Advance Data Structure Review of Chapter 2 張啟中

Review of Chapter 2 Arrays 1.3 Data Abstraction and Encapsulation 2.2 The Array As An abstract Data Type  2.5 The Representation of Arrays Example  2.3 The Polynomial Abstract Data Type  2.4 The Sparse Matrix Abstract Data Type  2.6 The String Abstract Data Type

定義： Data Type A data type is a collection of objects and a set of operations that act on those objects. 定義： Abstract Data Type An abstract data type(ADT) is a data type that is organized in such a way that the specification of the objects and the operations on the objects is separated from the representation of the objects and the implementation of the operations.

資料型態 Data Types Primitive Data Types Accumulated Data Types - Array - Structures - Union Abstract Data Types Examples

Example: Data Type of C++ 表一： C++ 的基本資料型態

Accumulated Data Type Array Struct Union

如何理解 Abstract Data Types ADTs 是一個物件的型態的抽象定義具有一般 化的特性，其中也包含這個物件相關的操作。 Data Structures 課程，旨在瞭解每一個 ADT ， 要用什麼樣的結構來表達與儲存，而這樣的表 達與儲存方式，又對於該物件的操作帶來什麼 樣的優缺點。亦即關注在二個重點：  物件儲存與表達方式（資料結構）  物件操作方式（演算法）

The Array as an Abstract Data Type Array  A collection of data of the same type  An array is usually implemented as a consecutive set of memory locations int list[5], *plist[5] ADT definition of an Array  More general structure than "a consecutive set of memory locations.“  An array is a set of pairs,, in mathematical, call correspondence or mapping

class GeneralArray { // objects: A set of pairs where for each value of index in // IndexSet there is a value of type float. IndexSet is a finite ordered set of one // or more dimensions, for example, {0, …, n - 1} for one dimension, {(0, 0), // (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)} for two dimensions, etc. public: GeneralArray(int j, RangeList list, float initValue = defaultValue) ; // The constructor GeneralArray creates a j dimensional array of floats; the // range of the kth dimension is given by the kth element of list. For each // index i in the index set, insert into the array. float Retrieve(index i) ; // if (i is in the index set of the array) return the float associated with i in the // array; else signal an error. void Store(index i, float x) ; // if (i is in the index set of the array) delete any pair of the form // present in the array and insert the new pair ; else signal an error. } ; // end of GeneralArray

Array （陣列） D 陣列是用來存放同樣型態的資料 D 陣列的大小必須在程式中預先設定 D 在程式執行中，陣列的大小無法改變 D 陣列中的資料是透過索引（ index ）來存取 const int ArraySize = 100; int iArray[ArraySize];

一維陣列和記憶體間的對應 int iArray[100]; m m + 2 m + 4 m + 6 m 假定 sizeof(int) = 2 Memory

多維陣列和記憶體間的對應 int mesh[7][11]; m m + 22 m + 44 m + 66 m + 88 m m 假定 sizeof(int) = 2 Memory

多維陣列的宣告 type array_name[arraySize 1 ] [arraySize n ]; 【範例】 int mesh[7][11]; float cube[6][8][3];

m m + 48 m + 96 m m m m 假定 sizeof(int) = 2

struct 和記憶體間的對應 struct studentType { char Name[20];// 姓名 char Address[30];// 地址 char PhoneNumber[10];// 電話 int Age;// 年齡 char Department[4];// 系別 int Year; // 年級 char Class;// 班級 }; studentType Student1, Student2; m m + 20 m + 50 m + 66 m + 60 m + 62 m + 68 假定 sizeof(int) = 2

