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Analysis of Algorithms
An algorithmis a step-by-step procedure forsolving a problem in a finite amount of time.
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Running Time
Most algorithms transforminput objects into outputobjects.
The running time of an
algorithm typically grows withthe input size.
Average case time is oftendifficult to determine.
We focus on the worst caserunning time. Easier to analyze
Crucial to applications such asgames, finance and robotics
0
20
40
60
80
100
120
RunningTime
1000 2000 3000 4000
Input Size
best case
average case
worst case
Analysis of Algorithms2
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Experimental Studies
Write a programimplementing the algorithm
Run the program with
inputs of varying size andcomposition
Use a function, like thebuilt-in clock()function, toget an accurate measure of
the actual running time
Plot the results0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 50 100
Input Size
Time(ms)
Analysis of Algorithms3
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Limitations of Experiments
Analysis of Algorithms4
It is necessary to implement the algorithm, whichmay 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
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Theoretical Analysis
Analysis of Algorithms5
Uses a high-level 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
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Pseudocode Details
Analysis of Algorithms7
Control flow
ifthen[else]
whiledo
repeatuntil fordo
Indentation replaces braces
Method declaration
Algorithm method(arg[, arg])Input
Output
Method/Function callvar.method (arg[, arg])
Return value
returnexpression ExpressionsAssignment
(like in C++)
Equality testing
(like in C++)n2Superscripts and other
mathematical formattingallowed
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The Random Access Machine (RAM)
Model
A CPU
An potentially unboundedbank of memorycells,
each of which can hold an
arbitrary number or
character
Analysis of Algorithms8
012
Memory cells are numbered and accessingany cell in memory takes unit time.
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Primitive Operations
Analysis of Algorithms9
Basic computations
performed by an algorithm
Identifiable in pseudocode
Largely independent from theprogramming language
Exact definition not important
(we will see why later)
Assumed to take a constant
amount of time in the RAM
model
Examples:
Evaluating an
expression
Assigning a value toa variable
Indexing into an
array
Calling a method
Returning from amethod
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Counting Primitive
Operations
Analysis of Algorithms10
By inspecting the pseudocode, we can determine themaximum number of primitive operations executed by analgorithm, as a function of the input size
AlgorithmarrayMax(A, n) # operationscurrentMaxA[0] 2fori1ton1do 2+n
ifA[i] currentMaxthen 2(n1)currentMaxA[i] 2(n1)
{ increment counter i} 2(n1)returncurrentMax 1
Total 7n1
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Running Time Calculation
Analysis of Algorithms11
Rule1ForLoops
Therunningtimeofaforloopisatmosttherunningtimeofthestatementinsidetheforloop(includingtests)timesthenumberofiterations
Rule2NestedLoopsAnalyzetheseinsideout.Thetotalrunning
timeofastatementinsideagroupofnestedloopsistherunningtimeofthestatementmultipliedbytheproductofthesizesofalltheloops
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Running Time Calculation
Example
Analysis of Algorithms12
Example1:sum = 0;
for (i=1; i
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Estimating Running Time
Analysis of Algorithms13
Algorithm arrayMaxexecutes 7n1 primitive
operations in the worst case. Define:
a= Time taken by the fastest primitive operation
b = Time taken by the slowest primitive operation
Let T(n)be worst-case time of arrayMax.Then
a (7n1) T(n)b(7n1)
Hence, the running time T(n)is bounded by two
linear functions
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Growth Rate of Running Time
Analysis of Algorithms14
Changing the hardware/ software environment
Affects T(n)by a constant factor, but
Does not alter the growth rate of T(n)
The linear growth rate of the running time T(n)is
an intrinsic property of algorithm arrayMax
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Growth Rates
Growth rates of
functions:
Linear n
Quadratic n2
Cubic n3
In a log-log chart,
the slope of the line
corresponds to the
growth rate of the
function
