Analysis of Algorithms

CPS212

Gordon College

Measuring the efficiency of algorithms

There are 2 algorithms: algo1 and algo2 that produce the same results.

How do you choose which algorithm to keep. Algorithm Efficiency

Measuring the efficiency of algorithms

How do you determine an algorithm’s inherent efficiency? Code them up and compare their running times on

a lab machine. How were they coded? We want to compare the

algorithms - not their implementations What computer do we use? What else is running? What data is a fair test of an algorithm’s efficiency.

Measuring the efficiency of algorithms

Goal: Measure the efficiency of an algorithm -

independent of implementation, hardware, or data. Solution:

Analyze and represent the number of operations the algorithm will perform as a function of input.

Example: Copying an array with n elements requires x

operations. What is x?

Growth rates Algo1 requires n2 / 2 operations to solve

a problem with input size n Algo2 requires 5n + 10 operations

Which is the better algorithm for the job?

Order of magnitude growth analysis A function f(x) is O(g(x)) if and only if

there exist 2 positive constants, c and n, such that | f(x) |≤ cg(x) for all x > n

size of input set

number ofoperations

cg(x)

f(x)

Order of magnitude growth analysis Asymptotic growth - as the growth approaches infinity An asymptote of a real-valued function y = f(x) is a

curve which describes the behavior of f as either x or y tends to infinity.

size of input set

number ofoperations

cg(x)

f(x)

Order of magnitude growth analysis Important points:

Focus only on the growth Shape of g(x) is essential As the input set grows large (ignore the shape for small x)

size of input set

number ofoperations

cg(x)

f(x)

Standard function shapes: constant O(1)

Examples?

size of input set

number ofoperations

= cg(x)f(x)

Standard function shapes: linear O(x)

Examples?

size of input set

number ofoperations

cg(x)

f(x)=ax+b

Standard function shapes: logarithmic O(log x) (base 2)

QuickTime™ and a decompressor

are needed to see this picture.

bc = a logba = c 23 = 8 log28 = 3

log(x*y) = log x + log y

log(xa) = a log x

Examples?

Standard function shapes: Polynomial O(x2)

Examples?

Polynomial vs. Linear

Polynomial vs. Linear

Example: 50n + 20 & n2

At what point will n2 surpass 50n+20?

n2 = 50n+20 Solve for x…quadratic formula

n2 - 50n - 20 = 0

n = 101/2 = 50.5 n = -1/2

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Standard function shapes: Exponential O(cx)

Examples?Hamiltonian CircuitTraveling SalesmanOptimization problems

Solution:Limit to small input setsIsolate special casesFind approximate solution

(near optimal)

Complexity in action

QuickTime™ and a decompressor

are needed to see this picture.

Real Examples Searching (sequential)

Unit of work: comparisons Best case: O(1) [theta] Worst case: O(n) [theta] Average Case: O(n) [theta]

Real Examples Sort (selection)

Unit of work: comparisons and exchanges Best case: O(n2) [theta] Worst case: O(n2) [theta] Average Case: O(n2) [theta]

Real Examples Search (binary)

Unit of work: comparisons Best case: O(1) [theta] Worst case: O(log n) [theta] Average Case: O(log n) [theta]