3dm下的GTA5 V1.33盗版，下载完成后，解压打开游戏，显示需要网际网络服务，更改时间也进不去_百度知道
色情、暴力
我们会通过消息、邮箱等方式尽快将举报结果通知您。
3dm下的GTA5 V1.33盗版，下载完成后，解压打开游戏，显示需要网际网络服务，更改时间也进不去
我有更好的答案
建议找最新升级档及破解补丁打一下，游戏文件夹路径不能有中文！我的文档默认位置为C盘下！先打开GTALauncher最小化或放一边别关！运行Launcher进入游戏！
采纳率：76%
盗版的需要联网？？？？？你逗我？
`你网速跟不上
是盗版的，怎么会需要联网
为您推荐：
其他类似问题
网际网络服务的相关知识
等待您来回答我从3大妈那里下载了v1.33版本的gtagta5，弄了三天进不去啊_百度知道
我从3大妈那里下载了v1.33版本的gtagta5，弄了三天进不去啊
我从3大妈那里下载了v1.33版本的gtagta5，弄了三天进不去啊没有打五号档显示的是0×0000142，打了五号档之后gtavlauncher变成了正版模式要我输入验证码，3大妈的launcher显示已停止工作，我弄了三天弄不了啊，求高手搭救
我有更好的答案
5.1，2复制Update文件夹下的升级文件到游戏目录全部下载完，解压缩第一个压缩包至D盘下（只要目录不含中文都可以）.8。我的文档-右键-属性-“还原默认值”。我的文档位置必须为C盘！，3复制Crack文件夹下的破解文件到游戏目录。可运行一次注册表一键恢复工具打开GTALauncher.exe最小化或放一边别关！！接下来安装升级档及破解补丁V5：1安装Installers文件夹下的Social Club v1
已经试过是不行的，打了五号档加补丁显示工作停止运行
我的文档-右键-属性-“还原默认值”。我的文档位置必须为C盘！！！
打开GTALauncher.exe最小化，运行Launcher.exe进入游戏。
还原完我的文档，必须重装Social Club
都试过了不可以
DX11 VC游戏运行库呢？
都更新了 而且试过修复 没查出问题
搜“GTA5PC版常见安装运行问题解决方法汇总”看看吧
采纳率：76%
为您推荐：
其他类似问题
等待您来回答车易保安卓版(车易保汽车后服务平台)V2.1 汉化版软件下载 - 绿色先锋下载 - 绿色软件下载站
赞助商链接
本类月下载排行
本类周下载排行
常用软件推荐
车易保安卓版(车易保汽车后服务平台)是一款专门为车主们打造的汽车后期服务应用，拥有车险比价、预约接送、租车、代驾等等特色功能更是轻松解决你的燃眉之急，涵盖了包括汽车美容、汽车保养、洗车、4S售后、快修钣喷等多种汽车服务功能于一体，让你足不出户先解决好自己爱车的问题，极大方便了你的生活，车险比价、预约接送、租车、代驾等等特色功能更是轻松解决你的燃眉之急，还在等什么，赶快来绿色先锋网下载吧！
车易保专业从事全国性汽车后服务平台，为数百万车主提供全方位的汽车养护服务。集合洗车、美容、保养、品牌维修、4S店售后、快修钣喷等主流优质商户，提供有品质保障的服务。并为车主提供汽车资讯、车险比价、预约接送、救援、租车、代驾、加油、停车等等多场景化的便捷服务，打造中国汽车后服市场最专业的、车主用户量最大的整合服务商！车易保利用移动互联网技术，解决车主需求的快速响应与对接，解决商户与车主需求不对称，商家展示过于封闭且难以形成品牌号召力等难点。
将用户的需求在互联网移动端采集，向目标商户进行推送，促进点对点接触机会，将商户信息进行发布便于用户车主寻找，同时推送各类活动促销在平台展示，便于车主在线了解与下单，提高到店率及活动有效率，同时将营业高低峰合理配置，提高人工与场地的产出比，将可能的空置价值挖掘，提升商户店面提升！ 车易保号召有品质的、注重商誉、有保障、具有互联网与客户服务意识的汽车服务性机构共同参与，加入车易保养护体系服务商。我们将向合作商户开放资源，实现资源互补、共享，分享！我们是中国领先有品质的汽车后服务平台！我们是最专业汽车服务的商户顾问！我们是车主出行的服务伴侣和管家
开发者简介
车易保网络科技成都有限公司位于成都市高新技术开发区成都市高新区天府大道北段１４８０号９号楼４栋６层１号交通便利。车易保网络科技成都有限公司本着&客户第一，诚信至上&的原则，欢迎国内外企业/公司/机构与本单位建立长期的合作关系。公司拥有一支较强的技术研发队伍并和国内多家机构强强联手，研发出拥有自主知识产权。热诚欢迎各界朋友前来参观、考察、洽谈业务。苏杰代表车易保网络科技成都有限公司欢迎新老客户来电咨询。
车：专业、清晰
易：容易、交易
保：保险、保养、保障
专业互联网汽车服务平台，oto
买车险找车易保
养护车找车易保
&&请点击以下链接下载该软件：&车易保安卓版(车易保汽车后服务平台)V2.1 汉化版
上一软件：
下一软件：
(评论内容只代表网友观点，与本站立场无关)
为了保证您快速的下载,推荐使用[] 、[] 等专业工具下载.
为确保软件能正常使用,请使用[
]解压本站软件.
目前不少软件都捆绑流氓插件，请在安装的之时务必留意每一个安装步骤.绿色先锋本身是不会捆绑任何插件在软件中的.
该软件为网上收集，若无意中侵犯了您的版权，.我们将在收信后24小时内删除侵权内
本站下载的软件中，部分软件经过压缩加密处理，解压密码为：
感谢您对绿色先锋的支持，请将网站地址放在您的博客，空间等地方，以便我们为您及您的朋友提供更好的服务.
