of Analysis Algorithms
Input
Algorithm
Output
An algorithm is a stepbystep procedure for solving a problem in a finite amount of time.
Running Time (1.1)
Most algorithms transform input objects into output objects. The running time of an algorithm typically grows with the input size. Average case time is often difficult to determine. We focus on the worst case running time.
best case average case worst case
120 100
Running Time
80 60 40 20 0 1000 2000 3000 4000
Easier to analyze Crucial to applications such as games, finance and robotics
Analysis of Algorithms
I nput Size
2
Experimental Studies ( 1.6)
Write a program implementing the algorithm Run the program with inputs of varying size and composition Use a method like System.currentTimeMillis( ) to get an accurate measure of the actual running time Plot the results
9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 50 100
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Time ( ms)
I nput Size
Analysis of Algorithms
Limitations of Experiments
It is necessary to implement the algorithm, which may be difficult Results may not be indicative of the running time on other inputs not included in the experiment. In order to compare two algorithms, the same hardware and software environments must be used
Analysis of Algorithms 4
Theoretical Analysis
Uses a highlevel description of the algorithm instead of an implementation Characterizes running time as a function of the input size, n. Takes into account all possible inputs Allows us to evaluate the speed of an algorithm independent of the hardware/ software environment
Analysis of Algorithms 5
Pseudocode (1.1)
Example: find max Highlevel description element of an array of an algorithm More structured than Algorithm arrayMax(A, n) English prose Input array A of n integers Less detailed than a Output maximum element of A program Preferred notation for currentMax A[0] for i 1 to n - 1 do describing algorithms if A[i] > currentMax then Hides program design currentMax A[i] issues return currentMax
Analysis of Algorithms 6
Pseudocode Details
Control flow
Method call Return value Expressions
var.method (arg [, arg...]) return expression Assignment (like = in Java) = Equality testing (like = = in Java) n2Superscripts and other mathematical formatting allowed
7
Method declaration
if ... then ... [else ...] while ... do ... repeat ... until ... for ... do ... Indentation replaces braces
Algorithm method (arg [, arg...]) Input ... Output ...
Analysis of Algorithms
The Random Access Machine (RAM) Model
A CPU An potentially unbounded bank of memory cells, each of which can hold an arbitrary number or character
0
1
2
Memory cells are numbered and accessing any cell in memory takes unit time.
Analysis of Algorithms 8
Primitive Operations
Basic computations performed by an algorithm Identifiable in pseudocode Largely independent from the programming language Exact definition not important (we will see why later) Assumed to take a constant amount of time in the RAM model
Analysis of Algorithms
Examples:
Evaluating an expression Assigning a value to a variable Indexing into an array Calling a method Returning from a method
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Counting Primitive Operations (1.1)
By inspecting the pseudocode, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size Algorithm arrayMax(A, n) currentMax A[0] for i 1 to n - 1 do if A[i] > currentMax then currentMax A[i] { increment counter i } return currentMax
Analysis of Algorithms
# operations 2 2+n 2(n - 1) 2(n - 1) 2(n - 1) 1 Total 7n - 1
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Estimating Running Time
Algorithm arrayMax executes 7n -1 primitive operations in the worst case. Define: Let T(n) be worstcase time of arrayMax. Then a (7n -1) T(n) b(7n -1) Hence, the running time T(n) is bounded by two linear functions
a = Time taken by the fastest primitive operation b = Time taken by the slowest primitive operation
Analysis of Algorithms
11
Growth Rate of Running Time
Changing the hardware/ software environment
The linear growth rate of the running time T(n) is an intrinsic property of algorithm arrayMax
Analysis of Algorithms 12
Affects T(n) by a constant factor, but Does not alter the growth rate of T(n)
Growth Rates
Growth rates of functions:
In a loglog chart, the slope of the line corresponds to the growth rate of the function
T (n )
