2.2 The array as an ADT

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2.2 The array as an ADT Chapter 2: Arrays Hsien-Chou Liao Depart. of Comp. Sci. and Info. Eng. Chaoyang Univ. of Tech. Many programmers view an array only as a consecutive set of memory locations.

1 2.2 The array as an ADT Chapter 2: Arrays Hsien-Chou Liao Depart. of Comp. Sci. and Info. Eng. Chaoyang Univ. of Tech. Many programmers view an array only as a consecutive set of memory locations. Is array defined as an ADT? Array is a set of pairs, <index, value> Also called correspondence or mapping 1 3 ADT (Abstract Data Type): NaturalNumber ADT NaturalNumber is objects: An ordered subrange of the integers starting at zero and ending at the maximum integer (MAXINT) on the computer. functions: for all x, y NaturalNumber; TRUE, FALSE Boolean and where +,,<,==, and = are the usual integer operations Zero(): NaturalNumber ::= IsZero(x): Boolean ::= if (x==) IsZero = TRUE else IsZero = FALSE Add(x, y): NaturalNumber ::= if (x+y<=maxint) Add = x + y Equal(x, y): Boolean else Add = MAXINT ::= if (x == y) Equal = true else Equal = false Successor(x): NaturalNumber::= if (x == MAXINT) Successor = x else Successor = x + 1 Subtract(x, y): NaturalNumber::= if (x < y) Subtract = else Subtract = x y end NaturalNumber ADT GeneralArray class GeneralArray // A set of pairs <index, value> public: GeneralArray(int j, RangeList list, float initvalue = defaultvalue); // constructor // j: dimension float Retrieve(index i); // return the float associated with i void Store(index i, float x); // replace the old value associate with i ; 2 4

2 Array 2.3 Polynomial ADT char A[3][4]; 1 2 logical structure A[2][1] Mapping: A[i][j]=A[][]+i*4+j // row-major physical structure A[][] A[][1] A[][2] A[][3] A[1][] A[1][1] A[1][2] A[1][3] Array can also be used to implement other ADT. Ordered (linear) list: for example: Days of the week (Sunday, Monday, ) Values in a deck of cards: (Ace, 2, 3, 4, ) Floors of a building 5 7 The Operations on 1-dim Array Store Replace the <index, oldvalue> pair with the <index, newvalue> pair. A[3] = 45 A Here is a problem requiring ordered list: Polynomial: a(x) = 3x 2 + 2x - 4 b(x) = x 8 1x 5-3x Coefficients and exponents a(x) is 3, 2, -4 and 2, 1, b(x) is 1, -1, -3, 1 and 8, 5, 3, 6 8

3 Polynomial Addition ADT: Polynomial class Polynomial // p(x) is a set of order pairs of <e i,a i > // where a i is a nonzero float coefficient and e i is a non-negative integer exponent. public: Polynomial( ); // Construct the polynomial p(x) = Polynomial Add(Polynomial poly); // return the sum of *this and poly Polynomial Mult(Polynomial poly); // return the product of *this and poly float Eval(float f ); // Evaluate the polynomial *this at f and return the result. ; a.termarray[].coef a.termarray[].exp Assume Representation 3 is used: Add the following two polynomials: a(x) = 2x b(x) = x 4 + 1x 3 + 3x c.termarray[].coef b.termarray[].coef b.termarray[].exp c.termarray[].exp Polynomial Representations Some representation decisions: Representation 1 int degree; // degree MaxDegree float coef [MaxDegree + 1]; Representation 2 int degree; float *coef; Polynomial::Polynomial(int d) degree = d; coef = new float [degree+1]; Representation 3 class Polynomial; // forward declaration class Term friend Polynomial; float coef; int exp; ; // coefficient // exponent term *termarray; int capacity; // size of termarray int terms; // number of nonzero terms C++ Program Program 2.8: The Add function Time complexity: O(m+n) Program 2.9: Adding a new term void Polynomial::NewTerm(const float thecoeff, const int theexp) // Add a new term to the end of termarray if (terms = = capacity) // double the capacity of termarray capacity *= 2; term *temp = new term[capacity]; // new array copy(termarray, termarray + terms, temp); delete [] termarray ; // deallocate old memory termarray = temp; termarray [terms].coef = thecoeff; termarray [terms++].exp = theexp; 1 12

