Download CS-240 Data Structures

Algorithm Analysis
(Big O)
1
Complexity
 In
examining algorithm efficiency we
must understand the idea of complexity
Space complexity
 Time Complexity

2
Space Complexity

When memory was expensive we focused on
making programs as space efficient as possible
and developed schemes to make memory appear
larger than it really was (virtual memory and
memory paging schemes)

Space complexity is still important in the field of
embedded computing (hand held computer
based equipment like cell phones, palm devices,
etc)
3
Time Complexity
 Is
the algorithm “fast enough” for my
needs
 How much longer will the algorithm
take if I increase the amount of data it
must process
 Given a set of algorithms that
accomplish the same thing, which is the
right one to choose
4
Algorithm Efficiency

a measure of the amount of resources consumed in
solving a problem of size n



Benchmarking: implement algorithm,



time
space
run with some specific input and measure time taken
better for comparing performance of processors than for
comparing performance of algorithms
Big Oh (asymptotic analysis)


associates n, the problem size,
with t, the processing time required to solve the problem
5
Cases to examine
Best case
 if the algorithm is executed, the fewest
number of instructions are executed
 Average case
 executing the algorithm produces path
lengths that will on average be the same
 Worst case
 executing the algorithm produces path
lengths that are always a maximum

6
Worst case analysis
 Of
the three cases, only useful case
(from the standpoint of program
design) is that of the worst case.
 Worst
case helps answer the software
lifecycle question of:

If its good enough today, will it be good
enough tomorrow?
7
Frequency Count
 examine
a piece of code and predict the
number of instructions to be executed
for each instruction predict how many times
 e.g. each will be encountered as the code runs
Inst
#
1
2
3
Code
F.C.
for (int i=0; i< n ; i++)
n+1
{ cout << i;
n
p = p + i;
n
}
____
3n+1
totaling the counts produces the F.C. (frequency count)
8
Order of magnitude

In the previous example:






best_case = avg_case = worst_case
Example is based on fixed iteration n
By itself, Freq. Count is relatively meaningless
Order of magnitude -> estimate of performance vs.
amount of data
To convert F.C. to order of magnitude:
 discard constant terms
 disregard coefficients
 pick the most significant term
Worst case path through algorithm ->

order of magnitude will be Big O (i.e. O(n))
9
Another example
Inst # Code
1
2
for (int i=0; i< n ; i++)
for int j=0 ; j < n; j++)
F.C.
F.C.
n+1
n+1
n(n+1)
n2+n
3
{ cout << i;
n*n
n2
4
p = p + i;
n*n
n2
____
}
3n2+2n+1
discarding constant terms produces :
3n2+2n
clearing coefficients :
Big O = O(n2)
n2+n
picking the most significant term: n2
10
What is Big O
 Big

rate at which algorithm performance
degrades as a function of the amount of
data it is asked to handle
 For

O
example:
O(n) -> performance degrades at a linear
rate O(n2) -> quadratic degradation
11
Common growth rates
12
Big Oh - Formal
Definition
Definition of "big oh":
 f(n)=O(g(n)), iff there exist constants c and n0
such that:
f(n) <= c  g(n) for all n>=n0
 Thus, g(n) is an upper bound on f(n)
 Note:
f(n) = O(g(n))
is NOT the same as
O(g(n)) = f(n)
 The '=' is not the usual mathematical operator
"=" (it is not reflexive)

13
Big-O Notation
Comparing Algorithms
and
ADT Data Structures
14
Algorithm Efficiency

a measure of the amount of resources consumed in
solving a problem of size n



benchmarking – code the algorithm, run it with some
specific input and measure time taken


time
space
better for measuring and comparing the performance of
processors than for measuring and comparing the performance
of algorithms
Big Oh (asymptotic analysis) provides a formula that
associates n, the problem size, with t, the processing
time required to solve the problem
15
big Oh

measures an algorithm’s growth rate


how fast does the time required for an algorithm to
execute increase as the size of the problem increases?
is an intrinsic property of the algorithm

independent of particular machine or code
based on number of instructions executed
 for some algorithms is data-dependent
 meaningful for “large” problem sizes

16
Computing

x * x * x .. * x
(n times)
recursive definition



for n >= 0
iterative definition


n
x
x0 = 1
xn = x * xn-1
(for n > 0)
another recursive definition



x0 = 1
xn = (xn/2)2
xn = x * (xn/2)2
(for n > 0 and n is even)
(for n > 0 and n is odd)
17
Iterative Power function
double IterPow (double X, int N) {
double Result = 1;
while (N > 0) {
Result *= X;
N--;
{
return Result;
}
Total instruction count:
1
n+1
n
n
critical region
1
3n+3
algorithm's computing time (t) as a function of n is: 3n + 3
t is on the order of f(n) - O[f(n)]
O[3n + 3] is n
18
Recursive Power function
double RecPow (double X, int N) {
if (N == 0)
return 1;
else
return X * RecPow(X, N - 1);
}
Base case
1
1
Recursive case
1
1 + T(n-1)
total: 2
2 + T(n-1)
Number of times base case is executed:
1
Number of times recursive case is executed: n
Algorithm's computing time (t) as a function of n is: 2n + 2
O[2n + 2] is n
19
Another Power Function
Base case
double Pow3 (double X, int N) {
if (N == 0)
1
return 1;
1
else {
double halfPower = Pow3(X, N/2);
if (N % 2 == 0)
return halfPower * halfPower;
else
return X * halfPower * halfPower;
}
}
total: 2
Recursive case
1
T(n/2)
1
1(even)
1(odd)
3 + T(n/2)
Number of times base case is executed:
1
Number of times recursive case is executed: log2 n
Algorithm's computing time (t) as a function of n is: 3 log2 n + 2
O[3 log2 n + 2] is log2 n
20
Computational Complexity

Computing time, T(n), of an algorithm is a function
of the problem size (based on instruction count)
T(n) for IterPow is: 3n + 3
 T(n) for RecPow is: 2n + 2
 T(n) for Pow3 is: 3 log2 n + 2


Computational complexity of an algorithm is the rate
at which T(n) grows as the problem size grows



is expressed using "big Oh" notation
growth rate (big Oh) of 3n+3 and of 2n+2 is: n
big Oh of 3 log2 n + 2 is: log2 n
21
Common big Ohs
 constant
 logarithmic
 linear
n
log n
 quadratic
 cubic
 exponential
O(1)
O(log2 N)
O(N)
O(N log2 N)
O(N2)
O(N3)
O(2N)
22
Comparing Growth Rates
2n
n2
n log2 n
n
T(n)
log2 n
Problem Size
23
An Experiment
Execution time (in seconds)
2^25
2^50
2^100
IterPow
.71
1.15
2.03
RecPow
3.63
7.42
15.05
.99
1.15
1.38
Pow3
(1,000,000 repetitions)
24
Uses of big Oh
 compare
algorithms which perform the
same function
search algorithms
 sorting algorithms

 comparing
data structures for an ADT
each operation is an algorithm and has a big
Oh
 data structure chosen affects big Oh of the
ADT's operations

25
Comparing algorithms
Sequential search
 growth rate is O(n)
 average number of
comparisons done is
n/2

n
100
500
1000
5000
n/2
50
250
500
2500
Binary search
 growth rate is O(log2 n)
 average number of
comparisons done is
2((log2 n) -1)

2((log2 n)-1)
12
16
18
24
26