Data Structure - 2nd Study

4
// Polynomial: A set of pairs, <e_i, a_i>
// a_i: coefficient (not 0, float type)
// e_i: exponent (not negative, int type)
class Polynomial
{
public:
// Create polynomial p(x) = 0
Polynomial();
// Add polynomial *this and poly
const Polynomial Add(const Polynomial& poly);
// Multiply polynomial *this and poly
const Polynomial Multiply(const Polynomial& poly);
// After assign f into x, evaluate polynomial and return result
const float Eval(const float& f);
};

5
private:
static const int MaxDegree = 100;
int degree; // degree <= MaxDegree
float coef[MaxDegree + 1]; // Coefficient array

6
private:
int degree;
float* coef; // Coefficient array
Polynomial::Polynomial(int d)
: degree(d), coef(new float[degree + 1]) { }

7
class Polynomial; // Forward Declaration
class Term
{
friend Polynomial;
private:
float coef; // Coefficient
int exp; // Exponent
};
class Polynomial
{
private:
Term* termArray; // Term array (not 0)
int capacity; // Capacity of termArray
int terms; // Number of terms (not 0)
...
};
3 2 -4coef
3 2 1exp
𝑎𝑎 𝑥𝑥 = 3𝑥𝑥3 + 2𝑥𝑥2 − 4𝑥𝑥
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8
− 10𝑥𝑥5
− 3𝑥𝑥3
+ 1

8
const Polynomial Polynomial::Add(const Polynomial& b)
{ // Return *this + b
Polynomial c;
int aPos = 0, bPos = 0;
while ((aPos < terms) && (bPos < b.terms)) {
if (termArray[aPos].exp == b.termArray[bPos].exp) {
float t = termArray[aPos].coef + b.termArray[bPos].coef;
if (t) c.NewTerm(t, termArray[aPos].exp);
aPos++; bPos++;
}
else if (termArray[aPos].exp < b.termArray[bPos].exp) {
c.NewTerm(b.termArray[bPos].coef, b.termArray[bPos].exp);
bPos++;
}
else {
c.NewTerm(termArray[aPos].coef, termArray[aPos].exp);
aPos++;
}
}

9
// Add rest terms of *this
for (; aPos < terms; aPos++)
// Add rest terms of b(x)
for (; bPos < b.terms; bPos++)
return c;
}

10
void Polynomial::NewTerm(const float& coef, const int& exp)
{ // Add new term into into last element of termArray
if (terms == capacity)
{ // Expand the size of termArray twice
capacity *= 2;
Term* temp = new Term[capacity];// New array
std::copy(termArray, termArray + terms, temp);
delete[] termArray;// Return original array
termArray = temp;
}
termArray[terms].coef = coef;
termArray[terms++].exp = exp;
}

11
Polynomial c;
int aPos = 0, bPos = 0; 3 2 -4coef
3 2 1exp
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
aPos
bPos
𝑐𝑐 𝑥𝑥 = 𝑥𝑥8
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1

12
3 2 -4coef
3 2 1exp
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
aPos++; bPos++;
}
bPos++;
}
else {
aPos++;
}
}
aPos
bPos
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1
a.termArray[aPos].exp = a.termArray[0].exp = 3
b.termArray[bPos].exp = b.termArray[0].exp = 8
∴ a.termArray[aPos].exp < b.termArray[bPos].exp
coef
8
1
exp

13
3 2 -4coef
3 2 1exp
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
aPos++; bPos++;
}
bPos++;
}
else {
aPos++;
}
}
aPos
bPos
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1
∴ a.termArray[aPos].exp < b.termArray[bPos].exp
-10coef
8
1
5exp

14
3 2 -4coef
3 2 1exp
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
aPos++; bPos++;
}
bPos++;
}
else {
aPos++;
}
}
aPos
bPos
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1
∴ a.termArray[aPos].exp == b.termArray[bPos].exp
-10coef
8
1
5exp
T = 3 – 3 = 0이므로 새 항은 만들어지지 않음

15
3 2 -4coef
3 2 1exp
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
aPos++; bPos++;
}
bPos++;
}
else {
aPos++;
}
}
aPos
bPos
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1
∴ a.termArray[aPos].exp > b.termArray[bPos].exp
-10 2coef
8
1
5 2exp

16
3 2 -4coef
3 2 1exp
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
aPos++; bPos++;
}
bPos++;
}
else {
aPos++;
}
}
aPos
bPos
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1
∴ a.termArray[aPos].exp > b.termArray[bPos].exp
-10 2 -4coef
8
1
5 2 1exp

