1. Algorithms and Growth Rates
Discrete Math CS 2800
Prof. Bart Selman
Module
Algorithms and Growth Rates

2. The algorithm problem Any legal Specification of input
all legal inputs
and
The algorithm
Specification of
desired output
as a function of
the input
The desired
output

3. Examples of algorithmic problems
Input: A list L, of integers
Output: The sum of the integers on L
Problem 3:
Input: A road map of cities
with distances attached to the
road map, and two designated
cities A and B
Output: A description of the
shortest path between A and B
Problem 2:
Input: Two texts A and B in English
Output: The list of common words
in both texts

4. Instance of an algorithmic problem Size of an instance
An instance of an algorithmic problem is a concrete case of such a problem with specific input. The size of an instance is given by the size of its input.
Examples of instances:
An instance of problem 1:
L= , 5, 26, 8, 170, 79, 1002
Problem 1:
Input: A list L, of integers
Output: The sum of the integers on L
Size of instance 
length of list
We use a “natural”
measure of input size.
Why generally ok?
Strictly speaking we
should count bits.
Size of instance = |L| = 7

5. The size of an instance is given by the size of its input.
Examples of instances
Problem 3:
Input: A road map of cities
with distances attached to the
road map, and two designated
cities A and B
Output: A description of the
shortest path between A and B
Size of instance 
Number of cities and roads
A particular instance: 1 2 3 4 5 6
Size of instance:
6 nodes
9 edges
The size of an instance is given by the size of its input.

6. Algorithm Definition:
An algorithm is a finite set of precise instructions for performing a computation or for solving a problem.
In general we describe algorithms using pseudocode: i.e., a language that is an intermediate step between an English language description of an algorithm and an implementation of this algorithm in a programming language

7. Properties of an Algorithm
Input: an algorithm has input values from a specified set.
Output: for each set of input values an algorithm produces output values from a specified set. The output values are the solution of the problem.
Definiteness: The steps of an algorithm must be defined precisely.
Correctness: An algorithm should produce the correct output values fro each set of input values.
Finiteness: an algorithm should produce the desired output after a finite (but perhaps large) number of steps for any input in the set.
Effectiveness: It must be possible to perform each step of an algorithm exactly and in a finite amount of time.
Generality: the procedure should be applicable for all the problems of the desired from, not just for a particular set of input values.
Distinction between: “problem” and “problem instance”
Quite confusing for folks outside CS. Alg. should work for all instances!

8. Algorithm: Finding the Maximum Element in a Finite Sequence
procedure max(a1,a2,…, an: integers)
max := a1
for i := 2 to n
if max < ai then max := ai
{max is the largest element}

9. Computer Programming computer execution Algorithm Programmer (human)
Compiler
(software)
Algorithm
programming
Program in high-level
language (C, Java, etc)
compilation
Equivalent program in
assembly language
machine code
computer execution

10. Searching Algorithms Searching problems:
the problem of locating an element in an ordered list.
Example: searching for a word in a dictionary.

11. Algorithm: The Linear Search Algorithm
procedure linear search( x: integer, a1,a2,…, an: distinct integers)
i := 1
while (i ≤ n and x  ai )
i := i +1
if i < n then location := i
else location := 0
{location is the subscript of the term that equals x, or is 0 if x is not found}

12. Binary search To search for 19 in the list
First split:
Second split
Third Split
19 is located as 14th item.

13. Algorithm: The Binary Search Algorithm
procedure binary search( x: integer, a1,a2,…, an: increasing integers)
i := 1 {i is left endpoint of search interval}
j := n {j is right endpoint of search interval}
while i < j
begin
m := (i +j)/2
if x > am then i := m + 1
else j := m
end
if x = ai then location := i
else location := 0
{location is the subscript of the term that equals x, or is 0 if x is not found}
This algorithm proceeds by comparing the element to be located to the middle term of the list. The list is then split into two smaller sublists of the same size, or where one of these smaller lists has one fewer term than the other. The search continues by restricting the search to the appropriate sublist based on the comparison of the element to be located and the middle term.

