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Arrays (1)

Arrays (1). Objectives. Arrays in C++ Implementation PolyNomial Sparse Matrix Representation String. Arrays in C++. Array: a set of index and value Data structure For each index, there is a value associated with that index. Representation Implemented by using consecutive memory.

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Arrays (1)

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  1. Arrays (1)

  2. Objectives • Arrays in C++ • Implementation • PolyNomial • Sparse Matrix • Representation • String

  3. Arrays in C++ • Array: a set of index and value • Data structure • For each index, there is a value associated with that index. • Representation • Implemented by using consecutive memory. • Example: int list[5]: list[0], …, list[4] each contains an integer

  4. Polynomial Abstract Data Type • Polynomial • requires ordered lists • the largest exponent is called degree • Example: • A(X)=3X2+2X+4, B(X)=X4+10X3+3X2+1 • sum and product of polynomials A(x) = aixi  and  B(x) = bixi

  5. Polynomial Abstract Data Type(Cont’) Class Polynomial {      // objects : p(x) = a0xe0 + … + anxe0 ; a set of ordered pairs of <ei, ai>, // where ai Coefficient and ei Exponent      // We assume that Exponent consists of integers ≥ 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

  6. Polynomial Abstract Data Type(Cont’) 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

  7. Polynomial Abstract Data Type(Cont’) Polynomial Representation • Principle • unique exponents are arranged in decreasing order

  8. Polynomial Abstract Data Type(Cont’) • Representation 1 (Array: static memory allocation) • define the private data members of Polynomial private : int degree ; // degree≤MaxDegree float coef[MaxDegree+1] ; • leads to a very simple algorithms for many of the operations on polynomials • wastes computer memory • for example, if a.degree << MaxDegree

  9. Polynomial Abstract Data Type(Cont’) • Representation 2 (Array: dynamic memory allocation) • define coef with size a.degree+1 • declare private data members private : int degree ; float *coef ; • add a constructor to Polynomial Polynomial::Polynomial(int d) { degree=d ; coef=newfloat[degree+1] ; } • wastes space for sparse polynomials • for example, x1000+1

  10. Polynomial Abstract Data Type(Cont’) • Representation 3 • previously, exponents are represented by array indices • now, (non-zero) exponents are stored • all Polynomials will be represented in a single array called termArray • termArray is shared by all Polynomial objects • it is declared as static • each element in termArray is of type term class term { friend Polynomial ; private : float coef ; // coefficient int exp ; // exponent };

  11. Polynomial Abstract Data Type(Cont’) • declare private data members of Polynomial • class Polynomial { private : static term termArray[MaxTerms]; static int free; int Start, Finish; public: Polynomial ADD(Polynomial poly); ... } • required definitions of the static class members outside the class definition term Polynomial::termArray[MaxTerms]; int Polynomial::free=0; // next free location in termArray • A(x)=2x1000+1, B(x)=x4+10x3+3x2+1

  12. Polynomial Abstract Data Type(Cont’) A.Start A.Finish B.Start B.Finish Free coef 2 1 1 10 3 1 exp 1000 0 4 3 2 0 0 1 2 3 4 5 6 Figure 2.1 : Array representation of two polynomials

  13. Polynomial Abstract Data Type(Cont’) • A(x) has n nonzero terms • A.Finish=A.Start+n-1 • comparison with Representation 2 • Representation 3 is superior when many zero terms are present as in A(x) • when all terms are nonzero, as in B(x), Representation 3 uses about twice as much space as Representation 2

  14. Polynomial Abstract Data Type(Cont’) 2.3.2 Polynomial Addition • Representation 3 is used • C(x)=A(x)+B(x)

  15. Polynomial Abstract Data Type(Cont’) Polynomial Polynomial::Add(Polynomial B) { Polynomial C; int a = Start; int b = B.Start; C.Start = free; float c; while ((a<=Finish) && (b<=B.Finish) switch (compare(termArray[a].exp, termArray[b].exp)){ case '=': c = termArray[a].coef + termArray[b].coef; if (c) NewTerm(c, termArray[a].exp); a++; b++; break; case '<': NewTerm(termArray[b].coef, termArray[b].exp); b++; break; case '>': NewTerm(termArray[a].coef, termArray[a].exp); a++; }// end of switch and while for (; a<=Finish; a++) // add in remaining terms of A(x) NewTerm(termArray[a].coef, termArray[a].exp); for (; b<=B.Finish; b++) // add in remaining terms of B(x) Newterm(termArray[b].coef, termArray[b].exp); C.Finish = free - 1; return C; } // end of Add Analysis: O(n+m) where n, m is the number of non-zeros in A, B.

