Asymptotic analysis

Analysis of Algorithms
CS 477/677
Asymptotic Analysis
Instructor: George Bebis
(Chapter 3, Appendix A)

2
Analysis of Algorithms
• An algorithm is a finite set of precise instructions
for performing a computation or for solving a
problem.
• What is the goal of analysis of algorithms?
– To compare algorithms mainly in terms of running
time but also in terms of other factors (e.g., memory
requirements, programmer's effort etc.)
• What do we mean by running time analysis?
– Determine how running time increases as the size
of the problem increases.

3
Input Size
• Input size (number of elements in the input)
– size of an array
– polynomial degree
– # of elements in a matrix
– # of bits in the binary representation of the input
– vertices and edges in a graph

4
Types of Analysis
• Worst case
– Provides an upper bound on running time
– An absolute guarantee that the algorithm would not run longer,
no matter what the inputs are
• Best case
– Provides a lower bound on running time
– Input is the one for which the algorithm runs the fastest
• Average case
– Provides a prediction about the running time
– Assumes that the input is random
Lower Bound RunningTime Upper Bound≤ ≤

5
How do we compare algorithms?
• We need to define a number of objective
measures.
(1) Compare execution times?
Not good: times are specific to a particular
computer !!
(2) Count the number of statements executed?
Not good: number of statements vary with
the programming language as well as the
style of the individual programmer.

6
Ideal Solution
• Express running time as a function of the
input size n (i.e., f(n)).
• Compare different functions
corresponding to running times.
• Such an analysis is independent of
machine time, programming style, etc.

7
Example
• Associate a "cost" with each statement.
• Find the "total cost“ by finding the total number of times
each statement is executed.
Algorithm 1 Algorithm 2
Cost Cost
arr[0] = 0; c1 for(i=0; i<N; i++) c2
arr[1] = 0; c1 arr[i] = 0; c1
arr[2] = 0; c1
... ...
arr[N-1] = 0; c1
----------- -------------
c1+c1+...+c1 = c1 x N (N+1) x c2 + N x c1 =
(c2 + c1) x N + c2

8
Another Example
• Algorithm 3 Cost
sum = 0; c1
for(i=0; i<N; i++) c2
for(j=0; j<N; j++) c2
sum += arr[i][j]; c3
------------
c1 + c2 x (N+1) + c2 x N x (N+1) + c3 x N2

9
Asymptotic Analysis
• To compare two algorithms with running
times f(n) and g(n), we need a rough
measure that characterizes how fast
each function grows.
• Hint: use rate of growth
• Compare functions in the limit, that is,
asymptotically!
(i.e., for large values of n)

10
Rate of Growth
• Consider the example of buying elephants and
goldfish:
Cost: cost_of_elephants + cost_of_goldfish
Cost ~ cost_of_elephants (approximation)
• The low order terms in a function are relatively
insignificant for large n
n4
+ 100n2
+ 10n + 50 ~ n4
i.e., we say that n4
+ 100n2
+ 10n + 50 and n4
have
the same rate of growth

11
Asymptotic Notation
• O notation: asymptotic “less than”:
– f(n)=O(g(n)) implies: f(n) “≤” g(n)
∀ Ω notation: asymptotic “greater than”:
– f(n)= Ω (g(n)) implies: f(n) “≥” g(n)
∀ Θ notation: asymptotic “equality”:
– f(n)= Θ (g(n)) implies: f(n) “=” g(n)

12
Big-O Notation
• We say fA(n)=30n+8 is order n, or O (n)
It is, at most, roughly proportional to n.
• fB(n)=n2
+1 is order n2
, or O(n2
). It is, at most,
roughly proportional to n2
.
• In general, any O(n2
) function is faster-
growing than any O(n) function.

