0 != s[j][1] - s[j][0]:#保证满足条件才往下走
            a = s[i][0]
            b = s[j][0]
            if s[i][0] >= s[j][1] - s[j][0]:#分析条件：i倒满j
                s[i][0] = s[i][0] + s[j][0] - s[j][1]
                s[j][0] = s[j][1]
            else:#分析条件：i倒空j未满
                s[j][0] += s[i][0]
                s[i][0] = 0
            if isin(s, trace):#本层更换选择;横向走
                #不能有重复状态,否则无限循环,i,j要恢复分析条件前的状态。
                s[i][0] = a
                s[j][0] = b
                continue 
            else:#(成功)局部成功，可以下一步。 
                trace.append(copy.deepcopy(s))#路径进栈;由第一选择开始进入下一层;纵向走
            f(s, trace, X, d)#纵向走 下一层
            #(工作未完成)弹出堆栈；原来的上一层更换为下一选择
            trace.pop()#回溯到上一层横向走。前面的for那里。
            #这里i,j也要恢复分析条件前的状态。
            s[i][0] = a
            s[j][0] = b
    #(全部工作未完成)因为有时六种倒法都不对。
    return X #输出结果

一般回溯方法的程序结构

把书上的每一步都写到代码里，没有什么能比这个说得再清除了。

打印

总共有13种倒法。总共回溯了212次。

>>> b=f([[0,5], [8,8], [0,3]], [[[0,5], [8,8], [0,3]]], [], 4)
>>> for x in b:print(b)

[[[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]]]
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[[[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]]]
>>> print(traceback)
212
>>> print(len(b))
13
>>>
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《算法与数据结构》
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“算法+数据结构=程序“。深爱着，想把你吃透。执着着，一连几天，闭眼都是你的影子。也迷茫过，按时交作业都难。你也有些小脾气，错个符号、漏个字的。找错误是很痛苦的，尖锐的笔刻向了你，血从我心底流出。你布满了我的印记，也不乏甜蜜的小涂鸦。结婚十四年，感谢老婆的一路陪伴。我是幸福的，会一直捧你们在手心！

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OpenClaw-遇到的问题-Health check failed: gateway closed (1006 abnormal closure (no close frame)): no clos OpenClaw网关服务安装成功但运行异常，显示健康检查失败（错误代码1006）。系统提供了Web控制界面（http://127.0.0.1:18789/）和带令牌的安全访问链接。文档建议参考故障排除指南，并提供了可选应用扩展功能（包括移动端支持）。注意运行本地代理存在安全风险，建议加固系统配置。若需启用网络搜索功能，需配置Brave搜索API密钥。安装完成后会自动打开仪表盘，需保持浏览器标签页以控制OpenClaw服务。 阅读详情

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自学笔记，分享给对统计学原理不太清楚但需要在论文中用到的小伙伴，欢迎大佬们补充或绕道。ps：本文不涉及公式讲解（文科生小白友好体质）～（部分定义等来源于知乎） 本文重点：多因素方差分析 【1.多因素方差分析简单原理】 【2.多因素方差分析步骤】 【3.多因素方差分析代码】

TUTO_TUTO的博客 4866

视频隐私脱敏工程化：CSRT/KCF追踪算法在本地推理中的实现与边界

在短视频内容合规审查中，动态遮挡（Dynamic Masking）是高频且高耗时的环节。传统手动关键帧打码不仅效率低下，且在目标快速移动或遮挡时极易出现遮罩漂移。本文从计算机视觉工程角度，解析基于CSRT与KCF算法的目标追踪方案在视频隐私处理中的技术路径、性能权衡及落地实践。

Mysssone的博客 255

HyperLynx（十五）多板仿真

在对话框的“Design file#1”下拉列表中选择“B00，Mainboard”，在其下方的“Reference designators”列表中选择元器件“J1”，然后在“Designfile#2”下拉列表中选择“BO1、RAM1”，在其下方的“Refer ence designators”列表中选择元器件“J1”，单击“Insert”按钮，为这两个元器件建立互连映射。在“Assign Models”对话框中，选择“B00 Motherboard”，将“U16.B12”设置为输出。

