UVA Problem 438 – The Circumference of the Circle Solution

UVA Problem 438 – The Circumference of the Circle Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

Given three points find the circumference of the circle. Points are non-collinear meaning they are not on a straight line.

Have a look at Circumscribed circle, Semi perimeter and Heron’s formula to find more about the formula and derivation.

Also related to this have a look at UVA 10432 – Polygon Inside a Circle.

 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

0.0 -0.5 0.5 0.0 0.0 0.5
0.0 0.0 0.0 1.0 1.0 1.0
5.0 5.0 5.0 7.0 4.0 6.0
0.0 0.0 -1.0 7.0 7.0 7.0
50.0 50.0 50.0 70.0 40.0 60.0
0.0 0.0 10.0 0.0 20.0 1.0
0.0 -500000.0 500000.0 0.0 0.0 500000.0

 


Output:

3.14
4.44
6.28
31.42
62.83
632.24
3141592.65

Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 438 - The Circumference of the Circle.
 * Technique: Finding diameter and circumference of Circumscribed
 *            circle or Circumcircle.
 *            Using Heron's formula to calculate semi perimeter
 *            and area of triangle from Three points.
 */

#include<stdio.h>
#include<string.h>
#include<math.h>


#define PI 3.141592653589793


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    double x1, y1;
    double x2, y2;
    double x3, y3;


    while( scanf("%lf%lf%lf%lf%lf%lf", &x1, &y1, &x2, &y2, &x3, &y3 ) == 6 ){

        // a dist between (x1,y1) and (x2,y2)
        // b dist between (x2,y2) and (x3,y3)
        // c dist between (x3,y3) and (x1,y1)

        double a = sqrt( pow(x1 - x2, 2) + pow(y1 - y2, 2) );
        double b = sqrt( pow(x2 - x3, 2) + pow(y2 - y3, 2) );
        double c = sqrt( pow(x3 - x1, 2) + pow(y3 - y1, 2) );


        // semi perimeter, s = (a+b+c)/2

        double s = ( a + b + c ) / 2;


        // Area using Heron's Formula

        double A = sqrt( s*(s-a)*(s-b)*(s-c) );


        // Diameter of circumscribed circle d = abc/2A

        double d = (a * b * c) / (2 * A);


        // Result circumference of the circumcircle or circumscribed circle

        double circumference = PI * d;


        printf("%.2lf\n", circumference );

    }


    return 0;
}


Unnecessary Complex Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 438 - The Circumference of the Circle.
 * Technique: Point and Line representation in structure.
 *            Finding diameter and circumference of Circumscribed
 *            circle or Circumcircle.
 *            Using Heron's formula to calculate semi perimeter
 *            and area of triangle from Three points.
 */

#include<stdio.h>
#include<string.h>
#include<math.h>


#define PI 3.141592653589793


struct Point{
    double x;
    double y;
};


struct Line{
    Point a;
    Point b;

    Point setPoints( Point _a = {0.0,0.0}, Point _b = {0.0,0.0} ){
        a = _a;
        b = _b;
    }

    double length(){
        return sqrt( pow(a.x - b.x, 2) + pow(a.y - b.y, 2) );
    }

};


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    Point point[3];
    Line  line [3];


    while( scanf("%lf%lf%lf%lf%lf%lf", &point[0].x, &point[0].y, &point[1].x, &point[1].y, &point[2].x, &point[2].y ) == 6 ){


        // Loop is same as code below.
        //line[0].setPoints(point[0], point[1]);
        //line[1].setPoints(point[1], point[2]);
        //line[2].setPoints(point[2], point[0]);

        int N = 3;
        for( int i = 0; i < N; ++i )
            line[i].setPoints( point[i], point[(i+1) % N] );



        // semi perimeter is half of perimeter.
        // Perimeter is distance around the shape in 2D.

        double s = ( line[0].length() + line[1].length() + line[2].length() ) / 2;


        // Area using Heron's Formula

        double A = sqrt( s * (s - line[0].length()) * (s - line[1].length()) * (s - line[2].length()) );


        // Diameter of circumscribed circle d = abc/2A

        double d = (line[0].length() * line[1].length() * line[2].length()) / (2 * A);


        // Result circumference of the circumcircle

        double circumference = PI * d;


        printf("%.2lf\n", circumference );

    }


    return 0;
}

UVA Problem 11716 – Digital Fortress Solution

UVA Problem 11716 – Digital Fortress Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

This is an easier string problem. The task is to arrange the characters within the string in a certain order.

If the input string length is not square of a number then print INVALID. Ex: “DAVINCICODE” has length of 11. Squaring no number results in 11.

“DTFRIAOEGLRSI TS” has length of 16 including the space character. Square root of 16 is 4 and 4 * 4 is 16. So this is valid input.

Now if the input is valid then next task is to rearrange them. Instead of following given solution technique in the question it can solved much faster in another way. Following instruction in the question gives intuition that first the string must be positioned on 2 dimensional matrix in row major order. Then traverse them column by column.

A better way is find out each group length from square rooting the original string length. Next starting from first group take a character then skip by group length to get next column major character. After its done do this from the next character of first group.


Visualization:
uva 11716 rearrange string in column major order
uva 11716 rearrange string in column major order

 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

3
WECGEWHYAAIORTNU
DAVINCICODE
DTFRIAOEGLRSI TS

 


Output:

WEAREWATCHINGYOU
INVALID
DIGITAL FORTRESS

Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 11716 - Digital Fortress
 * Technique: Square String Traverse from row major
 *            to column major by skipping.
 */

#include<stdio.h>
#include<string.h>
#include<math.h>


#define N 10002
static char s[N];
static char output[N];


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    int n;
    scanf("%d", &n);
    getchar();


    while( n-- ){
        gets(s);


        // Get the length of string and length of each string group.
        int len = strlen(s);
        int rc = sqrt(len);


        // Reset in case there are characters from previous iteration.
        memset(output, 0, sizeof output);



        // If the string length can be divided into equal parts
        // then traverse by skipping certain length.
        if( rc * rc == len ){

            int k = 0;

            for( int j = 0; j < rc; ++j ){
                for( int i = j; i < len; i = i + rc ){
                    output[k++] = s[i];
                }
            }

            puts(output);

        }
        else{
            puts("INVALID");
        }

    }


    return 0;
}

UVA Problem 913 – Joana and the Odd Numbers Solution

UVA Problem 913 – Joana and the Odd Numbers Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

Todo:
1. Reorganize.
2. check for mistakes.

This problem can be solved in quite a few ways. Brute force won’t work. I have given the brute force code anyway. It will give correct answer from 1 < n < 199999999 but the requirement is 1 < N < 1000000000, which is way way larger.

Another approach is to do mathematical calculation to come up with a formula to solve this. Multiple formula can be generated such as a single formula or, one formula complementing other or, multiple formula's combined. I have provided two formula in the codes below.


One thing to mention, the output or input must be long long data type. Other data types will give incorrect output.

Unfortunately I won’t be elaborate more than this. It took 5 pages to solve this many way with proof, but that’s a lot to write. I will try to explain a way for coming up with these formula.


