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Algorithmic Complexity
CMSC 132
Department of Computer Science
University of Maryland, College Park
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Algorithm Efficiency: Benchmarking and Asymptotic Analysis for Complexity - Prof. Nelson P, Study notes of Computer Science
An introduction to algorithmic complexity, focusing on efficiency, benchmarking, and asymptotic analysis. It covers the concept of measuring efficiency through time and space, the advantages and disadvantages of benchmarking, and the approach to asymptotic analysis. The document also includes examples of linear and binary search algorithms and their analysis.
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Algorithmic Complexity
CMSC 132
Department of Computer Science
University of Maryland, College Park
Algorithm Efficiency
Efficiency
Amount of resources used by algorithm
Time, space
Measuring efficiency
Benchmarking Asymptotic analysis
Benchmarking
Advantages
Precise information for given configuration
Implementation, hardware, inputs
Disadvantages
Affected by configuration
Data sets (usually too small) Hardware Software
Affected by special cases (biased inputs) Does not measure
intrinsic
efficiency
Asymptotic Analysis
Approach
Mathematically analyze efficiency Calculate time as function of input size
n
T
O[ f(
n
) ]
T is on the order of f(
n
“Big O” notation
Advantages
Measures intrinsic efficiency Dominates efficiency for large input sizes
Linear Search Algorithm
Algorithm
Guess number = 1
If incorrect, increment guess by 1
Repeat until correct
Example
Given number between 1…100 Pick 20 Guess sequence = 1, 2, 3, 4 … 20 Required 20 guesses
Linear Search Algorithm
Analysis of # of guesses needed for 1…n
If number = 1, requires 1 guess If number = n, requires n guesses On average, needs n/2 guesses Time = O( n ) =
Linear
time
Binary Search Algorithm
Example
Given number between 1…100 Pick 20 Guesses =
= 25, Answer = too large, subtract
= 12 , Answer = too large, subtract
Answer = too small, add
Answer = too small, add
Answer = too large, subtract
Answer = too large, subtract
Required 7 guesses
Binary Search Algorithm
Analysis of # of guesses needed for 1…n
If number = n/2, requires 1 guess If number = 1, requires log
2
( n ) guesses
If number = n, requires log
2
( n ) guesses
On average, needs log
2
( n ) guesses
Time = O( log
2
( n ) ) =
Log
time
Asymptotic Complexity
Comparing two linear functions
4n+
n/
Running Time
Size
Asymptotic Complexity
Comparing two functions
n/2 and 4n+3 behave similarly Run time roughly doubles as input size doubles Run time increases
linearly
with input size
For large values of n
Time(2n) / Time(n) approaches exactly 2
Both are O(
n
) programs
Asymptotic Complexity
Comparing two functions
log
2
( n )
and
5 * log
2
( n ) + 3
behave similarly
Run time roughly increases by constant as input sizedoubles Run time increases
logarithmically
with input size
For large values of n
Time(2n) – Time(n) approaches constant Base of logarithm does not matter
Simply a multiplicative factor
log
a
N = (log
b
N) / (log
b
a)
Both are O(
log(n)
) programs
Asymptotic Complexity
Comparing two quadratic functions
2 n
2
n
2
Running Time
Size
Big-O Notation
Represents
Upper bound on number of steps in algorithm Intrinsic efficiency of algorithm for large inputs
f(n)
O(…)
input size
# steps
Formal Definition of Big-O
Function f(
n
) is
( g(
n
) ) if
For some positive constants M, N
0
M
×
×
×
×
g(n)
f(n), for all n
N
0
Intuitively
For some coefficient M & all data sizes
N
0
M
××××
g(n) is always greater than f(n)