13
Visualizing Orders of Growth
• On a graph, as
you go to the
right, a faster
growing
function
eventually
becomes
larger...
fA(n)=30n+8
Increasing n →
fB(n)=n2
+1
Valueoffunction→

14
More Examples …
• n4
+ 100n2
+ 10n + 50 is O(n4
)
• 10n3
+ 2n2
is O(n3
)
• n3
- n2
is O(n3
)
• constants
– 10 is O(1)
– 1273 is O(1)

15
Back to Our Example
Algorithm 1 Algorithm 2
Cost Cost
arr[0] = 0; c1 for(i=0; i<N; i++) c2
arr[1] = 0; c1 arr[i] = 0; c1
arr[2] = 0; c1
...
arr[N-1] = 0; c1
----------- -------------
c1+c1+...+c1 = c1 x N (N+1) x c2 + N x c1 =
(c2 + c1) x N + c2
• Both algorithms are of the same order: O(N)

16
Example (cont’d)
Algorithm 3 Cost
sum = 0; c1
for(i=0; i<N; i++) c2
for(j=0; j<N; j++) c2
sum += arr[i][j]; c3
------------
c1 + c2 x (N+1) + c2 x N x (N+1) + c3 x N2
= O(N2
)

17
Asymptotic notations
• O-notation

18
Big-O Visualization
O(g(n)) is the set of
functions with smaller
or same order of
growth as g(n)

19
Examples
– 2n2
= O(n3
):
– n2
= O(n2
):
– 1000n2
+1000n = O(n2
):
– n = O(n2
):
2n2
≤ cn3
⇒ 2 ≤ cn ⇒ c = 1 and n0= 2
n2
≤ cn2
⇒ c ≥ 1 ⇒ c = 1 and n0= 1
1000n2
+1000n ≤ 1000n2
+ n2
=1001n2
⇒ c=1001 and n0 = 1000
n ≤ cn2
⇒ cn ≥ 1 ⇒ c = 1 and n0= 1

20
More Examples
• Show that 30n+8 is O(n).
– Show ∃c,n0: 30n+8 ≤ cn, ∀n>n0 .
• Let c=31, n0=8. Assume n>n0=8. Then
cn = 31n = 30n + n > 30n+8, so 30n+8 < cn.

21
• Note 30n+8 isn’t
less than n
anywhere (n>0).
• It isn’t even
less than 31n
everywhere.
• But it is less than
31n everywhere to
the right of n=8.
n>n0=8 →
Big-O example, graphically
Increasing n →
Valueoffunction→
n
30n+8
cn =
31n
30n+8
∈O(n)

22
No Uniqueness
• There is no unique set of values for n0 and c in proving the
asymptotic bounds
• Prove that 100n + 5 = O(n2
)
– 100n + 5 ≤ 100n + n = 101n ≤ 101n2
for all n ≥ 5
n0 = 5 and c = 101 is a solution
– 100n + 5 ≤ 100n + 5n = 105n ≤ 105n2
for all n ≥ 1
n0 = 1 and c = 105 is also a solution
Must find SOME constants c and n0 that satisfy the asymptotic notation relation

23
Asymptotic notations (cont.)
∀ Ω - notation
Ω(g(n)) is the set of functions
with larger or same order of
growth as g(n)

24
Examples
– 5n2
= Ω(n)
– 100n + 5 ≠ Ω(n2
)
– n = Ω(2n), n3
= Ω(n2
), n = Ω(logn)
∃ c, n0 such that: 0 ≤ cn ≤ 5n2 ⇒ cn ≤ 5n2
⇒ c = 1 and n0 = 1
∃ c, n0 such that: 0 ≤ cn2
≤ 100n + 5
100n + 5 ≤ 100n + 5n (∀ n ≥ 1) = 105n
cn2
≤ 105n ⇒ n(cn – 105) ≤ 0
Since n is positive ⇒ cn – 105 ≤ 0 ⇒ n ≤ 105/c
⇒ contradiction: n cannot be smaller than a constant

25
Asymptotic notations (cont.)
∀ Θ-notation
Θ(g(n)) is the set of functions
with the same order of growth
as g(n)

26
Examples
– n2
/2 –n/2 = Θ(n2
)
• ½ n2
- ½ n ≤ ½ n2
∀n ≥ 0 ⇒ c2= ½
• ½ n2
- ½ n ≥ ½ n2
- ½ n * ½ n ( ∀n ≥ 2 ) = ¼ n2
⇒ c1= ¼
– n ≠ Θ(n2
): c1 n2
≤ n ≤ c2 n2
⇒ only holds for: n ≤ 1/c1