The brute force way:

A simple way to do this is, for values greater than 1 set offset 10 and start multiplying with 5. For each step increase offset by 4 and add value to previous multiplying value.

Input -> Sum

3 -> 3 * 5
5 -> 3 * 15, [ 15 = 5 + 10 ]
7 -> 3 * 29, [ 29 = 15 + 14 ]
9 -> 3 * 47, [ 47 = 29 + 18 ]
11 -> 3 * 69, [ 69 = 47 + 22 ]
13 -> 3 * 95, [ 95 = 69 + 26 ]

Although this approach will not work since the range is huge. It is possible to pre-calculate for a certain range and anything outside that calculate with formula.


Trying to Come up with a Closed Form Expression:

First thing is to notice is that the numbers differ by 2 and are odd. So, if the first number is n, then the sum of n and next two numbers is,

n + ( n + 2 ) + ((n + 2) + 2)  = n + n + 2 + n + 4  = 3n + 6

Ex:
If n was 67, then result of 3n + 6 = 207.

The opposite is also true. For example, given sum of three numbers we can find the first number by,
if sum was 141 then,

3n + 6 = 141  => n = 45

Expanding and writing the sequence in format below,

Input – Values in the Row(Last three numbers bracketed) – Sum
1 - [ 1 ] - 1
3 - [ 3 5 7 ] - 15 
5 - 9 11 [ 13 15 17 ] - 45
7 - 19 21 23 25 [ 27 29 31 ] - 87  
9 - 33 35 37 39 41 43 [ 45 47 49 ] - 141
11 - 51 53 55 57 59 61 63 65 [ 67 69 71 ] - 207
13 - 73 75 77 79 81 83 85 87 89 91 [ 93 95 97 ] - 285

This means for a given value there are that many numbers in that row.


Trying to come up with a Brute Force Approach:

This is extra and not required to solve this problem but it will understand creation of the formula.

The first numbers of every row can be generated and memoized in a table sequentially using the formula below,
row – number break down – first number

1 -> 1 -> 1
3 -> 1 + ( 1 * 2 ) -> 3
5 -> 3 + ( 3 * 2 ) -> 9
7 -> 9 + ( 5 * 2 ) -> 19
9 -> 19 + ( 7 * 2 ) -> 33

Let n1 be equal to 1 and n2, n3 …, nk be next generated sequences. So the formula is,
n1 = n1
n2 = n1 + (1 * 2)
n3 = n2 + (3 * 2)
n4 = n3 + (5 * 2)
n5 = n4 + (7 * 2)
……………..
……………..
nk = n(k-1) + (i * 2)

this can be converted to an algorithm,
Let, T be table to store value of the first number.
n1 = 1
T[n1] = n1

i = 3

for i to size
T[i] = T[i-2] + (i-2) * 2
increment i by two
End


Generating Solution From the Output:

sum sequence calculation in terms of value
sum sequence calculation in terms of value

Similarly, The sum can also be calculated in terms of input by creating a sequence and finding out the formula.

1 -> 1
3 -> 15
5 -> 45
7 -> 87
9 -> 141
11 -> 207
13 -> 285

when they are written in a sequence,

1 + 15 + 45 + 87 + 141 + 207 + 285 + …………..

So, for 11 the output should be 207. The formula can be calculated from the picture above

207 = 141 + 66
207 = 141 + (54 + 12)
207 = 141 + ((12 * 5) – 6 + 12)
207 = 141 + ((12 * 6) – 6)
207 = 87 + 54 + ((12 * 6) – 6)
207 = 87 + ((12 * 5) – 6) + ((12 * 6) – 6)
207 = 45 + ((12 * 4) – 6) + ((12 * 5) – 6) + ((12 * 6) – 6)
207 = 15 + ((12 * 3) – 6) + ((12 * 4) – 6) + ((12 * 5) – 6) + ((12 * 6) – 6)
207 = 1 + ((12 * 2 – 6) – 4) + ((12 * 3) – 6) + ((12 * 4) – 6) + ((12 * 5) – 6) + ((12 * 6) – 6)
207 = (12 * 2 – 6) + ((12 * 3) – 6) + ((12 * 4) – 6) + ((12 * 5) – 6) + ((12 * 6) – 6) – 3

this can be written as summation formula,

-3 + \sum_{i = 2}^{ceil( \frac{n}{2} )} 12i - 6

this can be rewritten as,
-3 + 6 * \sum_{i = 2}^{ceil( \frac{n}{2} )} 2i - 1

The input for this formula is the input for the program,
for, n = 17 the sum of last three numbers should be 477.
Setting n = 17 in following the formula it will give,

-3 + 6 * ( 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 ) = 477

The formula above doesn’t quite work out for 1. Although we know for 1 output is 1 but it should be included in the formula. Expanding and changing values the summation will get the formula for all values.

Similarly, this is the formula to calculate the first number to sum from a row. It was calculated making a sequence of first summing numbers and find out the the summation formula,

First number,
N = 3 + 2 * \sum_{i = 2}^{floor( \frac{n}{2} )} 2i + 1

Separating the summation,
\sum_{i = 2}^{floor( \frac{n}{2} )} 2i + 1 = 2 * \sum_{i = 2}^{floor( \frac{n}{2} )} i + \sum_{i = 2}^{floor( \frac{n}{2} )} 1

= 2 * ( \sum_{i = 1}^{floor( \frac{n}{2} )} i - \sum_{i = 1}^{ 1 } i ) + \sum_{i = 1}^{floor( \frac{n}{2} )} 1 - \sum_{i = 1}^{ 1 } 1

= 2 * \sum_{i = 1}^{floor( \frac{n}{2} )} i - 2 + \sum_{i = 1}^{floor( \frac{n}{2} )} 1 - 1

= floor( \frac{n}{2} ) ( floor( \frac{n}{2} ) + 1 ) + floor( \frac{n}{2} ) - 3

I won’t simplify anymore.

Put the value of summation in the equation,
N = 3 + 2 * \sum_{i = 2}^{floor( \frac{n}{2} )} 2i + 1
because i only expanded the summation. Multiply result with 2 and add 3 to get the first number.

From, the equation above place the value of first number in 3n + 6 to get the summation of last three numbers for a given number.

The 3n + 6 and the summation formula can be combined into a single formula.


Generating Solution based on First Number of Last Three:

Similarly the first number to sum from every row can be calculated. If the first number from every row are written in a sequence,

1 + 3 + 13 + 27 + 45 + 67 + ………………..


The first number can be generated following this,

summation first number generation for an input
summation first number generation for an input

For input n = 11, task is to generate first number from the graphical representation above. Then the value can be placed in equation, 3N + 6 to generate sum of that value and next two (basically the answer).

