数据结构和算法

DSA - 主页 DSA - 概述 DSA - 环境设置 DSA - 算法基础 DSA - 渐近分析

数据结构

DSA - 数据结构基础 DSA - 数据结构和类型 DSA - 数组数据结构

链接列表

DSA - 链接列表数据结构 DSA - 双向链接列表数据结构 DSA - 循环链表数据结构

堆栈 &队列

DSA - 堆栈数据结构 DSA - 表达式解析 DSA - 队列数据结构

搜索算法

DSA - 搜索算法 DSA - 线性搜索算法 DSA - 二分搜索算法 DSA - 插值搜索 DSA - 跳跃搜索算法 DSA - 指数搜索 DSA - 斐波那契搜索 DSA - 子列表搜索 DSA - 哈希表

排序算法

DSA - 排序算法 DSA - 冒泡排序算法 DSA - 插入排序算法 DSA - 选择排序算法 DSA - 归并排序算法 DSA - 希尔排序算法 DSA - 堆排序 DSA - 桶排序算法 DSA - 计数排序算法 DSA - 基数排序算法 DSA - 快速排序算法

图形数据结构

DSA - 图形数据结构 DSA - 深度优先遍历 DSA - 广度优先遍历 DSA - 生成树

树数据结构

DSA - 树数据结构 DSA - 树遍历 DSA - 二叉搜索树 DSA - AVL 树 DSA - 红黑树 DSA - B树 DSA - B+ 树 DSA - 伸展树 DSA - 尝试 DSA - 堆数据结构

递归

DSA - 递归算法 DSA - 使用递归的汉诺塔 DSA - 使用递归的斐波那契数列

分而治之

DSA - 分而治之 DSA - 最大最小问题 DSA - 施特拉森矩阵乘法 DSA - Karatsuba 算法

贪婪算法

DSA - 贪婪算法 DSA - 旅行商问题(贪婪方法) DSA - Prim 最小生成树 DSA - Kruskal 最小生成树 DSA - Dijkstra 最短路径算法 DSA - 地图着色算法 DSA - 分数背包问题 DSA - 作业排序截止日期 DSA - 最佳合并模式算法

动态规划

DSA - 动态规划 DSA - 矩阵链乘法 DSA - Floyd Warshall 算法 DSA - 0-1 背包问题 DSA - 最长公共子序列算法 DSA - 旅行商问题(动态方法)

近似算法

DSA - 近似算法 DSA - 顶点覆盖算法 DSA - 集合覆盖问题 DSA - 旅行商问题(近似方法)

随机算法

DSA - 随机算法 DSA - 随机快速排序算法 DSA - Karger 最小割算法 DSA - Fisher-Yates 洗牌算法

DSA 有用资源

DSA - 问答 DSA - 快速指南 DSA - 有用资源 DSA - 讨论


Approximation Algorithms



Approximation Algorithms

Approximation algorithms are algorithms designed to solve problems that are not solvable in polynomial time for approximate solutions. These problems are known as NP complete problems. These problems are significantly effective to solve real world problems, therefore, it becomes important to solve them using a different approach.

NP complete problems can still be solved in three cases: the input could be so small that the execution time is reduced, some problems can still be classified into problems that can be solved in polynomial time, or use approximation algorithms to find near-optima solutions for the problems.

This leads to the concept of performance ratios of an approximation problem.

Performance Ratios

The main idea behind calculating the performance ratio of an approximation algorithm, which is also called as an approximation ratio, is to find how close the approximate solution is to the optimal solution.

The approximate ratio is represented using ρ(n) where n is the input size of the algorithm, C is the near-optimal solution obtained by the algorithm, C* is the optimal solution for the problem. The algorithm has an approximate ratio of ρ(n) if and only if −

$$max\left\{\frac{C}{C^{\ast} },\frac{C^{\ast }}{C} ight\}\leq ho \left ( n ight )$$

The algorithm is then called a ρ(n)-approximation algorithm. Approximation Algorithms can be applied on two types of optimization problems: minimization problems and maximization problems. If the optimal solution of the problem is to find the maximum cost, the problem is known as the maximization problem; and if the optimal solution of the problem is to find the minimum cost, then the problem is known as a minimization problem.

For maximization problems, the approximation ratio is calculated by C*/C since 0 ≤ C ≤ C*. For minimization problems, the approximation ratio is calculated by C/C* since 0 ≤ C* ≤ C.

Assuming that the costs of approximation algorithms are all positive, the performance ratio is well defined and will not be less than 1. If the value is 1, that means the approximate algorithm generates the exact optimal solution.

Examples

Few popular examples of the approximation algorithms are −