软件按字母排列：
中文按声母搜索：老外写了系列文章，&a href=&///?target=http%3A//cgnail.github.io/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&cgnail&i class=&icon-external&&&/i&&/a& 做了中文翻译，非常棒，每篇翻译都有原文的链接。&br&&ul&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-1/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&我的最爱Lambda演算——开篇 · cgnail's weblog&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-2/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&阿隆佐.丘奇的天才之作——lambda演算中的数字 · cgnail's weblog&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-3/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Lambda演算中的布尔值和选择 · cgnail's weblog&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-4/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&为什么是Y？ · cgnail's weblog&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-5/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&从Lambda演算到组合子演算 · cgnail's weblog&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-6/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Lambda演算的类型 · cgnail's weblog&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&///?target=http%3A//cgnail.github.io/academic/lambda-7/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&终章，Lambda演算建模——程序即证明！ · cgnail's weblog&i class=&icon-external&&&/i&&/a&&/li&&/ul&
老外写了系列文章， 做了中文翻译，非常棒，每篇翻译都有原文的链接。
Leveldb是一个 Google 实现的非常高效的kv数据库，目前的版本1.2能够支持billion级别的数据量了。 在这个数量级别下还有着非常高的性能，主要归功于它的良好的设计。特别是LSM算法。&ol&&li&&a href=&/?target=http%3A//qkldx.net/topic/18/leveldb%25E6%E7%25A8%258B1-%25E5%E8%25AF%2586leveldb& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程1 - 初识LevelDB&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/19/leveldb%25E6%E7%25A8%258B2-%25E6%%25E4%25BD%%259E%25B6%25E6%259E%2584& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程2 - 整体架构&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/20/leveldb%25E6%E7%25A8%258B3-log%25E6%E4%25BB%25B6& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程3 - log文件&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/21/leveldb%25E6%E7%25A8%258B4-sstable%25E6%E4%25BB%25B6& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程4 - SSTable文件&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/22/leveldb%25E6%E7%25A8%258B5-memtable%25E8%25AF%25A6%25E8%25A7%25A3& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程5：MemTable详解&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/23/leveldb%25E6%E7%25A8%258B6-%25E5%E5%%25E4%25B8%258E%25E5%%25E9%%25E8%25AE%25B0%25E5%25BD%2595& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程6：写入与删除记录&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/24/leveldb%25E6%E7%25A8%258B7-%25E8%25AF%25BB%25E5%258F%%25AE%25B0%25E5%25BD%2595& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDb教程7：读取记录&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/25/leveldb%25E6%E7%25A8%258B8-compaction%25E6%E4%25BD%259C& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程8：Compaction操作&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/26/leveldb%25E6%E7%25A8%258B9-leveldb%25E4%25B8%25AD%25E7%259A%2584cache& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程9：levelDB中的Cache&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/27/leveldb%25E6%E7%25A8%258B10-version-versionedit-versionset& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&LevelDB教程10：Version、VersionEdit、VersionSet&i class=&icon-external&&&/i&&/a&&/li&&/ol&&p&---&br&&/p&&p&精彩推荐&/p&&ul&&li&&a href=&/?target=http%3A//qkldx.net/topic/478& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&不增反降？比特币开发者提议降低区块大小至300K&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/410& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Vitalik Buterin : 去中心化的真正含义&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/382& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&HTTPS的七个误解（译文）&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/490& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&HTTPS工作原理和TCP握手机制&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/380& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&图解SSL/TLS协议&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/374& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Git内部原理&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/338& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&高性能队列——Disruptor&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/336& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Hyperledger Fabric V1.0– 开发者快速入门&i class=&icon-external&&&/i&&/a&&br&&/li&&li&&a href=&/?target=http%3A//qkldx.net/topic/335& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&分布式系统φ累计失败检测算法介绍&i class=&icon-external&&&/i&&/a&&br&&/li&&/ul&&p&加入区块连大学交流群请联系微信：isLishude&/p&
Leveldb是一个 Google 实现的非常高效的kv数据库，目前的版本1.2能够支持billion级别的数据量了。 在这个数量级别下还有着非常高的性能，主要归功于它的良好的设计。特别是LSM算法。
&figure&&img src=&/50/v2-e0f19ecc6fc8_b.jpg& data-rawwidth=&1083& data-rawheight=&610& class=&origin_image zh-lightbox-thumb& width=&1083& data-original=&/50/v2-e0f19ecc6fc8_r.jpg&&&/figure&&h2&&b&这篇文章讲了什么&/b&&/h2&&p&我将会在这篇文章中提到一点 Finger Tree 的设计思想、Finger Tree 的实现细节、 Finger Tree 的应用等内容。&/p&&h2&&b&我需要看这篇文章吗&/b&&/h2&&p&你需要对数据结构有着简单的了解，并且对 Finger Tree 有一定兴趣但没有深入了解。&/p&&p&你可能需要学会基础的 Haskell 语法&/p&&h2&&b&为什么是 Finger Tree&/b&&/h2&&p&它提供了高效的各种序列操作：&/p&&p&&br&&/p&&figure&&img src=&/v2-ac2cb9cc296d8cfaaf82d_b.jpg& data-caption=&& data-rawwidth=&1125& data-rawheight=&519& class=&origin_image zh-lightbox-thumb& width=&1125& data-original=&/v2-ac2cb9cc296d8cfaaf82d_r.jpg&&&/figure&&p&&br&&/p&&p&它在很多方面几乎达到了理论上的极限，这也是我们为什么在很多地方要使用 Finger Tree 来作为储存序列的东西。&/p&&h2&Finger Tree 是什么&/h2&&p&我们首先从简单的二叉树开始谈起&/p&&p&我们在 Haskell 中如何表示一颗二叉树？&/p&&p&这很简单，像这样就可以了&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&kr&&data&/span& &span class=&kt&&Tree&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Nil&/span&
&span class=&o&&|&/span& &span class=&kt&&Tree&/span& &span class=&p&&(&/span&&span class=&kt&&Tree&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&a&/span& &span class=&p&&(&/span&&span class=&kt&&Tree&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&当然我们也可以这样写（我们把它叫做 leafy 的二叉树）&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Tree&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Leaf&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&Tree&/span& &span class=&p&&(&/span&&span class=&kt&&Tree&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&kt&&Tree&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&我们可以发现第二种树是无法表达“空树”这个概念的，而且我们每次访问树上的值都需要到达叶子结点才能取到值。&/p&&p&那么如何表示一棵满的 leafy 的二叉树呢？我们可以借鉴自然数的定义。&/p&&p&自然数我们一般会写成这样&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Nat&/span& &span class=&ow&&=&/span& &span class=&kt&&Z&/span& &span class=&o&&|&/span& &span class=&kt&&S&/span& &span class=&kt&&Nat&/span&
