analysis of algorithms-day3

8/9/2019 Analysis of algorithms-Day3

1/45

Analysis of AlgorithmsDay 3


2/45

Copyright 2004, InfosysTechnologies Ltd

2ER/CORP/CRS/SE15/003

Version No: 2.0

Plan for Day-3

Analysis of well known algorithms

Algorithm Design Techniques Dynamic Programming

Intractable problems

Deterministic Vs Non-Deterministic machines

P Vs NP

NP Complete

NP Hard

Case Study ( for self study)


3/45



Version No: 2.0

Dynamic Programming

Dynamic Programming is a design principle which is used to solve problemswith overlapping sub problems

It solves the problem by combining the solutions for the sub problems

The difference between Dynamic Programming and Divide and Conquer is that

the sub problems in Divide and Conquer are considered to be disjoint anddistinct various in Dynamic Programming they are overlapping.


4/45



Version No: 2.0

Dynamic Programming Matrix multiplication

Matrix Multiplication:

Consider the matrices M1 x M2 x M3 x M4 The corresponding dimensions are (10,20), (20,50), (50,1), (1,100) respectively

In recursion, the multiplication will be done in the following order:

M1 x ( M2 x ( M3 x M4 ) )

M3

x M4

will need 50 x 100 = 5000 multiplications

M2 x ( M3 x M4 ) will need 20 x 50 x 100 = 100000 multiplications

M1 x ( M2 x ( M3 x M4 ) ) will need 10 x 20 x 100 = 20000 multiplications

So total number of multiplications = 125000


5/45



Version No: 2.0

Dynamic Programming Matrix multiplication

(Contd) The same matrix multiplication is done in the following order:

( M1 x ( M2 x M3 ) ) x M4 M2 x M3 will need 20 x 50 x 1 = 1000 multiplications

M1 x ( M2 x M3 ) will need 10 x 20 x 1 = 200 multiplications

( M 1 x ( M2 x M3 ) ) x M4 will need 10 x 1 x 100 = 1000 multiplications

Total number of multiplications = 2200.


6/45



Version No: 2.0

Dynamic Programming Computing a BinomialCoefficient

The Binomial Coefficient, denoted by C(n,k), is the number of combinations of

kelements from a set of n elements.

The Binomial Coefficient is computed using the following formula

C(n,k) = C(n-1,k-1) + C(n-1,k) for all n > k > 0 and C(n,0) = C(n,n) = 1


7/45



Version No: 2.0

Dynamic Programming Computing a BinomialCoefficient (Contd)

0 1 2 . . . k-1 k

0 1

1 1 1

2 1 2 1

.k 1 1

.

n-1 1 C(n-1,k-1) C(n-1,k)

n 1 C(n,k)


8/45



Version No: 2.0

Dynamic Programming Computing a BinomialCoefficient (Contd)

Computing a Binomial Coefficient:

1. Begin

2. For i = 0 to n do2.1 For j = 0 to min(i,k) do

2.1.1 If j = 0 or j = k then C(i,j] = 1else C[i,j] = C[i-1, j-1] + C[i-1, j]

3. Output C[n,k]

4. End


9/45

Analysis of AlgorithmsUnit 5 - Intractable Problems


10/45



Version No: 2.0

Intractable Problems

Tractable Problems vs. Intractable Problems

Polynomial Problems

NP Problems

P vs NP

NP Complete and NP Hard Problems


11/45



Version No: 2.0


An algorithm for a given problem is said to be a polynomial timealgorithm if itsworst case complexity belongs to O (nk) for a fixed integer k and an input size

of n.

The set of all problems that can be solved in polynomial amount of timearecalled Tractable Problems. These problems can run in a reasonable amountof time for even very large amounts of input data.

The set of all problems that cannot be solved in polynomial amount of time arecalled Intractable Problems. It is of type O( kn). Intractable problems requirehuge amounts of time for even modest input sizes.