67 = 45 + 22
67 = 45 + ( 18 + 4 )
67 = 45 + ( 14 + 4 + 4 )
67 = 45 + ( 10 + 4 + 4 + 4 )
67 = 45 + ( 10 + 4 + 4 + 4 )
67 = 27 + 18 + ( 10 + 4 + 4 + 4 )
67 = 27 + ( 10 + 4 + 4 ) + ( 10 + 4 + 4 + 4 )
67 = 13 + 14 + ( 10 + 4 + 4 ) + ( 10 + 4 + 4 + 4 )
67 = 13 + ( 10 + 4 ) + ( 10 + 4 + 4 ) + ( 10 + 4 + 4 + 4 )
67 = 3 + 10 + ( 10 + 4 ) + ( 10 + 4 + 4 ) + ( 10 + 4 + 4 + 4 )
67 = 3 + 10 + ( 10 + 4 ) + ( 10 + 4 + 4 ) + ( 10 + 4 + 4 + 4 )

this way the formula can be,

First \ number = 3 + \sum_{i = 1}^{ floor( \frac{n}{2} - 1 ) } 10 + \sum_{i = 1}^{ floor( \frac{n}{2} - 2 ) } 4i

First \ number = 3 + 10 * \sum_{i = 1}^{ floor( \frac{n}{2} - 1 ) } 1 + 4 * \sum_{i = 1}^{ floor( \frac{n}{2} - 2 ) } i

First \ number = 3 + 10 * ( floor( \frac{n}{2} ) - 1 ) + 2 * ( floor( \frac{n}{2} ) - 2 ) * ( floor( \frac{n}{2} ) - 1 )

This can be further simplified.

For n = 15 the first number should be 123. Setting n = 15 in equation above,

first number = 3 + 10 * ( 7 – 1 ) + 2 * ( 7 – 2 ) * ( 7 – 1 )
first number = 3 + 60 + 60 = 123

That’s it. There are other approaches but it will take a long time to write.

 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

3
5
7

 


Output:

15
45
87

Formula Generate sequence sum, first number & sum Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 913 Joana and Odd numbers
 * Technique: Generate Sequence Sum, First number
 *            then generate sum of next numbers
 *            including the generated number.
 */

#include<stdio.h>
#include<string.h>


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    long long n;

    while( scanf("%lld", &n) == 1 ){

        long long m = n >> 1;

        long long part = ( (n * (n + 1)) >> 1 ) - ( m * ( m + 1 ) ) - 4;

        long long firstNum = 2 * part + 3;

        printf("%lld\n", 3 * firstNum + 6);

    }


    return 0;
}


Reduced Formula Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 913 Joana and Odd numbers
 * Technique: Reduced formula.
 */

#include<stdio.h>
#include<string.h>


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);

    unsigned long long n;

    while( scanf("%lld", &n) == 1 ){

        long long ans =  3 * ( (n + 1) * ( n - (n >> 1) ) )  - 9;

        printf("%lld\n", ans);
    }


    return 0;
}


Brute Force Code( Won’t be Accepted ):


/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 913 - Joana and the odd numbers
 * Technique: Brute Force. Won't get Accepted.
 *            This is just for demonstration. This
 *            won't get accepted due to huge size that
 *            array can't process. But it will give correct
 *            answer till N.
 */

#include<stdio.h>
#include<string.h>


#define N 100000000
//#define M 1000000000


static long long value[N];


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    value[0] = 1;
    value[1] = 15;
    long long offset = 30;


    for( long long i = 2; i < N; ++i ){
        value[i] = value[i-1] + offset;
        offset = offset + 12;
    }


    long long n;
    while( scanf("%lld", &n) == 1 ){
        printf("%lld\n", value[n/2] );
    }


    return 0;
}

UVA Problem 10106 – Product Solution (Lattice Multiplication)

UVA Problem 10106 – Product Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

This problem is quite simple. Given two very large integers multiply them and print them. By large I mean almost 251 characters long for this problem. So string or character array must be used.

This problem can solved in many ways Big Integer, FFT, Grade School etc. But I have implemented lattice multiplication / Chinese multiplication to solve this problem.


Code Explanation:

The code below requires quite a bit of polish to understand easily ( A lot other post require the same. Maybe I will update this post sometimes in future or not. ).

This code is not quite space efficient. I have used several arrays to make it easy to understand but all of the tasks below can be done using the large array. Also the running time can further be improved to O(m*n) instead of O(n^2) . Where, m and n are length of two arrays and they may not be equal in size.


I have provided some graphical representations below to make it easier to understand.


lattice multiplication example
lattice multiplication example

This is an example of how the multiplication works in this code. As mentioned above the process can be sped up using rectangular matrix instead of square if the length is not equal.


lattice multiplication table fill
lattice multiplication table fill

This is a representation of how the table / square matrix is filled for further processing.

Multiplication starts with the last character of string 1 and proceeds to first character. For string 2 multiplication starts from the first character till last character. This way each character from string 1 and string 2 is first converted to a digit, then they are multiplied. If the result is two digits it is divided by 10.

The remainder is written as the lower left portion ( indicated by lower in the structure ) and the quotient is written as the upper left portion ( indicated by upper in the structure ).


lattice multiplication structure traversal
lattice multiplication structure traversal

This is graphical representation of how the structure is traversed. If you have trouble understanding how the recursive function traversing the structure, then take a look at UVA problem 264 – Count the Cantor Solution. The traversal is somewhat similar to that and I have provided explanation int that post.

Rest is explained in the code comments.

 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

12
12
2
222222222222222222222222

 


Output:

144
444444444444444444444444

Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   Lattice Multiplication.
 * Technique: High precision large number multiplication.
 *            Structure array representing upper left and
 *            and lower right corner in single cell.
 */

#include<stdio.h>
#include<string.h>


#define M 260

int N;


// Assuming both strings are of same length.
struct multiplicationTable{
    int upper;
    int lower;
};
struct multiplicationTable node[M][M];



static char string1[M];
static char string2[M];



// stores the diagonal sums;
int sum;

// decides whether to get the upper or lower
// value based on even or odd.
int m;


// Add diagonals.
int recursiveAdder( int i, int j ){

    // Termination condition.
    if( j < 0 || i >= N )
        return sum;

    int val;


    // Whether to get the upper left corner or
    // the lower right corner.
    if( m % 2 ){
        val = node[i][j].upper;
        j = j - 1;
    }
    else{
        val = node[i][j].lower;
        i = i + 1;
    }
    ++m;


    // store the sum of the whole diagonal.
    sum = sum + val;


    // recursively visit all row ans column
    // diagonally ( at least on pen and paper format ).
    // actually moves more like the snake in the snakes game.
    recursiveAdder(i,j);

    return sum;
}


// Debug.
// Print the matrix showing the multiplications.
// Please note left and right directions may be different.
void printMultiplicationMatrix(){

    printf("\n\n");
    for( int i = 0; i < N; ++i ){
        for( int j = N - 1; j >= 0; --j )
            printf("%d,%d, ", node[i][j].upper, node[i][j].lower );
        printf("\n");
    }

}







int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    while( gets(string1) ){
        gets(string2);

        int len1 = strlen(string1);
        int len2 = strlen(string2);

        // Fix length if both string are not equal size.
        // Adding leading zeros to smaller string.
        if( len1 > len2 ){
            N = len1;

            int shiftWidth = len1 - len2;

            for( int i = len1 - 1; i >= 0; --i )
                string2[i + shiftWidth] = string2[i];

            for(int j = 0; j < shiftWidth; ++j)
                string2[j] = '0';

        }
        else if(len2 > len1){
            N = len2;

            int shiftWidth = len2 - len1;

            for( int i = len2 - 1; i >= 0; --i )
                string1[i + shiftWidth] = string1[i];

            for(int j = 0; j < shiftWidth; ++j)
                string1[j] = '0';

        }
        else N = len1;


        //printf("%s \n%s \n", string1, string2);


        int k = N - 1;


        // Multiply the numbers digit by digit and set in the lattice.
        for( int i = 0; string2[i]; ++i ){
            for( int j = 0; string2[j]; ++j ){

                int num1 = string1[k] - '0';
                int num2 = string2[j] - '0';

                int multiply = num1 * num2;

                node[j][k].upper = multiply / 10;
                node[j][k].lower = multiply % 10;

            }
            --k;
        }

        //printMultiplicationMatrix();


        // Lattice is divided into two parts upper left half and
        // lower right half.


