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WEEK 2

CS 361: ADVANCED DATA STRUCTURES AND ALGORITHMS

Introduction to Algorithms

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Class Overview

Start thinking about analyzing a program or algorithm.

Understand algorithm efficiency and running-time complexity.

Analysis of an algorithm using Big-O notation.

Which Cost More to Feed?

Algorithm Efficiency

• There are often many approaches (algorithms) to solve a problem. • How do we choose between them?

• There are two (sometimes conflicting) goals at the heart of computer program design. To design an algorithm that:

1) is easy to understand, code, debug.

2) makes efficient use of the resources.

• Goal (1) is the concern of Software Engineering.

• Goal (2) is the concern of data structures and algorithm analysis.

• When goal (2) is important, • how do we measure an algorithm’s cost?

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Estimation Techniques

• Known as “back of the envelope” or “back of the napkin” calculation

1. Determine the major parameters that effect the problem.

2. Derive an equation that relates the parameters to the problem.

3. Select values for the parameters, and apply the equation to yield and estimated solution.

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Essentially, you need to understand the problem

Estimation Example

• How many library bookcases does it take to store books totaling one million pages?

• Estimate:• Pages/inch• Shelf/Feet• Shelves/bookcase

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Best, Worst, Average Cases

• Not all inputs of a given size take the same time to run.

• Sequential search for K in an array of n integers:• Begin at first element in array and look at each element in turn until

K is found

• Best case:• Worst case:• Average case:

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Time Analysis

Lower Bound Running Time Upper Bound

• Provides upper and lower bounds of running time.

• Different types of analysis:- Worst case- Best case- Average case

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Worst Case

• Provides an upper bound on running time.

• An absolute guarantee that the algorithm would not run longer, no matter what the inputs are.

Lower Bound Running Time Upper Bound

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Best Case

• Provides a lower bound on running time.

• Input is the one for which the algorithm runs the fastest.

Lower Bound Running Time Upper Bound

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Average Case

• Provides an estimate of “average” running time.

• Assumes that the input is random.

• Useful when best/worst cases do not happen very often• i.e., few input cases lead to best/worst cases.

Lower Bound Running Time Upper Bound

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Which Analysis to Use?

• While average time appears to be the fairest measure, It may be difficult to determine. For example, algorithms that are designed to operate on strings of

text.

• Why is the worst case time important?In some situations it may be necessary to use a pessimistic

analysis in order to guarantee safety.Recall the “bookcase” problem.

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How to Measure Efficiency?

• Critical resources:• Time, memory, programmer effort, user effort

• Factors affecting running time:• For most algorithms, running time depends on “size” of the input.• Running time is expressed as T(n) for some function T on input

size n.

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How do we analyze an algorithm?

• Need to define objective measures.

(1) Compare execution times? Not good:

times are specific to a particular machine.

(2) Count the number of statements? Not good:

number of statements varies with programming language and style.

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How do we analyze an algorithm? (cont.)

(3) Express running time T as a function of problem size n

(i.e., T=f(n) )

Asymptotic Algorithm Analysis

- Given two algorithms having running times f(n) and g(n), find which functions grows faster?

- Compare “rates of growth” of f(n) and g(n).

- Such an analysis is independent of machine time, programming style, etc.

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Understanding Rate of Growth

• Consider the example of feeding elephants and goldfish:

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Approximation: Total Cost ~ cost_of_feeding_elephants

Total Cost: (cost_of_feeding_elephants) + (cost_of_feeding_goldfish)

Understanding Rate of Growth (cont’d)

• The low order terms of a function are relatively insignificant for large n

n4 + 100n2 + 10n + 50

Approximation:n4

• Highest order term determines rate of growth!

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Visualizing Orders of Growth

• On a graph, as you go to the right, a faster growing function eventually becomes larger...

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Growth Rate Graph

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Common orders of magnitude

Orders of Magnitude

n log2n n log2n n2 n3 2n 2 1 2 4 8 4 4 2 8 16 64 16 8 3 24 64 512 256 16 4 64 256 4096 65536 32 5 160 1024 32768 4294967296 128 7 896 16384 2097152 3.4 x 1038

1024 10 10240 1048576 1073741824 1.8 x 10308

65536 16 1048576 4294967296 2.8 x 1014 Forget it!

Rate of Growth ≡ Asymptotic Analysis

• Using rate of growth as a measure to compare different functions implies comparing them asymptotically • i.e., as n

• If f(x) is growing faster than g(x), then f(x) always eventually becomes larger than g(x) in the limit• i.e., for large enough values of x

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Because we prefer the worst-case analysis !

Complexity

• Let us assume two algorithms A and B that solve the same class of problems.

