Analysis algorithm introduction Lecture# 1

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Analysis of AlgorithmsSpring 2008

Prepared byDr. Zahoor Jan

Department of Computer Science, University of Peshawar

Lecture 2 Introduction

IntroductionLec-1

11-02-08

Yahoo group

• Group name: Algo_UOP_Spring08 Group home

• page: http://groups.yahoo.com/group/Algo_UOP_Spring08

• Group email: [email protected]

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Lecture 2 Introduction

11-02-08

http://groups.yahoo.com/group/Algo_UOP_Spring08

mailto:[email protected]

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Analysis of Algo

Steps involved in writing a computer program:Steps involved in writing a computer program:

• Problem formulation & specification

• Design of solution

• Implementation

• Testing

• Documentation

• Evaluation of the solution11-02-08

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Problem formulation & specificationProblem formulation & specification

• Half the battle is knowing what problem to solve.

• Most problems have no simple precise specification.

• Some problems are impossible to formulate.

• Example: Gourmet recipe, World peace, Etc.

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How to formulate & specify a problem?How to formulate & specify a problem?

• Identify the problem parameters.

• Expressing the problem by a formal model.

• Looking for a solution for the model.

• In the absence of a solution, discover about the model.

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How to model and solve a problem?How to model and solve a problem?

• Using any branch of Mathematics or Science.

• Identifying problems of numerical nature.

• With a suitable model, attempt to find a solution in terms of that model

• Simultaneous linear equations for electrical circuits.

• Differential equations for predicting chemical reaction rates.

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AlgorithmAlgorithm

• Finite sequence of instructions.

• Each instruction having a clear meaning.

• Each instruction requiring finite amount of effort.

• Each instruction requiring finite time to complete.

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AlgorithmAlgorithm

• Finite sequence of instructions. An input should not take the program in an infinite loop

• Each instruction having a clear meaning. Very subjective. What is clear to me, may not be clear to you.

• Each instruction requiring finite amount of effort. Very subjective. Finite on a super computer or a P4?

• Each instruction requiring finite time to complete. Very subjective. 1 min, 1 hr, 1 year or a lifetime?

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Algorithm Definition

A finite set of statements that guarantees an optimal solution in finite interval of time

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Using a model to solve a complicated traffic Using a model to solve a complicated traffic light problemlight problem

GIVEN: A complex intersection.

OBJECTIVE: Traffic light with minimum phases.

SOLUTION:• Identify permitted turns, going straight is a “turn”.

• Make group of permitted turns.

• Make the smallest possible number of groups.

• Assign each phase of the traffic light to a group.

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AB

C

D

E

An intersection

Charsadda

Roads C and E are one way, others are two way.

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There are 13 permitted turns.

Some turns such as AB (from A to B) and AC can be carried out simultaneously.

Other like AD and EB cross each other and can not be carried out simultaneously.

The traffic light should permit AB and EC simultaneously, but should not allow AD and EB.

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AB

C

D

E

An intersection

AB & AC

AD & EB

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SOLUTION:• We model the problem using a structure called graph G(V,E).

• A graph consists of a set of points called vertices V, and lines connecting the points, called edges E.

•Drawing a graph such that the vertices represent turns.

• Edges between those turns that can NOT be performed simultaneously.

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AD


AB

C

D

E

An intersection

BD

AB AC

BA BC

DA DB DC

EA EC ED

Partial graph of incompatible turns

EB

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Partial table of incompatible turns


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SOLUTION:

The graph can aid in solving our problem.

A coloring of a graph is an assignment of a color to each vertex of the graph, so that no two vertices connected by an edge have the same color.

Our problem is of coloring the graph of incompatible turns using as few colors as possible.

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More on SOLUTION (Graph coloring):

Graph coloring problem has been studied for decades.

The theory of algorithms tells us a lot about it.

Unfortunately this belongs to a class of problems called as NP-Complete problems.

For such problems, all known solutions are basically “try all possibilities”

In case of coloring, try all assignments of colors.

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Approaches to attempting NP-Complete problems:

1. If the problem is small, might attempt to find an optimal solution exhaustively.

2.Look for additional information about the problem.

3.Change the problem a little, and look for a good, but not necessarily optimal solution.

An algorithm that quickly produces good but not necessarily optimal solutions is called a heuristic.

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Using model to solve complicated traffic light problemUsing model to solve complicated traffic light problem

A reasonable heuristic for graph coloring is the greedy “algorithm”.

Try to color as many vertices as possible with the first color, and then as many uncolored vertices with the second color, and so on.

The approach would be:

1. Select some uncolored vertex, and color with new color.

2. Scan the list of uncolored vertices.

3. For each uncolored vertex, determine whether it has an edge to any vertex already colored with the new color.

4. If there is no such edge, color the present vertices with the new color.11-02-08

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This is called “greedy”, because it colors a vertex, whenever it can, without considering potential drawbacks.

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Using model to solve complicated traffic light problemUsing model to solve complicated traffic light problem

Three colors used

Two colors used(optimum)(optimum)

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Work SMARTnot

HARD

Analysis of Algo

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Analysis algorithm introduction Lecture# 1

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