Analysis Tools
What is an Algorithm?
An algorithm is a finite set of instructions that
specify a sequence of operations to be carried
out in order to solve a specific problem or class
of problems.
Algorithm Properties
An algorithm must possess the following properties:
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Finiteness: Algorithm must complete after a finite number of
instructions have been executed.
Absence of ambiguity: Each step must be clearly defined,
having only one interpretation.
Definition of sequence: Each step must have a unique defined
preceding & succeeding step. The first step (start step) & last
step (halt step) must be clearly noted.
Input/output: There must be a specified number of input
values, and one or more result values.
Feasibility: It must be possible to perform each instruction.
Analyzing an Algorithm
In order to learn more about an algorithm, we can analyze it. That
is, draw conclusions about how the implementation of that
algorithm will perform in general. This can be done in various
ways.
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determine the running time of a program as a function of its inputs;
determine the total or maximum memory space needed for program
data;
determine the total size of the program code;
determine whether the program correctly computes the desired
result;
determine the complexity of the program--e.g., how easy is it to
read, understand, and modify; and,
determine the robustness of the program--e.g., how well does it
deal with unexpected or erroneous inputs?
Analyzing an Algorithm
Different ways of analyzing the algorithm with
render different results. An algorithm that runs
fast is not necessarily robust or correct. An
algorithm that has very little lines of code does
not necessarily use less resources.
So, how can we measure the efficiency of an
algorithm?
Design Considerations
Given a particular problem, there are typically a
number of different algorithms that will solve that
problem. A designer must make a rational choice
among those algorithms:
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To design an algorithm that is easy to understand,
implement, and debug (software engineering).
To design an algorithm that makes efficient use
of the available computational resources (data
structures and algorithm analysis)
Example
Replaced by
Easy to understand,
but slow.
Gauss summation
sum  N(N+1)/2
More efficient, but
need to be as smart
as 10-year-old
Gauss.
Gauss Summation: http://www.cut-the-knot.org/Curriculum/Algebra/GaussSummation.shtml
Algorithm Analysis in CSCI 3333
But, how do we measure the efficiency of
an algorithm? Note that the number of
operations to be performed and the space
required will depend on the number of input
values that must be processed.
Benchmarking algorithms
It is tempting to measure the efficiency of an algorithm by
producing an implementation and then performing benchmarking
analysis by running the program on input data of varying sizes and
measuring the “wall clock” time for execution.
However, there are many factors that affect the running time of a
program. Among these are the algorithm itself, the input data, and
the computer system used to run the program. The performance
of a computer is determined by
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the hardware:
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

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processor used (type and speed),
memory available (cache and RAM), and
disk available;
the programming language in which the algorithm is specified;
the language compiler/interpreter used;
the computer operating system software.
Asymptotic Analysis
Therefore, the goal is to have a way of describing the
inherent complexity of a program, independent of
machine/compiler considerations. This means not
describing the complexity by the absolute time or
storage needed.
Instead, focus should be concentrated on a
"proportionality" approach, expressing the complexity
in terms of its relationship to some known function
and the way the program scales as the input gets
larger. This type of analysis is known as asymptotic
analysis.
Asymptotic Analysis
There is no generally accepted set of rules
for algorithm analysis. In some cases, an
exact count of operations is desired; in other
cases, a general approximation is sufficient.
Therefore, we strive to setup a set of rules to
determine how operations are to be counted.
Analysis Rules
To do asymptotic analysis, start by counting the primitive
operations in an algorithm and adding them up.
Assume that primitive operations will take a constant
amount of time, such as:
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Assigning a value to a variable
Calling a function
Performing an arithmetic operation
Comparing two numbers
Indexing into an array
Returning from a function
Following a pointer reference
Example of Counting Primitive
Operations
Inspect the pseudocode to count the primitive
operations as a function of the input size (n)
Algorithm arrayMax(A,n):
currentMax  A[0]
Count
Array indexing + Assignment
2
for i  1 to n – 1 do
Initializing i
Verifying i<n
if currentMax < A[i] then
currentMax  A[i]
return currentMax
Best case: 2+1+n+4(n–1)+1 = 5n
1
n
Array indexing + Comparing 2(n-1)
Array indexing + Assignment 2(n-1) worst
Incrementing the counter
2(n-1)
Returning
1
Worst case: 2+1+n+6(n–1)+1 = 7n-2
Best, Worst, or Average Case
Analysis
An algorithm may run faster on some input data
than on others.