Self-Referential Structures One or more of its components is a pointer to itself. typedef struct list { char data; list *link; } list item1, item2, item3; item1.data=‘a’; item2.data=‘b’; item3.data=‘c’; item1.link=item2.link=item3.link=NULL; Construct a list with three nodes item1.link=&item2; item2.link=&item3; malloc: obtain a node abc

union 的宣告 union { field1 declaration ; field2 declaration ; } variable_name ; typedef union { field1 declaration ; field2 declaration ; } type_name ; 【範例】 union { char charValue; int intValue; float floatValue; } dataCell; typedef union { char charValue; int intValue; float floatValue; } dataCellType; 宣告變數宣告型態

union 和記憶體間的對應 union { char charValue;/* 1 byte */ int intValue;/* 2 byte */ float floatValue;/* 4 byte */ } dataCell; m m + 4 typedef struct { char opcode; union { int intValue; char strValue[256]; } data; } instruction; m m m+1 m+5

Example Ordered List Polynomial ADT Sparse Matrix ADT String ADT

Ordered List Examples (MONDAY, TUEDSAY, WEDNESDAY, THURSDAY, FRIDAY, SATURDAYY, SUNDAY) (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, ACE) (1941, 1942, 1943, 1944, 1945) (a1, a2, a3, …, an-1, an) () ordered (linear) list: (item1, item2, item3, …, itemn)

Operations on Ordered List Find the length, n, of the list. Read the items from left to right (or right to left). Retrieve the i’th element. Store a new value into the i’th position. Insert a new element at the position i, causing elements numbered i, i+1, …, n to become numbered i+1, i+2, …, n+1 Delete the element at position i, causing elements numbered i+1, …, n to become numbered i, i+1, …, n-1 array (sequential mapping) ? (1)~(4) O (5)~(6) X

Polynomial （多項式） Addition:Multiplication:

class Polynomial { // objects: p(x) = a 0 xe0 + … + a n xen ; a set of ordered pairs of, where // a i ∈ Coefficient and e i ∈ Exponent // We assume that Exponent consists of integer ≥ ０ public: Polynomial() ; // return the polynomial p(x) = 0 int operator!() ; // if *this is the zero polynomial, return 1; else return 0; Coefficient Coef(Exponent e) ; // return the coefficient of e in *this Exponent LeadExp() ; // return the largest exponent in *this Polynomial Add(Polynomial poly) ; // return the sum of the polynomials *this and poly Polynomial Mult(Polynomial poly); // return the product of the polynomials *this and poly float Eval(float f) ; //Evaluate the polynomial *this at f and return the result. }; // end of Polynomial

Polynomial : Representation 1 Representation by Degrees private int degree; // degree  MaxDegree float coef [MaxDegree + 1]; Example: Let A(x)= Σ a i x i, then a.degree= n, a.coef[i]=a n-i,,0<= i<= n

Polynomial : Representation 1 012n-1n n degree coef

Polynomial : Representation 2 Representation by Degrees, but dynamic allocation space private: degree int degree; //degree <= MaxDegree coef float *coef; polynomial::polynomial(int d) { degree= d, coef=new float[degree+1]; }

Polynomial : Representation 3 Representation by Terms. Class polynomial; Class term { Friend polynomial; private: int exp coef float coef; }; In polynomial class In Implement file of polynomial class private: static term termArray[MaxTerms]; // MaxTerms is a constant. // termArray[MaxTerms] shared by all polynomial objects. static int free; int start, finish; term Polynomial::termArray[MaxTerms]; int polynomial::free = 0;

A(X)=2X B(X)=X 4 +10X 3 +3X 2 +1 coef exp A.start A.finish B.start B. finish free Polynomial: Representation 3-1 In general, A.Finish = A.Start + n –1. For zero polynomial, A.Finish = A.Start – 1

Polynomial : Representation k-1 k numTerm coef ? 012k-1 expon ?

The Representations of Polynomials Compare Representation 1Representation 2Representation 3 Advantages Simply algorithms for most operations Save space than R1 Save space when polynomial is sparse than R1, R2 Disadvantages Waste spaces if a.degree << MaxDegree Waste spaces when polynomial is sparse Waste twice space when all terms are nonzero than R2 ， and algorithms is complex than R1, R2

操作分析 多項式相加  As Representation 3 ， we take O(m+n) at time complexity ， if A(x) has m terms, B(x) has n terms. See Book pp 多項式相乘

col1 col2 col3 col4 col5 col6 row0 row1 row2 row3 row4 row5 (a) (b) *Figure 2.3:Two matrices 8/36 6*65*3 15/15 Sparse Matrix sparse matrix data structure?