&/code&&/pre&&/div&&p&借助它我们可以表达类型的嵌套层数&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Nat&/span& &span class=&n&&s&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Z&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&S&/span& &span class=&p&&(&/span&&span class=&kt&&Nat&/span& &span class=&n&&s&/span& &span class=&p&&(&/span&&span class=&n&&s&/span& &span class=&n&&a&/span&&span class=&p&&))&/span&
&/code&&/pre&&/div&&p&于是我们只需要定义 leafy 二叉树的结点类型&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Node&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Node&/span& &span class=&n&&a&/span& &span class=&n&&a&/span&
&/code&&/pre&&/div&&p&就有&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&type&/span& &span class=&kt&&Tree&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Nat&/span& &span class=&kt&&Node&/span& &span class=&n&&a&/span&
&/code&&/pre&&/div&&p&&code&Nat&/code& 每推进一层，&code&Node&/code& 就会多嵌套一层，而 &code&Node&/code& 的定义保证了它一定是一棵满的 leafy 二叉树，比如&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kt&&Z&/span& &span class=&mi&&1&/span& &span class=&ow&&::&/span& &span class=&kt&&Tree&/span& &span class=&kt&&Int&/span&
&span class=&kt&&S&/span& &span class=&p&&(&/span&&span class=&kt&&Z&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&mi&&1&/span& &span class=&mi&&2&/span&&span class=&p&&))&/span& &span class=&ow&&::&/span& &span class=&kt&&Tree&/span& &span class=&kt&&Int&/span&
&span class=&kt&&S&/span& &span class=&p&&(&/span&&span class=&kt&&S&/span& &span class=&p&&(&/span&&span class=&kt&&Z&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&mi&&1&/span& &span class=&mi&&2&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span&&span class=&p&&))))&/span&
&/code&&/pre&&/div&&p&而不满的二叉树会产生类型错误&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kt&&S&/span& &span class=&p&&(&/span&&span class=&kt&&S&/span& &span class=&p&&(&/span&&span class=&kt&&Z&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&mi&&1&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&mi&&2&/span& &span class=&mi&&3&/span&&span class=&p&&))))&/span& &span class=&c1&&-- error&/span&
&/code&&/pre&&/div&&p&然后我们把二叉树结点换成 2-3 树的结点&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Node23&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Node2&/span& &span class=&n&&a&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&Node3&/span& &span class=&n&&a&/span& &span class=&n&&a&/span& &span class=&n&&a&/span&
&span class=&kr&&data&/span& &span class=&kt&&Tree23&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Nat&/span& &span class=&kt&&Node23&/span& &span class=&n&&a&/span&
&/code&&/pre&&/div&&p&这就得到了 Finger Tree 的前身，我们给一个例子 (这个例子是 Paper 上的)&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kt&&S&/span&
&span class=&p&&(&/span&&span class=&kt&&S&/span&
&span class=&p&&(&/span&&span class=&kt&&S&/span&
&span class=&p&&(&/span&&span class=&kt&&Z&/span&
&span class=&p&&(&/span&&span class=&kt&&Node2&/span&
&span class=&p&&(&/span&&span class=&kt&&Node3&/span&
&span class=&p&&(&/span&&span class=&kt&&Node2&/span& &span class=&n&&t&/span& &span class=&n&&h&/span&&span class=&p&&)&/span&
&span class=&p&&(&/span&&span class=&kt&&Node2&/span& &span class=&n&&i&/span& &span class=&n&&s&/span&&span class=&p&&)&/span&
&span class=&p&&(&/span&&span class=&kt&&Node2&/span& &span class=&n&&i&/span& &span class=&n&&s&/span&&span class=&p&&)&/span&
&span class=&p&&)&/span&
&span class=&p&&(&/span&&span class=&kt&&Node3&/span&
&span class=&p&&(&/span&&span class=&kt&&Node3&/span& &span class=&n&&n&/span& &span class=&n&&o&/span& &span class=&n&&t&/span&&span class=&p&&)&/span&
&span class=&p&&(&/span&&span class=&kt&&Node2&/span& &span class=&n&&a&/span& &span class=&n&&t&/span&&span class=&p&&)&/span&
&span class=&p&&(&/span&&span class=&kt&&Node3&/span& &span class=&n&&r&/span& &span class=&n&&e&/span& &span class=&n&&e&/span&&span class=&p&&)&/span&
&span class=&p&&)&/span&
&span class=&p&&)&/span&
&span class=&p&&)&/span&
&span class=&p&&)&/span&
&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&画出来是这个样子&/p&&p&&br&&/p&&figure&&img src=&/v2-99efb528caa536e49c98e6de2a596995_b.jpg& data-caption=&& data-rawwidth=&1079& data-rawheight=&229& class=&origin_image zh-lightbox-thumb& width=&1079& data-original=&/v2-99efb528caa536e49c98e6de2a596995_r.jpg&&&/figure&&p&&br&&/p&&p&这棵树储存的序列是 &thisisnotatree&。&/p&&p&树形结构在访问时对效率很大的一个限制就是深度。大多数情况，平衡树都是单纯通过限制深度来保证访问时间能达到
，在这里的这棵树中也是一样的。但我们想要更快的访问头尾结点，同时不牺牲随机访问其它结点的效率，于是我们想到的方法是缩短树的头尾结点到根结点的距离。&/p&&p&我们把树最左端和最右端的两条链提起来，变成下面这个样子&/p&&figure&&img src=&/v2-a040f0f43a_b.jpg& data-caption=&& data-rawwidth=&698& data-rawheight=&413& class=&origin_image zh-lightbox-thumb& width=&698& data-original=&/v2-a040f0f43a_r.jpg&&&/figure&&p&&br&&/p&&p&然后把这两条链粘在一起，就得到了一棵能
访问头尾结点的树了。&/p&&p&被提起来的那两条链，我们把它们叫做 Digit,
被粘起来之后合成的那些结点我们把它们叫做 Deep，于是我们下面给出 Finger Tree 的定义&/p&&h2&&b&Finger Tree 的定义&/b&&/h2&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&Empty&/span&
&span class=&o&&|&/span& &span class=&kt&&Single&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&Deep&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&a&/span&&span class=&p&&))&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&span class=&kr&&type&/span& &span class=&kt&&Digit&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Digit&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&]&/span&
&/code&&/pre&&/div&&p&Digit 这里我们使用 List 来表示。&/p&&p&可以看出，Finger Tree 中的 a 是一个嵌套了 n 次的 &code&Node&/code& ，所以 Finger Tree 每向下一层，结点的数量就会至少增加一倍，所以这棵树的深度显然是 &img src=&/equation?tex=O%28%5Clog+n%29& alt=&O(\log n)& eeimg=&1&& 的。&/p&&p&下面是 &a href=&/?target=http%3A//www.staff.city.ac.uk/%7Eross/papers/FingerTree.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&这里&i class=&icon-external&&&/i&&/a& 给出的一些例子，其中红色是 &code&Deep&/code& ，绿色是 &code&Node&/code& ，蓝色是数据&/p&&figure&&img src=&/v2-42ed25ba850e8c60efb828d9ea5986d7_b.jpg& data-rawwidth=&109& data-rawheight=&142& class=&content_image& width=&109&&&figcaption&Deep (Digit [a]) (Single (Node3 b c d)) (Digit [e])&/figcaption&&/figure&&figure&&img src=&/v2-9bb9f1f5b49dfe62c43e6d_b.jpg& data-caption=&& data-rawwidth=&209& data-rawheight=&198& class=&content_image& width=&209&&&/figure&&figure&&img src=&/v2-e6e32fa7cd4b135f9421_b.jpg& data-caption=&& data-rawwidth=&787& data-rawheight=&271& class=&origin_image zh-lightbox-thumb& width=&787& data-original=&/v2-e6e32fa7cd4b135f9421_r.jpg&&&/figure&&p&&br&&/p&&p&&br&&/p&&p&为了保证 Finger Tree 的操作效率，我们限制了 &code&Digit&/code& 的大小在 1 到 4 之间。&/p&&p&下面我们就可以开始实现 Finger Tree 的各种操作了。&/p&&h2&Finger Tree 上的操作&/h2&&h2&&b&头尾插入单个结点&/b&&/h2&&p&第一个是 push-front 和 push-back ，即从头/尾插入 &/p&&p&（迫真安利：使用 Fira-Code 可以达到更好的符号显示效果）&/p&&p&push-front 的实现: &/p&&p&&br&&/p&&figure&&img src=&/v2-bb889a069b25ed697e807_b.jpg& data-caption=&& data-rawwidth=&772& data-rawheight=&267& class=&origin_image zh-lightbox-thumb& width=&772& data-original=&/v2-bb889a069b25ed697e807_r.jpg&&&/figure&&p&&br&&/p&&p&在插入时我们优先插在 &code&Digit&/code& 上，当 &code&Digit&/code& 满了我们会把 &code&Digit&/code& 分裂并向下一级插入一个新的 &code&Node&/code& 结点。每向下一级需要的 &code&a&/code& 的数量会翻