        // result of upper half
        int upperHalfResult[N];

        // Add upper half
        int i = N - 1;
        for(; i >= 0; --i){
            sum = 0;
            m = 1;
            upperHalfResult[i] = recursiveAdder(0, i);
        }


        // result of upper half
        int lowerHalfResult[N];

        // Add upper half
        i = 0;
        for(; i < N; ++i){
            sum = 0;
            m = 0;
            lowerHalfResult[i] = recursiveAdder(i, N - 1);
        }



        // Combine upper and lower left half to a single array to fix addition
        // problems.
        int combinedRawResult[N + N];
        i = 0;
        for(; i < N; ++i )
            combinedRawResult[i] = upperHalfResult[i];
        for(k = 0; i < N + N; ++i, ++k )
            combinedRawResult[i] = lowerHalfResult[k];



        // If a cell has more than 9 then it should be added to the next cell.
        for( int i = N + N - 1; i >= 0; --i ){

            if( combinedRawResult[i] > 9 ){
                combinedRawResult[i - 1] = combinedRawResult[i - 1] + combinedRawResult[i] / 10;
                combinedRawResult[i] = combinedRawResult[i] % 10;
            }

        }


        // Discard leading zeros.
        for(i = 0; i < N + N; ++i)
            if(combinedRawResult[i]) break;


        // print if the result can be printed or its zero.
        bool zero = true;
        for(; i < N + N; ++i){
            printf("%d", combinedRawResult[i]);
            zero = false;
        }

        // If the result is zero.
        if( zero )
            printf("0");

        printf("\n");



    }

    return 0;
}

UVA Problem 11988 – Broken Keyboard (a.k.a. Beiju Text) Solution

UVA Problem 11988 – Broken Keyboard (a.k.a. Beiju Text) Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

The input may be classified into three things. One is append to beginning represented with ‘[‘, another is append to end represented with ‘]’ and the other types any other character except above mentioned. The other character are appended to the next position of last inserted character.

So task is move all characters following ‘[‘ to beginning until ‘]’ is found or end of string is reached. Similar logic applies to ‘]’ character. Also it is a recursive definition, keep applying this logic for all the following third brackets.


Other ideas:

Some other ideas to solve this problem can be, formatting the input to remove useless brackets then create a tree from inputs and perform in-order traversal to get the result. Another idea is create a linked list of strings.


Code Explanation:

I may later ( takes quite bit of time to create graphics ) provide a graphical representation of inserting in the doubly linked list ( 2nd code ). Meanwhile I provided a commented version ( 2nd code ) to make it a little easier to understand.

Here I have provided two codes. Both are linked list implementations. First one is singly linked list with no tail pointer( rather inefficient although I try to reduce some complexity by Boolean flag checking to reduce traversing the list to find tail pointer ). It is not able to keep track of tail pointer efficiently. Need to do some traversing ( some times almost the whole linked list ) to find the tail pointer.

The second implementation is doubly linked list with tail node as the tail pointer. The tail pointer has character has 0 which is ASCII decimal code for NULL character. I use this to stop printing otherwise a space character is printed at the end of output.

 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

This_is_a_[Beiju]_text
[[]][][]Happy_Birthday_to_Tsinghua_University

 


Output:

BeijuThis_is_a__text
Happy_Birthday_to_Tsinghua_University

Code Singly Linked list Without Tail:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 11988 - broken keyboard
 * Technique: Singly Linked List with no
 *            tail Pointer(slow).
 */

#include<stdio.h>
#include<string.h>


#define N 100000

static char input[N];


struct vertex{
    char c;
    struct vertex* next;
};

typedef struct vertex node;


node* insertNode(node* n, char c){

    node* temp = new node();
    temp->c = c;
    temp->next = n->next;
    n->next = temp;

    return temp;
}


void printList( node* dummy ){

    node* tmp = dummy;

    while( tmp = tmp->next )
        putchar( tmp->c );

    printf("\n");
}


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    while( gets(input) ){

        node* dummy = new node();
        dummy->next = NULL;
        node* cur = dummy;


        node* tail = dummy;

        bool rightopen = false;

        for(int i = 0; input[i]; ++i){

            if( input[i] == '[' ){
                cur = dummy;
                rightopen = true;
                continue;
            }
            if( input[i] == ']' ){
                rightopen = false;

                node *tmp = tail;

                while( tmp->next != NULL )
                    tmp = tmp->next;

                tail = cur = tmp;
                continue;
            }

            cur = insertNode(cur, input[i] );
            if( !rightopen )
                tail = cur;
        }

        printList( dummy );

    }


    return 0;
}

Code Doubly Linked list With Tail Pointer:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 11988 - broken keyboard
 * Technique: Doubly Linked List with
 *            dummy tail node as Pointer.
 */

#include<stdio.h>
#include<string.h>


#define N 100000


static char input[N];



// Doubly linked list.
struct vertex{
    char c;
    struct vertex* next;
    struct vertex* prev;
};
typedef struct vertex node;



node* insertNode(node* n, char c){

    // Create the new node to hold current character.
    node* temp = new node();
    temp->c = c;


    // Create forward connection from this created node
    // to the next node of passed in node (n).
    // Also create backward connection from the passed
    // in node to current node.
    temp->next = n->next;
    n->next->prev = temp;


    // Create forward connection from passed in node (n) to the
    // newly created node (temp).
    // Also create backward connection from created node to the
    // passed in node.
    n->next = temp;
    temp->prev = n;


    // return the pointer to the current node.
    return temp;
}



void printList( node* dummy ){

    // Dummy (head) is not a part of output.
    node* tmp = dummy->next;


    // Since I use tail pointer, check if its tail.
    while( tmp->c ){
        putchar( tmp->c );
        tmp = tmp->next;
    }

    printf("\n");
}



int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    while( gets(input) ){

        // Dummy node is just to keep track of beginning.
        // It does not contain any input character.
        // Tail is used to insert before the last node.
        // Tail is also another dummy node keeps track of
        // the end.
        node* dummy = new node();
        node* tail = new node();


        // NULL may not be necessary since I use tail character
        // checking to stop printing.
        tail->prev = dummy;
        dummy->next = tail;
        tail->next = NULL;
        dummy->prev = NULL;


