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Arrays

Arrays. 2014, Fall Pusan National University Ki-Joune Li. a k , a k +1. Most Basic Data Structure. Primitive Elements : Integer, Float, String, etc.. Container of elements Ordered or not Duplicated or Not How to define the same object

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Arrays

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  1. Arrays 2014, Fall Pusan National University Ki-Joune Li

  2. ak, ak+1 Most Basic Data Structure • Primitive Elements : Integer, Float, String, etc.. • Container of elements • Ordered or not • Duplicated or Not • How to define the same object • In fact, any container of elements is stored in ordered way. • Only the interface makes it different

  3. Operations • Operation defines the interface (or nature) to the users • Should be implemented, whatever the internal implementation • Operations • Unordered • Maintenance : create a new container, Insert, Delete, Update (?) • Search : search by atttributes • Information or statistics : number of elements, max, min, etc. • Ordered • Operations for Unordered Container + • Scan : get the k-th, the last, or the next elements • Sorting • Ordered container can be used as an unordered container

  4. Arrays • Array • Contains elements • Ordered (or unordered) • Set (No Order) • Example : Polynomial • Representation 1: float Coef[MaxDegree+1]; • Representation 2: int degree; float *Coef; Coef=new float[degree+1];

  5. 3.0 2.0 1.4 0.4 9.6 1.0 -2.4 2 101 0 1 15 0 2 Class Term { friend Polynomial; private: float coef; float exp;}; free a.Start a.Finish b.Start b.Finish MaxTerms Representation of Array • Representation 3: • Array of (Coefficient, Exponent): ((am,m),(am-1,m), … (a0,0)) • Sparse Array : Example. • 3.0x101-2.4x2+1.0x+9.6 : ((3.0,101),(-2.4,2),(1.0,1),(9.6,0)) • MaxTerms Class Polynomial { private: static Term termArray[MaxTerms]; static int free; int Start, Finish;};

  6. 3.0 -2.4 1.0 9.6 101 2 1 0 Example : Adding two polynomials A = 3.0x101 - 2.4x2 + 1.0x + 9.6 4.0 15 1.4 2 0.4 0 B = 4.0x15 + 1.4x2 + 0.4 C = 3.0 101 4.0 15 -1.0 2 1.0 1 10.0 0 Termination Condition - Aptr > A.finish or Bptr > B.finish If Terminated by Aptr > A.finish Append the rest of B to the tail of B Time Complexity : O(LenA + LenB )

  7. Sparse Matrix • Sparse Matrix • Matrix with many zero elements • Two Representations Class MatrixElement { friend SparseMatrix; private: int row,col; int value;}; vs. But Row Major ! Class SparseMatrix { private: int nRows,nCols,nElements; MatrixElement smArray[MaxElements];};

  8. A’ row col value Sort by (row, col) 1 1 1 0 4 4 1 4 2 Exchange row  col 5 5 1 2 6 6 AT row col value 0 4 4 1 1 1 1 4 2 2 6 6 5 5 1 Transposing a Matrix: An Algorithm A Time Complexity : O(nElements+ nElements log nElements) = O(nElements log nElements)

  9. Transposing a Matrix: another Algorithm A A.nCols= 6, A.nRows=7A.nElements=5 AT Algorithm MatrixTranspose(SparseMatrix A) SparseMatrix B; swap(A.nRows,A.nCols); countB=0; if(A.nElements>0) { // for non-empty matrix for(c=0;c<A.nCols;c++) // for each element c of A for(i=0;i<A.nElements;i++) { // find elements in column c if(A.smArrays[i].col==c) { B.smArray[countB].row=c; B.smArray[countB].col=A.smArray[i].row; B.smArray[countB].value=A.smArrary[countB].value; ++countB; } } } return B;end Algorithm Time Complexity : O(nCols·nElements) = O(nCols2·nRows) > O(nCols·nRows)

  10. AT index row col value 0 1 3 5 1 0 0 2 2 1 3 3 5 0 4 0 3 1 22.5 4.5 82.5 1.5 3.5 12.5 22.5 2.5 2 3 4 5 6 7 0 1 2 3 4 5 0 1 2 3 4 5 2 2 2 1 0 1 0 2 4 6 7 7 AT Row Size ATRow Start Transposing a Matrix: Another Improved Algorithm A Time Complexity : O(ncol+ ncol + nElements)

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