Performance

1
Performance O(f(n))

• What is the problem?

2
Quality software

• Correctness
• Performance Efficiency
• Space how much memory must be allocated?
• Time how much time elapses during execution?

3
Measuring performance

• Testing
• Analysis of algorithms
• How much space / time is required for an
  algorithm?
• How does one algorithm compare to another?
• How do requirements change depending on input?
• ? Need a scale for describing performance

4
Measuring performance

• Big Oh notation O(f(n))
• Efficiency of performance
• expressed as
• an approximate function of
• the number of data elements processed

5
Three simple methods
space
n 6 Memory space

• public void s0(int n)
• int val n

public void s1(int n) int val new
intn
public void s2(int n) int val new
intnn
6
Three simple methods

• public void s0(int n)
• int val n

public void s1(int n) int val new
intn
public void s2(int n) int val new
intnn
7

• We will mainly consider
• TIME efficiency

8
Three simple methods
time
n 6 of executions

• public void m0(int n)
• System.out.println(n)

1
public void m1(int n) for(int i 0 iltn i)
System.out.println(n)
6
public void m2(int n) for(int i 0 iltn i)
for(int j 0 jltn j) System.out.println(n)

36
9
Three simple methods

• public void m0(int n)
• System.out.println(n)

O(1)
public void m1(int n) for(int i 0 iltn
i) System.out.println(n)
O(n)
public void m2(int n) for(int i 0 iltn
i) for(int j 0 iltn i)
System.out.println(n)
O(n2)
10
Approximate measure
public void m2(int n) for(int i 0 iltn
i) for(int j 0 iltn i)
System.out.println(n)
public void m2a(int n) for(int i 0 iltn
i) for(int j i iltn i)
System.out.println(n)
Better performance? NO
11
Approximate measure
1 2 6 10 100
exec 1 4 36 100 10000
exec 1 3 21 55 5050
n
n2 O(n2)
n2n 2 O(n2)
public void m2(int n) for(int i 0 iltn
i) for(int j 0 iltn i)
System.out.println(n)
public void m2a(int n) for(int i 0 iltn
i) for(int j i iltn i)
System.out.println(n)
12
Approximate measure

• // linear search (unsorted array)
• public int search(double n, double arr)
• int i 0
• while (i lt arr.length arri ! n)
• i
• return i
• From 1 to n iterations
• Failed search n
• Average for successful search n/2
• ? Performance O(n)

13
BUT Different measure

• // binary search (sorted array)
• public int search(double n, double arr)
• int i, min 0, max arr.length-1
• while (min lt max)
• i (min max ) / 2
• if (arri n) return i
• if (arri lt n) min i1 else max i-1
• return arr.length
• From 1 to log n iterations
• Failed search log n
• Average for successful search log n - 1
• ? Performance O(log n)

14
O(log n) performance
15
O(log n) performance

• examples
• binary search in sorted arrays
• binary search tree operations
• retrieve, update, insert, delete
• B-trees
• heaps
• insert, remove

16
Sorting algorithms

• two nested repeat operations
• both O(n) ? O(n2)
• bubble, insertion, selection,
• one O(n) and one O(log n) ? O(n log n)
• mergesort, quicksort, heapsort,
• Shellsort ??, O(n (log n)2), O(n1.25)

17
Analyzing an algorithm

• rule of thumb
• depth of nested structure
• (or recursion)
• number of factors
• in performance measure
• some algorithms are difficult to analyze (e.g.
  hash table processing)

18
Comparing polymomial performance
n factors O(1) 0 O(log n) 1 O(n) 1 O(n log n) 2 O(n2) 2 O(n3) 3
2 1 1 2 2 4 8
8 1 3 8 24 64 512
32 1 5 64 320 1024 32768
128 1 7 128 896 16384 2097152
1024 1 8 1024 8192 1048576 1073741824
19
Combinatorial problems

• Example list the possible orderings of S
• 1,2,3 123, 132, 213, 231, 312, 321
• S 2 3 4 5 n
• orderings 2 6 24 120 n!

20
Combinatorial problems
P Q PQ
T T T
T F F
F T F
F F F

• Example generate
• truth table for PQ

number of propositions 2 3 4 5 n rows in table
4 8 16 32 2n
21
Combinatorial problems

• only known solutions
• have n levels of nesting
• for n data values
• performance is exponential O(en)
• includes n!, 2n, 10n,

22
Comparing polymomial (P) andexponential (NP)
performance
n factors O(log n) 1 O(n) 1 O(n log n) 2 O(n3) 3 O(en), e.g. 2n n
2 1 2 2 8 4
8 3 8 24 512 256
32 5 64 320 32768 4294967296
128 7 128 896 2097152 3.4028 x 1038
1024 8 1024 8192 1073741824 1.7977 x 10308
Non-deterministic Polynomial
23
NP Non-deterministic polynomial

• if a solution is known, it can be verified in
  O(nk) time
• e.g., SAT
• a solution (T,T,F,T,F,,T,F,T,F,F,F)
• can be tested by substituting in the fitness
  function.

24
P and NP problems

• P NP?
• (Are there polynomial algorithms
• for combinatorial problems?)
• YES ? there are but we are too stupid to find
  them
• NO ? we need to find other ways to solve
  these problems
• (in 50 years of trying, no answer to the
  question)

25
P and NP problems
NP-hard
NP
P
NP-complete