12/45



Version No: 2.0

Tractable Problems vs. Intractable Problems(Contd)

1.1 trillion1.1 billion1 million10242n

640002700080001000n3

1600900400100n2

2121478633nlogn

40302010n

5.34.94.33.3logn

11111


13/45



Version No: 2.0

Tractable Problems vs. Intractable Problems(Contd)

All problems

Solvableproblems

Tractablesolutions

Intractable solutions


14/45


14 ER/CORP/CRS/SE15/003

Version No: 2.0


Deterministic machines:

Conventional Digital machines are Deterministic in nature.

Serialization of resource access

Non Deterministic machines: Hypothetical machine.

More than one job can be done in one unit of time.


15/45



Version No: 2.0

P-Class problems

Polynomial problems are the set of problems which have polynomial timealgorithms

A formal definition for the same is given below

The class of decision problems that can be solved in polynomial time by

deterministic algorithms is called the P class or Polynomial problems.

Polynomial problems

O(1) -- Constant

O(log n) -- Sub-linear

O(n

) -- LinearO(n log n) -- Nearly linear

O(n2) -- Quadratic


16/45



Version No: 2.0

NP Problems (Contd)

NP problems are the set of problems which have nondeterministic polynomialtime algorithms


P NP

NP P

The class of decision problems that can be solved in polynomial time by

nondeterministic algorithms is called the NP class or Nondeterministic

Polynomial problems.

Not known

NP

P


17/45



Version No: 2.0

P Vs NP Deterministic algorithm for searching an

element

d:depth

A binary tree

No of elements 2d+1

-1 Searching an element.


18/45



Version No: 2.0

P Vs NP (Contd)- Non deterministic algorithm for

searching an element start

Machine having special capability- will check all numbers atone level and it will take one unit of time. Hence this will takeonly 4 unit of time


19/45



Version No: 2.0

NP Complete problems

A NP Complete problem is one which belongs to the NP class and which has a

surprising property. Every problem in NP class can be reduced to this problemin polynomial time on a deterministic machine


A decision problem Dis said to NP Complete if

1. If it belongs to NP class2. Every problem in the NP class is polynomially reducible to D


20/45



Version No: 2.0

Examples for NP Complete Problems

Traveling Salesman Problem (TSP): Given n cities and a salesman who is

required to visit each city once, find such a route.

Printed Circuit Board Problem (PCB): Consider a PCB (Printed Circuit Board)

manufacturing unit that churns out several thousands of etched and drilled PC

boards of the same type every day. The final stage of the process is drilling. To

minimize the drilling time required for each board, we need to decide on the

sequence in which holes will be drilled (The drilling time can be cut down by as

much as 50% by sequencing the holes using sophisticated algorithms rather than

relying on an intuitive or visually judged sequence).


21/45



Version No: 2.0

Examples for NP Complete Problems ctd.,

Bin Packing Problem: Given a set of items, each with a certain weight and a set

of bins, each with a constant weight capacity. You are required to pack the items

using the minimum number of bins.

Knapsack Problem: Given a set of items, each with a certain weight and value

and a knapsack with a certain weight capacity. You are required to pack the

knapsack with items so as to maximize the value of the packed items.

Node Cover Problem: You are given a network that consists of a set nodes, and

a set of edges. An edge is a connection between two nodes. You are required to

find a minimal set of nodes S such that every other edge in this network is

connected to at least one node in S.