Best case –
the data is distributed so that the
algorithm runs fastest
Worst case –
the data distribution causes the
slowest running time
Average case – very difficult to calculate
For our purposes, will concentrate on analyzing
algorithms by identifying the running time for
the worst case data.
Estimating the Running Time
The actual running time depends on the
speed of the primitive operations—some of
them are faster than others
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Let t = speed of the slowest primitive operation
= worst case scenario
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Let f(n) = the worst-case running time of arrayMax
f(n) = t (7n – 2)
Growth Rate of a Linear Loop
600
Slow PC
10(7n-2)
Growth rate of arrayMax 500
is linear. Changing the
400
hardware alters the value
of t, so that arrayMax willf(n) 300
run faster on a faster
200
computer. However,
growth rate is still linear. 100
Fast PC
5(7n-2)
Fastest PC
1(7n-2)
0
0
2
4
n
6
8
Growth Rate of a Linear Loop
What about the following loop?
for (i=0; i<n; i+=2 )
do something
Here, the number of iterations is half of n.
However, higher the factor, higher the number
of loops. So, although f(n) = n/2, if you were to
plot the loop, you would still get a straight line.
Therefore, this is still a linear growth rate.
Growth Rate of a Logarithmic
Loop
What about the following segments?
for (i=1; i<n; i*=2 )
do something
for (i=n; i>=1; i/=2 )
do something
When n=1000, the loop will iterate only 10
times. It can be seen that the number of
iterations is a function of the multiplier or
divisor. This kind of function is said to have
logarithmic growth, where f(n) = log n
Growth Rate of a Linear
Logarithmic Loop
What about the following segments?
for (i=1; i<n; i++ )
for (j=1; j<n; j*=2 )
do something
Here, the outer loop has a linear growth, and
the inner loop has a logarithmic growth.
Therefore:
f(n) = n log n
Growth Rates of
Common Classes of Functions
Exponentially
Quadratically
Running time
f(n)
Linearly
Logarithmically
Constant
Input size n
Is This Really Necessary?
Is it really important to find out the exact number of
primitive operations performed by an algorithm? Will
the calculations change if we miss out one primitive
operation?
In general, each step of pseudo-code or statement
corresponds to a small number of primitive operations
that does not depend on the input size. Thus, it is
possible to perform a simplified analysis that estimates
the number of primitive operations, by looking at the
“big picture”.
What exactly is Big-O?
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Big-O expresses an upper bound on the
growth rate of a function, for sufficiently
large values of n.
Upper bound is not the worst case. What is
being bounded is not the running time
(which can be determined by a given value
of n), but rather the growth rate for the
running time (which can only be
determined over the range of values for n).
Big-Oh Notation
Definition:
Let f(n) and g(n) be functions mapping nonnegative integers to
real numbers.
Then, f(n) is O(g(n)) ( f(n) is big-oh of g(n) ) if there is a real
constant c>0 and an integer constant n01, such that
f(n)  cg(n) for every integer nn0
By the definition above, demonstrating that a function f is big-O
of a function g requires that we find specific constants C and N
for which the inequality holds (and show that the inequality
does, in fact, hold).
Big-O Theorems
For all the following theorems, assume that f(n) is a function of n
and that K is an arbitrary constant.
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Theorem1: K is O(1)
Theorem 2: A polynomial is O(the term containing the highest power
of n)
f(n) = 7n4 + 3n2 + 5n + 1000 is O(7n4)
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Theorem 3: K*f(n) is O(f(n)) [that is, constant coefficients can be
dropped]
g(n) = 7n4 is O(n4)
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Theorem 4: If f(n) is O(g(n)) and g(n) is O(h(n)) the f(n) is O(h(n)).
[transitivity]
Big-O Theorems
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Theorem 5: Each of the following functions is strictly
smaller
big-O of its successors:
K [constant]
logb(n) [always log base 2 if no base is shown]
n
nlogb(n)
n2
n to higher powers
2n
3n
larger constants to the n-th power
n! [n factorial]
nn
For Example: f(n) = 3nlog(n) is O(nlog(n)) and O(n2) and O(2n)
larger
Big-O Theorems
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Theorem 6: In general, f(n) is big-O of the
dominant term of f(n), where “dominant” may
usually be determined from Theorem 5.
f(n) = 7n2+3nlog(n)+5n+1000 is O(n2)
g(n) = 7n4+3n+106 is O(3n)
h(n) = 7n(n+log(n)) is O(n2)
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Theorem 7: For any base b, logb(n) is
O(log(n)).
