Analysis

Running Time (§1.1) w Most algorithms transform

Analysis of Algorithms

Input

Algorithm

Output

n

An algorithm is a step-by-step procedure for solving a problem in a finite amount of time.

n

best case average case worst case 120 100

Running Time

input objects into output objects. w The running time of an algorithm typically grows with the input size. w Average case time is often difficult to determine. w We focus on the worst case running time.

80 60 40 20

Easier to analyze Crucial to applications such as games, finance and robotics

0

1000

2000

3000

Analysis of Algorithms

Experimental Studies (§ 1.6) w Write a program

w It is necessary to implement the

8000

algorithm, which may be difficult

7000

Time (ms)

2

Limitations of Experiments

9000

implementing the algorithm w Run the program with inputs of varying size and composition w Use a method like System.currentTimeMillis() to get an accurate measure of the actual running time w Plot the results

4000

Input Size

w Results may not be indicative of the

6000

running time on other inputs not included in the experiment. w In order to compare two algorithms, the same hardware and software environments must be used

5000 4000 3000 2000 1000 0 0

50

100

Input Size Analysis of Algorithms

3

Theoretical Analysis

4

Pseudocode (§1.1) Example: find max 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 currentMax ← A[0] Preferred notation for for i ← 1 to n − 1 do describing algorithms if A[i] > currentMax then Hides program design currentMax ← A[i] issues return currentMax

w High-level description

w Uses a high-level description of the algorithm instead of an implementation w Characterizes running time as a function of the input size, n. w Takes into account all possible inputs w Allows us to evaluate the speed of an algorithm independent of the hardware/software environment Analysis of Algorithms

Analysis of Algorithms

5

w w w w

Analysis of Algorithms

6

The Random Access Machine (RAM) Model

Pseudocode Details w Method call

w Control flow n n n n n

if … then … [else …] while … do … repeat … until … for … do … Indentation replaces braces

w Method declaration Algorithm method (arg [, arg…]) Input … Output …

w A CPU

var.method (arg [, arg…])

w Return value

w An potentially unbounded

return expression

w Expressions

bank of memory cells, each of which can hold an arbitrary number or character

← Assignment (like = in Java) = Equality testing (like == in Java) n 2 Superscripts and other mathematical formatting allowed

w Memory cells are numbered and accessing any cell in memory takes unit time.

Analysis of Algorithms

7

w w w w

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

8

w By inspecting the pseudocode, we can determine the

w Examples: n

n

n

n n

maximum number of primitive operations executed by an algorithm, as a function of the input size

Evaluating an expression Assigning a value to a variable Indexing into an array Calling a method Returning from a method

Analysis of Algorithms

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

# operations 2 2 +n 2(n − 1) 2(n − 1) 2(n − 1) 1 Total

9

Estimating Running Time

7n − 1

Analysis of Algorithms

10

Growth Rate of Running Time

w Algorithm arrayMax executes 7n − 1 primitive

w Changing the hardware/ software

operations in the worst case. Define:

environment

a = Time taken by the fastest primitive operation b = Time taken by the slowest primitive operation

n

w Let T(n) be worst-case time of arrayMax. Then a (7n − 1) ≤ T(n) ≤ b(7n − 1)

n

Affects T(n) by a constant factor, but Does not alter the growth rate of T(n)

w The linear growth rate of the running

w Hence, the running time T(n) is bounded by two linear functions

Analysis of Algorithms

Analysis of Algorithms

Counting Primitive Operations (§1.1)

Primitive Operations w Basic computations

2

1

0

11

time T(n) is an intrinsic property of algorithm arrayMax Analysis of Algorithms

12

Growth Rates

Constant Factors

functions: n n

Linear ≈ n Quadratic ≈ n2 Cubic ≈ n 3

T (n )

n

w In a log-log chart,

the slope of the line corresponds to the growth rate of the function

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

w The growth rate is

Cubic

not affected by

Quadratic

constant factors or lower-order terms

n

Linear

n

w Examples

T ( n)

w Growth rates of

102 n + 105 is a linear function 105 n2 + 108 n is a quadratic function

n

n

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

Quadratic Quadratic Linear Linear

1E+0 1E+2

1E+4

1E+6

1E+8

1E+2

1E+4

1E+10

Analysis of Algorithms

13

f(n) ≤ cg(n) for n ≥ n0 w Example: 2n + 10 is O(n) n n n n

2n + 10 ≤ cn (c − 2) n ≥ 10 n ≥ 10/(c − 2) Pick c = 3 and n 0 = 10

1,000,000

3n

w Example: the function

2n+10

n2

n

n n n

10

10

100

n 2 ≤ cn n≤c The above inequality cannot be satisfied since c must be a constant

100n 10n n

10,000 1,000

1,000

100

1

10

100

1,000

n 15

Analysis of Algorithms

16

Big-Oh and Growth Rate w The big-Oh notation gives an upper bound on the

7n-2 is O(n) need c > 0 and n 0 ≥ 1 such that 7n-2 ≤ c•n for n ≥ n 0 this is true for c = 7 and n 0 = 1

growth rate of a function

w 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)

+ 20n2 + 5

3n 3 + 20n2 + 5 is O(n3 ) need c > 0 and n 0 ≥ 1 such that 3n3 + 20n 2 + 5 ≤ c•n3 for n ≥ n 0 this is true for c = 4 and n 0 = 21

log n + log log n

3 log n + log log n is O(log n) need c > 0 and n 0 ≥ 1 such that 3 log n + log log n ≤ c•log n for n ≥ n 0 this is true for c = 4 and n 0 = 2 Analysis of Algorithms