  16. Polynomial Abstract Data Type(Cont’) void Polynomial::NewTerm(float c, int e) // Add a new term to C(x). { if (free>=MaxTerms) { cerr << "Too many terms in polynomials" << endl; exit(1); } termArray[free].coef = c; termArray[free].exp = e; free++; }// end of NewTerm Program 2.9 : Adding a new term

  17. Polynomial Abstract Data Type(Cont’) • Analysis of Add • m and n are number of nonzero-terms in A and B, respectively • the asymptotic computing time is O(n+m) • Disadvantages of representing polynomials by arrays • space must be reused for unused polynomials • linked lists in chapter 4 provide a solution

  18. Sparse Matrix • A general matrix consists of m rows and n columns of numbers • An m×n matrix • It is natural to store a matrix in a two-dimensional array, say A[m][n] • A matrix is called sparse if it consists of many zero entries • Implementing a spare matrix by a two-dimensional array waste a lot of memory • Space complexity is O(m×n)

  19. Sparse Matrix

  20. Sparse Matrices(Cont’) Class SparseMatrix { // objects : A set of triples, <row, column, value>, where row // and column are integers and form a unique combination; value // is also an integer. public: SparseMatrix(int MaxRow, int MaxCol); // the constructor function creates a SparseMatrix // that can hold up to MaxItems=MaxRow x MaxCol and whose // maximum row size is MaxRow and whose maximum // column size is MaxCol SparseMatrix Transpose(); // returns the SparseMatrix obtained by interchanging the // row and column value of every triple in *this

  21. Sparse Matrices(Cont’) SparseMatrix Add(SparseMatrix b); // if the dimensions of a(*this) and b are the same, then // the matrix produced by adding corresponding items, // namely those with identical row and column values is // returned else error. SparseMatrix Multiply(SparseMatrix b); // if number of columns in a (*this) equals number of // rows in b then the matrix d produced by multiplying a // by b according to the formula // d[i][j]= (a[i][k]·b[k][j]), // where d[i][j] is the (i, j)th element, is returned. // k ranges from 0 to the number of columns in a-1 // else error. }; ADT 2.3 : Abstract data type SparseMatrix

  22. Sparse Matrices(Cont’) 2.4.2 Sparse Matrix Representation • Representation • use the triple <row, col, value> to represent an element • store the triples by rows • for each row, the column indices are in ascending order • store the number of rows, columns, and nonzero elements

  23. Sparse Matrices(Cont’) • C++ code class SparseMatrix; // forward declaration class MatrixTerm { friend class SparseMatrix private: int row, col, value; }; class SparseMatrix { private: int Rows, Cols, Terms; MatrixTerm smArray[MaxTerms]; public: SparseMatrix Transpose(); ... }

  24. Sparse Matrices(Cont’) • Representation of the matrix of Figure 2.2(b) using smArray row col value row col value smArray[0] 0 0 15 smArray[0] 0 0 15 [1] 0 3 22 [1] 0 4 91 [2] 0 5 -15 [2] 1 1 11 [3] 1 1 11 [3] 2 1 3 [4] 1 2 3 [4] 2 5 28 [5] 2 3 -6 [5] 3 0 22 [6] 4 0 91 [6] 3 2 -6 [7] 5 2 28 [7] 5 0 -15 • (b) Figure 2.3 : Sparse matrix and its transpose stored as triples

  25. Sparse Matrices(Cont’) Transposing a Matrix • n element at [i][j] will be at [j][i] for (each row i) take element (i, j, value) and store it in (j, i, value) of the transpose; • difficulty: where to put <j, i, value> • (0, 0, 15)  (0, 0, 15) • (0, 3, 22)  (3, 0, 22) • (0, 5, -15)  (5, 0, -15) • (1, 1, 11)  (1, 1, 11) • need to insert many new triples, elements are moved down very often • Find the elements in the order for (all elements in column j) place element (i, j, value) in position (j, i, value);

  26. Sparse Matrices(Cont’) SparseMatrix SparseMatrix::Transpose() // return the transpose of a (*this) { SparseMatrix b; b.Rows = Cols; // rows in b = columns in a b.Cols = Rows; // columns in b = rows in a b.Terms = Terms; // terms in b = terms in a if (Terms > 0) // nonzero matrix { int CurrentB = 0; for (int c = 0; c < Cols; c++)// transpose by columns for (int i = 0; i < Terms; i++) // find elements 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; CurrentB++; } } // end of if (Terms > 0) return b; } // end of transpose Time Complexity: O(terms*columns)

  27. Sparse Matrices(Cont’) • Analysis of transpose • the number of iterations of the for loop at line 12 is terms • the number of iterations of the for loop at line 11 is columns • total time is O(terms*columns) • total space is O(space for a and b) • Using two-dimensional arrays for(int j=0; j<columns; j++) for(int i=0; i<rows; i++) B[j][i]=A[i][j]; • total time is O(rows*columns)

  28. [0] [1] [2] [3] [4] [5] RowSize = 3 2 1 0 1 1 RowStart = 0 3 5 6 6 7 Sparse Matrices(Cont’) • FastTranspose algorithm • Determine the number of elements in each column of the original matrix. • Determine the starting positions of each row in the transpose matrix.

