BASICS 1

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Prof. Michael Tsai

2012/02/21

(2)

Variables

• 變數是什麼 ?

• 變數 : Names(x 和 y) 對應到某個 data

• 每個變數可能有不同的資料型態 (data type)

•

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Data Type

• What is a data type?

• A data type in a programming language is a set of data with value s having pre-defined characteristics.

• A data type is a collection of objects and a set of operations that act on those objects.

• 每種 data type 有所占的記憶體大小及可表示的資料數值範圍

• Data types in C

• char, int, float, long, double (unsigned, signed, …)

• Array

• Structure

(User-defined)

struct { int a;

int b;

char str[16];

int * iptr;

} blah;

int iarray[16];

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Data Types’ Operations

• Operations

• +, -, *, /, %, ==

• =, +=, -=

• ? :

• sizeof, - (negative)

• giligulu(int a, int b)

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Data Type

• Representation of the objects of the data type

• Example: char

• char blah=‘A’; (‘A’: ASCII code is 65(dec), or 0x41 (hex))

Q: The maximum number which can be represented with a char variable?

A: 255.

• How about char, int, long, float?

01000001 01000001 1 byte of memory:

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Data type

• Q: Do we need to know about the representation of a data typ e?

• A: It depends.

• Algorithms could be made more efficient.

• But, If the representation of the data type is changed, the program needs to be verified, revised, or completely

re- written

^.

囧

• Porting to a different platform (x86, ARM, embedded system, …)

• Changing the specification of a program or a library (ex. 16-bit int  3 2-bit long)

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Abstract Data Type

• “Abstract Data Type” (ADT):

• Separate the specifications from the representation an d the implementation

Representation and Implementation Specification (Interface)

User

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Abstract Data Type

• Specifications:

• Operations:

• Name of the function and the description of what the function does

• The type of the argument(s)

• The type of the result(s) (return value)

• Data (usually hidden)

• Function categories:

• Creator/constructor

• Transformers

• Observer/reporter

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What is a Data Structure?

•

An organization of information, usually in memory, for better algorith m efficiency. ( 演算法跑起來比較快 )

•

Or, a way to store and organize data in order to facilitate access and m odifications. ( 比較好存取和更改 )

•

例子 : 存多項式 :

•

又可分為 :

• Linear data structure: 必須循序地存取 ( 如 linked list, stack, queue)

• Non-linear data structure: 可以不循序的存取 ( 如 tree, graph)

0 1 2 3 4 5 6

27 11 -3 0 0 2 0

2 �

⁵

− 3 �

²

+11 �+27

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What is an algorithm?

• “The step-by-step instructions to solve a given problem”

• 解決某一個問題的詳細、一步一步地指令

• “A computable set of steps to achieve a desired result.”

• All algorithm must satisfy the following criteria:

• Input: 外部給的資訊 ( 可以是零個或多個 )

• Output: 產生的結果 ( 至少一個 )

• Definiteness: 每一個指令都是清楚而不模糊的

• Finiteness: 所有的狀況下 ( 所有的 input), 演算法會在有限步驟之後結束

• Effectiveness: 每一個指令都必須是簡單可以直接執行的 ( 必須可以執行 )

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例子

• Statement 1: “Is n=2 the largest value of n for whic h there exist positive integers x, y, and z such th at has a solution?”

• Statement 2: “Store 5 divided by zero into x and go to statement ㄆ .”

• Which criterion do they violate?

• Input

• Output

• Definiteness

• Finiteness

• Effectiveness

•

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How do we describe an algorithm?

• Human language (English, Chinese, …)

• Programming language

• A mix of the above

1. 拿平底鍋 2. 拿沙拉油

1. 我們有油嗎 ?

1. 有的話 , 倒一茶匙的沙拉油到鍋子裡 2. 沒有的話 , 我們想要買油嗎 ?

1. 是的話 , 就去全聯買一罐沙拉油 2. 如果不想的話 , 只好先不煮了 . 3. 打開火爐 , …

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Example: Selection Sort

•

Integers are stored in an array, list. The i-th integer is stored in list [i], 0<i<n.

•

Goal: Devise a program to sort a set of integers

•

Solution: From those integers that are currently unsorted, find the smallest and place it next in the sorted list.

