upload Algorithm.

Author: Seven-Streams

2025_Spring/Algorithm.md

@@ -0,0 +1,60 @@
+# Algorithm
+
+## Introduction
+
+Turing machine is a Triple $(\Gamma,Q,\delta)$
+
+Church-Turing Thesis: an effective method iff it's computable by a Turing machine.
+
+Halt Problem
+
+- Divide and Conquer
+
+- Graphs
+
+- Greedy
+
+- Dynamic Programming
+
+- Max Flow
+
+- NP-hardness
+
+- Advanced Problems
+
+NP and P
+
+NP-complete problem
+
+Approximation Algorithms
+
+Online Algorithm: how to evaluate:$\forall \text{ input }A(\sigma)\leq \Gamma\cdot OPT(\sigma)$
+
+Ski-rental
+
+## Divide and Conquer and Running Time Analysis
+
+### Integer Multiplication
+
+Divide a multiplication problem into two smaller problems!
+
+#### Karatsuba Algorithm
+
+$$
+
+(a*10^\frac{n}{2}+b)(c*10^\frac n2+d)=ac*10^n+(ad+bc)*10^\frac n2+bd\\
+
+=ac*10^n+((a+b)(c+d)-ac-bd)*10^\frac n2+bd
+$$
+
+Try to down the calculation!
+
+#### Better algorithms
+
+Toom-Cook: $O(n^{1.465})$，break it into 5 $\frac n3$-size problems
+
+### Matrix Problems
+
+#### Srassen's magical idea
+
+