14. Just because we know how to solve a given problem – we have an algorithm - that does not mean that the problem can be solved.
The procedure (algorithm) may be so inefficient that it would not be possible to solve the problem within a useful period of time.
So we would like to have an idea in terms of the “complexity” of our algorithm.

15. Complexity of Algorithms

16. Complexity of Algorithms
The complexity of an algorithm is the number of steps that it takes to transform the input data into the desired output.
Each simple operation (+,-,*,/,=,if,etc) and each memory access corresponds to a step.(*)
The complexity of an algorithm is a function of the size of the input (or size of the instance). We’ll denote the complexity of algorithm A by CA(n), where n is the size of the input.
What does this mean for the complexity of, say, chess?
Complexity: CA(n) = O(1)
Two issues: (1) fixed input size
(2) Memory access just 1 step
So, model/defn. not always “useful”!
(*) This model is a simplification but still valid to give us a good idea of the complexity of algorithms.

17. Example: Insertion Sort
From: Introduction to Algorithms
Cormen et al

18. Different notions of complexity
Worst case complexity of an algorithm A –
the maximum number of computational steps required for the execution of Algorithm A, over all the inputs of the same size, s. It provides an upper bound for an algorithm. The worst that can happen given the most difficult instance – the pessimistic view.
Best case complexity of an algorithm A -the minimum number of computational steps required for the execution of Algorithm A, over all the inputs of the same size, s. The most optimistic view of an algorithm– it tells us the least work a particular algorithm could possibly get away with for some one input of a fixed size – we have the chance to pick the easiest input of a given size.
Linear search: Worst cost? Best cost?

19. Practical difficulty: What is the distribution of “real-world”
Average case complexity of an algorithm A - i.e., the average amount of resources the algorithm consumes assuming some plausible frequency of occurrence of each input.
Figuring out the average cost is much more difficult than figuring out either the worst-cost or best-cost  e.g., we have to assume a given probability distribution for the types of inputs we get.
Practical difficulty: What is the distribution of “real-world”
problem instances?

20. Different notions of complexity
In general this
is the notion
that we use to
characterize the
complexity of
algorithms
We perform
upper bound
analysis on
algorithms.

21. Algorithm “Good Morning”
For I = 1 to n
For J = I+1 to n
ShakeHands(student(I), student(J))
Running time of “Good Morning”
Time = (# of HS) x (time/HS) + some overhead
We want an expression for T(n), running time of “Good Morning” on input of size n.

22. Growth Rates J I Algorithm “Good Morning” For I = 1 to n
For J = I+1 to n
ShakeHands(student(I), student(J)) J
How many handshakes? 1 2 3 4 5 n 1 2 3 4 5 n 1 2 3 4 5 n n2
- n I 2

23. Growth Rates Algorithm “Good Morning” For I = 1 to n For J = I+1 to n
ShakeHands(student(I), student(J))
T(n) = s(n2- n)/2 + t
Where s is time for one HS, and t is time for getting organized.
But do we always characterize the complexity of algorithms with such a detail? What is the most important aspect that we care about?

24. Comparing algorithms wrt complexity
Let us consider two algorithms A1 and A2,
with complexities:
CA1(n) = 0.5 n2
CA2(n) = 5 n
Which one is has larger complexity?

25. CA2(n) = 5 n ≥ CA1(n) = 0.5 n2 for n ≤ 10

26. CA1(n) = 0.5 n2 >CA2(n) = 5 n for n >10
When we look at the complexity of algorithms we think asymptotically –
i.e., we compare two algorithms as the problem sizes tend to infinity!

27. Growth Rates In general we only worry about growth rates because:
Our main objective is to analyze the cost performance of algorithms asymptotically.
(reasonable in part because computers get faster and faster every year.)
Another obstacle to having the exact cost of algorithms is that sometimes the algorithms are quite complicated to analyze.
When analyzing an algorithm we are not that interested in the exact time the algorithm takes to run – often we only want to compare two algorithms for the same problem – the thing that makes one algorithm more desirable than another is its growth rate relative to the other algorithm’s growth rate.