倍，所以插入这个过程是均摊 &img src=&/equation?tex=%5CTheta%281%29& alt=&\Theta(1)& eeimg=&1&&
的。&/p&&h2&&b&头尾删除单个结点&/b&&/h2&&p&然后是删除头/尾结点。这个过程的效率和插入一样，也是 &img src=&/equation?tex=%5CTheta%281%29& alt=&\Theta(1)& eeimg=&1&& 的。&/p&&p&我们(&del&Paper 上&/del&)实现了一个数据类型 &code&ViewL&/code& （实际上是可以使用 &code&Maybe&/code& 的）来储存提取的结果&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&kr&&data&/span& &span class=&kt&&ViewL&/span& &span class=&n&&s&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Nil&/span&
&span class=&o&&|&/span& &span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&p&&(&/span&&span class=&n&&s&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&然后有&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&
&span class=&n&&viewL&/span& &span class=&ow&&::&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&ViewL&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span&
&span class=&n&&viewL&/span& &span class=&kt&&Empty&/span& &span class=&ow&&=&/span& &span class=&kt&&Nil&/span&
&span class=&n&&viewL&/span& &span class=&p&&(&/span&&span class=&kt&&Single&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&kt&&Empty&/span&
&span class=&n&&viewL&/span& &span class=&p&&(&/span&&span class=&kt&&Deep&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&l&/span&&span class=&p&&)&/span& &span class=&n&&m&/span& &span class=&n&&r&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kt&&Cons&/span& &span class=&p&&(&/span&&span class=&n&&head&/span& &span class=&n&&l&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&n&&deepL&/span& &span class=&p&&(&/span&&span class=&n&&tail&/span& &span class=&n&&l&/span&&span class=&p&&)&/span& &span class=&n&&m&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&这里的 &code&deepL&/code& 是处理左 &code&Digit&/code& 不一定存在时的 Finger Tree 构造函数&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&deepL&/span& &span class=&ow&&::&/span&
&span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&]&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span&
&span class=&kt&&Digit&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&n&&a&/span&
&span class=&n&&deepL&/span& &span class=&kt&&[]&/span& &span class=&n&&m&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&r&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kr&&case&/span& &span class=&n&&viewL&/span& &span class=&n&&m&/span& &span class=&kr&&of&/span&
&span class=&kt&&Nil&/span& &span class=&ow&&-&&/span& &span class=&n&&toTree&/span& &span class=&n&&r&/span&
&span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&n&&as&/span& &span class=&ow&&-&&/span& &span class=&kt&&Deep&/span& &span class=&p&&(&/span&&span class=&n&&nodeToDigit&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&as&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&span class=&n&&deepL&/span& &span class=&n&&l&/span& &span class=&n&&m&/span& &span class=&n&&r&/span& &span class=&ow&&=&/span& &span class=&kt&&Deep&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&l&/span&&span class=&p&&)&/span& &span class=&n&&m&/span& &span class=&n&&r&/span&
&/code&&/pre&&/div&&p&如果为空，就再从中间的子树中分离出一个结点（但在这里是 &code&Node a&/code&），转化为 &code&Digit&/code& 后再放在左边。(转化为 &code&Digit&/code& 的过程是 trivial 的)&/p&&p&有了 &code&viewL&/code& 我们就能得到诸如 &code&headL&/code& , &code&tailL&/code& 这样的函数。&/p&&p&到这里我们就已经拥有一个合格的 &code&Deque&/code& 了，我们不妨封装一下&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&newtype&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Deque&/span& &span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&span class=&n&&pushFront&/span& &span class=&ow&&::&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span&
&span class=&n&&pushFront&/span& &span class=&n&&a&/span& &span class=&p&&(&/span&&span class=&kt&&Deque&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kt&&Deque&/span& &span class=&p&&(&/span&&span class=&n&&a&/span& &span class=&o&&&|&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span&
&span class=&n&&pushBack&/span& &span class=&ow&&::&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span&
&span class=&n&&pushBack&/span& &span class=&p&&(&/span&&span class=&kt&&Deque&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Deque&/span& &span class=&p&&(&/span&&span class=&n&&xs&/span& &span class=&o&&|&&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&span class=&n&&popFront&/span& &span class=&ow&&::&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&View&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span&
&span class=&n&&popFront&/span& &span class=&p&&(&/span&&span class=&kt&&Deque&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kr&&case&/span& &span class=&n&&viewL&/span& &span class=&n&&xs&/span& &span class=&kr&&of&/span&
&span class=&kt&&Nil&/span& &span class=&ow&&-&&/span& &span class=&kt&&Nil&/span&
&span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&n&&as&/span& &span class=&ow&&-&&/span& &span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&p&&(&/span&&span class=&kt&&Deque&/span& &span class=&n&&as&/span&&span class=&p&&)&/span&
&span class=&n&&popBack&/span& &span class=&ow&&::&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&View&/span& &span class=&kt&&Deque&/span& &span class=&n&&a&/span&
&span class=&n&&popBack&/span& &span class=&p&&(&/span&&span class=&kt&&Deque&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kr&&case&/span& &span class=&n&&viewR&/span& &span class=&n&&xs&/span& &span class=&kr&&of&/span&
&span class=&kt&&Nil&/span& &span class=&ow&&-&&/span& &span class=&kt&&Nil&/span&
&span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&n&&as&/span& &span class=&ow&&-&&/span& &span class=&kt&&Cons&/span& &span class=&n&&a&/span& &span class=&p&&(&/span&&span class=&kt&&Deque&/span& &span class=&n&&as&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&看上去还不错。&/p&&p&但还不够，我们需要让两个 Finger Tree 连起来。&/p&&h2&&b&把两棵 Finger Tree 连起来&/b&&/h2&&p&考虑两个 Finger Tree ， &code&Deep l1 m1 r1&/code&
和 &code&Deep l2 m2 r2&/code& ，我们需要把它们拼起来。&/p&&p&自然的想法就是把 &code&l1&/code& 和 &code&r2&/code& 拿出来，把中间的部分再组装一次成为一棵新树。&/p&&p&那么我们需要这样一个函数：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&nf&&appendDigits&/span& &span class=&ow&&::&/span&
&span class=&kt&&FingerTree&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span&
&span class=&kt&&Digit&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&kt&&Digit&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&但事实上我们可以把 &code&Digit&/code& 都塞到一起（因为它们本质上还是 &code&[a]&/code&），来组成一些 &code&Node a&/code&
( &code&Node a&/code& 是中间那棵 Finger Tree 的数据类型)，这样我们只需要实现这样两个函数：(conc for concat)&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&conc&/span& &span class=&ow&&::&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&]&/span& &span class=&ow&&-&&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span&