Examples
In general, in Big-Oh analysis, we focus on the “big
picture,” that is, the operations that affect the
running time the most – the loops
Simplify the count:
1.
2.
3.
Drop all lower-order terms
7n – 2
7n
Eliminate constants
7n
n
Remaining term is the Big-Oh
7n – 2 is O(n)
More Examples
Example: f(n) = 5n3 – 2n2 + 1
1.
Drop all lower order terms
5n3 – 2n2 + 1
2.
Eliminate the constants
5n3
3.
5n3
n3
The remaining term is the Big-Oh
f(n) is O(n3)
Determining Complexities in
General
1.
2.
We can drop the constants
sum rule: for a sequential loops add their Big-Oh values
statement 1; statement 2; ... statement k;
total time = time(statement 1) + time(statement 2) + ... +
time(statement k)
•
if-then-else statements: the worst-case time is the slowest of
the two possibilities
if (cond) {
sequence of statements 1
}
else {
sequence of statements 2
}
Total time = max(time(sequence 1), time(sequence 2))
Determining Complexities in
General
4.
for loops: The loop executes N times, so the sequence of
statements also executes N times. Since we assume the
statements are O(1),
total time = N * O(1) = O(N)
5.
Nested for loops: multiply their Big-Oh values
for (i = 0; i < N; i++) {
for (j = 0; j < M; j++) {
sequence of statements
}
}
total time = O(N) * O(M) = O(N*M)
6.
In a polynomial, the term with the highest degree establishes
the Big-Oh
Growth Rate Examples
Efficiency
Constant
O(1)
Logarithmic
O(log n)
Linear
O(n)
Quadratic
O(n2)
Exponential
O(2n)
Example
Accessing an element of an array
Binary search
Pushing a collection of elements onto a
stack
Bubble sort
Towers of Hanoi (Goodrich, 198)
Why Does Growth Rate Matter?
Constant
O(1)
n
Logarithmic
O(logn)
Linear
O(n)
Quadratic
O(n2)
10
0.4 nsec
1.33 nsec
4.0 nsec
40.0 nsec
1,000
0.4 nsec
3.99 nsec
0.4 msec
0.4 msec
100,000
0.4 nsec
6.64 nsec
0.04 msec
4.0 sec
10,000,000
0.4 nsec
9.30 nsec
0.004 sec
1.11 hr
Assuming that it takes 0.4 nsec to process one element
Finding the Big-Oh
for( int i=0; i<n; ++i ) {
for(int j=0; j<n; ++j) {
myArray[i][j] = 0;
}
}
sum = 0;
for( int i=0; i<n; ++i ) {
for(int j=0; j<i; ++j) {
sum++;
}
}
Example that shows why the BigOh is so important.
Write a program segment so that, given an array X,
compute array A such that, each number A[i] is the
average of the numbers in X from X[0] to X[i]
X
1
3
5
7
9
4
5
X[0] + X[1] + X[2] = 9
9/3=3
A
1
2
3
Solution #1 – Quadratic time

Algorithm prefixAverages1
…
for i  0 to n-1 do
a  0
for j  0 to i do
a  a + X[j]
A[i]  a/(i+1)
…
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Two nested loops
Inner loop – loops
through X, adding the
numbers from element 0
through element i
Outer loop – loops
through A, calculating
the averages and
putting the result
into A[i]
Solution #2 - Linear Time

Algorithm prefixAverages2
…
s  0
for i  0 to n-1 do
s  s + X[i]
A[i]  s/(i+1)
…

Only one loop
Sum – keeps track of the
sum of the numbers in X
so that we don’t have to
loop through X every time
Loop – loops
through A, adding
to the sum, calculating
the averages, and
putting the result
into A[i]
Lessons Learned
1.
Both algorithms correctly solved the problem
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2.
Lesson – There may be more than one way to write your
program.
One of the algorithms was significantly faster
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Lesson – The algorithm that we choose can have a big
influence on the program’s speed.
Evaluate the solution that you pick, and ask whether it is
the most efficient way to do it.