• The time complexity of A is 5,000n, T = f(n) = 5000*n • the one for B is 2n for an input with n elements, T= g(n) = 2n

• For n = 10, • A requires 5*104 steps, • but B only 1024, • so B seems to be superior to A.

• For n = 1000, • A requires 5*106 steps, • while B requires 1.07*10301 steps.

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Asymptotic Notation

O notation: asymptotic “less than”:

f(n) = O(g(n)) implies: f(n) “≤” c*g(n) in the limit, c is a constant

worst-case analysisc is a constant

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In English: “ f(n) grows asymptotically no faster than g(n) ”

Asymptotic Notation

notation: asymptotic “greater than”:

f(n) = (g(n)) implies: f(n) “≥” c*g(n) in the limit , c is a constant

best-case analysis

*formal definition in CS477/677

c is a constant

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In English: “ f(n) grows asymptotically faster than g(n) ”

Asymptotic Notation

notation: asymptotic “equality”:

f(n)= (g(n)) implies: f(n) “=” c*g(n) in the limit , c is a constant

tight bound analysis (best and worst cases are same)

*formal definition in CS477/677

c is a constant

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In English: “ f(n) grows asymptotically as fast as g(n) ”

Common Misunderstanding

Worst case & Upper bound

Upper bound refers to a limit for the run-time of that algorithm.

Worst case refers to the worst input among the choices for possible inputs of a given size.

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Big O in practice

1. Figure out T=f(n): run-time (number of basic operations) required on an input of size n

2. Remove low-order terms

More on big-O

f(n) = O(g(n)) if “f(n)≤c*g(n)”

O(g(n)) can be related to a set of functions f(n)

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Big-O notation provides a machine independent means for determining the efficiency of an algorithm.

Names of Orders of Magnitude

O(1) bounded (by a constant) time

O(log2N) logarithmic time

O(N) linear time

O(N*log2N) N*log2N time

O(N2) quadratic time

O(N3) cubic time

O(2N ) exponential time

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Constant Time Algorithms

• An algorithm is O(1) when its running time is independent of the number of data items. The algorithm runs in constant time.

The storing of the element involves a simple assignment statement and thus has efficiency O(1).

fro nt rear

D irec t In se r t a t R ear

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Linear Time Algorithms

• An algorithm is O(n) when its running time is proportional to the size of the list.

• When the number of elements doubles, the number of operations doubles.

S e quential S earch fo r the Minim um E le m e nt in an A rray

32 46 8 12 3

m in im u m elem en t fou n d in th e list a fter n com p a rison s

n = 51 2 3 4 5

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Logarithmic Time Algorithms

• The logarithm of n, base 2, is commonly used when analyzing computer algorithms. For example, sorting algorithms.

Ex. log2(2) = 1

log2(75) = 6.2288

• When compared to the functions n and n2, the function log2n grows very slowly.

nn 2

lo g 2n

How do we calculate T=f(n) for a program/algorithm?

1) Associate a "cost" with each statement

2) Find total number of times each statement is executed

3) Add up the costs

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i = 0;

while (i<N)

{

X=X+Y; // O(1)

result = mystery(X); // O(N)

i++; // O(1)

}

Running Time Examples34

• The body of the while loop: O(N)• Loop is executed: N times

Running time of the entire iteration? N x O(N) = O(N2)

if (i<j)

for ( i=0; i<N; i++ )

X = X+i;

else

X=0;

O(N)

O(1)

Running Time Examples (cont.’d)

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Max (O(N), O(1)) = O(N)

Running time of the entire if-else statement?

Complexity Examples

What does the following algorithm compute?

returns the maximum difference between any two numbers in the input array

# of Comparisons: n-1 + n-2 + n-3 + … + 1 = (n-1)n/2 = 0.5n2 - 0.5n

Time complexity is O(n2)36

int who_knows(int a[n]) {

int m = 0;

for {int i = 0; i<n; i++}for {int j = i+1; j<n; j++}

if (abs(a[i] – a[j]) > m ) m = abs(a[i] – a[j]);

return m;}

Complexity Examples

Another algorithm solving the same problem:

# of Comparisons: 2n - 2

Time complexity is O(n).37

int max_diff(int a[n]) {

int min = a[0];int max = a[0];

for {int i = 1; i<n; i++} {

if (a[i] < min ) min = a[i];else if (a[i] > max ) max = a[i];

}

return max-min;}

Running time of various statements38

Examples (cont.’d)

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Examples (cont.’d)

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Analyze the complexity of the following code segments

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Homework #2: Algorithm analysis

• Already assigned on BB, due on 9/14/2014, 11:59PM

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Next class & Reading

• Next class: ADTs of Lists, Stacks, and Queues

• Book Chapter 3: “Lists, Stacks, and Queues”