&span class=&n&&nodes&/span& &span class=&ow&&::&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&]&/span& &span class=&ow&&-&&/span& &span class=&p&&[&/span&&span class=&kt&&Node&/span& &span class=&n&&a&/span&&span class=&p&&]&/span&
&/code&&/pre&&/div&&p&第二个函数的实现也没什么，每隔 2 / 3 个元素切一刀就好了。&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&nf&&nodes&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&n&&b&/span&&span class=&p&&]&/span& &span class=&ow&&=&/span& &span class=&p&&[&/span&&span class=&kt&&Node2&/span& &span class=&n&&a&/span& &span class=&n&&b&/span&&span class=&p&&]&/span&
&span class=&n&&nodes&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&n&&b&/span&&span class=&p&&,&/span& &span class=&n&&c&/span&&span class=&p&&]&/span& &span class=&ow&&=&/span& &span class=&p&&[&/span&&span class=&kt&&Node3&/span& &span class=&n&&a&/span& &span class=&n&&b&/span& &span class=&n&&c&/span&&span class=&p&&]&/span&
&span class=&n&&nodes&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&n&&b&/span&&span class=&p&&,&/span& &span class=&n&&c&/span&&span class=&p&&,&/span& &span class=&n&&d&/span&&span class=&p&&]&/span& &span class=&ow&&=&/span& &span class=&p&&[&/span&&span class=&kt&&Node2&/span& &span class=&n&&a&/span& &span class=&n&&b&/span&&span class=&p&&,&/span& &span class=&kt&&Node2&/span& &span class=&n&&c&/span& &span class=&n&&d&/span&&span class=&p&&]&/span&
&span class=&n&&nodes&/span& &span class=&p&&(&/span&&span class=&n&&a&/span&&span class=&kt&&:&/span& &span class=&n&&b&/span&&span class=&kt&&:&/span& &span class=&n&&c&/span&&span class=&kt&&:&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kt&&Node3&/span& &span class=&n&&a&/span& &span class=&n&&b&/span& &span class=&n&&c&/span& &span class=&kt&&:&/span& &span class=&n&&nodes&/span& &span class=&n&&xs&/span&
&span class=&n&&nodes&/span& &span class=&kr&&_&/span& &span class=&ow&&=&/span& &span class=&ne&&error&/span& &span class=&s&&&What the fuck??&&/span&
&/code&&/pre&&/div&&p&第一个函数则有一些细节需要考虑。&/p&&p&当其中一颗树是空树或者 &code&Single&/code& 时，我们可以直接把 &code&List&/code& 中的元素一个一个塞进去（这个函数的调用者传进来的 &code&List&/code& 大小是常数，所以不会影响时间复杂度）。&/p&&p&当两颗树都是 &code&Deep&/code& 时，情形和直接连接两颗树差不多，只需要把 &code&l2&/code& ，&code&r2&/code& 和加进来的 &code&list&/code& 一起转成 &code&list&/code& 就行了。&/p&&p&所以 &code&conc&/code& 的实现也就很自然了&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&conc&/span& &span class=&kt&&Empty&/span& &span class=&n&&as&/span& &span class=&n&&xs&/span& &span class=&ow&&=&/span& &span class=&n&&listConTree&/span& &span class=&n&&as&/span& &span class=&n&&xs&/span&
&span class=&n&&conc&/span& &span class=&n&&xs&/span& &span class=&n&&as&/span& &span class=&kt&&Empty&/span& &span class=&ow&&=&/span& &span class=&n&&treeConList&/span& &span class=&n&&xs&/span& &span class=&n&&as&/span&
&span class=&n&&conc&/span& &span class=&p&&(&/span&&span class=&kt&&Single&/span& &span class=&n&&x&/span&&span class=&p&&)&/span& &span class=&n&&as&/span& &span class=&n&&xs&/span& &span class=&ow&&=&/span& &span class=&n&&x&/span& &span class=&o&&&|&/span& &span class=&n&&listConTree&/span& &span class=&n&&as&/span& &span class=&n&&xs&/span&
&span class=&n&&conc&/span& &span class=&n&&xs&/span& &span class=&n&&as&/span& &span class=&p&&(&/span&&span class=&kt&&Single&/span& &span class=&n&&x&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&treeConList&/span& &span class=&n&&xs&/span& &span class=&n&&as&/span& &span class=&o&&|&&/span& &span class=&n&&x&/span&
&span class=&n&&conc&/span& &span class=&p&&(&/span&&span class=&kt&&Deep&/span& &span class=&n&&l1&/span& &span class=&n&&m1&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&r1&/span&&span class=&p&&))&/span& &span class=&n&&as&/span& &span class=&p&&(&/span&&span class=&kt&&Deep&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&l2&/span&&span class=&p&&)&/span& &span class=&n&&m2&/span& &span class=&n&&r2&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span&
&span class=&kt&&Deep&/span& &span class=&n&&l1&/span& &span class=&p&&(&/span&&span class=&n&&conc&/span& &span class=&n&&m1&/span& &span class=&p&&(&/span&&span class=&n&&nodes&/span& &span class=&p&&(&/span&&span class=&n&&r1&/span& &span class=&o&&++&/span& &span class=&n&&as&/span& &span class=&o&&++&/span& &span class=&n&&l2&/span&&span class=&p&&))&/span& &span class=&n&&m2&/span&&span class=&p&&)&/span& &span class=&n&&r2&/span&
&/code&&/pre&&/div&&p&listConTree和treeConList是两个辅助函数，暴力地把list中的东西一个一个塞到树里&/p&&p&于是我们可以有：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&kr&&infixl&/span& &span class=&mi&&5&/span& &span class=&o&&&&&/span&
&span class=&p&&(&/span&&span class=&o&&&&&/span&&span class=&p&&)&/span& &span class=&ow&&::&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&a&/span&
&span class=&n&&a&/span& &span class=&o&&&&&/span& &span class=&n&&b&/span& &span class=&ow&&=&/span& &span class=&n&&conc&/span& &span class=&n&&a&/span& &span class=&kt&&[]&/span& &span class=&n&&b&/span&
&/code&&/pre&&/div&&p&显然的，每一次我们调用 &code&conc&/code& 时的 &code&list&/code& 的长度不会大于 ，所以每次调用是均摊
的 ，而每一次会使两颗树同时向下一层，在其中一颗树触底时递归结束，所以递归层数不会超过
。&/p&&p&好的，我们的树现在可以实现 &code&concat&/code& 了，めでたしめでたし。&/p&&h2&实现其它操作的基础&/h2&&p&但上面的数据结构也仅仅只能做到这一步了，如果想要实现其它的功能，我们需要为它加上一些东西。&/p&&p&我们重写 Finger Tree 的各组成部分并添加标注 &code&v&/code&：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Node&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Node2&/span& &span class=&n&&v&/span& &span class=&n&&a&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&Node3&/span& &span class=&n&&v&/span& &span class=&n&&a&/span& &span class=&n&&a&/span& &span class=&n&&a&/span&
&span class=&kr&&data&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&Empty&/span&
&span class=&o&&|&/span& &span class=&kt&&Single&/span& &span class=&n&&a&/span&
&span class=&o&&|&/span& &span class=&kt&&Deep&/span& &span class=&n&&v&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&))&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&这里的 &code&v&/code& 是 Finger Tree 需要维护的&i&某种&/i&东西，而我们将会限制它的类型为 &code&Monoid&/code& 。&/p&&h2&&b&为什么是 &code&Monoid&/code& ？&/b&&/h2&&p&Finger Tree 事实上维护的是一个序列，而这里的 &code&v&/code& 是序列的某种性质，那么它应该对&i&空序列&/i&有对应的值，而且能正确的处理 Finger Tree 中频繁的序列合并的情况。所以我们需要一个有幺元和有结合律的东西。&/p&&p&为了用到我们添加的 &code&v&/code& ，我们可以引入一个 type class :&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&class&/span& &span class=&kt&&Monoid&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&kr&&where&/span&
&span class=&n&&measure&/span& &span class=&ow&&::&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&n&&v&/span&
&/code&&/pre&&/div&&p&直接写是会报错的，需要打开 GHC 扩展&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&cm&&{-# LANGUAGE MultiParamTypeClasses #-}&/span&
