1. 2017, Fall Pusan National University Ki-Joune Li
Arrays
2017, Fall
Pusan National University
Ki-Joune Li

2. 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
ak, ak+1

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

6. Example : Adding two polynomials
A = 3.0x x x + 9.6
3.0
101
-2.4 2
1.0 1
9.6
B = 4.0x x
4.0 15
1.4 2
0.4
C =
3.0
101
4.0 15
-1.0 2
1.0 1
10.0
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 1 2 3 4 5 6 7
Row
Col
Value 2 1 5 4 6 7 3
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. Transposing a Matrix: An Algorithm6 2 1 5 4
value
col
row A’
row
col
value 1 4 2 5 6
Sort by (row, col)
Exchange row  col 1 5 6 2 4
value
col
row AT
Time Complexity : O(nElements + nElements log nElements) = O(nElements log nElements)

9. Transposing a Matrix: another Algorithm
A.nCols= 6, A.nRows=7 A.nElements=5 AT
row
col
value 1 4 2 5 6
row
col
value 4 1 1 4 2 2 6 5 1
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. Transposing a Matrix: Another Improved Algorithm7 6 5 4 3 2 1
index
value
col
row AT
index
row
col
value
4.5 1
1.5 2
2.5 3
3.5 4
12.5 5
82.5 6
22.5 7
62.5
4.5 3
3.5 1
1.5 1
2.5 2 3
12.5 2 5
22.5 3
82.5 5 4
22.5 1 5 2 4 3
A Col Size 7 5 6 4 2 3 1
AT Row Start
Time Complexity : O(nElements + ncol + nElements)