算法时间复杂度分析 学习笔记

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Big-O analysis 大O符号

The Big-O Asymptotic Notation gives us the Upper Bound上界 Idea, mathematically described below:

f(n) = O(g(n)) if there exists a positive integer n0 and a positive constant c, such that f(n)≤c.g(n) ∀ n≥n0

The general step wise procedure for Big-O runtime analysis is as follows
基本步骤:

• Figure out what the input is and what n represents. 分析输入和n
• Express the maximum number of operations, the algorithm performs in terms of n. 最大的操作次数
• Eliminate all excluding the highest order terms. 保留高阶，去除其他的
• Remove all the constant factors. 去除常量

The algorithms can be classified as follows from the best-to-worst performance (Running Time Complexity)
从好到坏排序:

• Constant Running Time – O(1)
• A logarithmic algorithm – O(logn)
• A linear algorithm – O(n)
• A superlinear algorithm – O(nlogn)
• A polynomial algorithm – O(n^c)
• A exponential algorithm – O(cⁿ)
• A factorial algorithm – O(n!)

不同时间复杂度的比较

渐近下限与渐近紧约束：

• f(n) = O(g(n)) 可以理解为渐近上限
• 渐近下限：如果g(n) = O(f(n))，则f(n) = Ω(g(n))
• 渐近紧约束：如果f(n) = O(g(n))且f(n) = Ω(g(n))，则f(n) = Θ(g(n))

非递归算法分析

非递归算法分析

递归算法分析

迭代法与递归树

主方法

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引用：
Analysis of Algorithms | Big-O analysis
算法导论------递归算法的时间复杂度求解