&/code&&/pre&&/div&&p&我们利用 &code&Measure&/code& 来&i&计算&/i&出我们需要的值&/p&&p&Finger Tree 的各部分都可以实现为 &code&Measured&/code& 的一个实例&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&instance&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&kt&&Measured&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&v&/span& &span class=&kr&&where&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Node2&/span& &span class=&n&&v&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&v&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Node3&/span& &span class=&n&&v&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&v&/span&
&span class=&kr&&instance&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&kt&&Measured&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&v&/span& &span class=&kr&&where&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Node2&/span& &span class=&n&&v&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&v&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Node3&/span& &span class=&n&&v&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&v&/span&
&span class=&kr&&instance&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&kt&&Measured&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&v&/span& &span class=&kr&&where&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&])&/span& &span class=&ow&&=&/span& &span class=&n&&measure&/span& &span class=&n&&a&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&p&&(&/span&&span class=&n&&a&/span& &span class=&kt&&:&/span& &span class=&n&&as&/span&&span class=&p&&))&/span& &span class=&ow&&=&/span&
&span class=&n&&measure&/span& &span class=&n&&a&/span& &span class=&p&&`&/span&&span class=&n&&mappend&/span&&span class=&p&&`&/span& &span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&as&/span&&span class=&p&&)&/span&
&span class=&kr&&instance&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&kt&&Measured&/span& &span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&v&/span& &span class=&kr&&where&/span&
&span class=&n&&measure&/span& &span class=&kt&&Empty&/span& &span class=&ow&&=&/span& &span class=&n&&mempty&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Single&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&measure&/span& &span class=&n&&a&/span&
&span class=&n&&measure&/span& &span class=&p&&(&/span&&span class=&kt&&Deep&/span& &span class=&n&&v&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&v&/span&
&/code&&/pre&&/div&&p&为了简化 &code&mappend&/code& 的使用，我们把 Semi Group 中的 &code&&&&/code& 借过来用一下&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&infixl&/span& &span class=&mi&&5&/span& &span class=&o&&&&&/span&
&span class=&p&&(&/span&&span class=&o&&&&&/span&&span class=&p&&)&/span& &span class=&ow&&::&/span& &span class=&kt&&Monoid&/span& &span class=&n&&a&/span& &span class=&ow&&=&&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&n&&a&/span&
&span class=&p&&(&/span&&span class=&o&&&&&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&mappend&/span&
&/code&&/pre&&/div&&p&然后我们就可以愉快的重写之前的代码了 XD&/p&&p&首先是 &code&Node&/code& 的新构造函数需要把 &code&v&/code& 一并计算(只写 node2)：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&node2&/span& &span class=&ow&&::&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&Node&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&
&span class=&n&&node2&/span& &span class=&n&&a&/span& &span class=&n&&b&/span& &span class=&ow&&=&/span& &span class=&kt&&Node2&/span& &span class=&p&&(&/span&&span class=&n&&measure&/span& &span class=&n&&a&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&b&/span&&span class=&p&&)&/span& &span class=&n&&a&/span& &span class=&n&&b&/span&
&/code&&/pre&&/div&&p&&code&Deep&/code& 也一样&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&deep&/span& &span class=&ow&&::&/span&
&span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span&
&span class=&kt&&Digit&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&p&&(&/span&&span class=&kt&&Node&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span&
&span class=&kt&&Digit&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&
&span class=&n&&deep&/span& &span class=&n&&l&/span& &span class=&n&&m&/span& &span class=&n&&r&/span& &span class=&ow&&=&/span& &span class=&kt&&Deep&/span& &span class=&p&&(&/span&&span class=&n&&measure&/span& &span class=&n&&l&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&m&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&r&/span&&span class=&p&&)&/span& &span class=&n&&l&/span& &span class=&n&&m&/span& &span class=&n&&r&/span&
&/code&&/pre&&/div&&p&其它的部分也没有很多变化，照着写即可。&/p&&h2&&b&把树分裂成两棵&/b&&/h2&&p&有了 &code&Measured&/code& 我们就能够实现 &code&split&/code& 了。为了合理的表示 &code&split&/code& 的结果，我们定义这么一个数据结构&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Split&/span& &span class=&n&&s&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Split&/span& &span class=&p&&(&/span&&span class=&n&&s&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&a&/span& &span class=&p&&(&/span&&span class=&n&&s&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&&code&Split&/code& 的三个数据域分别存放 &code&split&/code& 目标的左侧部分、目标、右侧部分。&/p&&p&当然这里用 Tuple 也是可以的&/p&&p&假设给出一个函数 &code&f :: v -& Bool&/code& ，对 Finger Tree 所维护的序列使用时结果是单调的，即在越过某个结点之前结果都是 &code&False&/code&
，我们就能将 Finger Tree 在这个结点分成 3 份。&/p&&p&我们要实现这么一个函数：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&splitTree&/span& &span class=&ow&&::&/span&
&span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span&
&span class=&p&&(&/span&&span class=&n&&v&/span& &span class=&ow&&-&&/span& &span class=&kt&&Bool&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span&
&span class=&n&&v&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&kt&&Split&/span& &span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&n&&a&/span&
&/code&&/pre&&/div&&p&显然的，我们不应该在一棵空树上 &code&split&/code& ，所以&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&splitTree&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&kt&&Empty&/span& &span class=&ow&&=&/span& &span class=&ne&&error&/span& &span class=&s&&&You should not split on an empty tree&&/span&
&/code&&/pre&&/div&&p&&code&Single&/code& 的话很简单，问题就在于 &code&Deep&/code& 该如何处理，方法和线段树一样。&/p&&p&分
种情况，分界点在左 &code&Digit&/code& 上、分界点在子树上、分界点在右 &code&Digit&/code& 上。 ,
实际上是一种，我们放在一起考虑。&/p&&p&如果分界点在 &code&Digit&/code& 上，我们可以单独使用一个函数来处理这种情形&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&splitDigit&/span& &span class=&ow&&::&/span& &span class=&kt&&Measure&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span& &span class=&p&&(&/span&&span class=&n&&v&/span& &span class=&ow&&-&&/span& &span class=&kt&&Bool&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span& &span class=&n&&v&/span& &span class=&ow&&-&&/span& &span class=&kt&&Digit&/span& &span class=&ow&&-&&/span& &span class=&kt&&Split&/span& &span class=&kt&&[]&/span& &span class=&n&&a&/span&