4 2.4 Sparse Matrices ADT SparseMatrix A matrix is a mathematical object that arises in many physical problems. It is natural to store a matrix in a two-dimensional array, a[m][n]. For example: (sparse matrix) Q: What is the problem to represent a sparse matrix in a[m][n]? 13 class SparseMatrix public: SparseMatrix(int r, int c,int t); // constructor SparseMatrix Transpose( ); //returns the SparseMatrix obtained by interchanging the row and column value SparseMatrix Add(SparseMatrix b); // SparseMatrix Multiply(SparseMatrix b); // ; 15 Sparse Matrix Representation Sparse Matrix Representation (Cont.) Use triple <row, column, value> Store triples row by row For all triples within a row, their column indices are in ascending order. To ensure that the operations terminate: Must know the number of rows and columns and the number of nonzero elements The class definition: (a) Sparse matrix stored as triples (b) The transpose class SparseMatrix class MatrixTerm friend class SparseMatrix int row, col, value; ; int rows, cols, terms, capacity; MatrixTerm *smarray; 14 16

5 Transposing a Matrix Intuitive way: for (each row i) take element (i, j, value) and store it in (j, i, value) of the transpose Q: What is the difficulty of the above way? It cannot know the place of an element in the transpose. More efficient way: for (all elements in column j) place element (i, j, value) in position (j, i, value) Please see the example in the previous slide. 17 Fast Matrix Transpose If matrix is represented in 2-dim arrays. The time complexity is O(rows cols). To sort terms is of the order of rows cols, the time complexity is O(rows cols). Therefore, O(terms cols) becomes O(rows cols 2 ). O(rows cols 2 ) is worse than O(rows cols), what does it means? Conserve space but trading away too much time. A fast transpose function in Program Determine the number of elements in each column. It gives the number of element in each row of b Obtain the starting point of each row of b. Move the elements of a one by one into correct position in b. Q: Time complexity? 19 Program 2.1: Transposing a Matrix SparseMatrix SparseMatrix::Transpose( ) // return the transpose of *this SparseMatrix b(cols, rows, terms); // capacity of b.smarray is terms if (terms > ) // nonzero matrix int currentb = ; for (int c = ; c < cols ; c++) // transpose by columns for (int i = ; i < terms ; i++) // find and move terms in column c if (smarray[i].col = = c) b.smarray[currentb].row = c; b.smarray[currentb].col = smarray[i].row; b.smarray[currentb++].value = smarray[i].value; // end of if (terms > ) return b; Q: Time complexity? 18 Program 2.11 Fast Matrix Transposing SparseMatrix SparseMatrix::FastTranspose( ) SparseMatrix b(cols, rows, terms); if (terms > ) int *rowsize = new int[cols]; int *rowstart = new int[cols]; fill(rowsize, rowsize + cols, ); for (int i = ; i < terms ; i ++) rowsize[smarray[i].col]++; O(terms) rowstart[] = ; for (int i = 1 ; i < cols ; i++) rowstart[i] = rowstart[i-1] + rowsize[i-1]; O(cols-1) for (int i = ; i < terms ; i++) O(terms) int j = rowstart[smarray[i].col]; b.smarray[j].row= smarray[i].col; b.smarray[j].col = smarray[i].row; b.smarray[j].value = smarray[i].value; rowstart[smarray[i].col]++; delete [] rowsize; delete [] rowstart; return b; Overall complexity is O(cols + terms) 2

6 Time complexity of FastTranspose Is really only O(cols + terms)? To sort terms is of the order of rows cols, the time complexity is O(rows cols). Therefore, O(cols + terms) becomes O(rows cols). It is the same as when 2-dim arrays are in use. Two Dimensional Array Row Major Order Figure 2.6: row row 1 row u 1-1 u 2 elements u 2 elements col col 1 col 2 col u 2-1 X X X X X X X X X X X X row row 1 row i row u 1-1 i * u 2 element Representation of Arrays Multidimensional arrays are usually implemented by one dimensional array via either row major order or column major order. Example: one dimensional array α: the address of a[] α α+1 α+2 α+3 α+4 α+5 α+6 α+7 α+8 α+9 α+1 α+12 a[] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[1] a[11] The address of elements in 2-dim array For a two-dim array, A[r][c], assume every element occupies len bytes and the address of the first element is α. The array is stored in row major order. Therefore, The address of A[][3] is α+3*len The address of A[3][] is α+(3*c+)*len... The address of A[m][n] is α+(m* c + n)*len 22 24