•

ㄅㄆ 1

1 ㄆ 2 ㄅ

1 2 ㄆㄅ

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Example: Selection Sort

• First attempt:

for (i=0; i<n; ++i) {

Examine list[i] to list[n-1] and suppose that the smallest integer is at list[min];

Interchange list[i] and list[min];

}

Task 1

Task 2

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Task 2

void swap(int *x, int *y) { int temp = *x;

*x=*y;

*y=temp;

}

Or

#define SWAP(x,y,t) ((t)=(x), (x)=(y), (y)=(t)) Task 2

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Task 1

min=i;

for(j=i;j<n;++j)

if (list[j]<list[min]) min=j;

Task 1

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How do we prove that it is correct?

• ( 這一頁回家自己看 )

• [Theorem] Function sort(list,n) correctly sorts a set of n>=1 integers. The res ult remains in list[0], …, list[n-1] such that .

• Proof: When the outer for loop completes its iteration for i=q, we have . Fur ther, on subsequent iterations, i>q and list[0] through list[q] are unchanged.

Hence following the last iteration of the outer for loop (i.e., i=n-2), we have .

•

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Example: Binary Search

• Input:

• searchnum: the number to be found

• list: sorted array, size n, and

• Output:

• -1 if searchnum is not found in list

• the index of searchnum in list if searchnum is found

•

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Example:

1 3 4 4 6 7 11 13 13 13 18 19

0 1 2 3 4 5 6 7 8 9 10 11

searchnum=13;

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Example:

1 3 4 4 6 7 11 13 13 13 18 19

0 1 2 3 4 5 6 7 8 9

searchnum=13;

lef middle right

middle=(left+right)/2;

left=middle+1;

10 11

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Example:

1 3 4 4 6 7 11 13 13 13 18 19

0 1 2 3 4 5 6 7 8 9

searchnum=5;

return -1;

10 11

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int binsearch(int list[], int searchnum, int left, int right) {

int middle;

while(left<=right) { middle=(left+right)/2;

switch(COMPARE(list[middle], searchnum)) { case -1: left=middle+1; break;

case 0: return middle;

case 1: right=middle-1;

} }

return -1;

}

list: 存 sort 好數字的 array searchnum: 要找的數字

lef, right: 正在找的範圍左邊和右邊邊界

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怎麼評估一個程式寫得好不好 ?

1.Does the program meet the original specifications of the task?

2.Does it work correctly?

3.Does the program contain documentation that shows how to u se it and how it works?

4.Does the program effectively use functions to create logical uni

ts?

5.Is the program’s code readable?

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怎麼評估一個程式寫得好不好 ?

6.Does the program efficiently use primary and secondary stor age?

Primary storage: memory?

Secondary storage: Hard drive, flash disk, etc.

7.Is the program’s running time acceptable for the task?

Example: Network intrusion detection system

(1) 99.8% detection rate, 50 minutes to finish analysis of a minute of traffic

(2) 85% detection rate, 20 seconds to finish analysis of a minute of traffic

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怎麼評估一個程式寫得好不好 ?

6. 程式是否有效地使用主要及次要的儲存 ?

7.程式的執行時間是否適合所需解決的工作內容 ?

Time complexity

Space complexity

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空間及時間複雜度

• 程式的空間複雜度 :

• 程式執行完畢所需使用的所有空間 ( 記憶體 )

• 程式的時間複雜度 :

• 程式執行完畢所需使用的 ( 執行 ) 時間

• Goal: 找出執行時間 / 使用空間”如何”隨著 input size 變長 ( 成長的有多快 )

• 什麼是 input size?

• 問題給的 input 的”元素數量” , 如 :

• Array 大小

• 多項式最高項的次方

• 矩陣的長寬

• 二進位數的位元數目

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空間複雜度

• 程式所需空間 :

1. 固定的空間

• 和 input/output 的大小及內容無關

2. 變動的空間

• 和待解問題 P 的某個 input instance I( 某一個 input) 有關

• 跟 recursive function 會使用到的額外空間有關

•

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時間複雜度

• 一個程式 P 所需使用的時間 :

• Compile 所需時間

• 執行時間 (execution time or run time)

• Compile 時間 : 固定的 . ( 例外 ?)

• C (and other compiled programming languages)

 One Compilation  Multiple Executions

• Run time:

• 和 input instance 的特性有關 !

•

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如何得到 ?

1. 總執行時間

2. 所執行的程式指令的數目

3. 比較數學的方法 ( 使用 function 來代表程式執行的時間 )

(30)

如何得到 ?

• 方法 1:

• 數出整個程式每個 operation 各總共花了多少時間 .

Add Subtract Load Store

ADD(n) SUB(n) LDA(n) STA(n)

Add Subtract Load Store

ADD(n) SUB(n) LDA(n) STA(n)

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Is it good to use?

( 方法 1)

Is it good to use?

( 方法 1)

(31)

如何得到 ?