1

n

7n-2

n3

is not O(n)

10 1

More Big-Oh Examples

n 3n 3

14

n^2

100,000

1

Analysis of Algorithms

n

1E+10

Big-Oh Example

10,000

g(n), we say that f(n) is 1,000 O(g(n)) if there are positive constants 100 c and n0 such that

1E+8

Analysis of Algorithms

Big-Oh Notation (§1.2) w Given functions f(n) and

1E+6 n

n

17

w We can use the big-Oh notation to rank functions according to their growth rate

g(n) grows more f(n) grows more Same growth

f(n) is O(g(n))

g(n) is O(f(n))

Yes No Yes

No Yes Yes

Analysis of Algorithms

18

Big-Oh Rules

Asymptotic Algorithm Analysis w The asymptotic analysis of an algorithm determines

w If is f(n) a polynomial of degree d, then f(n) is O(n d), i.e., n n

n

Drop lower-order terms Drop constant factors

n

Say “2n is O(n)” instead of “2n is

n

O(n2 )”

n

w Use the simplest expression of the class n

Analysis of Algorithms

eventually dropped anyhow, we can disregard them when counting primitive operations

19

Computing Prefix Averages asymptotic analysis with two algorithms for prefix averages w The i-th 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)

w Computing the array A of

35

quadratic time by applying the definition

X A

30

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

25 20 15 10

0 1

2

3

4

5

6

Analysis of Algorithms

7

21

Arithmetic Progression

There is a simple visual proof of this fact

w Thus, algorithm

prefixAverages1 runs in O(n 2) time

20

Prefix Averages (Quadratic)

5

prefix averages of another array X has applications to financial analysis

n

Analysis of Algorithms

w The following algorithm computes prefix averages in

w We further illustrate

prefixAverages1 is O(1 + 2 + …+ n) w The sum of the first n integers is n(n + 1) / 2

We determine that algorithm arrayMax executes at most 7n − 1 primitive operations We say that algorithm arrayMax “runs in O(n) time”

w Since constant factors and lower-order terms are

Say “3n + 5 is O(n)” instead of “3n + 5 is O(3n)”

w The running time of

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

w Example:

w Use the smallest possible class of functions n

the running time in big-Oh notation

w To perform the asymptotic analysis

22

Prefix Averages (Linear) w The following algorithm computes prefix averages in

7

linear time by keeping a running sum

6 5 4 3 2 1 0

Analysis of Algorithms

Analysis of Algorithms

1

2

3

4

5

6 23

Algorithm prefixAverages2(X, n) Input array X of n integers Output array A of prefix averages of X A ← new array of n integers s←0 for i ← 0 to n − 1 do s ← s + X[i] A[i] ← s / (i + 1) return A

#operations

n 1 n n n 1

w Algorithm prefixAverages2 runs in O(n) time Analysis of Algorithms

24

Relatives of Big-Oh

Math you need to Review w Summations (Sec. 1.3.1) w Logarithms and Exponents (Sec. 1.3.2)

w big-Omega

w w

log b(xy) = logbx + logby log b (x/y) = log bx - log by log bxa = alogbx log ba = logx a/log x b w properties of exponentials: a(b+c) = aba c abc = (ab)c ab /ac = a(b-c) Proof techniques (Sec. 1.3.3) b = a loga b b c = a c*log a b Basic probability (Sec. 1.3.4) Analysis of Algorithms

25

Analysis of Algorithms

26

Example Uses of the Relatives of Big-Oh

Intuition for Asymptotic Notation Big-Oh n f(n) is O(g(n)) if f(n) is asymptotically less than or equal to g(n) big-Omega n f(n) is Ω(g(n)) if f(n) is asymptotically greater than or equal to g(n) big-Theta n f(n) is Θ(g(n)) if f(n) is asymptotically equal to g(n) little-oh n f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n) little-omega n f(n) is ω(g(n)) if is asymptotically strictly greater than g(n)

Analysis of Algorithms

f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant n0 ≥ 1 such that f(n) ≥ c•g(n) for n ≥ n0 w big-Theta n f(n) is Θ(g(n)) if there are constants c ’ > 0 and c’’ > 0 and an integer constant n 0 ≥ 1 such that c’•g(n) ≤ f(n) ≤ c’’•g(n) for n ≥ n0 w little-oh n f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant n 0 ≥ 0 such that f(n) ≤ c•g(n) for n ≥ n 0 w little-omega n f(n) is ω(g(n)) if, for any constant c > 0, there is an integer constant n 0 ≥ 0 such that f(n) ≥ c•g(n) for n ≥ n 0 n

w properties of logarithms:

27

n

5n 2 is Ω(n2 )

n

f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant n 0 ≥ 1 such that f(n) ≥ c•g(n) for n ≥ n0 let c = 5 and n0 = 1 5n 2 is Ω(n)

n

f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant n 0 ≥ 1 such that f(n) ≥ c•g(n) for n ≥ n0 let c = 1 and n0 = 1 2 5n is ω(n) f(n) is ω(g(n)) if, for any constant c > 0, there is an integer constant n0 ≥ 0 such that f(n) ≥ c•g(n) for n ≥ n0 need 5n02 ≥ c•n0 → given c, the n0 that satifies this is n0 ≥ c/5 ≥ 0 Analysis of Algorithms

28