  29. Sparse Matrices(Cont’)

  30. Sparse Matrices(Cont’)

  31. Sparse Matrices(Cont’)

  32. Sparse Matrices(Cont’) SparseMatrix SparseMatrix::FastTranspose() // The transpose of a (*this) is placed in b and is found in // O(terms+columns) time. { int *RowSize = new int[Cols]; int *RowStart = new int[Cols]; SparseMatrix b; b.Rows = Cols; b.Cols = Rows; b.Terms = Terms; if (Terms>0) // nonzero matrix { // compute RowSize[i] = number of terms in row i of b for (int i = 0; i<Cols; i++) RowSize[i] = 0; // initialize for (i = 0; i<Terms; i++) RowSize[smArray[i].col]++; // RowStart[i] = starting position of row i in b RowStart[0] = 0; O(Columns) O(Terms)

  33. Sparse Matrices(Cont’) for (i =1; i<Cols; i++) RowStart[i] = RowStart[i-1] + RowSize[i-1]; for (i = 0; i<Terms; i++) // move from a to b { 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]++; } // end of for } // end of if delete [] RowSize; delete [] RowStart; return b; } // end of FastTranspose O(Columns) O(Terms) Total: O(Columns+Terms) Program 2.11 : Transposing a matrix faster

  34. Sparse Matrices(Cont’) • By using sparse matrix • Pick a row of A and find all elements in column j of B for j = 0, 1, .., B.col-1 , where B.col is the number of columns in B • By using the transpose of B • We can avoid the scanning all of B to find all the elements in column j • All column elements are in consecutive order in B Matrix Multiplication A : m x n B : n x p

  35. Sparse Matrices(Cont’) int SparseMatrix::StoreSum(int sum,int& LastInResult,int r,int c){ if (sum != 0) { if (LastInResult < MaxTerms -1) { LastInResult++; smArray[LastInResult].row = r; smArray[LastInResult].col = c; smArray[LastInResult].value = sum; return 0; } else{ cerr << "Number of terms in product exceeds MaxTerms" << endl; return 1; } } else return 0; }

  36. SparseMatrix SparseMatrix::Multiply(SparseMatrix b) if (Cols != b.Rows) { cout << "Incompatible matrices" << endl; return EmptyMatrix(); } if ((Terms == MaxTerms) || (b.Terms == MaxTerms)) { cout << "One additional space in a or b needed" << endl; return EmptyMatrix(); } SparseMatrix bXpose = b.FastTranspose(); SparseMatrix result; int currRowIndex = 0, LastInResult = -1, currRowBegin = 0, currRowA = smArray[0].row; smArray[Terms].row = Rows; bXpose.smArray[b.Terms].row = b.Cols; bXpose.smArray[b.Terms].col = -1; int sum = 0;

  37. while (currRowIndex<Terms) { int currColB = bXpose.smArray[0].row; int currColIndex = 0; while (currColIndex <= b.Terms){ if (smArray[currRowIndex].row != currRowA){ if (result.StoreSum(sum, LastInResult, currRowA, currColB)) return EmptyMatrix(); else sum = 0; // sum currRowIndex = currRowBegin; while (bXpose.smArray[currColIndex].row==currColB) currColIndex++; currColB = bXpose.smArray[currColIndex].row; }else if (bXpose.smArray[currColIndex].row != currColB){ if (result.StoreSum(sum, LastInResult, currRowA, currColB)) return EmptyMatrix(); else sum = 0; currRowIndex = currRowBegin; currColB = bXpose.smArray[currColIndex].row; }

  38. else switch (compare(smArray[currRowIndex].col, bXpose.smArray[currColIndex].col)){ case '<' : currRowIndex++; break; case '=' : sum += smArray[currRowIndex].value *bXpose.smArray[currColIndex].value; currRowIndex++; currColIndex++; break; case '>' : currColIndex++; } // switch }// while(currCollindex <=b.Terms) while (smArray[currRowIndex].row == currRowA) currRowIndex ++ ; currRowBegin = currRowIndex; currRowA = smArray[ccurrRowIndex].row; }//while(currRowIndex<Terns) result.Rows = Rows; result.Cols = bCols; return result; }//Multiply

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