22/45



Version No: 2.0

NP Hard Problems

A problem Pis said to be a NP Hard problem if any problem (not necessarily inNP) is polynomially reducible to P

NP Hard problems are basically the optimization versions of the problems in

NP Complete class

The NP Hard problems are not mere yes/no problems. They are problemswhere in we need to find the optimal solution


23/45



Version No: 2.0

Summary of Unit-5


Need for polynomial time solutions

Polynomial Problems

Deterministic vs. Non Deterministic Algorithms

NP Problems

P vs. NP

NP Complete and NP Hard Problems


24/45

Analysis of Algorithms

Case study


25/45



Version No: 2.0

SDLC

Requirementsgathering

Design

Build

Test

Deployment

Maintenance


26/45



Version No: 2.0

Writing a computer program

Problem formulation

& specification

implementation

Design

Testing

Documentation


27/45



Version No: 2.0

Models

Formalize a problem

Solve this problem

with the existingsolution

Discover what is

known about thismodel.

SolutionExists?

YesNo

Use the properties

to get a goodsolution


28/45



Version No: 2.0

Solving a Problem

(i) Get a mathematical model

(ii) Construct an algorithm

(iii) Analyse the Algorithm


29/45



Version No: 2.0

Analysis of algorithms

Help us in:

How to solve a problem?

How efficiently to solve the problem?

Converting the problem into a formal model will help in doing the so.


30/45



Version No: 2.0

Exercise

Write a program which accepts a string as input and returns 1 if the number ofcharacters in the string is even, else returns 0

Design an algorithm to solve the above problem.

Analyse the algorithm in terms of the resources required.


31/45



Version No: 2.0

Traffic signal design system

Are all the traffic signal systems alike?

If so, what is the generalised algorithm?

If not, how to design one?


32/45



Version No: 2.0

TSDS ( Cont.)

What is the input to TSDS?

(i) The intersection (junction) of roads.

(ii) Left / Right Drive.

Objective: To make the maximum

flow of traffic at the intersection

without any permissible collision.


33/45



Version No: 2.0

TSDS ( Cont.)

The roads A,B,C are two way roads and

D is a one way.

Here there are 9 possible turns

B

A

C

D


34/45



Version No: 2.0

TSDS (Cont.)

Among the 9 possible turns certain turns can be donesimultaneously like AB and CD.

For simplicity, take AB as AB.

List of possible turns:

AB, AC, AD, BA, BC, BD, CA, CB, CDConsider the Left drive as in our Indian Roads.

B

A

C

D


35/45


35 ER/CORP/CRS/SE15/003Version No: 2.0

Graph representation of incompatible turns

AB

CDCBCA

BDBCBA

ADAC

B

A

C

D


36/45



Graph coloring

B

A

C

D

Graph coloring: the assignment of color to eachnode such that no two nodes connected by an

edge have the same color.

CA, CBYellow

BA, BDBlue

AB, AC, AD, BC, CDRed

TurnsColor


37/45


38/45



Analysis

Where is the complexity of the problem Traffic Signal Design System lies?

Number of roads intersecting?

Right / Left drive?

Finding the incompatible turns?

Graph coloring?


39/45



Analysis Number of roads intersecting

If n roads are intersecting,

there will be n * (n-1) turns possible.

Complexity in Big oh notation: O(n2)It is polynomial in nature.


40/45



Analysis Left / Right drive

For the same junction the change of the drive will affect the number ofincompatible turns, but within the n* (n-1) turns.

So, this wont matter much


41/45



Analysis finding the incompatible turns

How to check for the incompatible turns?

Take two possible turns and check for the compatibility ( i.e. can be turnedsimultaneously)

Do the above process with all possible combinations taking into consideration the natureof the road (two/one way) and the drive (left/right).

Its Brute force.

Inefficient !


42/45


43/45



Analysis Graph coloring

A reasonable technique that can be applied here is greedy technique

A possible algorithm is:

Step 1: select an uncolored vertex and color it with a new color.

Step 2: check the uncolored vertices. For each uncolored vertices check if

there is any edge to a new colored vertex. If not, then color the present vertexwith the new color.


44/45



Summary of Day-3

Analysis of well known algorithms (continued)

Intractable problems


P Vs NP

NP Complete

NP Hard

Case Study


45/45



Thank You!

analysis of algorithms-day3

Documents