17
1 -10 -3 1coef
8 5 3 0exp
𝑏𝑏 𝑥𝑥 = 𝑥𝑥8 − 10𝑥𝑥5 − 3𝑥𝑥3 + 1
// Add rest terms of *this
for (; aPos < terms; aPos++)
// Add rest terms of b(x)
for (; bPos < b.terms; bPos++)
bPos
− 10𝑥𝑥5
+ 2𝑥𝑥2
− 4𝑥𝑥 + 1
-10 12 -4coef
08
1
5 2 1exp
3 2 -4coef
3 2 1exp
aPos

21
// Three elements: <row, column, value>
// row, column: int (not negative)
// <row, column> is a unique pair
class SparseMatrix
{
public:
// Create SparseMatrix with r rows, c columns, t terms (not 0)
SparseMatrix(int r, int c, int t);
// Transpose all elements of *this and return it
SparseMatrix Transpose();
// If the dimension of *this equals to b, add/subtract *this and b and return result
// Otherwise, throw exception
SparseMatrix Add(const SparseMatrix& b);
SparseMatrix Subtract(const SparseMatrix& b);
// If the column of *this equals to the row of b,
// multiply *this and b and return result
// Otherwise, throw exception
SparseMatrix Multiply(const SparseMatrix& b);
};

22
class SparseMatrix; // Forward Declaration
class MatrixTerm
{
friend class SparseMatrix;
private:
int row, col, value;
};
class SparseMatrix
{
private:
int rows, cols, terms, capacity;
MatrixTerm* smArray;
...
};
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28

23
2 × 2 2 × 2 2 × 2
=
A의 1행, B의 1열
A의 2행, B의 1열 A의 2행, B의 2열
A의 1행, B의 2열

25
SparseMatrix SparseMatrix::Transpose()
{ // Return the transpose matrix of *this
SparseMatrix b(cols, rows, terms); // the size of b.smArray is terms
if (terms > 0)
{ // This is not 0-matrix
int currentB = 0;
for (int c = 0; c < cols; c++)
{ // Transpose each column
for (int i = 0; i < terms; i++)
{ // Find element from column c and move it
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 > 0)
return b;
}

27
SparseMatrix SparseMatrix::FastTranspose()
{ // Return the transpose matrix of *this in O(terms + cols)
SparseMatrix b(cols, rows, terms);
if (terms > 0)
{ // This is not 0-matrix
int* rowSize = new int[cols];
int* rowStart = new int[cols];
std::fill(rowSize, rowSize + cols, 0); // Initialization
for (int i = 0; i < terms; i++) rowSize[smArray[i].col]++;
// Start point of row i of rowStart[i] = b
rowStart[0] = 0;
for (int i = 1; i < cols; i++)
rowStart[i] = rowStart[i - 1] + rowSize[i - 1];

28
{ // Copy *this 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
delete[] rowSize;
delete[] rowStart;
} // End of if
return b;
}

29
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 0 0 0 0 0 0
rowStart =

30
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 1 0 0 0 0 0
rowStart =
i = 0 → smArray[0].col = 0 → rowSize[0]++;

31
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 1 0 0 1 0 0
rowStart =

32
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 1 0 0 1 0 1
rowStart =

33
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 1 1 0 1 0 1
rowStart =

34
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 1 1 1 1 0 1
rowStart =

35
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 1 1 1 2 0 1
rowStart =

36
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 1 2 0 1
rowStart =

37
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart =

38
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
rowStart[0] = 0;
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 0

39
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
rowStart[0] = 0;
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 0 2
i = 1 → rowStart[1] = rowStart[0] + rowSize[0];

40
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
rowStart[0] = 0;
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 0 2 3

41
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
rowStart[0] = 0;
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 0 2 3 5

42
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
rowStart[0] = 0;
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 0 2 3 5 7

43
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
rowStart[0] = 0;
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 0 2 3 5 7 7

44
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 1 2 3 5 7 7
} // End of for
row col value
smArray[0] 0 0 15
smArray[1]
smArray[2]
smArray[3]
smArray[4]
smArray[5]
smArray[6]
smArray[7]
i = 0 → smArray[0].col = 0 → j = rowStart[0] = 0
rowStart[0]++;

45
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 1 2 3 6 7 7
} // End of for
row col value
smArray[0] 0 0 15
smArray[1]
smArray[2]
smArray[3]
smArray[4]
smArray[5] 3 0 22
smArray[6]
smArray[7]
rowStart[3]++;

46
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 1 2 3 6 7 8
} // End of for
row col value
smArray[0] 0 0 15
smArray[1]
smArray[2]
smArray[3]
smArray[4]
smArray[5] 3 0 22
smArray[6]
smArray[7] 5 0 -15
rowStart[5]++;

47
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 1 3 3 6 7 8
} // End of for
row col value
smArray[0] 0 0 15
smArray[1]
smArray[2] 1 1 11
smArray[3]
smArray[4]
smArray[5] 3 0 22
smArray[6]
smArray[7] 5 0 -15
rowStart[1]++;

