[Algorithm Design and Analysis] Chapter 1 Introduction

About introduction

(Algorithm + data structure) = program

Turing-the father of computer science and artificial intelligence
Turing Award-the Nobel Prize in the computer world
Turing Test

Algorithm related properties: input, output, certainty, finiteness

Algorithm: a method or process to solve a problem

Algorithm correctness

An algorithm is correct if it eventually stops for every input and produces the correct output.

Program debugging can only prove that the program is wrong, not that it is error-free!

Algorithm complexity

Time complexity

The time required for an algorithm to run on an abstract computer
(the number of atomic operations or "steps" required to produce a result on a specific input)
$$T=T(N,I)$$

N: Scale of the problem
I: Input

1. The time complexity in the worst case 2. The time complexity in the
best case 3. The time complexity in the
average case

Time complexity token

f(N) and g(N) are positive functions defined on the set of positive numbers.
If there is a normal number $C$ and natural number$N_0$

1, upper bounded $THE$

$f\left(N\right)$ is bounded when N is sufficiently large: when$N\ge N_0$When, $f(N)\le Cg(N)$，$g\left(N\right)$ isf (N) f(N)an upper bound of$f$$($$N$$)$

f(N)=O(g(N))

$$O\left(1\right)<O({log}_{2}n)<O\left(n\right)<O(nlo{g}_{2}n<O(n^2)<\cdot \cdot \cdot <O(2^n)<O(n!)<O(n^n)$$

2. Lower bound $\Omega$

$f\left(N\right)$ is bounded when N is sufficiently large: when$N\ge N_0$When, $f(N)\ge Cg(N)$，$g\left(N\right)$ isf (N) f(N)a lower bound of$f$$($$N$$)$

f(N)=Ω(g(N))

Polynomial complexity

Efficient algorithm, stacking hardware

Exponential complexity

Inefficient algorithms, stacking hardware is useless

3. The same order $\theta$

$f(N)\; is$ equivalent to$g\left(N\right)$ same order: if and only if$f(N)=O(g(N))$且$f\left(N\right)=\Omega \left(g\right(N\left)\right)$

f(N)=θ(g(N))

Algorithm

O(f) + O(g) = O(max(f,g))
O(f) + O(g) = O(f + g)
O(f)O(g) = O(fg)
若g(N) = O(f(N))，则O(f) + O(g) = O(f)
O(Cf(N)) = O(f(N))，其中C是一个正的常数
f = O(f)

Space complexity

The amount of storage space required by an algorithm to produce results for a specific input
$$S=S(N,I)$$