27
Examples
– 6n3
≠ Θ(n2
): c1 n2
≤ 6n3
≤ c2 n2
⇒ only holds for: n ≤ c2 /6
– n ≠ Θ(logn): c1 logn ≤ n ≤ c2 logn
⇒ c2 ≥ n/logn, ∀ n≥ n0 – impossible

28
• Subset relations between order-of-growth sets.
Relations Between Different Sets
R→R
Ω( f )O( f )
Θ( f )
• f

31
Logarithms and properties
• In algorithm analysis we often use the notation “log n”
without specifying the base
nn
nn
elogln
loglg 2
=
= =y
xlogBinary logarithm
Natural logarithm
)lg(lglglg
)(lglg
nn
nn kk
=
=
xy log
=xylog yx loglog +
=
y
x
log yx loglog −
logb x =
ab
xlog
=xb
alog
log
log
a
a
x
b

32
More Examples
• For each of the following pairs of functions, either f(n) is
O(g(n)), f(n) is Ω(g(n)), or f(n) = Θ(g(n)). Determine
which relationship is correct.
– f(n) = log n2
; g(n) = log n + 5
– f(n) = n; g(n) = log n2
– f(n) = log log n; g(n) = log n
– f(n) = n; g(n) = log2
n
– f(n) = n log n + n; g(n) = log n
– f(n) = 10; g(n) = log 10
– f(n) = 2n
; g(n) = 10n2
– f(n) = 2n
; g(n) = 3n
f(n) = Θ (g(n))
f(n) = Ω(g(n))
f(n) = O(g(n))
f(n) = Ω(g(n))
f(n) = Ω(g(n))
f(n) = Θ(g(n))
f(n) = Ω(g(n))
f(n) = O(g(n))

33
Properties
• Theorem:
f(n) = Θ(g(n)) ⇔ f = O(g(n)) and f = Ω(g(n))
• Transitivity:
– f(n) = Θ(g(n)) and g(n) = Θ(h(n)) ⇒ f(n) = Θ(h(n))
– Same for O and Ω
• Reflexivity:
– f(n) = Θ(f(n))
– Same for O and Ω
• Symmetry:
– f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n))
• Transpose symmetry:
– f(n) = O(g(n)) if and only if g(n) = Ω(f(n))

34
Asymptotic Notations in Equations
• On the right-hand side
Θ(n2
) stands for some anonymous function in Θ(n2
)
2n2
+ 3n + 1 = 2n2
+ Θ(n) means:
There exists a function f(n) ∈ Θ(n) such that
2n2
+ 3n + 1 = 2n2
+ f(n)
• On the left-hand side
2n2
+ Θ(n) = Θ(n2
)
No matter how the anonymous function is chosen on
the left-hand side, there is a way to choose the
anonymous function on the right-hand side to make
the equation valid.

35
Common Summations
• Arithmetic series:
• Geometric series:
– Special case: |x| < 1:
• Harmonic series:
• Other important formulas:
2
)1( +nn
∑=
=+++=
n
k
nk
1
...21
( )1
1
11
≠
−
−+
x
x
xn
=++++=∑=
n
n
k
k
xxxx ...1 2
0
x−1
1
=∑
∞
=0k
k
x
nln≈∑=
+++=
n
k nk1
1
...
2
1
1
1
∑=
n
k
k
1
lg nnlg≈
1
1
1 +
+
p
n
p∑=
≈+++=
n
k
pppp
nk
1
...21

36
Mathematical Induction
• A powerful, rigorous technique for proving that a
statement S(n) is true for every natural number n,
no matter how large.
• Proof:
– Basis step: prove that the statement is true for n = 1
– Inductive step: assume that S(n) is true and prove that
S(n+1) is true for all n ≥ 1
• Find case n “within” case n+1

37
Example
• Prove that: 2n + 1 ≤ 2n
for all n ≥ 3
• Basis step:
– n = 3: 2 ∗ 3 + 1 ≤ 23
⇔ 7 ≤ 8 TRUE
• Inductive step:
– Assume inequality is true for n, and prove it for (n+1):
2n + 1 ≤ 2n
must prove: 2(n + 1) + 1 ≤ 2n+1
2(n + 1) + 1 = (2n + 1 ) + 2 ≤ 2n
+ 2 ≤
≤ 2n
+ 2n
= 2n+1
, since 2 ≤ 2n
for n ≥ 1

Asymptotic analysis

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