48
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 1 3 4 6 7 8
} // End of for
row col value
smArray[0] 0 0 15
smArray[1]
smArray[2] 1 1 11
smArray[3] 2 1 3
smArray[4]
smArray[5] 3 0 22
smArray[6]
smArray[7] 5 0 -15
rowStart[2]++;

49
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 1 3 4 7 7 8
} // End of for
row col value
smArray[0] 0 0 15
smArray[1]
smArray[2] 1 1 11
smArray[3] 2 1 3
smArray[4]
smArray[5] 3 0 22
smArray[6] 3 2 -6
smArray[7] 5 0 -15
rowStart[3]++;

50
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 2 3 4 7 7 8
} // End of for
row col value
smArray[0] 0 0 15
smArray[1] 0 4 91
smArray[2] 1 1 11
smArray[3] 2 1 3
smArray[4]
smArray[5] 3 0 22
smArray[6] 3 2 -6
smArray[7] 5 0 -15
rowStart[0]++;

51
row col value
smArray[0] 0 0 15
smArray[1] 0 3 22
smArray[2] 0 5 -15
smArray[3] 1 1 11
smArray[4] 1 2 3
smArray[5] 2 3 -6
smArray[6] 4 0 91
smArray[7] 5 2 28
[0] [1] [2] [3] [4] [5]
rowSize = 2 1 2 2 0 1
rowStart = 2 3 5 7 7 8
} // End of for
row col value
smArray[0] 0 0 15
smArray[1] 0 4 91
smArray[2] 1 1 11
smArray[3] 2 1 3
smArray[4] 2 5 28
smArray[5] 3 0 22
smArray[6] 3 2 -6
smArray[7] 5 0 -15
rowStart[2]++;

54
class String {
public:
// Constructor with content *init, length m
String(char* init, int m);
// Return true if *this's string is equal to t's. Otherwise, return false
bool operator==(String t);
// Return true if *this's string is empty. Otherwise, return false
bool operator!();
// Return the number of *this's characters
int Length();
// Return *this's string + t's
String Concat(String t);
// Return substring with index (i, i+1, ..., i+j-1) of *this's string
String Substr(int i, int j);
// Return index i if *this's substring matches pat's string
// Return -1 if pat is empty or not *this's substring
int Find(String pat);
};

56
int String::Find(String pat)
{ // If pat is found in *this, then return start position of pat in *this
// Otherwise, return -1
for (int start = 0; start <= Length() - pat.Length(); start++)
{ // Check pattern match at str[start]
int j;
for (j = 0; j < pat.Length() && str[start + j] == pat.Substr[j]; j++)
if (j == pat.Length()) return start; // Found match
}
return -1; // pat is empty or not exist in *this
}

60
𝑝𝑝𝑝𝑝𝑝𝑝
𝑝𝑝𝑝𝑝𝑝𝑝 𝑏𝑏와 같으므로,
𝑠𝑠𝑖𝑖+1 ≠ 𝑎𝑎라는 것을 알기 때문에 𝑝𝑝𝑝𝑝𝑝𝑝의 첫 문자와 𝑠𝑠𝑖𝑖+1을 비교할 필요가 없음
𝑠𝑠 = - 𝑎𝑎 𝑏𝑏 ? ? ? . . . . ?
𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑏𝑏

61
𝑝𝑝𝑝𝑝𝑝𝑝 𝑏𝑏를 비교
𝑠𝑠 = - 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 ? ? . . . ?
𝑠𝑠 = - 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 ? ? . . . ?

62
𝑗𝑗 0 1 2 3 4 5 6 7 8 9
𝑝𝑝𝑝𝑝𝑝𝑝 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑏𝑏
𝑓𝑓 −1 −1 −1 0 1 2 3 −1 0 1

63
int String::FastFind(String pat)
{ // Determine whether pat is substring of s or not
int posP = 0, posS = 0;
int lengthP = pat.Length(), lengthS = Length();
while ((posP < lengthP) && (posS < lengthS)) {
if (pat.str[posP] == pat.str[posS]) { // String match
posP++; posS++;
}
else {
if (posP == 0) posS++;
else posP = pat.f[posP - 1] + 1;
}
}
if (posP < lengthP) return -1;
else return posS - lengthP;
}

66
void String::FailureFunction()
{ // Calculate failure function about *this pattern
int lengthP = Length();
f[0] = -1;
for (int j = 1; j < lengthP; j++)
{
int i = f[j - 1];
while ((*(str + j) != *(str + i + 1)) && (i >= 0)) i = f[i];
if (*(str + j) == *(str + i + 1)) f[j] = i + 1;
else f[j] = -1;
}
}

Data Structure - 2nd Study

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Data Structure - 2nd Study