&/code&&/pre&&/div&&p&因为 &code&Digit&/code& 实际上就是个 &code&list&/code& ，所以只需要沿着每个结点判断一次就可以了&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&splitDigit&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&p&&[&/span&&span class=&n&&a&/span&&span class=&p&&])&/span& &span class=&ow&&=&/span& &span class=&kt&&Split&/span& &span class=&kt&&[]&/span& &span class=&n&&a&/span& &span class=&kt&&[]&/span&
&span class=&n&&splitDigit&/span& &span class=&n&&f&/span& &span class=&n&&i&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&p&&(&/span&&span class=&n&&a&/span& &span class=&kt&&:&/span& &span class=&n&&as&/span&&span class=&p&&))&/span&
&span class=&o&&|&/span& &span class=&n&&f&/span& &span class=&p&&(&/span&&span class=&n&&i&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kt&&Split&/span& &span class=&kt&&[]&/span& &span class=&n&&a&/span& &span class=&n&&as&/span&
&span class=&o&&|&/span& &span class=&n&&otherwise&/span& &span class=&ow&&=&/span&
&span class=&kr&&let&/span& &span class=&kt&&Split&/span& &span class=&n&&l&/span& &span class=&n&&x&/span& &span class=&n&&r&/span& &span class=&ow&&=&/span& &span class=&n&&splitDigit&/span& &span class=&n&&f&/span& &span class=&p&&(&/span&&span class=&n&&i&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&kt&&Digit&/span& &span class=&n&&as&/span&&span class=&p&&)&/span&
&span class=&kr&&in&/span&
&span class=&kt&&Split&/span& &span class=&p&&(&/span&&span class=&n&&a&/span& &span class=&kt&&:&/span& &span class=&n&&l&/span&&span class=&p&&)&/span& &span class=&n&&x&/span& &span class=&n&&r&/span&
&span class=&n&&splitDigit&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&ow&&=&/span& &span class=&ne&&error&/span& &span class=&s&&&What the fuck ??&&/span&
&/code&&/pre&&/div&&p&而如果分界点在树上，我们就可以递归调用这个过程，对子树直接使用 &code&splitTree&/code& ，会拿到一个 &code&Node a&/code& 回来，再把它转成 &code&Digit&/code& 并再使用一次 &code&splitDigit&/code& 就能拿到那个元素了。&/p&&p&再者就是 &code&splitDigit&/code& 拿回来的可能是一个空 &code&list&/code& ，所以这里要使用前面提到的 &code&deepL&/code& 和 &code&deepR&/code& 来构造新的 Finger Tree&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&nf&&splitTree&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&kt&&Empty&/span& &span class=&ow&&=&/span& &span class=&ne&&error&/span& &span class=&s&&&Split on an empty tree!&&/span&
&span class=&n&&splitTree&/span& &span class=&kr&&_&/span& &span class=&kr&&_&/span& &span class=&p&&(&/span&&span class=&kt&&Single&/span& &span class=&n&&x&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&kt&&Split&/span& &span class=&kt&&Empty&/span& &span class=&n&&x&/span& &span class=&kt&&Empty&/span&
&span class=&n&&splitTree&/span& &span class=&n&&f&/span& &span class=&n&&i&/span& &span class=&p&&(&/span&&span class=&kt&&Deep&/span& &span class=&kr&&_&/span& &span class=&n&&l&/span& &span class=&n&&m&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&span class=&o&&|&/span& &span class=&n&&f&/span& &span class=&n&&iml&/span& &span class=&ow&&=&/span& &span class=&kr&&let&/span& &span class=&kt&&Split&/span& &span class=&n&&l'&/span& &span class=&n&&x&/span& &span class=&n&&r'&/span& &span class=&ow&&=&/span& &span class=&n&&splitDigit&/span& &span class=&n&&f&/span& &span class=&n&&i&/span& &span class=&n&&l&/span& &span class=&kr&&in&/span&
&span class=&kt&&Split&/span& &span class=&p&&(&/span&&span class=&n&&toTree&/span& &span class=&n&&l'&/span&&span class=&p&&)&/span& &span class=&n&&x&/span& &span class=&p&&(&/span&&span class=&n&&deepL&/span& &span class=&n&&r'&/span& &span class=&n&&m&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&span class=&o&&|&/span& &span class=&n&&f&/span& &span class=&n&&imm&/span& &span class=&ow&&=&/span& &span class=&kr&&let&/span& &span class=&kt&&Split&/span& &span class=&n&&l'&/span& &span class=&n&&xs&/span& &span class=&n&&r'&/span& &span class=&ow&&=&/span& &span class=&n&&splitTree&/span& &span class=&n&&f&/span& &span class=&n&&iml&/span& &span class=&n&&m&/span&
&span class=&kt&&Split&/span& &span class=&n&&l''&/span& &span class=&n&&x&/span& &span class=&n&&r''&/span& &span class=&ow&&=&/span& &span class=&n&&splitDigit&/span& &span class=&n&&f&/span& &span class=&p&&(&/span&&span class=&n&&iml&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&l'&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&n&&nodeToDigit&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&kr&&in&/span&
&span class=&kt&&Split&/span& &span class=&p&&(&/span&&span class=&n&&deepR&/span& &span class=&n&&l&/span& &span class=&n&&l'&/span& &span class=&n&&l''&/span&&span class=&p&&)&/span& &span class=&n&&x&/span& &span class=&p&&(&/span&&span class=&n&&deepL&/span& &span class=&n&&r''&/span& &span class=&n&&r'&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&span class=&o&&|&/span& &span class=&n&&otherwise&/span& &span class=&ow&&=&/span&
&span class=&kr&&let&/span& &span class=&kt&&Split&/span& &span class=&n&&l'&/span& &span class=&n&&x&/span& &span class=&n&&r'&/span& &span class=&ow&&=&/span& &span class=&n&&splitDigit&/span& &span class=&n&&f&/span& &span class=&n&&imm&/span& &span class=&n&&r&/span& &span class=&kr&&in&/span&
&span class=&kt&&Split&/span& &span class=&p&&(&/span&&span class=&n&&deepR&/span& &span class=&n&&l&/span& &span class=&n&&m&/span& &span class=&n&&l'&/span&&span class=&p&&)&/span& &span class=&n&&x&/span& &span class=&p&&(&/span&&span class=&n&&toTree&/span& &span class=&n&&r'&/span&&span class=&p&&)&/span&
&span class=&kr&&where&/span& &span class=&n&&iml&/span& &span class=&ow&&=&/span& &span class=&n&&i&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&l&/span&
&span class=&n&&imm&/span& &span class=&ow&&=&/span& &span class=&n&&iml&/span& &span class=&o&&&&&/span& &span class=&n&&measure&/span& &span class=&n&&m&/span&
&/code&&/pre&&/div&&p&于是 &code&split&/code& 也很显然了&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&split&/span& &span class=&ow&&::&/span& &span class=&kt&&Measured&/span& &span class=&n&&a&/span& &span class=&n&&v&/span& &span class=&ow&&=&&/span&
&span class=&p&&(&/span&&span class=&n&&v&/span& &span class=&ow&&-&&/span& &span class=&kt&&Bool&/span&&span class=&p&&)&/span& &span class=&ow&&-&&/span&
&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span&
&span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&kt&&FingerTree&/span& &span class=&n&&v&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&span class=&n&&split&/span& &span class=&kr&&_&/span& &span class=&kt&&Empty&/span& &span class=&ow&&=&/span& &span class=&p&&(&/span&&span class=&kt&&Empty&/span&&span class=&p&&,&/span& &span class=&kt&&Empty&/span&&span class=&p&&)&/span&
&span class=&n&&split&/span& &span class=&n&&f&/span& &span class=&n&&xs&/span&
&span class=&o&&|&/span& &span class=&n&&f&/span& &span class=&p&&(&/span&&span class=&n&&measure&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&p&&(&/span&&span class=&n&&l&/span&&span class=&p&&,&/span& &span class=&n&&x&/span& &span class=&o&&&|&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&span class=&o&&|&/span& &span class=&n&&otherwise&/span& &span class=&ow&&=&/span& &span class=&p&&(&/span&&span class=&n&&xs&/span&&span class=&p&&,&/span& &span class=&kt&&Empty&/span&&span class=&p&&)&/span&
