近年来，随着互联网和人工智能的广泛应用，算法作为其关键技术的内核，备受学校和企业的重视，算法竞赛更成为算法领域的一颗明珠。本书依托编著者多年算法竞赛的教学积累，全方位介绍了竞赛中常用的算法及近年来算法竞赛领域最新的研究成果，基于算法竞赛中广泛使用的在线评测网站——洛谷，着重介绍线性数据结构，基础算法，搜索算法，动态规划等方面的知识。本书适合对算法竞赛感兴趣的青少年阅读，也可作为相关领域教师、计算机专业学生的参考用书。

作者梁博：2019年至今，负责北大附中初中信息学奥赛教学工作。2022年学生成绩：入选省队：陈凯丰，谢梓涵，梁嘉文。北京20个省队名额中占3人。（根据往年情况，进入省队基本都可以保送清华或者北大）2021年学生成绩：吕彦哲银牌。陈凯丰（初二）作为夏令营选手获得银牌，签约北大，成为历史上签约北大最小的选手。2020年学生成绩：学生中初中组（普及组）参赛人数60人，一等奖获奖20人，二等奖25人，北京市7个满分同学中占3人。省一等奖比例30%远超全市平局值，总获奖率68%初二即有两名同学获得高中组（提高组）一等奖。2014-2021在小米负责签名与编译系统研发，科研成果转化为32项专利，如：CN201510549837.6一种应用软件预装次数的控制方法及装置CN201510547599.5应用版本信息的获取方法、设备和系统CN201510857770.2终端系统升级方法及装置CN201610694577.6数字签名方法及装置其余28项专利不列出，可以在相关网站检索到.2009-2021 浙江大学竺可桢学院计算机科学与技术专业

第 0 章 一些不那么常识的常识 ············································································.1
0.1 本地编程环境的配置··············································································.1
0.1.1 在 Windows 系统上安装使用 Dev C++···············································.1
0.1.2 在 MacOS 系统上安装 Xcode ··························································.4
0.2 在线评测系统—洛谷···········································································.7
0.2.1 注册洛谷 ···················································································.8
0.2.2 提交题目 ···················································································.9
0.2.3 团队管理 ···················································································11
第 1 章 线性数据结构 ························································································15
1.1 数据结构·····························································································15
1.1.1 数据结构的定义 ··········································································15
1.1.2 数据结构的运算 ··········································································17
1.1.3 线性数据结构 ·············································································17
1.2 栈······································································································18
1.2.1 栈的定义 ···················································································18
1.2.2 栈的作用 ···················································································20
1.2.3 栈的固定数组实现 ·······································································21
1.2.4 STL 中的栈 ················································································24
1.2.5 括号匹配问题 ·············································································26
1.2.6 前缀、中缀、后缀表达式 ······························································30
1.2.7 后缀表达式的计算 ·······································································32
1.2.8 中缀表达式转换为后缀表达式 ························································36
1.2.9 中缀表达式的计算 ·······································································41
1.3 队列···································································································43
1.3.1 队列的定义 ················································································44
1.3.2 队列的作用 ················································································46
1.3.3 队列的固定数组实现 ····································································46
1.3.4 STL 中的队列 ·············································································47
1.3.5 基数排序（Radix Sorting）·····························································50
1.3.6 结构体的构造函数 ·······································································56
1.3.7 队列的应用 ················································································59
1.4 前缀和·······························································································.66
1.4.1 前缀和的引入 ············································································.66
1.4.2 一维数组前缀和 ·········································································.66
1.5 动态数组····························································································.75
1.5.1 动态数组 vector ··········································································.75
1.5.2 STL 中的动态数组 ······································································.75
1.5.3 vector 的缺点 ·············································································.76
1.5.4 vector 与迭代器 iterator·································································.76
1.5.5 vector 与 C++11 ··········································································.78
1.5.6 vector 的实现原理 ·······································································.80
1.6 树·····································································································.82
1.6.1 树的相关概念 ············································································.82
1.6.2 树的性质 ··················································································.84
1.6.3 特殊的树—二叉树 ···································································.84
1.6.4 完全二叉树的性质 ······································································.85
1.6.5 树的存储方式 ············································································.86
1.6.6 树的遍历 ··················································································.89
1.6.7 知二求一 ··················································································101
1.6.8 树的宽度优先遍历 ······································································105
1.7 本章习题····························································································105
第 2 章 基础算法 ·····························································································106
2.1 贪心算法····························································································106
2.1.1 贪心算法的概念 ·········································································106
2.1.2 基础贪心问题举例 ······································································107
2.1.3 线段覆盖问题 ············································································111
2.2 高精度计算·························································································113
2.2.1 C++语言中的数据类型·································································113
2.2.2 高精度加法 ···············································································115
2.2.3 高精度减法 ···············································································118
2.2.4 高精度乘法 ···············································································121
2.2.5 高精度除法取余数 ······································································126
2.3 归并排序····························································································129
2.3.1 归并 ························································································130
2.3.2 归并排序的时间复杂度分析 ··························································134
2.3.3 归并排序的应用 ·········································································135
2.4 快速排序··························································································.143
2.4.1 快速排序的思想················································································.143
2.4.2 快速排序的时间复杂度分析 ························································.146
2.5 STL ································································································.151
2.5.1 algorithm 头文件中的函数 ···························································.152
2.5.2 容器 ······················································································.156
2.6 本章习题··························································································.159
第 3 章 搜索算法 ···························································································.160
3.1 深度优先搜索····················································································.160
3.1.1 迷宫寻路与烤鸡问题 ·································································.160
3.1.2 全排列问题与回溯 ····································································.166
3.1.3 洪水填充（Flood Fill）算法·························································.168
3.1.4 八皇后问题与剪枝 ····································································.171
3.1.5 数独问题 ················································································.174
3.1.6 剪枝 ······················································································.177
3.2 宽度优先搜索····················································································.181
3.2.1 找眼镜 ···················································································.181
3.2.2 马的遍历 ················································································.182
3.2.3 01 迷宫···················································································.185
3.2.4 八数码问题 ·············································································.188
3.3 本章习题··························································································.190
第 4 章 动态规划 ···························································································.191
4.1 动态规划入门····················································································.191
4.1.1 斐波那契数列 ··········································································.191
4.1.2 数字三角形 ·············································································.193
4.1.3 递推+填表···············································································.195
4.2 动态规划解题步骤··············································································.199
4.2.1 分解子问题 ·············································································.199
4.2.2 确定状态 ················································································.200
4.2.3 状态转移 ················································································.200
4.2.4 动态规划能解决的问题的特点 ·····················································.201
4.3 线性动态规划····················································································.201
4.3.1 最长上升子序列问题（LIS）·······················································.201
4.3.2 最长公共子序列问题（LCS）······················································.209
4.4 背包类动态规划·················································································.213
4.4.1 01 背包问题···············································································213
4.4.2 多重背包问题 ············································································225
4.4.3 完全背包问题 ············································································228
4.4.4 分组背包问题 ············································································231
4.4.5 二维费用背包问题 ······································································241
4.5 区间动态规划与多维动态规划·············································································243
4.5.1 区间动态规划 ············································································243
4.5.2 多维动态规划 ············································································248
4.6 本章习题····························································································256