This document summarizes the contents of the CS110 lecture on modeling computation. It discusses solving quadratic equations, including calculating discriminants and square roots. It provides sample code for solving quadratic equations and multiplying polynomials. It also presents an algorithm for designing a communication pattern such that all people can share information in log2n steps.

1. CS110: Models of Computing
Lecture 6
V. Kamakoti
14th January 2008

2. Today’s Lecture Contents
• Solving Quadratic Equations
• Given ax2 + bx + c = 0
• The roots are
• (-b + SQRT(b2 - 4ac))/2a
• (-b - SQRT(b2 - 4ac))/2a
• New things
•
•
•
•
Check if ‘a’ is 0
Check if ‘b’ is 0
Check if discriminant is < 0
Calculate SQRT

3. Calculate SQRT
• Input a double precision float and get a
double precision float as output
• The function “sqrt” is in <math.h> like how
“printf” is in <stdio.h>
• There is another function “sqrtl” also - find out
what it does?
• Use “Manual pages” in Unix
– Type the command “man sqrt” and you shall get
the details.

4. The Program
#include <stdio.h>
#include <math.h>
main() {
double a,b,c,DISC,R1,R2;
printf(“Enter a ?n”);
scanf(“%f”,&a); //Similarly for b and c

5. The Program
if (a == 0) {
printf(“The equation is not Quadratic n”);
if (b == 0)
printf(“No equation at alln”);
else
printf(“Root is %fn”, -c/b);
}
else

6. Program
{
DISC = b*b - 4*a*c; //Operator Precedence
if (DISC < 0)
{
printf(“No real solution existsn”);
}
else
{
//Paranthesis to handle precedence
R1 = (-b + sqrt(DISC))/(2*a));
R2 = (-b - sqrt(DISC))/(2*a));
//Print the roots
}
}
}

7. Polynomial Multiplication
• Input: Coefficients of two polynomials
P1 and P2 of degree 5
• Output: Coefficients of the polynomial,
which is the product of P1 and P2.
• Do not forget: There are Twelve Inputs
and Eleven Outputs

8. Polynomial multiplication
• (a1x + a0 ) (b1x + b0 )
– a1b1x2 + (a0b1 +a1b0 ) x + a0b0
• The coefficient of xk is sum of aibj’s such that i
+j=k
• In the program you have A0, A1,… A5 and
B0, B1,…, B5 for the coefficients of the input
polynomials and C0, C1, …., C10 for the
coefficients of the output polynomial.

9. For your problem
•
•
•
•
•
•
C0 = A0 * B0;
C1 = A0*B1 + B1*A0
…….
C8 = A3*B5 + A4*B4 + A5*B3
C9 = A4*B5 + B4*A5
C10 = A5 * B5

10. The problem
• Suppose there are n = 2k persons, each with
a certain item of information. In each step,
each person can communicate with another
person and share all the information he or
she knows (including information learned in
previous steps). A person cannot
communicate with more than one person in
any step. Design a communication (gossip)
pattern such that after log2n (= k) steps,
everyone knows everything.

11. Solution
• Numbering problem
– Number “N” people using integers 0 to N-1.
– N = 2k
– Group pairs of two - Number each 0,1
– Group pairs of these and prefix 0 to
numbers in one pair and prefix 1 to
numbers in the other pair.
– Repeat till only one group exist

12. Solution
0
0
0
0
0
0
0
0

13. Solution - Step 1
0
1
0
1
0
Groups of two
1
0
1

14. Solution - Step 2
00
01
10
11
00
Groups of four
01
10
11

15. Solution - Step 3
000
001
010
011
100
Groups of eight
101
110
111

16. Gossip - Step 1
0
1
0
1
0
Groups of two
1
0
1

17. Gossip - Step 2
00
01
10
11
00
Groups of four
01
10
11

18. Gossip - Step 3
000
001
010
011
100
101
110
Groups of eight - k-bits for n = 2k
111

19. Solution at a glance - Step 1
000
001
010
011
100
101
110
111

20. Solution at a glance - Step 2
000
001
010
011
100
101
110
111

21. Solution at a glance - Step 3
000
001
010
011
100
101
110
111

22. Creative Question - 3
• You are given a BLACK BOX - nothing is
known what is inside it - but it does the
following for you
– Give a sequence of n real numbers and an integer
k, the box will output “yes” or “no” indicating
whether there is a subset of the numbers whose
sum is exactly “k”. It will not tell you the subset.
– Use this black box to find the subset whose sum is
“k”, if it exists.
– You are allowed to use the black box exactly “n”
times, where “n” is the length of the sequence.

23. Example
(1.3,.7,2.5,1.7) and
k=3
Yes
The problem is using the above box exactly four times you
Should find out the subset that adds to 3, namely (1.3,1.7)

24. Example
(1.3,.7,2.5,1.7) and
k=3
Yes
All elements distinct
Remove an element and ask if it solves
Yes: element not in subset
No: element is in the subset

25. Example
(1.5,2,1.5,1.5) and
k=5
Yes
All elements not distinct

26. Thank You