        // Set current node to dummy (beginning). Set tail
        // to NULL character equivalent ASCII value.
        node* cur = dummy;
        tail->c = 0;


        for(int i = 0; input[i]; ++i){

            // Update node pointer to beginning based on bracket.
            // If not bracket use previous pointer location.
            if( input[i] == '[' ){
                cur = dummy;
                continue;
            }
            if( input[i] == ']' ){
                cur = tail->prev;
                continue;
            }

            cur = insertNode(cur, input[i] );
        }


        // Print output characters one by one.
        printList( dummy );

    }


    return 0;
}

UVA Problem 12646 – Zero or One Solution

UVA Problem 12646 – Zero or One Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

The input is all possible combination of 3 bits. If A, B, C are thought as bits then possible binary combination are,

000 \rightarrow * \\  001 \rightarrow C \\  010 \rightarrow B \\  011 \rightarrow A \\  ---- \\  100 \rightarrow A \\  101 \rightarrow B \\  110 \rightarrow C \\  111\rightarrow * \\

This problem can be solved in multiple ways such as using string, checking equality, performing xor equality etc. My code below is exactly same as equality checking but it perform xor operation between them.

XOR Table,

\begin{tabular}{l*{6}{c}r}  X & Y & X XOR Y \\ \hline  0 & 0 & 0 \\  0 & 1 & 1 \\  1 & 0 & 1 \\  1 & 1 & 0 \\  \end{tabular}
 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

1 1 0
0 0 0
1 0 0

 


Output:

C
*
A

Code:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 12646 - zero or one
 * Technique: XOR Bits to check equality.
 */

#include<stdio.h>


int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    int A, B, C;
    while( scanf("%d%d%d", &A, &B, &C) == 3 ){

        if( !( A ^ B ) && !( A ^ C ) )
            putchar('*');
        else{
            if( A ^ B ){
                if( B ^ C )
                    putchar('B');
                else
                    putchar('A');
            }
            else
                putchar('C');
        }
        putchar('\n');
    }


    return 0;
}

UVA Problem 10189 – Minesweeper Solution

UVA Problem 10189 – Minesweeper Solution:


Click here to go to this problem in uva Online Judge.

Solving Technique:

Given a mine field that is a matrix / 2D array, produce an output that contains count of adjacent mines for each squares.

In a 2D array for each squares there are at most adjacent 8 squares. If the current position is i and j then,

\begin{bmatrix}  (i-1,j-1) & (i-1,j) & (i-1,j+1) \\  (i+0,j-1) &  (i,j)  & (i+0,j+1) \\  (i+1,j-1) & (i-1,j) & (i+1,j+1) \\   \end{bmatrix}

Just traverse the matrix row-column wise and check its adjacent squares for getting mine count for current position. The adjacent squares check can be implemented with 8 if conditions for each one.

Another technique is to store the co-ordinates of adjacent squares and using a for loop to check them. The illustration below shows how the 2nd code is implemented using arrays and for loops,
uva 10189 matrix adjacent squares check with saved co-ordinates and for loop
uva 10189 matrix adjacent squares check with saved co-ordinates and for loop

Here the arrow from the center shows where checking starts. The 1D arrays drow and dcol hold the row and column values the way shown above. It can be changed by modifying drow and dcol arrays.

 

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

4 4
*...
....
.*..
....
3 5
**...
.....
.*...
0 0

 


Output:

Field #1:
*100
2210
1*10
1110

Field #2:
**100
33200
1*100

Code Using Multiple if Conditions:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 10189 - Minesweeper
 * Technique: 2D Array / Matrix Boundary checking using
 *            if conditions.
 */

#include<stdio.h>
#include<string.h>


#define MAXSIZE 101


static char MineField[MAXSIZE][MAXSIZE];



int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    int n, m;

    int FieldNumber = 0;

    while( scanf("%d%d", &n, &m), n ){

        getchar();

        for(int i = 0; i < n; ++i)
            scanf("%s", &MineField[i]);


        if( FieldNumber )
            printf("\n");


        for(int i = 0; i < n; ++i){
            for(int j = 0; j < m; ++j){

                if( MineField[i][j] == '*' )
                    continue;

                int temp = 0;

                if( i + 1 < n && MineField[i + 1][j] == '*' )
                    ++temp;
                if( i + 1 < n && j + 1 < m && MineField[i + 1][j + 1] == '*' )
                    ++temp;
                if( j + 1 < m && MineField[i][j + 1] == '*' )
                    ++temp;
                if( i - 1 >= 0 && j + 1 < m && MineField[i - 1][j + 1] == '*' )
                    ++temp;
                if( i - 1 >= 0 && MineField[i - 1][j] == '*' )
                    ++temp;
                if( i - 1 >= 0 && j - 1 >= 0 && MineField[i - 1][j - 1] == '*' )
                    ++temp;
                if( j - 1 >= 0 && MineField[i][j - 1] == '*' )
                    ++temp;
                if( i + 1 < n && j - 1 >= 0 && MineField[i + 1][j - 1] == '*' )
                    ++temp;


            MineField[i][j] = temp + '0';

            }
        }


        printf("Field #%d:\n", ++FieldNumber);


       for(int i = 0; i < n; ++i){
            for(int j = 0; j < m; ++j)
                putchar(MineField[i][j]);
            printf("\n");
       }

    }

    return 0;
}

Code Bound checking using Array & for Loop:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 10189 - Minesweeper
 * Technique: 2D Array / Matrix Boundary checking using
 *            co-ordinate array and for loop.
 */

#include<stdio.h>
#include<string.h>


#define MAXSIZE 101


static char MineField[MAXSIZE][MAXSIZE];


// Co-ordinates / directions of adjacent 8 squares.
// W, SW, S, SE, E, NE, N, NW
static const int drow[] = {0, 1, 1, 1, 0, -1, -1, -1};
static const int dcol[] = {-1, -1, 0, 1, 1, 1, 0, -1};



int main(){

    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);


    int n, m;

    int FieldNumber = 0;

    while( scanf("%d%d", &n, &m), n ){

        getchar();

        for(int i = 0; i < n; ++i)
            scanf("%s", &MineField[i]);


        if( FieldNumber )
            printf("\n");


        for(int i = 0; i < n; ++i){
            for(int j = 0; j < m; ++j){

                int temp = 0;

                // If mine found do nothing.
                if( MineField[i][j] == '*' )
                    continue;


                // For each adjacent squares of the current square calculate mine count.
                // and set the count in current square.
                for(int k = 0; k < 8; ++k){

                    // Check if out of bound of the 2D array or matrix.
                    if( i + drow[k] < 0 || j + dcol[k] < 0 || i + drow[k] >= n || j + dcol[k] >= m )
                        continue;

                    // Check the appropriate co-ordinate for mine, if mine found increase count.
                    if( MineField[i + drow[k] ][j + dcol[k]] == '*' )
                        ++temp;

                }

                // All adjacent squares checked set the mine count for current squares.
                MineField[i][j] = temp + '0';

            }
        }


        printf("Field #%d:\n", ++FieldNumber);


       for(int i = 0; i < n; ++i){
            for(int j = 0; j < m; ++j)
                putchar(MineField[i][j]);
            printf("\n");
       }

    }

    return 0;
}

UVA Problem 11565 – Simple Equations Solution

UVA Problem 11565 – Simple Equations Solution:


Click here to go to this problem in uva Online Judge.