Linear n Quadratic n2 Cubic n3
1E+30 1E+28 1E+26 1E+24 1E+22 1E+20 1E+18 1E+16 1E+14 1E+12 1E+10 1E+8 1E+6 1E+4 1E+2 1E+0 1E+0
Cubic Quadratic Linear
1E+2
1E+4
1E+6
1E+8
1E+10
n
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Analysis of Algorithms
Constant Factors
The growth rate is not affected by
Examples
102n + 105 is a linear function 105n2 + 108n is a quadratic function
T (n )
constant factors or lowerorder terms
1E+26 1E+24 1E+22 1E+20 1E+18 1E+16 1E+14 1E+12 1E+10 1E+8 1E+6 1E+4 1E+2 1E+0 1E+0
Quadratic Quadratic Linear Linear
1E+2
1E+4 n
1E+6
1E+8
1E+10
Analysis of Algorithms
14
BigOh Notation (1.2)
Given functions f(n) and g(n), we say that f(n) is 1,000 O(g(n)) if there are positive constants 100 c and n0 such that f(n) cg(n) for n n0 Example: 2n + 10 is O(n)
10,000
3n 2n+ 10 n
10
2n + 10 cn (c - 2) n 10 n 10/(c - 2) Pick c = 3 and n0 = 10
1 1 10
n
100
1,000
Analysis of Algorithms
15
BigOh Example
Example: the function 100,000 n2 is not O(n)
1,000,000
n^ 2 100n 10n n
n2 cn n c The above inequality be cannot satisfied since c must be a constant
10,000 1,000 100 10 1 1
10
n
100
1,000
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Analysis of Algorithms
More BigOh Examples
7n2 is O(n) need c > 0 and n0 1 such that 7n2 cn for n n0 this is true for c = 7 and n0 = 1
7n2
3n3 + 20n2 + 5 is O(n3) need c > 0 and n0 1 such that 3n3 + 20n2 + 5 cn3 for n n0 this is true for c = 4 and n0 = 21
3n3 + 20n2 + 5
3 log n + log log n
3 log n + log log n is O(log n) need c > 0 and n0 1 such that 3 log n + log log n clog n for n n0
this is true for c = 4 and n0 = 2
Analysis of Algorithms
17
BigOh and Growth Rate
The bigOh notation gives an upper bound on the growth rate of a function The statement "f(n) is O(g(n))" means that the growth rate of f(n) is no more than the growth rate of g(n) We can use the bigOh notation to rank functions according to their growth rate f(n) is O(g(n)) g(n) grows more f(n) grows more Same growth
g(n) is O(f(n)) No Yes Yes
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Yes No Yes
Analysis of Algorithms
BigOh Rules
If is f(n) a polynomial of degree d, then f(n) is O(nd), i.e.,
1. 2.
Use the smallest possible class of functions
Drop lowerorder terms Drop constant factors
Use the simplest expression of the class
Analysis of Algorithms
Say "2n is O(n)" instead of "2n is O(n2)"
Say "3n + 5 is O(n)" instead of "3n + 5 is O(3n)"
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Asymptotic Algorithm Analysis
The asymptotic analysis of an algorithm determines the running time in bigOh notation To perform the asymptotic analysis
Example:
We find the worstcase number of primitive operations executed as a function of the input size We express this function with bigOh notation
Since constant factors and lowerorder terms are eventually dropped anyhow, we can disregard them when counting primitive operations
Analysis of Algorithms
We determine that algorithm arrayMax executes at most 7n - 1 primitive operations We say that algorithm arrayMax "runs in O(n) time"
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Computing Prefix Averages
We further illustrate asymptotic analysis with two algorithms for prefix averages The ith prefix average of an array X is average of the first (i + 1) elements of X: A[i] = ( X[0] + X[1] + ... + X[i])/
(i+1)
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X A
Computing the array A of prefix averages of another array X has applications to financial analysis
Analysis of Algorithms
21
Prefix Averages (Quadratic)
The following algorithm computes prefix averages in quadratic time by applying the definition Algorithm prefixAverages1(X, n) Input array X of n integers Output array A of prefix averages of X #operations A new array of n integers n for i 0 to n - 1 do n s X[0] n for j 1 to i do 1 + 2 + ...+ (n - 1) s s + X[j] 1 + 2 + ...+ (n - 1) A[i] s / (i + 1) n return A 1
Analysis of Algorithms 22
Arithmetic Progression
The running time of prefixAverages1 is O(1 + 2 + ...+ n) The sum of the first n integers is n(n + 1) / 2
7 6 5 4 3 2 1 0 1 2 3 4 5
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Thus, algorithm prefixAverages1 runs in O(n2) time
There is a simple visual proof of this fact
6
Analysis of Algorithms
Prefix Averages (Linear)
The following algorithm computes prefix averages in linear time by keeping a running sum Algorithm prefixAverages2(X, n) Input array ...