&span class=&kr&&where&/span& &span class=&kt&&Split&/span& &span class=&n&&l&/span& &span class=&n&&x&/span& &span class=&n&&r&/span& &span class=&ow&&=&/span& &span class=&n&&splitTree&/span& &span class=&n&&f&/span& &span class=&n&&mempty&/span& &span class=&n&&xs&/span&
&/code&&/pre&&/div&&p&在整个序列都不满足条件的时候我们直接返回空树&/p&&p&它的时间复杂度与合并两颗 Finger Tree 类似，也是均摊 &img src=&/equation?tex=%5CTheta%28%5Cmin%28n%2C+n+-+m%29%29& alt=&\Theta(\min(n, n - m))& eeimg=&1&&&/p&&h2&&b&下标访问&/b&&/h2&&p&为了提供 &code&Seq&/code& 的 &code&index&/code& 功能，我们需要为它定制一个 &code&Monoid&/code& 来实现取 size 的功能&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&newtype&/span& &span class=&kt&&Sized&/span& &span class=&ow&&=&/span& &span class=&kt&&Size&/span& &span class=&p&&{&/span&&span class=&n&&getSize&/span& &span class=&ow&&::&/span& &span class=&kt&&Int&/span&&span class=&p&&}&/span& &span class=&kr&&deriving&/span&&span class=&p&&(&/span&&span class=&kt&&Eq&/span&&span class=&p&&,&/span& &span class=&kt&&Ord&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&这里的 &code&Sized&/code& 就是封装过的 &code&Int&/code& ，而整数本身是一个幺半群。&/p&&p&于是有：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&instance&/span& &span class=&kt&&Monoid&/span& &span class=&kt&&Sized&/span& &span class=&kr&&where&/span&
&span class=&n&&mempty&/span& &span class=&ow&&=&/span& &span class=&kt&&Size&/span& &span class=&mi&&0&/span&
&span class=&kt&&Size&/span& &span class=&n&&a&/span& &span class=&p&&`&/span&&span class=&n&&mappend&/span&&span class=&p&&`&/span& &span class=&kt&&Size&/span& &span class=&n&&b&/span& &span class=&ow&&=&/span& &span class=&kt&&Size&/span& &span class=&o&&$&/span& &span class=&n&&a&/span& &span class=&o&&+&/span& &span class=&n&&b&/span&
&/code&&/pre&&/div&&p&然后可以把它套到 &code&Measure&/code& 上，但这里我们需要对 &code&Seq&/code& 的原始数据进行一次封装，塞到 &code&Elem&/code& 中。&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&newtype&/span& &span class=&kt&&Elem&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Elem&/span& &span class=&p&&{&/span&&span class=&n&&getElem&/span& &span class=&ow&&::&/span& &span class=&n&&a&/span&&span class=&p&&}&/span&
&span class=&kr&&newtype&/span& &span class=&kt&&Seq&/span& &span class=&n&&a&/span& &span class=&ow&&=&/span& &span class=&kt&&Seq&/span& &span class=&p&&(&/span&&span class=&kt&&FingerTree&/span& &span class=&kt&&Size&/span& &span class=&p&&(&/span&&span class=&kt&&Elem&/span& &span class=&n&&a&/span&&span class=&p&&))&/span&
&/code&&/pre&&/div&&p&然后实现 &code&Measure&/code& 的 &code&instance&/code&&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span&&span class=&kr&&instance&/span& &span class=&kt&&Measure&/span& &span class=&p&&(&/span&&span class=&kt&&Elem&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&kt&&Sized&/span& &span class=&kr&&where&/span&
&span class=&n&&measure&/span& &span class=&kr&&_&/span& &span class=&ow&&=&/span& &span class=&kt&&Size&/span& &span class=&mi&&1&/span&
&/code&&/pre&&/div&&p&于是在 &code&FingerTree Size (Elem a)&/code& 的数据域中，存放的就是当前树的大小，所以我们可以轻松地实现 &code&length&/code& ：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&length&/span& &span class=&ow&&::&/span& &span class=&kt&&Seq&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&Int&/span&
&span class=&n&&length&/span& &span class=&p&&(&/span&&span class=&kt&&Seq&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&getSize&/span& &span class=&o&&$&/span& &span class=&n&&measure&/span& &span class=&n&&xs&/span&
&/code&&/pre&&/div&&p&还记得我们之前实现的 &code&FingerTree v a&/code& 作为 &code&Measure&/code& 的实例吗，在这里就派上了用场。&/p&&p&然后是 &code&splitAt&/code& ，在特定的下标处分离两个序列&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&splitAt&/span& &span class=&ow&&::&/span& &span class=&kt&&Int&/span& &span class=&ow&&-&&/span& &span class=&kt&&Seq&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&p&&(&/span&&span class=&kt&&Seq&/span& &span class=&n&&a&/span&&span class=&p&&,&/span& &span class=&kt&&Seq&/span& &span class=&n&&a&/span&&span class=&p&&)&/span&
&span class=&n&&splitAt&/span& &span class=&n&&i&/span& &span class=&p&&(&/span&&span class=&kt&&Seq&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&p&&(&/span&&span class=&kt&&Seq&/span& &span class=&n&&l&/span&&span class=&p&&,&/span& &span class=&kt&&Seq&/span& &span class=&n&&r&/span&&span class=&p&&)&/span&
&span class=&kr&&where&/span& &span class=&p&&(&/span&&span class=&n&&l&/span&&span class=&p&&,&/span& &span class=&n&&r&/span&&span class=&p&&)&/span& &span class=&ow&&=&/span& &span class=&n&&split&/span& &span class=&p&&(&/span&&span class=&kt&&Size&/span& &span class=&n&&i&/span& &span class=&o&&&&/span&&span class=&p&&)&/span& &span class=&n&&xs&/span&
&/code&&/pre&&/div&&p&在满足 &code&Size i & Size x&/code& 之前的一共有
个元素，所以就是相当于从
处切开。&/p&&p&同时 &code&index&/code& 也类似&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&n&&index&/span& &span class=&ow&&::&/span& &span class=&kt&&Seq&/span& &span class=&n&&a&/span& &span class=&ow&&-&&/span& &span class=&kt&&Int&/span& &span class=&ow&&-&&/span& &span class=&n&&a&/span&
&span class=&n&&index&/span& &span class=&p&&(&/span&&span class=&kt&&Seq&/span& &span class=&n&&xs&/span&&span class=&p&&)&/span& &span class=&n&&i&/span& &span class=&ow&&=&/span& &span class=&n&&getElem&/span& &span class=&n&&x&/span&
&span class=&kr&&where&/span& &span class=&kt&&Split&/span& &span class=&kr&&_&/span& &span class=&n&&x&/span& &span class=&kr&&_&/span& &span class=&ow&&=&/span& &span class=&n&&splitTree&/span& &span class=&p&&(&/span&&span class=&kt&&Size&/span& &span class=&n&&i&/span& &span class=&o&&&&/span&&span class=&p&&)&/span& &span class=&n&&mempty&/span& &span class=&n&&xs&/span&
&/code&&/pre&&/div&&h2&&b&实现优先队列&/b&&/h2&&p&类似于 &code&Sized&/code& 这个 Monoid ，我们同样可以借助其它的 Monoid 来使我们的 Finger Tree 支持其它的操作，优先队列就是一个例子。&/p&&p&我们维护的序列性质就变成了序列中的最大值：&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&data&/span& &span class=&kt&&Priority&/span& &span class=&n&&a&/span&
&span class=&ow&&=&/span& &span class=&kt&&MinusInf&/span&
&span class=&o&&|&/span& &span class=&kt&&Priority&/span& &span class=&n&&a&/span&
&span class=&kr&&deriving&/span&&span class=&p&&(&/span&&span class=&kt&&Eq&/span&&span class=&p&&,&/span& &span class=&kt&&Ord&/span&&span class=&p&&)&/span&
&/code&&/pre&&/div&&p&显然，它也是一个 Monoid :&/p&&div class=&highlight&&&pre&&code class=&language-haskell&&&span&&/span& &span class=&kr&&instance&/span& &span class=&kt&&Ord&/span& &span class=&n&&a&/span& &span class=&ow&&=&&/span& &span class=&kt&&Monoid&/span& &span class=&p&&(&/span&&span class=&kt&&Priority&/span& &span class=&n&&a&/span&&span class=&p&&)&/span& &span class=&kr&&where&/span&
&span cl}