28. Growth of Rates Algorithm analysis is concerned with:
Type of function that describes run time (we ignore constant factors since different machines have different speed/cycle)
Large values of n

29. Growth of functions Important definition: For functions f and g from
Will be applied to running time, so you’ll usually consider T(n)
(>= 0)
Important definition:
For functions f and g from
the set of integers to the set of real numbers we say
f(x) is O(g(x))
to denote
 C,k so that  n>k, |f(n)|  C |g(n)|
Recipe for proving f(n) = O(g(n)): find a constant C and k (called witnesses to the fact that f(x) is O(g(x))) so that the inequality holds.
We say “f(n) is big O of g(n)”
Note: when C and k are found, there are infinitely many pairs of witnesses. Sometimes it is also said f(x) = O(g(x)), even
though this is not a real equality. Also, in general we perform this analysis for runtime, therefore the functions are positive.

30. k
f(x) is O(g(x))

31. x2 + 2x + 1 is O(x2)
C = 4
k = 1
also
C = 3
k = 2

32. Note:
When f(x) is O(g(x)), and h(x) is a function that has larger absolute values than g(x) does for sufficiently large values of x, it follows that f(x) is O(h(x)). In other words, the function g(x) in the relationship can be replace by a function with larger absolute values. This can be seen given that:
|f(x)| ≤ C|g(x)| if x > k
and if (h(x)| > |g(x)| for all x > k, then
|f(x)| ≤ C|h(x)| if x > k
Therefore f(x) is O(h(x))

33. Growth of functions (examples)
f(x) = O(g(x))
iff
 c,k so that  x>k, |f(x)|  Cg(x)|
There’s k
There’s C
3n = O(15n) since  n>0, 3n  1  15n

34. The complexity of A2 is of lower order than that of A1. While A1 grows
quadratically O(n2) A1 only grows linearly O(n).

35. x2 vs. (x2 + x) (x <=20)

36. x2 vs. (x2 + x) (x2 + x) is O(n2) (oh of n-squared)

37. Growth of functions (examples)
f(x) = O(g(x))
iff
 c,k so that  x>k, |f(x)|  C|g(x)|
a) Yes, and I can prove it.
b) Yes, but I can’t prove it.
c) No, x=1/2 implies x2 > x3
d) No, but I can’t prove it.
x2 = O(x3) ?
Yes, since  x> __, x2  x3 1
C = 1
k = 1

38. Growth of functions (examples)
f(x) = O(g(x))
iff
 c,k so that  x>k, |f(x)|  C|g(x)|
1000
1000x2 is O(x2) since  x> __, x2  ____ ·x2
C = 1000
k = 0

39. Growth of functions (examples)
f(x) = O(g(x))
iff
 c,k so that  x>k, |f(x)|  C|g(x)|
Prove that x x = O((1/100)x2)
100x  100x2
x x  201x2 when x > 1
100  100x2
 20100·(1/100)x2
k = 1,
C = 20100

40. Growth of functions (examples)
Similar problem, different technique.
Prove that 5x = O(x/2)
Try c = 10
Need  x> ___, 5x  ___ · x/2
 x> ___, 5x  10 · x/2
Nothing works for k
Try c = 11
 x> ___, 5x  5x + x/2
200
 x> __ _, 100  x/2
k = 200,
c = 11

41. Theorem 1
Then

42. (where did we use this?) Proof:
Assume the triangle inequality that states: |x| + |y|  |x + y|
(where did we use this?)

43. Estimating Functions Example1:
Estimate the sum of the first n positive integers

44. Estimating Functions
Example2:
Estimate f(n) = n! and log n!

45. Growth of functions Guidelines:
In general, only the largest term in a sum matters.
a0xn + a1xn-1 + … + an-1x1 + anx0 = O(xn)
n dominates lg n.
n5lg n = O(n6)
List of common functions in increasing O() order:
n (n lg n) n2 n3 … 2n n!
Linear time
Exponential time
Constant time
Quadratic time