7 2.6 The String ADT A string is defined as the form: S=s, s 1,, s i, where s i are characters. H e l l o W o r l d \ Operations: Create a new empty string Reading a string or printing it out Concatenation: Appending two string together Copying a string 25 String pattern matching Two string, s and pat, where pat is a pattern to be searched for in s. s.find(pat): return an index i, or return -1 The easiest but least efficient method: int String::Find(String pat) for (int start = ; start <= Length( )-pat.length(); star ++) int j; for (j = ; j < pat.length( ) && str [start+j] = = pat.str[j]; j++) if (j = = pat.length( )) return start; // match found // no match at position start return -1 ; // pat is empty or does not occur in s Q: Time complexity? O(lengthP lengths) 27 Abstract data type String String Matching: The Knuth-Morris-Pratt Algorithm class String public: String(char *init, int m); // Constructor bool operator = = (String t); // if the string represented by *this equals t, return true; else return false. bool operator!( ); // If *this is empty then return true; else return false. int Length( ); // Return the number of characters in *this. String Concat(String t); // Return a string whose elements are those of *this followed by those of t. String Substr(int i, int j); // Return a string containing the j characters of *this at positions i, i+1,..., i+j 1 // if there are valid positions of *this; otherwise, throw an exception. int Find(String pat); // Return an index i such that pat match the substring of *this that begins // at position i. Return 1 if pat is either empty or not a substring of*this 26 Definition: If p = p p 1 p n-1 is a pattern, then its failure function, f, is defined as f ( j) largest k < j such that p... pk = p j k... p j if such a k exists = 1 otherwise If a partial match is found such that s i-j s i-1 = p 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. If j =, then we may continue by comparing s i+1 and p. 28

8 Fast Matching Example 1 Suppose exists a string s and a pattern pat = abcabcacab, let s try to match pattern pat in string s. Failure function is shown below: j pat a b c a b c a c a b f for j=, since k < j, therefore f()=-1 for j=1, k=, p =a p 1 =b, therefore f(1)=-1 for j=4, k=, a b k=1, a b = a b k=2, a b c c a b k=3, a b c a b c a b, therefore f(4)=1 Fast Matching Example 2 Failure function is shown below: j pat a b c a b c a c a b f s = - a b c a b c a?.? pat = a b c a b c a c a b j = 7, f (j-1) + 1 = = 4 a b c a b c a c a b New start matching point Fast Matching Example 1 Suppose exists a string s and a pattern pat = abcabcacab, let s try to match pattern pat in string s. Failure function is shown below: j pat a b c a b c a c a b f s = - a b c a??...? pat = a b c a b c a c a b j = 4, f (j-1)+1 = 1 a b c a b c a c a b Easier Matching for Example 2 Failure function is shown below: j pat a b c a b c a c a b f s = - a b c a b c a?.? pat = a b c a b c a c a b a.. a.. a b c a b.. The above matching can be skipped using the fast matching method. New start matching point 3 32

9 Pattern-matching with failure function 1 int String::FastFind(String pat) 2 //Determine if pat is a substring of s 3 int posp =, poss = ; 4 int lengthp = pat.length(), lengths = Length(); 5 while((posp < lengthp) && (poss < lengths)) 6 if (pat.str[posp] = = str[poss]) // character match 7 posp++; poss++; 8 9 else 1 if (posp = = ) 11 poss++; 12 else posp = pat. f [posp-1] + 1; 13 if (posp < lengthp) return 1; 14 else return poss-lengthp; 15 Thanks for your attention! Q & A Q: Time complexity? 33 Computing the failure function 1 void String::FailureFunction() 2 // Compute the failure function for the pattern *this 3 int lengthp = Length( ); 4 f[] = -1; 5 for (int j = 1; j < lengthp; j++) // compute f [j] 6 7 int i = f [j-1]; 8 while ((*(str+j)!= *(str+i+1)) && (i >= )) i = f [i]; 9 if (*(str+j) = = *(str+i+1)) 1 f [j] = i+1; 11 else f [j] = -1; Q: Time complexity of computing failure function? Q: The total time complexity of FastFind? 34

Arrays. 2.2 The Array as an Abstract Data Type. Array:

Arrays. 2.2 The Array as an Abstract Data Type. Array: Arrays 2.2 The Array as an Abstract Data Type Array: A set of pairs: (correspondence or mapping) Two operations: retrieve, store Now we will use the C++ class to define an ADT. 1 GeneralArray