小幽余生不加糖 5061

【算法拾光】链表 哈希表｜K个一组翻转链表 + 两数之和

K个一组翻转链表需分段反转，借助哑节点与前驱指针维护连接，避免断链；两数之和利用哈希表实现O(1)查找，空间换时间，高效定位目标组合。前者考察链表指针操作，后者聚焦映射关系构建，均为面试高频题。

wuqingyigexiaoba的博客 324

19、硬件工程师笔试面试——无线通讯模块（2025版本）

例如，如果你的项目需要传输大量数据，如视频流或大数据文件，那么你可能需要一个高速率的模块，如支持802.11ac或802.11ax（WiFi 6）的模块。如果你的数据传输需求较小，如传感器数据，那么一个较低速率的模块，如蓝牙或Zigbee，可能就足够了。较高的频率（如5GHz）提供更大的带宽，支持更高的数据传输速率，但信号覆盖范围较小，穿透能力较弱。例如，在蓝牙通信中，一台设备作为主机，另一台设备作为从机，它们通过频率跳频的方式确定连接，并在连接建立后通过特定的协议进行数据交换。

XU157303764的博客 3226

4.Simulink基础建模操作——加减乘除运算

加减乘除运算

FPGA/MATLAB学习教程/源码/项目合作开发 9550

把 KV Cache 压到 FP4：BitDecoding-FP4 的算法-系统协同设计复盘

长上下文推理卡在KV Cache的显存与带宽上，而现有FP4方案要么精度崩、要么速度废。这篇文章复盘BitDecoding-FP4如何用三个算法补丁加一个系统重构，在B300上把服务吞吐提升2.4–3.0倍，精度几乎不掉。

helloagent的博客 302

Carla运行显示The following component(s) are required to run this program:DirectX Runtime |nvapi64.dll没有被

Carla运行显示The following component(s) are required to run this p 参考： 【1】EPIC 启动游戏提示The following component(s) are required to run this program:... - 『悬赏问答区』 - 吾爱破解 - LCG - LSG |安卓破解|病毒分析|www.52pojie.cnhttps://www.52pojie.cn/thread-1485651-1-1.html ...

anananajiushiwo的博客 5万+

排序算法深度笔记（下）：非比较类排序的线性突破与工程边界

排序算法时间复杂度额外空间复杂度稳定性是否基于比较选择排序O(N^2)(O(1))不稳定是冒泡排序O(N^2)(O(1))稳定是插入排序O(N^2)(O(1))稳定是归并排序O(N\log N)(O(N))稳定是快速排序O(N\log N)O(log N)不稳定是堆排序O(N\log N)O(1)不稳定是计数排序(O(N + M))O(M))稳定否基数排序O(N)稳定否桶排序平均O(N) / 最坏O(N^2)

2401_87069538的博客 598

LinePhone3.2.7(android端最新版本) sip客户端

SIP客户端For android：LinePhone3.2.7(最新版本)，官网地址：http://www.linphone.org/

win10 跳过U8环境检测的IE WebControls.rar

win10 安装u8的时候 提示IE WebControls 不通过，这个工具可以欺骗u8的检查。 跳过U8环境检测的IE WebControls。提示导入失败后，重新检测通过。

优选算法：分治-归并排序

交易逆序对的总数

Wangshijie1015的博客 142

文档：亲测可用解决vnc灰屏，黑屏，鼠标大黑叉情况

主要解决vnc-viewer灰屏，黑屏，鼠标大黑叉等现象，详细见文档。

算法日常・每日刷题--＜动态规划＞42

本题为“不相交的线”，与最长公共子序列问题相似。使用动态规划求解：定义 dp[i][j] 表示 nums1[0:i] 与 nums2[0:j] 的最大不相交连线数。若元素相等，则继承左上值加1；否则取上方或左方最大值。最终返回 dp[m][n]，即两数组的最大不相交连线数。时间复杂度 O(mn)，空间复杂度 O(mn)。

2503_92261635的博客 23

温度PID控制PLC程序

温度PID稳定控制程序，适用于欧姆龙PLC，对其他PLC也具有很大参考价值

408 算法 06｜堆与 Top-K：最小的 10 个数，到底该用数组还是堆

2022(42) 找 100 万个数里最小的 10 个，官方给了"10 元素有序数组"和"大根堆"两种答案。我把两种都实现并数了比较次数：k=10 时两者几乎打平，k=10000 时堆少 111 倍。另附 2021(42) 计数排序的比较次数与稳定性实测。

m0_56226131的博客 358

牛客周赛165C题(前后缀性质，递推，排序，枚举)

