Lecture 1 Introduction + Mathematical Induction

Algorithms + Data Structures = Programs

一、What is an algorithm?

• A sequence of precise and concise instructions that guide you (or a computer) to solve a class of specific problem

最简单的（straightforward）算法（暴力算法brute force algorithm）可能会运行得很慢，因此我们需要更高效的解决方案

 

 

通过观察这个序列是有序的，可以使用更高效的方法——二分查找 

 

二、Well-known Computational Problems

1. Sorting        排序是将一组元素按照某种顺序（通常是升序或降序）排列的过程。

2. Searching        从数据集中查找某一特定元素的过程

3. Graph and Combinatorial Problems 

        – Shortest paths in a graph        如Dijkstra算法

        – Minimum spanning tree        计算一个加权图的最小生成树的问题Prim 和Kruskal算法      

        – Traveling salesman problem

旅行商问题是寻找一条访问所有节点一次并最终返回到起点的最短路径的问题。

        – Knapsack problem

背包问题是选择一组物品，使得它们的总重量不超过限制且总价值最大。

4. Problem in Number Theory

        – Primality testing        素数检验是判断一个数是否为素数的问题。

                                                                                                                                                                                                                                                                        

三、Computing the n-th power

Input: a number x & a non-negative integer n

Output: the n-th power of x

Algorithm:

1. Set a temporary variable p to 1.

2. Repeat the multiplication p = p * x for n times.

3. Output the result p.

1. Pseudo Code

（1）Pseudo Code: statement

Assignment statement：variable = expression

e.g., assign the expression 3 * 4 to the variable result：

result = 3 * 4

compound statements

begin

statement1

statement2

end

（2）Pseudo Code: conditional

Conditional statement

if condition then

statement

if a < 0 then

a = -a

abs = a

output abs

if condition then

statement

else

statement

if a > 0 then

abs = a

else

abs = -a 

output abs

 （3）Pseudo Code: iterative (loop)

Iterative statement

(a) for loop

for var = start_value to end_value do

statement

 

（b) while loop 

 condition to CONTINUE the loop 

while condition do

statement

 

this loop is executed for undetermined number of times

 

(c) repeat-until loop

condition to STOP the loop 

repeat

statement

until condition

condition for while loop is NEGATION of condition for repeat-until loop

2. exercise 

(1) Find the product of all integers in the interval [x, y], assuming that x and y are both integers

product = 1

For i = x to y do

begin

        product = x * product

end

output product

product = 1

i = x

while i <=y do

begin

        product = x * product

end

output product

product = 1

i = x

repeat

begin

        product = x * product

end

output product

until i>=y

(2) List all factors of a given positive integer x

– e.g., if x is 60, then the output should be 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

input x

for i = 1 to x do

begin

        if x%i = 0 then

        output i

end

input x

i = 1

while i<=x do

begin

        if x%i = 0 then

        output i

end

input x

i = 1

repeat

begin

        if x%i = 0 then

        output i

end

until i>x