This document summarizes a lecture on analyzing algorithms. It discusses defining problems precisely, modeling problems mathematically, and designing algorithms to solve them. A specific example given is modeling a complex traffic intersection as a graph to determine the minimum number of traffic light phases. Colorings of the graph's vertices represent groups of permitted simultaneous turns. While finding optimal solutions is NP-complete, heuristics like greedy graph coloring can produce good approximations.

1. Lecture 2 Introduction
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Analysis of Algorithms
Spring 2008
Introduction
Lec-1
Prepared by
Dr. Zahoor Jan
Department of Computer Science, University of Peshawar
11-02-08

2. Lecture 2 Introduction
Yahoo group
• Group name: Algo_UOP_Spring08 Group home
• page: http://groups.yahoo.com/group/Algo_UOP_Spring08
• Group email: Algo_UOP_Spring08@yahoogroups.com
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3. 3
Analysis of Algo
Steps involved in writing aa ccoommppuutteerr pprrooggrraamm::
• Problem formulation & specification
• Design of solution
• Implementation
• Testing
• Documentation
• Evaluation of the solution
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4. Problem formulation && ssppeecciiffiiccaattiioonn
• 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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Analysis of Algo
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5. How to formulate && ssppeecciiffyy aa pprroobblleemm??
• 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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Analysis of Algo
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6. How ttoo mmooddeell aanndd ssoollvvee aa pprroobblleemm??
• 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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Analysis of Algo
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7. 7
Analysis of Algo
AAllggoorriitthhmm
• 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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8. AAllggoorriitthhmm
• 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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9. Algorithm Definition
A finite set of statements that guarantees an
optimal solution in finite interval of time
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10. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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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Analysis of Algo
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11. Analysis of Algo
Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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A
C
B
D
E
An intersection
Charsadda
Roads C and E are one way, others are two way.
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12. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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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Analysis of Algo
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13. Analysis of Algo
Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
13
A
C
B
D
E
An intersection
AB & AC
AD & EB
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14. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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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Analysis of Algo
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15. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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AD
A
C
B
D
E
An intersection
BD
AB AC
BA BC
DA DB DC
EA EB
EC ED
Partial graph of incompatible turns
Analysis of Algo
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16. Analysis of Algo
Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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Partial table of incompatible turns
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17. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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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Analysis of Algo
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18. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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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19. Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
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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20. 20
Analysis of Algo
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21. Using model to solve complicated ttrraaffffiicc lliigghhtt pprroobblleemm
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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22. UUssiinngg mmooddeell ttoo ssoollvvee ccoommpplliiccaatteedd ttrraaffffiicc lliigghhtt pprroobblleemm
This is called “greedy”, because it colors a vertex,
whenever it can, without considering potential
drawbacks.
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1 5
3
4
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1 5
3
4
2
Three colors used
Two colors used
((ooppttiimmuumm))
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23. Analysis of Algo
Using a mmooddeell ttoo ssoollvvee aa ccoommpplliiccaatteedd ttrraaffffiicc
lliigghhtt pprroobblleemm
Work SMART
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not
HARD 
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