• 方法 2:

• 把程式分解為一個一個步驟

• 每個步驟的執行時間變成跟 input 的特性無關了

• 然後數數看總共有幾個步驟

• 但是…

• 不同的程式步驟可能有不同的執行時間

• a=2;

• a=2*b+3*c/d-e+f/g/a/b/c;

• 要數出所有步驟的數目需要解某一個 input instance

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Example

float sum(float list[], int n) {

float tempsum = 0; count++; //assignment int i;

for (i=0;i<n;i++) {

count++; // for loop

tempsum+=list[i]; count++; //assignment

}

count++; //last iteration of for count++; return tempsum;

}

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Example

float sum(float list[], int n) {

float tempsum = 0;

int i;

for (i=0;i<n;i++) {

count+=2;

}

count+=3;

return tempsum;

}

count = 2n+3 (steps)

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Example 2

float rsum(float list[], int n) {

count++; // if if (n) {

count++; //return

return rsum(list,n-1)+list[n-1];

}

count++; //return return list[0];

}

count = 2n+2 (steps)

2n+2 < 2n+3.

Does this mean rsum is faster than sum ?

No!

(35)

比較數學的方法— Asymptotic analysis

• “ 預測”當 input 的性質改變 ( 通常是 input size 改變 ) 時 , 執行時間的成長速度 (growth of run time)

• 比較兩個做相同事情的程式的時間複雜度

• “ 程式步驟”不是那麼精確 :

• “3n+3” 比“ 3n+5” 快 ?

• 通常 “ 3n+3”, “7n+2”, or “2n+15” 執行時間都相差不遠 .

•  我們將使用一個比較不準確的方式來描述執行時間…

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Example

• Program P and Q

• N 很大的時候 , Q 會比 P 快 , 不管的數值是什麼 .

• Example:

• , then for .

• 需要知道的數值嗎 ?

•

Break even point constants are

close).

constants are close).

(37)

Asymptotic Notation – Big OH

• Definition [Big “oh”]:

•

• “f of n is big oh of g of n” ( 是集合的成員 )

• “=“ is “is” not “equal”

•

• 可以想成是 Upper Bound

( 王先生打了太多全壘打 , 所以是 upper bo und)

•

(38)

Example

• ?

• Yes, since for all .

• ?

•

for all

(39)

Example

• ?

• No. Cannot find

•

for all

(40)

The World of Big Oh

• constant

• linear

• quadratic

• cubic

• exponential

• , , , ,

•

Faster Slower

(41)

Big Oh 是 Upper Bound

• 但是沒有說它是多好的 upper bound

• 通常我們會把它取得越小 ( 緊 ) 越好 .

•

(42)

A Useful Theorem

• Theorem: If , then .

• Proof: ( 自己閱讀 )

, for all . So,

•

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BASICS 1

Prof. Michael Tsai

2012/02/21

Variables

Data Type

Data Types’ Operations

Data Type

Data type

re- written

囧

Abstract Data Type

• “Abstract Data Type” (ADT):

• Separate the specifications from the representation an d the implementation

Abstract Data Type

• Specifications:

• Function categories:

What is a Data Structure?

An organization of information, usually in memory, for better algorith m efficiency. ( 演算法跑起來比較快 )

Or, a way to store and organize data in order to facilitate access and m odifications. ( 比較好存取和更改 )

例子 : 存多項式 :

又可分為 :

2 �

− 3 �

+11 �+27

What is an algorithm?

例子

How do we describe an algorithm?

Example: Selection Sort

Integers are stored in an array, list. The i-th integer is stored in list [i], 0<i<n.

Goal: Devise a program to sort a set of integers

Solution: From those integers that are currently unsorted, find the smallest and place it next in the sorted list.

Example: Selection Sort

• First attempt:

Task 2

Task 1

How do we prove that it is correct?

Example: Binary Search

Example:

Example:

Example:

怎麼評估一個程式寫得好不好 ?

ts?

怎麼評估一個程式寫得好不好 ?

怎麼評估一個程式寫得好不好 ?

Time complexity

Space complexity

空間及時間複雜度

空間複雜度

• 程式所需空間 :

1. 固定的空間

2. 變動的空間

時間複雜度

如何得到 ?

如何得到 ?

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Is it good to use?

( 方法 1)

Is it good to use?

( 方法 1)

如何得到 ?

Example

Example

Example 2

比較數學的方法— Asymptotic analysis

Example

Asymptotic Notation – Big OH

Example

for all

for all

Example

for all

for all

The World of Big Oh

Big Oh 是 Upper Bound

A Useful Theorem

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