Prof. Michael Tsai 2013/2/19 INTRODUCTION

Prof. Michael Tsai 2013/2/19

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 access 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)

0 1 2 3 4 5 6

27 11 -3 0 0 2 0

2𝑛⁵ − 3𝑛² + 11𝑛 + 27

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 computational problem.

• Computational problem  input/output relationship

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

What is an algorithm?

• Example:

• Sorting problem:

• Input: A sequence of n numbers 𝑎₁, 𝑎₂, … , 𝑎_𝑛

• Output: A permutation (reordering) 𝑎₁^′ , 𝑎₂^′ , … , 𝑎_𝑛^′ of the input sequence such that 𝑎₁^′ ≤ 𝑎₂^′ ≤ ⋯ ≤ 𝑎_𝑛^′

• An instance of the sorting problem: 31,41,59,26,41,58

• A sorting algorithm should return as output the sequence 26,31,41,41,58,59 .

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)

Example

• Statement 1: “Is n=2 the largest value of n for which there exist positive integers x, y, and z such that 𝑥^𝑛 + 𝑦^𝑛 = 𝑧^𝑛 has a solution?”

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

• Which criterion do they violate?

• Input

• Output

• Definiteness

• Finiteness

• Effectiveness

Definiteness

Effectiveness

important?

• 它們被用在生活中的每個層面:

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!

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, …

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.

ㄅ ㄆ 1

1 ㄆ 2 ㄅ

1 2 ㄆ ㄅ

Sorting problem:

Input: A sequence of n numbers 𝑎₁, 𝑎₂, … , 𝑎_𝑛

Output: A permutation (reordering) 𝑎₁^′, 𝑎₂^′, … , 𝑎_𝑛^′ of the input sequence such that 𝑎₁^′ ≤ 𝑎₂^′ ≤ ⋯ ≤ 𝑎_𝑛^′

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

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

Task 1

min=i;

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

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

Task 1

#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]);

}

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);

} }

How do we prove that it is correct?

• [Theorem] Function sort(list,n) correctly sorts a set of n≥1 integers. The result remains in list[0], …, list[n-1] such that 𝑙𝑖𝑠𝑡 0 ≤ 𝑙𝑖𝑠𝑡 1 ≤ ⋯ ≤ 𝑙𝑖𝑠𝑡[𝑛 − 1].

• Proof:

When the outer for loop completes its iteration for i=q, we have 𝑙𝑖𝑠𝑡 𝑞 ≤ 𝑙𝑖𝑠𝑡[𝑟], 𝑞 < 𝑟 < 𝑛. Further, 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 𝑙𝑖𝑠𝑡 0 ≤ 𝑙𝑖𝑠𝑡 1 ≤ ⋯ ≤ 𝑙𝑖𝑠𝑡[𝑛 − 1].

Example: Binary Search

• Input:

• searchnum: the number to be found

• list: sorted array, size n, and 𝑙𝑖𝑠𝑡 0 ≤ 𝑙𝑖𝑠𝑡 1 ≤ ⋯ ≤ 𝑙𝑖𝑠𝑡[𝑛 − 1]

• Output:

• -1 if searchnum is not found in list

• the index of searchnum in list[] if searchnum is found

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;

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;

left middle right

middle=(left+right)/2;

left=middle+1;

10 11

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

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;

}

searchnum: 要找的數字

left, right: 正在找的範圍左邊和右邊邊界

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];

Operations of Data Types

• Operations

• +, -, *, /, %, ==

• =, +=, -=

• ? :

• sizeof, - (negative)

• giligulu(int a, int b)

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 1 byte of memory:

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-bit long)

Abstract Data Type

•

“Abstract Data Type” (ADT):

•

Separate the specifications from the representation and the implementation

Representation and Implementation Specification (Interface)

User

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

Example

ADT NaturalNumber is objects:

an ordered subrange of the integers starting at zero and ending at the maximum 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

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 use it and how it works?

4.Does the program effectively use functions to create logical units?

5.Is the program’s code readable?

怎麼評估一個程式寫得好不好?

6.Does the program efficiently use primary and secondary storage?

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

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?

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

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

• 𝑆 𝑃 = 𝑐 + 𝑆

(𝐼)

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: 𝑇_𝑃

• 和input instance的特性有關!

Reading materials for this lecture

• Cormen Chapter 1

• Horowitz Chapter 1.3-1.4