UPDATE:

As a commenter below pointed out the solution is wrong due to the fact I made some wrong assumptions. Kindly check other sources for correct solution. This was accepted due to weak test cases. Also check udebug for different test cases.

Problem Statement Description:

Given input of A, B, C where 0 \leqslant A, B, C \leqslant 10000 and three equations,

\mathbf{ x + y + z = A }
\mathbf{ x.y.z = B }
\mathbf{ x^2 + y^2 + z^2 = C }

The task is to find values of x, y, z where x, y, z are three different integers which is ( x != y != z ). Meaning x can not be equal to y, t can not be equal to z and z can not be equal to x.

The problem asks to output the least value of x then y then z meaning the output will be in x < y < z order.


Things to notice:

One very important thing to notice is that for all three equation the values of x, y, z can be interchanged and it will still satisfy the problem.

For example,
\mathbf{ if, x = 5, y = 0, z = 9 }

then it can also be written as,
\mathbf{ x = 0, y = 5, z = 9 }
\mathbf{ x = 9, y = 5, z = 0 }

etc. all possible combinations. But the problem asks for least value of x then y then z.

So if after calculation the result is x = 9, y = 0, z = 5 which is supposed to be x = 0, y = 5, z = 9. In that case the values can be sorted which will result in x = 0, y = 5, z = 9 and this still satisfies the equations.


Solution Technique Explanation:

The simple approach to solve this problem is try all possible combinations of x, y, z and see if it produces a correct result.

The values of x, y, z being negative also satisfies this problem. Forgetting the value of negatives for the moment and focusing on the positive values. To get the naive loop count limit see x or y or z can be at least 0 and at most 100.

Same goes for the negative portion so looping for -100 to 100 for each 3 nested loops seem easy ( I haven’t given the 3 nested loop solution here. It can be found on other sites ). So from -100 to 0 to 100 there are 201 number in between. So naive solution requires 201*201*201 which is more than 8 million (8120601). This program should do 117 * 45 = 5265 iterations per test case.


This problem can be solve much faster using some algebraic substitution. Notice,

\mathbf{ x + y + z = A ........ (i) \\ }
\mathbf{ x.y.z = B ........ (ii) \\ }
\mathbf{ x^2 + y^2 + z^2 = C ........ (iii) \\ }

from equation (ii),

\mathbf{ x.y.z = B }

This can be rewritten as,

\mathbf{ z = \frac{B}{x.y} ........ (iv) }

from equation (i),

\mathbf{ x + y + z = A }

substituting value of z from (iv) to (i) gives,

\mathbf{ x + y + \frac{B}{x.y} = A }

This can be rewritten as,

\mathbf{ \frac{B}{x.y} = A - x - y ........ (v) }

similarly from (iii) and (iv),

\mathbf{ x^2 + y^2 + z^2 = C }

plugging in the value of z and solving for x and y,

\mathbf{ x^2 + y^2 + ( \frac{B}{x.y} )^2 = C ........ (vi) }

Now from (v) and (vi),
\mathbf{ x^2 + y^2 + (A - x - y)^2 = C }

Notice this equation above does not contain z. Now z can be calculated in terms of B, x, y.

This will completely remove one of the nested loops. Further improvements can be made by cutting down number of loop iterations.


Important point to notice that A, B, C can be at most 10000. Also x, y, z can differ by 1 and satisfy the equation. From above equations,

\mathbf{ x + y + z = A ........ (i) }
\mathbf{ x.y.z = B ........ (ii) }
\mathbf{ x^2 + y^2 + z^2 = C ........ (iii) }

So if x, y, z differ by 1 then,

\mathbf{ y = x + 1 }
\mathbf{ z = x + 2 }

from equation (i),

\mathbf{ x + (x + 1) + (x + 2) = A }

but A can be at most 10000.

\mathbf{ x + (x + 1) + (x + 2) = 10000 }
\mathbf{ 3x + 3 = 10000 }

So, x = 3332.3333….

But to keep it simple, notice the lower order term 3 does not make much difference. By setting x = y = z the equation can be,

\mathbf{ x + x + x = 10000 }
\mathbf{ 3x = 10000 }
\mathbf{ x = 3333.3333.... }


from equation (iii) since x is squared so it to get its iteration just calculate the square root. or, setting x = y = z and B = 10000 in equation (iii) gives,

\mathbf{ x^2 + x^2 + x^2 = 10000 }
\mathbf{ 3x^2 = 10000 }
\mathbf{ x = 57.735.... }

after rounding of,
x = 58

So the loop range for x is [-58…58].


Again from equation (ii) the iteration limit for y can be counted using x = y = z,

\mathbf{ y.y.y = 10000 }
\mathbf{ y^3 = 10000 }
\mathbf{ y = 21.544.... }

Rounded of to,
y = 22

So iteration limit for y will be from [-22…22].

Calculating using y + 1 result in rounded range [-21…21]. Although I have not tested it.


Doing further algebraic simplification will remove another loop as z can be calculated in terms of A,B,C using,

\mathbf{ (A-z)^2 - 2.\frac{B}{z} = C - z^2 }

where,

\mathbf{ xy = \frac{B}{z} }

and,

\mathbf{ x + y = A - z }

using the main equation above, the value of z can be found and by solving the other equations the values of x and y can be calculated.


Also the loops can be input dependent. If max of A, B, C is bigger than 58 than choose 58 other wise choose the max as the loop counter.

There may be more improvements possible but that’s all I can come up with now. One final improvement can be made by breaking from all loops at once once the values are found. Although goto isn’t recommended c++ doesn’t support label to break out of multiple loops unlike java.

Important:  Be sure to add or print a new line after each output unless otherwise specified. The outputs should match exactly because sometimes even a space character causes the answer to be marked as wrong answer. Please compile with c++ compiler as some of my codes are in c and some in c++.


More Inputs of This Problem on uDebug.


