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CMSC 132:
Object-Oriented Programming II
Algorithmic Complexity II
Department of Computer Science
University of Maryland, College Park
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Overview
Critical sections
Comparing complexity
Types of complexity analysis
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Understanding Critical Sections and Algorithmic Complexity in Computer Science, Study notes of Computer Science
An overview of critical sections in algorithms and discusses the importance of analyzing their complexity. It covers the goal of algorithmic complexity analysis, the approach to finding critical sections, and examples of critical sections in various algorithms. The document also introduces recursive algorithms and the concept of asymptotic complexity categories.
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1
CMSC 132:
Object-Oriented Programming II
Algorithmic Complexity II
Department of Computer Science
University of Maryland, College Park
2
Overview
Critical sections
Comparing complexity
Types of complexity analysis
3
Analyzing Algorithms
Goal
Find asymptotic complexity of algorithm
Approach
Ignore less frequently executed parts of algorithm Find critical section of algorithm Determine how many times critical section is executed as function of problem size
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Critical Section of Algorithm
Heart of algorithm
Dominates overall execution time
Characteristics
Operation central to functioning of program Contained inside deeply nested loops Executed as often as any other part of algorithm
Sources
Loops Recursion
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Critical Section Example 2
Code (for input size n)
1. A
- for (int i = 0; i < n; i++)
- B
- for (int j = 0; j < n; j++)
- C
- D
Code execution
A โ
B โ
Time โ
C โ
D โ
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Critical Section Example 2
Code (for input size n)
1. A
- for (int i = 0; i < n; i++)
- B
- for (int j = 0; j < n; j++)
- C
- D
Code execution
A โ once B โ n times
Time โ 1 + n + n 2 + 1 = O(n^2 )
critical
section
C โ n^2 times D โ once
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Critical Section Example 3
Code (for input size n)
1. A
- for (int i = 0; i < n; i++)
- for (int j = i+1; j < n; j++)
- B
Code execution
A โ
B โ
Time โ
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Critical Section Example 3
Code (for input size n)
1. A
- for (int i = 0; i < n; i++)
- for (int j = i+1; j < n; j++)
- B
Code execution
A โ once B โ ยฝ n (n-1) times
Time โ 1 + ยฝ n^2 = O(n 2 )
critical
section
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Critical Section Example 5
Code (for input size n)
- for (int i = 0; i < n; i++)
- for (int j = 0; j < n; j++)
- A
- for (int i = 0; i < n; i++)
- for (int j = 0; j < n; j++)
- B
Code execution
A โ
B โ
Time โ
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Critical Section Example 5
Code (for input size n)
- for (int i = 0; i < n; i++)
- for (int j = 0; j < n; j++)
- A
- for (int i = 0; i < n; i++)
- for (int j = 0; j < n; j++)
- B
Code execution
A โ n^2 times B โ n^2 times
Time โ n^2 + n 2 = O(n 2 )
critical
sections
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Critical Section Example 6
Code (for input size n)
- i = 1
- while (i < n)
- A
- i = 2 ร i
- B
Code execution
A โ
B โ
Time โ
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Critical Section Example 6
Code (for input size n)
- i = 1
- while (i < n)
- A
- i = 2 ร i
- B
Code execution
A โ log(n) times B โ 1 times
Time โ log(n) + 1 = O( log(n) )
critical
section
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Recursive Algorithms
Definition
An algorithm that calls itself
Components of a recursive algorithm
- Base cases
Computation with no recursion
- Recursive cases
Recursive calls Combining recursive results
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Recursive Algorithm Example
Code (for input size n)
- DoWork (int n)
- if (n == 1)
- A
- else
- DoWork(n/2)
- DoWork(n/2)
recursive
cases
base
case
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Asymptotic Complexity Categories
Complexity Name Example
O(1) Constant Array access O(log(n)) Logarithmic Binary search O(n) Linear Largest element O(n log(n)) N log N Optimal sort O(n^2 ) Quadratic 2D Matrix addition O(n^3 ) Cubic 2D Matrix multiply O(nk) Polynomial Linear programming O(kn^ ) Exponential Integer programming
From smallest to largest
For size n, constant k > 1
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Comparing Complexity
Compare two algorithms
f(n), g(n)
Determine which increases at faster rate
As problem size n increases
Can compare ratio
If โ, f() is larger
If 0, g() is larger If constant, then same complexity
lim nโ โ
f(n)
g(n)
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2D Matrix Multiplication Example
Problem
C = A * B
Lower bound
ฮฉ(n^2 ) Required to examine 2D matrix
Upper bounds
O(n^3 ) Basic algorithm O(n2.807^ ) Strassenโs algorithm (1969) O(n2.376^ ) Coppersmith & Winograd (1987)
Improvements still possible (open problem)
Since upper & lower bounds do not match
ร (^) =
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Additional Complexity Categories
Name Description
NP Nondeterministic polynomial time (NP) PSPACE Polynomial space EXPSPACE Exponential space Decidable Can be solved by finite algorithm Undecidable Not solvable by finite algorithm
Mostly of academic interest only
Quadratic algorithms usually too slow for large data Use fast heuristics to provide non-optimal solutions
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NP Time Algorithm
Polynomial solution possible
If make correct guesses on how to proceed
Required for many fundamental problems
Boolean satisfiability Traveling salesman problem (TLP) Bin packing
Key to solving many optimization problems
Most efficient trip routes Most efficient schedule for employees Most efficient usage of resources
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NP Time Algorithm
Properties of NP
Can be solved with exponential time Not proven to require exponential time Currently solve using heuristics
NP-complete problems
Representative of all NP problems Solution can be used to solve any NP problem Examples Boolean satisfiability Traveling salesman