若以二維陣列的表示法： #define MAX_ROW 100 #define MAX_COL 100 typedef int matrix[MAX_ROW][MAX_COL]; matrix m1, m2; 來儲存稀疏矩陣，那麼，矩陣中含有許多的 0 。有沒有必要 儲存這些 0 呢？有沒有比較節省記憶體空間的另類表示法？

Abstract Data Type Sparse Matrix class SparseMatrix { //objects: a set of triples,, where row and column are integers and // form a unique combination, and value comes from the set item. public: SparseMatrix(int MaxRow, int MaxCol); //create a SparseMatrix that can hold up to MaxItems= MaxRow*MaxCol and whose //maximum row size is MaxRowand whose maximum column size is MaxCol Transpose SparseMatrix Transpose(); // return the matrix produced by interchanging the row and column value of every triple. Add SparseMatrix Add(SparseMatrix b); //if the dimensions of a(*this) and b are the same, return the matrix produced by adding //corresponding items, namely those with identical row and column values. else return //error. Multiply SparseMatrix Multiply(a, b); //if number of columns in a equals number of rows in b return the matrix d produced by //multiplying a by b according to the formula: d[i][j]= Sum(a[i][k](b[k][j]), //where d(i, j) is the (i, j)th element, k=0 ~ ((columns of a) –1) else return error. };

Representation of Sparse Matrix class SparseMatrix; class MatrixTerm { friend class SparseMatrix private: int col, row, value; }; In class SparseMatrix private: int col, row,Terms; MatrixTerm smArray[MaxTerms]; rowcolumns // Note: triples are ordered by row and within rows by columns

rowcolvalue a[0]0015 a[1]0322 a[2]05-15 a[3]1111 a[4]123 a[5]23-6 a[6]4091 a[7] row 0 row 1 row 2 row 3 row 4 row term col

Sparse Matrix 運算分析 轉置 (transpose) 相加 相乘

Class String { //objects: a finite set of zero or more characters. public: String (char *init, int m); // Constructor that initializes *this to string init of length is m. int operator==(string t); // if the string represented by *this equal string t return 1(true) else return 0(false) int operator!(); // if *this is empty then return 1(TRUE); else return 0 (FALSE). Length int Length(); //return the number of characters in *this. Concat String Concat(String t); //return a string whose elements are those of *this followed by those of t. Substr String Substr(int i, int j); //return the string containing j characters of *this at positions i, i+1,..., i+j-1, //if these are valid positions of *this; else return empty string. int Find(String pat); //return an index i such that pat matches the substring of *this that begins at //position i. Return –1 if pat is either empty or not a substring of *this. }; Abstract Data Type String

Pattern Matching Given two strings, string and pat, where pat is a pattern to be searched for in string Two methods  a simple algorithm O(S*P)  O(n 2 )  optimal algorithm (by Knuth-Morris-Pratt) Linear complexity O(S+P)  O(n)

aabpattern ababba a aaa no match ababba a baa match O(n*m) A simple algorithm aabpattern

The Failure Function Definition  If p= p 0 p 1...p n-1 is a pattern, then its failure function, f is defined as: f(j) = largest i = 0 = -1, otherwise Example j pat a b c a b c a c a b f

Rule for Optimal Pattern Matching If a partial match is found such that s i-j...s i-1 = p 0 p 1...p j-1 and s i <>p j then matching may be resumed by comparing s i and p f(j-1) +1 if j<>0. If j= 0, continue by comparing s i+1 and p 0 string a b c a ? ?... ? pat a b c a b c a c a b f Continue here

Example of Optimal Pattern Matching j pat a b c a b c a c a b f str c b a b c a a b c a b c a b c a c a b