Input:

2
1 2 3
6 6 14

Output:

No solution.
1 2 3

Optimized Code without goto:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 11565 - Simple Equations
 * Technique: Mathematical elimination and substitution
 */

#include<cstdio>
#include<algorithm>
#include<iostream>

int main(){
    std::ios_base::sync_with_stdio(false);
    std::cin.tie(NULL);

    int n;
    scanf("%d", &n);

    int A, B, C;

    while( n-- ){
        scanf("%d%d%d", &A, &B, &C);

        bool solutionFound = false;
        int x, y, z;

        for( x = -58; x <= 58; ++x ){
            for( y = -22; y <= 22; ++y ){

                if( x != y && ( (x * x + y * y) + (A - x - y) * (A - x - y) == C )  ){

                    int temp = x * y;

                    if( temp == 0 ) continue;

                    z = B / temp;

                    if( z != x && z != y && x + y + z == A   ){
                        if(!solutionFound){
                            int tmpArray[3] = {x, y, z};
                            std::sort(tmpArray, tmpArray + 3);
                            std::cout << tmpArray[0] << " " << tmpArray[1] << " " << tmpArray[2] << "\n";

                            solutionFound = true;
                            break;
                        }
                    }

                }
            }
        }

        if(!solutionFound) std::cout << "No solution." << "\n";

    }

    return 0;
}

Optimized Code without goto:

/**
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   UVA 11565 - Simple Equations
 * Technique: Mathematical elimination and substitution
 */

#include<cstdio>
#include<algorithm>
#include<iostream>

int main(){
    std::ios_base::sync_with_stdio(false);
    std::cin.tie(NULL);

    int n;
    scanf("%d", &n);

    int A, B, C;

    while( n-- ){
        scanf("%d%d%d", &A, &B, &C);

        bool solutionFound = false;
        int x, y, z;

        for( x = -58; x <= 58; ++x ){
            for( y = -22; y <= 22; ++y ){

                if( x != y && ( (x * x + y * y) + (A - x - y) * (A - x - y) == C )  ){

                    int temp = x * y;

                    if( temp == 0 ) continue;

                    z = B / temp;

                    if( z != x && z != y && x + y + z == A   ){
                        if(!solutionFound){
                            int tmpArray[3] = {x, y, z};
                            std::sort(tmpArray, tmpArray + 3);
                            std::cout << tmpArray[0] << " " << tmpArray[1] << " " << tmpArray[2] << "\n";

                            solutionFound = true;
                            goto done;
                        }
                    }

                }
            }
        }

        if(!solutionFound) std::cout << "No solution." << "\n";
        done:;

    }

    return 0;
}

 

Grade School High Precision Floating Point Number Adder Implementation in C++

Warning: This program has not been thoroughly tested. So it may produce incorrect results.

How the Code Works:

Note this problem calculates the integer and fractional portion separately in array as opposed to techniques learned in numerical analysis. To test if outputs are correct check against other high precision calculator such as this calculator.

Similar to this problem or same techniques used in uva problem 424 integer inquiry solution, uva problem 10035 primary arithmetic solution, uva problem 713 adding reversed numbers solution. The technique from this problem can also be applied to the mentioned problems.


Solution Steps:
Example Input:
25.401
534.2914
710.84
9.7808
980.7223

Note here are two separate arrays with one for decimal part and other for fraction. The dot is not stored anywhere, it is just shown to make it easy to understand. First the numbers are aligned with padding functions to create this,

025 . 4010
534 . 2914
710 . 8400
009 . 7808
980 . 7223

Although not used in this problem below, this pic that shows how I calculated sum column-row (column major) wise which is not cache friendly.

uva 424 integer inquiry order of operation
uva 424 integer inquiry order of operation

As shown in the pic above the technique is almost same but this time row and columns are interchanged and kept in a separate array which looks like this,

New Decimal Array:
05709 
23108 
54090 
New Fraction Array:
42877
09482
11002
04083

Next the operation is carried out as shown below,

row wise addition of high precision floating point numbers
row wise addition of high precision floating point numbers

Sample Test Input:

3.6692156156
65652.6459156456415616561
33.54615616
1.1
9854949494968498.49684984444444444444444443213615312613216512331918565

Output:

Output: 9854949495034189.45813726568600610054444443213615312613216512331918565

Code:

/********************************************************************************************
 * Author:    Asif Ahmed
 * Site:      https://quickgrid.wordpress.com
 * Problem:   Naive High Precision Floating Point Number Adder
 *
 * Warning:   This Code is mostly useless and hard to read. It is not represented in
 *            usual sign, exponent sign, exponent magnitude, mantissa magnitude format.
 *            The Program should produce correct output till given FRACTION_LIMIT after
 *            decimal point for unsigned numbers (Negative numbers not supported).
 *
 *            Also it may not necessary produce correct result as it is not throughly
 *            tested. Use it with caution.
 *
 *            Please follow input format. I have not handled any exception for incorrect
 *            input formats.
 *
 *******************************************************************************************/



#include<stdio.h>
#include<string.h>



// Increase the TEST_NUMBERS to calculate more float numbers.
#define TEST_NUMBERS  5

#define DECIMAL_SIZE  100
#define FRACTION_SIZE 100




// Function Prototypes
void sumDecimal(char **, int, int);
void sumFractions(char **, int);


// Array for holding decimal and fractional portion
static char decimal[TEST_NUMBERS][DECIMAL_SIZE];
static char fraction[TEST_NUMBERS][FRACTION_SIZE];


// Pointers reversed decimal and fractional result array.
char *resDec, *resFrac;
int resDecLen, resFracLen;



// Align decimal portion of the numbers and zeros.
void fixPaddingDecimal(int carry){

    // Holds calculated length of all strings.
    static int memo[TEST_NUMBERS];


    // Find out the max length of largest decimal.
    int maxlength = memo[0] = strlen(decimal[0]);


    // Get the max length for padding or aligning.
    for(int i = 1; i < TEST_NUMBERS; ++i){
        int len = memo[i] = strlen(decimal[i]);
        if( len > maxlength )
            maxlength = len;
    }



    // Shift elements by appropriate positions or, create padding.
    // Look at my other linked pages from the post to understand this part.
    for(int i = 0; i < TEST_NUMBERS; ++i){
        int len = memo[i];

        if( len != maxlength ){
            int padding = maxlength - len;

            for(int j = len - 1; j >= 0; --j)
                decimal[i][padding + j] = decimal[i][j];

            for(int j = 0; j < padding; ++j)
                decimal[i][j] = '0';
        }
    }



    // Create a new 2D array that will hold values in the new changed order.
    char **cacheFrindlyFloatOrganization = new char*[maxlength + 1];
    for(int i = 0; i < maxlength; ++i)
        cacheFrindlyFloatOrganization[i] = new char[TEST_NUMBERS + 1];


    // Copy decimal portion to the mew changed order array.
    for(int j = 0; j < maxlength; ++j){
        for(int i = 0; i < TEST_NUMBERS; ++i){
            cacheFrindlyFloatOrganization[j][i] = decimal[i][j];
        }
    }


    // Debug
    /*
    for(int i = 0; i < maxlength; ++i){
        for(int j = 0; j < TEST_NUMBERS; ++j){
            printf("%c ", cacheFrindlyFloatOrganization[i][j]);
        }
        printf("\n");
    }
    */


    // After fixing alignment add the decimal portion.
    sumDecimal( cacheFrindlyFloatOrganization, maxlength, carry);

}



// Add all the decimal numbers.
void sumDecimal(char **cacheFrindlyFloatOrganization, int maxlength, int carry){

    // Holds the output decimal part in reversed order.
    char *resultDecimal = new char[maxlength + 1];

    int z = 0;


    // Again look at the pages linked from this post.
    // I have explained this in uva 424, 10035, 713 etc.
    for( int i = maxlength - 1; i >= 0; --i ){

        int sum = 0;

        for(int j = 0; j < TEST_NUMBERS; ++j)
            sum = sum +  cacheFrindlyFloatOrganization[i][j] - '0';

        sum = sum + carry;
        resultDecimal[z++] = sum % 10 + '0';
        carry = sum / 10;
    }

