Prof. Michael Tsai2013/2/19 INTRODUCTION

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Prof. Michael Tsai 2013/2/19

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What is a data structure?

• An organization of information, usually in memory, for better algorithm efficiency.

• Or, a way to store and organize data in order to facilitate acce ss and modifications.

• 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)

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2�⁵− 3�²+11�+27

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What is an algorithm?

• An algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output.

• An algorithm is a tool for solving a well-specified computationa l problem.

• Computational problem  input/output relationship

• The algorithm describes a specific computational procedure for achieving that input/output relationship.

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What is an algorithm?

• Example:

• Sorting problem:

• Input: A sequence of n numbers

• Output: A permutation (reordering) of the input sequence suc h that

• An instance of the sorting problem:

• A sorting algorithm should return as output the sequence .

•

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What is an algorithm?

•

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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Example

• Statement 1: “Is n=2 the largest value of n for wh ich there exist positive integers x, y, and z suc h that has a solution?”

• Statement 2: “Store 5 divided by zero into x and g o to statement ㄅ .”

• Which criterion do they violate?

• Input

• Output

• Definiteness

• Finiteness

• Effectiveness

•

Definiteness

Effectiveness

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Why are algorithms/data structure important?

• 它們被用在生活中的每個層面 :

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Why are algorithms/data structure important?

• Q: 如果電腦無限快 / 記憶體免錢，我們還需要研究資料結構與演算法嗎 ?

• A: Yes. 我們仍然需要確認我們想出來的解法會停止 ( 不會無窮地執行下去 ) ，而且每次都產生正確的答案。

• 在這個假想的世界中，任何正確的解法都適用，因此我們通常會選最容易實作的方法。

• 但是在真實的世界裡 :

• 電腦不是無限快 ( 計算需要時間 )

• 記憶體不是免錢 ( 儲存資料需要空間 )

• 因此我們需要學習如何好好利用這些資源來解決問題 !

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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.

• 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

Sorting problem:

Input: A sequence of n numbers

Output: A permutation (reordering) of the input sequence such that

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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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#define MAX-SIZE 101

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

void sort(int [],int); /*selection sort */

void main(void) {

int i,n;

int list[MAX-SIZE];

printf("Enter the number of numbers to generate: ");

scanf (" %d", &n) ;

if( n < 1 I In> MAX-SIZE) {

fprintf(stderr, "Improper value of n\n");

exit(EXIT_FAILURE);

}

for (i = 0; i < n; i++) {/*randomly generate numbers*/

list[i] = rand() % 1000;

printf("%d ",list[i]);

}

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printf("\n");

}

void sort(int list[],int n) {

int i, j, min, temp;

for (i = 0; i < n-1; i++) { min = i;

for (j = i+1; j < n; j++) if (list[j] < list[min])

min = j;

SWAP(list[i],list[min],temp);

} }

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How do we prove that it is correc t?

• [Theorem] Function sort(list,n) correctly sorts a set of n1 intege rs. The result remains in list[0], …, list[n-1] such that .

• Proof:

When the outer for loop completes its iteration for i=q, we hav e . Further, on subsequent iterations, i>q and list[0] through list [q] are unchanged. Hence following the last iteration of the out er 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:

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searchnum=13;

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Example:

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searchnum=13;

lef middle right

middle=(left+right)/2;

left=middle+1;

10 11

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Example:

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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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What is a data Type?

• 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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Operations of Data Types

• 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: 我們需要知道 data type 的 representation 嗎 ?

• A: 不一定 .

• 知道 representation 可能可以設計出更有效率的 algorithm .

• 但是當 data type 的 representation 被修改以後，程式

可能必須重新確認、修正、或

完全重寫

。 .

囧

• 移植到不同的平台上 (x86, ARM, embedded system, …)

• 改變 program 或 library 的 specification( 規格 ) (ex. 16-bit int  32-bi t 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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Example

ADT NaturalNumber is objects:

an ordered subrange of the integers starting at zero and ending at the maxim um integer (lNT-MAX) on the computer

functions:

for all x, Y E NaturalNumber; TRUE, FALSE E Boolean

and where +, -, <, and == are the usual integer operations

NaturalNumber Zero() ::=0

Boolean IsZero(x) ::= if (x) return FALSE

else return TRUE Boolean Equal(x, y) ::= if (x == y) return TRUE else return FALSE

NaturalNumber Successor(x) ::= if (x == INT-MAX) return x else return x + 1

NaturalNumber Add(x, y) ::= if ((x + y) <= INT-MAX) return x + y

else return INT-MAX NaturalNumber Subtract(x, y) ::= if (x < y) return 0 else return x-y end NaturalNumber

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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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e

• Cormen Chapter 1

• Horowitz Chapter 1.3-1.4