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CSE 20
DISCRETE MATH
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CA
Prof. Shachar Lovett
http://cseweb.ucsd.edu/classes/wi15/cse20-a/
Todays topics
• Algorithms: what are they all about?
• Sections 1.1-1.2 in Jenkyns, Stephenson
What’s an algorithm?
A. A step by step process
B. Any way of solving complex problems
C. Computer code
D. I don’t know
An algorithm?
• Definition: a step-by-step process
• Each basic step is simple
• Together they can compute very complicated things
• What properties do we care about?
A. It should have the correct format (“compile”)
B. It should always terminate
C. It should give the correct answer
D. All of the above
Two algorithms
Alg1(real a, pos int n)
Begin
x  1.0;
i  1;
Repeat
x  x * a;
i  i+1;
Until i > n;
Output x
End.
Alg2(real a, pos int n)
Begin
x  1.0;
i  n;
While i>0 Do
If (i is odd) Then
x  x * a;
End;
i  i/2;
If (i > 0) Then
a  a * a;
End;
Output x
End
What happens when we run Alg1 with a = 4 and n = 5? What about Alg2?
Two algorithms
Alg1(real a, pos int n)
Begin
x  1.0;
i  1;
Repeat
x  x * a;
i  i+1;
Until i > n;
Output x
End.
A. Alg1 doesn’t always work.
B. Alg2 doesn’t always work.
C. Alg1 is better than Alg2 because
it’s simpler.
D. Alg2 is better than Alg1 because
it’s faster.
Alg2(real a, pos int n)
Begin
x  1.0;
i  n;
While i>0 Do
If (i is odd) Then
x  x * a;
End;
i  i/2;
If (i > 0) Then
a  a * a;
End;
Output x
End.
Complexity of algorithms
• Simplistic model: each basic operation
(addition, multiplication, etc) takes unit time
• Complexity of an algorithm =
number of steps =
running time
• Depends on the input
Complexity of algorithms
Alg1(real a, pos int n)
Begin
x  1.0;
i  1;
Repeat
x  x * a;
i  i+1;
Until i > n;
Output x
End.
• How many steps does
this algorithm make?
A. Constant number
B. ~ n
C. ~ log(n)
D. ~ a*n
E. Other
Complexity of algorithms
Alg1(real a, pos int n)
Begin
x  1.0;
i  1;
Repeat
x  x * a;
i  i+1;
Until i > n;
Output x
End.
• How many steps does
this algorithm make?
A. Constant number
B. ~ n
C. ~ log(n)
D. ~ a*n
E. Other
Running for n=1000000 takes ~3000000 steps
Complexity of algorithms
• How many steps does
this algorithm make?
A. Constant number
B. ~ n
C. ~ log(n)
D. ~ a*n
E. Other
Alg2(real a, pos int n)
Begin
x  1.0;
i  n;
While i>0 Do
If (i is odd) Then
x  x * a;
End;
i  i/2;
If (i > 0) Then
a  a * a;
End;
Output x
End.
Complexity of algorithms
• How many steps does
this algorithm make?
A. Constant number
B. ~ n
C. ~ log(n)
D. ~ a*n
E. Other
Alg2(real a, pos int n)
Begin
x  1.0;
i  n;
While i>0 Do
If (i is odd) Then
x  x * a;
End;
i  i/2;
If (i > 0) Then
a  a * a;
End;
Output x
End.
Running for n=1000000 takes <= 120 steps
Pseudocode
• Begin, End delimit list of steps.
• Allowed types of steps
• Assignment
• Repeat-loop
• If-Then-Else-End
• While-loop
• Walkthrough / Trace
JS p. 10, 15
Casting out 9s
• Goal: check if a number n is divisible by 9
• Input: positive integer n
• Algorithm:
• While n >= 10:
• Let b1,…,bk be the digits of n in base 10
• n  b1+…+bk
• Return TRUE if n=9, return FALSE otherwise
Does it always terminate?
Does it give the right answer?
What is the complexity of this algorithm?
Notation and terms
JS p. 9
• n DIV d
• n MOD d
• d divides evenly into n
d is a factor (divisor) of n, n is a multiple of d
d | n,
n MOD d =0,
If n is any integer and d is any positive integer, then
there are (unique) integers q and r where
n = d*q + r and 0 <= r < d
Solve the equation
For what integer x do we have
x DIV 5 = 14
and
x MOD 5 = 2 ?
A. No integers satisfy these equations at the same time.
B. Any multiple of 5 works.
C. Any non-multiple of 5 works.
D. Only x = 72 works.
E. None of the above / more than one of the above.
Solve the equation
For what integer x do we have
x MOD 5 = 2 ?
A. -3 (and others)
B. -2 (and others)
C. All and only the multiples of 5
D. All and only the non-multiples of 5
E. None of the above / more than one of the above.
Why do we care?
Round robin scheduling, secret sharing, parallel computation
Notation and terms
JS p. 18
• Common divisor of two integers
• GCD (x,y)
Useful Facts
• Lower bound
• Upper bound
• If y | x then …
• Otherwise …
GCD (x,y) >= 1 … even for negative x and y!
GCD (x,y) <= MIN(|x|,|y|)
How can we compute GCD?
• GCD(20,30) =
A. 20
B. 30
C. 5
D. 10
E. Other
How can we compute GCD?
• One way: factoring to prime factors
• Example: GCD(20, 30)
• Factor 20 to prime factors: 20 = 2 * 2 * 5
• Factor 30 to prime factors: 30 = 2 * 3 * 5
• Common factor: 2 * 5 = 10
How can we compute GCD?
• GCD(42,70) =
A. 7
B. 10
C. 14
D. 28
E. Other
How can we compute GCD?
• GCD(1967343712, 82321118) =
A. 762372922
B. 637432093
C. 343848344
D. 7672322
E. Other
How can we compute GCD?
• GCD(1967343712, 82321118) = …
• Factoring numbers to prime factors is HARD
• We don’t know any efficient way to factor large numbers
• The security of e-commerce relies on this!!!
(RSA encryption)
Euclid’s Algorithm
JS p. 19
for computing GCD
Euclid(int x, int y)
Begin
A  x;
B  y;
R  A MOD B;
While ( R>0 ) Do
A  B;
B  R;
R  A MOD B;
End;
Output B
End.
Why on Earth does this work???
Euclid’s Algorithm
JS p. 19
for computing GCD
Euclid(int x, int y)
Begin
A  x;
B  y;
R  A MOD B;
While ( R>0 ) Do
A  B;
B  R;
R  A MOD B;
End;
Output B
End.
Why on Earth does this work???
Key fact:
GCD(A,B) = GCD(B, A MOD B)
Euclid’s Algorithm
JS p. 19
for computing GCD
Euclid(int x, int y)
Begin
A  x;
B  y;
R  A MOD B;
While ( R>0 ) Do
A  B;
B  R;
R  A MOD B;
End;
Output B
End.
Why on Earth does this work???
Key fact:
GCD(A,B) = GCD(B, A MOD B)
Correctness:
GCD(A,B) start of loop =
GCD(A,B) end of loop
Termination: numbers get smaller
This week
• Read sections 1.1-1.3 in Jenkyns, Stephenson
• Ask any questions you have about the course,
expectations, requirements either in person or via TED
• Next class … number representations (Section 1.3)