    // Stored in reversed order
    if(carry)
        resultDecimal[z++] = carry + '0';
    resultDecimal[z] = '\0';


    // Store the address and length of resultDecimal
    resDec = resultDecimal;
    resDecLen = z;


}



// As explained above this ALMOST does the same thing as function decimal code.
void fixPaddingFraction(){

    // Holds calculated length of all strings
    static int memo[TEST_NUMBERS];


    // Find out the max length of largest fraction
    int maxlength = memo[0] = strlen(fraction[0]);

    for(int i = 1; i < TEST_NUMBERS; ++i){
        int len = memo[i] = strlen(fraction[i]);
        if( len > maxlength )
            maxlength = len;
    }


    // Add zeros to empty positions.
    for(int i = 0; i < TEST_NUMBERS; ++i){
        int len = memo[i];

        if( len != maxlength ){
            for(int j = len ; j < maxlength; ++j)
                fraction[i][j] = '0';
            fraction[i][maxlength] = '\0';
        }
    }


    // Explained above.
    char **cacheFrindlyFloatOrganization = new char*[maxlength + 1];
    for(int i = 0; i < maxlength; ++i)
        cacheFrindlyFloatOrganization[i] = new char[TEST_NUMBERS + 1];


    // Interchanging row and columns
    for(int j = 0; j < maxlength; ++j){
        for(int i = 0; i < TEST_NUMBERS; ++i){
            cacheFrindlyFloatOrganization[j][i] = fraction[i][j];
        }
    }


    // Debug
    /*
    for(int i = 0; i < maxlength; ++i){
        for(int j = 0; j < TEST_NUMBERS; ++j){
            printf("%c ", cacheFrindlyFloatOrganization[i][j]);
        }
        printf("\n");
    }
    */

    // Sum the fractional array part.
    sumFractions(cacheFrindlyFloatOrganization, maxlength);
}




// Does ALMOST the same thing for decimal function code.
void sumFractions(char **cacheFrindlyFloatOrganization, int maxlength){

    char *resultFraction = new char[maxlength + 1];

    int z = 0, carry = 0;

    for( int i = maxlength - 1; i >= 0; --i ){

        int sum = 0;

        for(int j = 0; j < TEST_NUMBERS; ++j)
            sum = sum +  cacheFrindlyFloatOrganization[i][j] - '0';

        sum = sum + carry;
        resultFraction[z++] = sum % 10 + '0';
        carry = sum / 10;
    }


    // Stored in reversed order
    resultFraction[z] = '\0';


    // Hold the address and length resultFraction array.
    resFrac = resultFraction;
    resFracLen = z;


    // Call the padding function for by sending the carry from fraction part.
    fixPaddingDecimal(carry);
}



// The output decimal and fraction array are in reversed order.
// They need to be reversed before outputting.
void reverseString(char *tmpString, int tmpStringLen){

    for(int i = 0, j = tmpStringLen - 1; i < j; ++i, --j ){
        int temp = tmpString[i];
        tmpString[i] = tmpString[j];
        tmpString[j] = temp;
    }
}



int main(){


    // Comment freopen lines below to input and output from console.
    // In order to use freopen create a file named input_file.txt
    // in same directory as your cpp file. The output of the program
    // will be in output_file.txt file.

    /*
    freopen("input_file.txt", "r", stdin);
    freopen("output_file.txt", "w", stdout);
    */



    for(int i = 0; i < TEST_NUMBERS; ++i){
        //printf("Enter float:\n");
        scanf("%[0-9].%[0-9]", &decimal[i], &fraction[i]);
        getchar();
    }


    // This is where it all starts.
    fixPaddingFraction();



    reverseString(resDec, resDecLen);
    reverseString(resFrac, resFracLen);


    // Debug
    printf("Output: %s.%s\n", resDec, resFrac );



    return 0;
}

Digital Logic: Binary to Gray Code Converter

Before starting with this read Digital Logic: Designing Decimal to 4 bit Gray Code Converter if you are unfamiliar.


Binary, Decimal and Gray Code Table:

Here I am representing gray code bits with A, Decimal values with D and Binary bits with B.

Binary Decimal Gray Code
B3 B2 B1 B0 D A3 A2 A1 A0
0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 1
0 0 1 0 2 0 0 1 1
0 0 1 1 3 0 0 1 0
0 1 0 0 4 0 1 1 0
0 1 0 1 5 0 1 1 1
0 1 1 0  6 0 1 0 1
0 1 1 1 7 0 1 0 0
1 0 0 0 8 1 1 0 0
1 0 0 1 9 1 1 0 1
1 0 1 0 10 1 1 1 1
1 0 1 1 11 1 1 1 0
1 1 0 0 12 1 0 1 0
1 1 0 1 13 1 0 1 1
1 1 1 0 14 1 0 0 1
1 1 1 1 15 1 0 0 0
Binary to Gray Code Converter:

This same technique can be applied to make gray to binary converter. There will be 4 input bits, which represent binary and 4 output bits which represent equivalent gray code.

Since we are creating binary to gray code converter so, we need to find expressions for each gray code output in terms of input binary bits.

So, there will be four output bits A_3, A_2, A_1, A_0 . For these output bits the input will be different combinations of B_3, B_2, B_1, B_0 based on minimized expression.


Karnaugh Maps:

A3:

4 bit binary to gray code A3 expression
4 bit binary to gray code A3 expression

Minimized expression from the above k map,
A_3 = B_3


A2:

4 bit binary to gray code A2 k map
4 bit binary to gray code A2 k map

Minimized expression from the above k map,
A_2 = B_3 \overline{B_2} + B_2 \overline{B_3}

But, the expression of XOR Gate is (Let, A and B be the inputs),
A.\overline{B} + B.\overline{A} = A \oplus B
[Note: This is not related to this problem but, only showing how XOR can be made from And, Or, Not gates]

Similarly,it can be shown that,
B_3 \overline{B_2} + B_2 \overline{B_3} = B_2 \oplus B_3

If XOR gate is not available then the former expression can be used with and, or and not gates to design the gray code converter.


A1:

4 bit binary to gray code A1 k map
4 bit binary to gray code A1 k map

Minimized expression from the above k map,
A_1 = B_1 \overline{B_2} + B_2 \overline{B_1}

Similarly,it can be shown that,
B_1 \overline{B_2} + B_2 \overline{B_1} = B_1 \oplus B_2


A0:

4 bit binary to gray code A0 k map
4 bit binary to gray code A0 k map

Minimized expression from the above k map,
A_0 = B_0 \overline{B_1} + B_1 \overline{B_0}

Similarly,it can be shown that,
B_0 \overline{B_1} + B_1 \overline{B_0} = B_0 \oplus B_1


Circuit Diagram using XOR gates:

Download the logisim circ file from github.

4 bit binary to Gray Code Converter using XOR gates
4 bit binary to Gray Code Converter using XOR gates

Circuit Diagram without using XOR gates:

Download the logisim circ file from github.

4 bit binary to Gray Code Converter without using XOR gates
4 bit binary to Gray Code Converter without using XOR gates