Analysis of Algorithms15
1E-1
1E+1
1E+31E+5
1E+7
1E+9
1E+11
1E+13
1E+15
1E+171E+19
1E+21
1E+23
1E+25
1E+27
1E+29
1E-1 1E+1 1E+3 1E+5 1E+7 1E+9
T(n)
n
Cubic
Quadratic
Linear
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Constant Factors
Analysis of Algorithms16
The growth rate is
not affected by
constant factors or
lower-order terms
Examples
102n 105is a linear
function
10
5n2
10
8nis aquadratic function
1E-1
1E+11E+3
1E+5
1E+7
1E+9
1E+11
1E+13
1E+15
1E+17
1E+19
1E+21
1E+23
1E+25
1E-1 1E+2 1E+5 1E+8
T(n)
n
Quadratic
Quadratic
Linear
Linear
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Big-Oh Notation
Analysis of Algorithms17
Given functions f(n) and
g(n), we say that f(n) is
O(g(n))if there are
positive constants
cand n0such that
f(n)cg(n) for n n0
Example: 2n+10is O(n)
2n+10cn
(c2) n 10
n 10/(c2)
Pick c 3 and n0 10
1
10
100
1,000
10,000
1 10 100 1,000
n
3n
2n+10
n
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Big-Oh Example
Analysis of Algorithms18
Example: the function
n2is not O(n)
n2cn
n c The above inequality
cannot be satisfied since
cmust be a constant
1
10
100
1,000
10,000
100,000
1,000,000
1 10 100 1,000n
n^2
100n
10n
n
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Analysis of Algorithms19
More Big-Oh Examples
7n-27n-2 is O(n)
need c > 0 and n01 such that 7n-2 cn for n n0
this is true for c = 7 and n0= 1
3n3+ 20n2+ 53n3+ 20n2+ 5 is O(n3)
need c > 0 and n01 such that 3n3+ 20n2+ 5 cn3for n n0
this is true for c = 4 and n0= 21
3 log n + log log n3 log n + log log n is O(log n)
need c > 0 and n01 such that 3 log n + log log n clog n for n n0
this is true for c = 4 and n0= 2
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Big-Oh and Growth Rate
Analysis of Algorithms20
The big-Oh 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 big-Oh notation to rank functions
according to their growth rate
f(n) is O(g(n)) g(n) is O(f(n))
g(n) grows more Yes No
f(n) grows more No Yes
Same growth Yes Yes
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Asymptotic Algorithm Analysis
Analysis of Algorithms22
The asymptotic analysis of an algorithm determinesthe running time in big-Oh notation
To perform the asymptotic analysis We find the worst-case number of primitive operations
executed as a function of the input size We express this function with big-Oh notation
Example: We determine that algorithm arrayMaxexecutes at most 7n1 primitive operations
We say that algorithm arrayMaxruns in O(n) time Since constant factors and lower-order terms are
eventually dropped anyhow, we can disregard themwhen counting primitive operations
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Math you need to Review
properties of logarithms:
logb(xy) = logbx + logby
logb(x/y) = logbx - logbylogbxa = alogbx
logba = logxa/logxb
properties of exponentials:a(b+c)= aba c
abc= (ab)cab/ac= a(b-c)b = a logabbc= a c*logab
Analysis of Algorithms23
Summations
Logarithms and Exponents
Proof techniques
Basic probability
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Analysis of Algorithms24
Relatives of Big-Oh
big-Omega
f(n) is (g(n)) if there is a constant c > 0
and an integer constant n01 such that
f(n) cg(n) for n n0
big-Theta f(n) is (g(n)) if there are constants c > 0 and c > 0 and an
integer constant n01 such that cg(n) f(n) cg(n) for n n0
little-oh
f(n) is o(g(n)) if, for any constant c > 0, there is an integerconstant n
00 such that f(n) cg(n) for n n
0little-omega
f(n) is (g(n)) if, for any constant c > 0, there is an integerconstant n00 such that f(n) cg(n) for n n0
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Intuition for Asymptotic
Notation
Analysis of Algorithms25
Big-Oh
f(n) is O(g(n)) if f(n) is asymptotically less than or equalto g(n)
big-Omega
f(n) is (g(n)) if f(n) is asymptotically greater than or equalto g(n)big-Theta
f(n) is (g(n)) if f(n) is asymptotically equalto g(n)
little-oh
f(n) is o(g(n)) if f(n) is asymptotically strictly lessthan g(n)little-omega
f(n) is (g(n)) if is asymptotically strictly greaterthan g(n)
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Analysis of Algorithms26
Example Uses of theRelatives of Big-Oh
f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n00 such that f(n) cg(n) for n n0
need 5n02cn0given c, the n0that satisfies this is n0c/5 0
5n2is (n)
f(n) is (g(n)) if there is a constant c > 0 and an integer constant n01such that f(n) cg(n) for n n0
let c = 1 and n0= 1
5n2is (n)
f(n) is (g(n)) if there is a constant c > 0 and an integer constant n01such that f(n) cg(n) for n n0
let c = 5 and n0= 1
5n2is (n2)