46. Note: log scale on y axis.

47. Combination of Growth of functions
Theorem:
If f1(x) = O(g1(x)) and f2(x)=O(g2(x)), then f1(x) + f2(x) is O(max{|g1(x)|,|g2(x)|})
Proof: Let h(x) = max{|g1(x)|,|g2(x)|}
Need to find constants c and k so that x>k, |f1(x) + f2(x)|  c |h(x)|
We know |f1(x) | c1| g1(x)|and |f2(x)|  c2 |g2(x)| and using triangle inequality |f1(x) + f2(x)| ≤ |f1(x)| + |f2(x)|
c = c1+c2,
k = max{k1,k2}
And |f1(x)| + |f2(x)|  c1|g1(x)| + c2|g2(x)|
 c1|h(x)| + c2|h(x) |
= (c1+c2)•h(x)

48. Growth of functions – two more theorems
If f1(x) = O(g1(x)) and f2(x)=O(g2(x)), then f1(x)·f2(x) = O(g1(x)·g2(x))
Theorem:
If f1(x) = O(g (x)) and f2(x)=O(g (x)), then (f1+f2)(x) = O(g (x))

49. Growth of functions - two definitions
“g is big-omega of f”
“lower bound”
If f(x) = O(g(x)) then we write g(x) = (f(x)).
What does this mean?
If  c,k so that  x>k, f(x)  c·g(x), then:
 k,c’ so that  x>k, g(x)  c’f(x)
c’ = 1/c
If f(x) = O(g(x)), and f(x) = (g(x)), then f(x) = (x)
“f is big-theta of g”
When we write f=O(g), it is like f  g
When we write f= (g), it is like f  g
When we write f= (g), it is like f = g.

50. Growth of functions - other estimates
“f is little-o of g”
For functions f and g, f = o(g) if
c>0 k so that  n>k, f(n)  c·g(n),
What does this mean?
No matter how tiny c is, cg eventually dominates f.
Example: Show that n2 = o(n2log n)
Proof foreshadowing: find a k (possibly in terms of c) that makes the inequality hold.

51. Growth of functions - other estimates
“f is little-o of g”
For functions f and g, f = o(g) if
c>0 k so that  n>k, f(n)  c·g(n),
Example: Show that n2 = o(n2log n)
Proof foreshadowing: find a k (possibly in terms of c) that makes the inequality hold.
Choose c arbitrarily. How large does n have to be so that n2  c n2log n?
This inequality holds when n > 21/c.
1  c log n
1/c  log n
21/c  n
So, k = 21/c.

52. Growth of functions - other estimates
“f is little-o of g”
For functions f and g, f = o(g) if
c>0 k so that  n>k, f(n)  c·g(n),
Example: Show that 10n2 = o(n3)
Proof foreshadowing: find a k (possibly in terms of c) that makes the inequality hold.
Choose c arbitrarily. How large does n have to be so that 10n2  c n3?
This inequality holds when n > 10/c.
10/c  n
So, k = 10/c.

53. Growth of functions - other estimates
“g is little-omega of f”
For functions f and g, if f = o(g) then g = (f)

54. Growth of functions - other estimates
“g is little-omega of f”
For functions f and g, if f = o(g) then g = (f)
A thought to ponder:
What if f = o(g) and f = (g)?

55. How do computer scientists differentiate between
good (efficient) and bad (not efficient) algorithms?

56. How do computer scientists differentiate between good (efficient) and bad (not efficient) algorithms?
The yardstick is that any algorithm that runs in no more than polynomial time is an efficient algorithm; everything else is not.

57. c lg n lgc n n n2 n3 nc,c≥1 rn, r>1 Efficient algorithms
Ordered functions by their growth rates c
Order
constant 1
logarithmic 2
polylogarithmic 3
nr ,0<r<1 n
sublinear 4
linear 5
nr ,1<r<2
subquadratic 6
quadratic 7
cubic 8
nc,c≥1
rn, r>1
polynomial 9
exponential 10
lg n
lgc n n3 n2
Efficient algorithms
Not efficient algorithms

59. Polynomial vs. exponential growth (Harel 2000)
Binary B&B alg.
exponential
polynomial
LP’s interior point
Min. Cost Flow Algs
Transportation Alg
Assignment Alg
Dijkstra’s alg. N2

60. Growth of functions f(x) = O(g(x)) iff
 c,k so that  x>k, f(x)  c·g(x) k
c·g(x)
f(x)
g(x)
We give an eventual upper bound on f(x) x