Bipartite Graph Edge Coloring Approach to Course Timetabling

K.LaxmiKanth U.Phanindra

R.V.R & J.C College Of Engg. R.V.R & J.C College Of Engg. laxmikanthkandi1234 @gmail.com phanindra.cse@gmail.com

Overview

• Overview on the TimeTabling• Keywords.• Various Approaches. • Proposed System.• Conclusion.

What is Scheduling?

What is Scheduling?

• Scheduling is a process of pre planning.

• Example: Tour Planning

What is TIME TABLE ?

What is TIME TABLE ?

Process of assigning limited RESOURCES to a

set of EVENTS without violating the set of

CONSTRAINTS.

Types Of Time Table

• Course Time Table• Examination Time Table

History

• Automated Timetabling is not todays problem, it already stated in past 40 years.

• From Bardadym`s 1995 survey ,– Interest on Timetabling was rapid growth in

1960s to 70s.–Lowering during 70`s–Again rapid growth from 1970`s to 1980`s.–And reaches peak in 1995

• Only in 1995, 60 papers are published.

1960 1970 1980 1990 2000

Inte

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Defination of Timetabling Problem

• ParametersTime slots (T).Resources (R).Constraints (C).

• Problem is the arrangement between Timeslots and Resources without violating Constraints.

What Happens if Constraints are Violated ????

• What happens in Real Life,consider Lab Record, it must be submitted before deadline, if not……

• In same way Penalties are increased based on the type of constraints .

Constraints Efiicency

Types of CONSTRAINTS

1. Hard constraint: Constraints that cannot be violated.

e.g., A group of students cannot be assigned to more than one course at the same time.

Types of CONSTRAINTS

2. Soft constraint: Preferences that do not contract with any time conflict and have lower penalty associated with them.

e.g., Lecturer should have maximum of four hours of classes in a day.

How we develop timetable?

There are several ways ,

Using graphs Heuristic methods and so on…..

KEYWORDS

• Graph.• Time Tabling.• Scheduling.• Bipartite Graph.• Coloring.

Graph

• The field of mathematics plays vital role in various fields.

• One of the important areas in mathematics is graph theory.

• Graph theoretical ideas are highly utilized by Computer science applications only.

Graph

• The field graph theory started its journey in 1735.

• In 1840, idea of complete graph and bipartite graph.

• The concept of tree was implemented by “Gustav Kirchhoff” in 1845, and he employed graph theoretical ideas in the calculation of currents in electrical networks or circuits.

• In 1852, coloring

Bipartite Graph

• Bipartite Graph (or BIGRAPH) is a graph whose

vertices can be divided into two disjoint sets U

and V such that every edge connects a vertex in U

to one in V; that is, U and V are independent sets.

Coloring

• Def: A coloring of a simple graph is assignment of colors to each vertex(edge, face)of a graph so that no two adjacent vertices are assigned the same color.

Types of coloring• 1.Vertex coloring: The vertices must be

colored differently if they are joined by an edge. No 2 adjacent vertices should get same color

Types Of coloring(Cont..)

• 2.Edge coloring: Edges with vertces in common must be colored differently.No 2 adjacent edges shoul get same color

Types Of coloring(cont..)

• 3.Face coloring: Faces with adjacent edges are always colored differently.

Applications of Graph Coloring

1. Coloring models to number of scheduling problems to schedule without conflicts.

2. In compilers, for code optimization.3. We use registers at processor for faster

execution, for REGISTER ALLOCATION coloring used.

4. Pattern Matching.5. Sudoku Problem solving.

Various methods Applied to the Course Timetabling Problem

How to start?

Techniques Applied to the Course Timetabling Problem

The EXISTING system uses “HEURISTIC methods”

The most fundamental heuristic is trial and error

Basic Idea of Heuristics

• Step 1: Generate an initial solution (based

on the computational history so far).

• Step 2: Apply (generalized) local search

to find a good locally optimal solution.

• Step 3: Halt if convergence condition is

met, after outputting the best solution

found so far. Otherwise return to Step 1.

Proposed System

Using Bipartite Graphs and Coloring techniques, we can easily solve this Time Tabling Problem

Proposed System

COMMON TERMS..

Course: Course is a subject that is taught byonly one lecturer.Class: A group of students.Timeslot: Start time and end time for the event to take place..

Algorithm

Input to the algorithm :

V1 is set of LecturersV2 is set of CoursesV3 is set of StudentsSet of Timeslots

AlgorithmFor every vertex in V2 do{While there are more edges to color ending at v1 {//getinitialTimeSlot();Get initial color for the edgeWhile color of the adjacent edges are distinct is false {Search for existing colour at edges from v2 ending at V1;Compare the initial color with the existing color ofadjacent edges;if (initial color == existing color of adjacent edges) {search for another color; //getAvailableTimeSlot(v1)Update the color with the new available color }Else { color = initial color; }

Algorithm// second graph, color edges from v2 to adjacent edge, ends at V3While (there are more edges to color from v2 ending at v3 and color of the adjacent edges are distinct is true) {Search for existing color at edges from v2 ending at V3;Compare the color with the existing color of adjacentedges;if (color == existing color of adjacent edges) {Update color of the adjacent edges are distinct is false }Else {Update color of the adjacent edges are distinct is true;// color is distinct}}}}Update the color to respective edge;}

Proposed System

Expected OUTPUTS

Expected OUTPUTS

LUNCH

Conclusion

There are two main differences between existing system and present system ,

1.Time complexity.2.Efficiency.

Thank you for your attention

Any Queries ?