题目大意：给定长度为 `n` 的数组，选择 `k ∈ [1,n]`，将前 `k` 个元素整体翻转，后面的元素不变。要求操作后的整个数组非递减，统计满足条件的 `k` 的数量。题目链接：https://ac.nowcoder.com/acm/contest/141721/C。时间复杂度：O(nlogn)

2501_94316951的博客 172

408 算法 04｜主元素与三元组最小距离：两种“扫描时做决定“的思路

2013(41) 的主元素用摩尔投票一遍选出候选，但真正的考点是"选完必须验票"；2020(41) 的三元组最小距离，三指针每次只挪最小的那个，靠的是 D=2×(max−min) 这个恒等式。两套完整 C 代码加真实输出，附官方评分档位表。

m0_56226131的博客 416

【数据结构-3】算法美学之降复杂度

在while循环的循环体内，首先用临时变量保存最后一个位置的数据，再通过for循环把所有元素往后移动一位，最后把tmp中保存的值放到下标为0的位置上去。方法二的思路是：创建新数组，让原数组中后k个数据移动到新数组中前面的位置，让原数组中前numsSize-k个数据移动到新数组中后面的位置，这。这段代码的思路是：三次逆置，第一次逆置前numsSize-k个数据，第二次逆置后k个数据，最后整体逆置。这篇博文分享一道解法不一的算法题，求解过程的目的在于降低算法的时间复杂度，题源力扣。，属于空间换时间的操作。

a bit 89

算法61，判定是否为字符重排

两字符串能互为重排 ⇔。

sheshibenniaodan的博客 220

算法日常・每日刷题--＜动态规划＞32

给定严格递增的正整数数组，求最长斐波那契子序列长度。使用二维DP，dp[i][j]表示以arr[i]、arr[j]为最后两项的序列长度，初始化均为2。通过哈希表快速查找前一项target = arr[j] - arr[i]，若存在且下标小于i，则状态转移：dp[i][j] = dp[k][i] + 1。遍历更新全局最大长度，最终若结果小于3则返回0。时间复杂度O(n²)，空间复杂度O(n²)。

2503_92261635的博客 180

【前缀和+复合离散化】题解：双色矩形(two)_数据结构_算法优化_预处理_二维前缀和_STL_C++算法竞赛

给定一个 $n \times n$ 的黑白棋盘，初始按 $i+j$ 奇偶染色，经 $k$ 个格子颜色翻转后，求子矩形中黑白格子数差的绝对值最大值。由于 $n$ 可达 $10^9$，但 $k \leq 50$，可离散化关键点。枚举所有由修改点构成的矩形及其四边向外扩展（共 $16$ 种扩展方式），利用二维前缀和快速计算变化量，结合奇偶性判断多出的格子颜色，最终取最大绝对差。时间复杂度为 $O(k^4)$，在 $k \leq 50$ 下可行。

YLCHUP的博客 183

算法札记：几何分布在算法题UVA1262的运用

集卡问题求解核心：将集齐n种优惠券的过程分为n个阶段，每个阶段期望购买次数为 $ \frac{n}{k} $（k为剩余种类数），总期望为 $ n \cdot H_n $（调和数）。通过自洽方程推导单阶段期望，无需几何分布或无穷级数，逻辑清晰且适用于大一水平。例如，n=5时期望约11.42，非5。题目要求以带分数形式输出精确结果，算法上需累加分数并化简。

Keven_11的博客 368

SSL证书选RSA还是ECDSA：公钥算法、CA签名与兼容性怎么核对

申请SSL证书时，RSA和ECDSA该怎么选？先分清站点公钥算法、CA签名算法和TLS实际兼容性，再用隔离OpenSSL实验读取两类材料，核对申请、续期和部署交接，避免只看Signature Algorithm就选错。

谁呀！的博客 401

考拉兹猜想 - 7

本文通过引入虚数单位重构Collatz猜想的迭代逻辑，指出偶数减半与奇数“乘3加1”的过程在虚数坐标下等价于归结操作。关键在于：乘3加1使奇数跃升至偶数，随后持续减半，而虚数单位代表“无限减1”，实现无限向有限的转化。无论初始值如何，减半始终主导，最终归结为4-2-1循环。该视角消解了正向思维中的速度竞争疑虑，证明归结是必然结果。

杨铸人的博客 48
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