COLOGNE UNIVERSITY OF APPLIED SCIENCES

March 9, 2015

Mohammad Aminul Islam (MatrNo: 01110381212)

Operations Research

Dynamic Programming Application

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Inhalt

1 Introduction ...................................................................................................................... 1

2 Basic Concept about dynamic programmingError! Bookmark not defined.

3 Application (Shortest Path calculation) ...................................................... 3

3.1 Greedy Algorithm ........................................................................................................ 4

3.2 Brute Force ................................................................................................................. 6

3.3 Dynamic Programming...............................................................7

4 Conclusion .............................................................................9

5 References ....................................................................................................................... 10

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1 Introduction

Dynamic programming is a method for solving difficult problem by breaking it down into

similar smaller problems. To solve a given problem by using dynamic programming need to

divide the problem, solve the different parts and combine the solution. Dynamic programming

overcome the overlapping problem, time consuming problem for shortest path calculation

between two points. Its practical application is huge. It introduces some new techniques such

as memorization, subdivision which have big impact on algorithm design, programming and

many more complex problems. Dynamic programming was introduced by Richard Bellman in

1940 for finding the best solution one after one. Now every programmer, designer considers

this as a technique for problem solution.1

2 Basic Concept about dynamic programming

Now we will see an example for calculating the Fibonacci number with Naive Recursive

algorithm and dynamic programming for getting a basic concept about it.

Normal Fibonacci number

1,1,2,3,5,8,13,21,........................................

In mathematical term Fibonacci represent by

Fn= Fn-1 +Fn-2 and F1=1, F2=1

Naive Recursive Algorithm:

Fib (n):

if n

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Why dynamic programming is efficient?

Fn

Figure 1: Recursion tree of Fibonacci number

According the picture, there is two times of Fn-3. If we use Naive Recursive algorithm

then need to calculate two times the same things. It is wastage of time. If we use Dynamic

Programming then dont need to calculate the same thing again because it is already in memory, so just need to call the result.

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3 Application (Shortest path calculation)

For an application of Dynamic Programming we will calculate the shortest path from a

multi stage diagram. We will also describe the problem using the other algorithms.

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1 2 6

9 2 5 4

7 7 4

3 2

3

11 5 5

2

11

8 6

Figure 2: Multi stage graph

According to the figure there are 5 stages and 12 nodes. We will calculate the shortest

path from node 1 to node 12 by using greedy algorithm, brute force and dynamic

programming. From node 1 to node 2 weight is 9, Node 1 to 3, 4, 5 weights are 7, 3, and 2

respectively. Node 1 stands on stage one, node 2,3,4,5 on stage two, node 6, 7, 8 on stage

three, node 9,10,11 on stage four and the last stage is node 12 on stage five.3

3 (Shortest path)

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8 11

10 12

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3.1 Greedy Algorithm

Now we will solve the shortest path calculation problem according to the greedy

algorithm. For visit one stage to another stage we have to select the smallest edge.

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1 2 6

9 2 5 4

7 7 4

3 2

3

11 5 5

2

11

8 6

Figure 3: shortest path according to the greedy algorithm

According to the greedy choice from stage 1 to stage 2 we have to select shortest edge.

From stage one to stage two the shortest path is 2 then stage two to three is 8, three to four is

5 and four to 5 is 2. The total weight is 17.

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9

8 11

10 12

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4

1 2 6

9 2 5 4

7 7 4

3 2

3

11 5 5

2

11

8 6

Figure 4: Opposite of the greedy algorithm

According to the figure 4, shows totally opposite of the greedy algorithm. From stage

one to stage two it visited the biggest edge, the weight is 9. This is totally opposite the greedy

algorithm. Then stage two to stage three weight is 2 , stage three to four weight is 3 and

finally stage four to stage five weight is 2. Total weight is 16. It is opposite the greedy

algorithm but it shows the shortest path. So, greedy algorithm fails here.

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3.2 Brute Force

Now we will calculate the shortest path by the help of brute force. According to the

brute force we have to calculate the every possible combination.

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2 3 4 5

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7 8 6 7

8 7 8

9 10

9 10 10 11 9 10 9 10 10 11 9 10 10 11

Figure 5: All possible combination tree

According to the Figure 5, there are lot of calculation thats mean time wasting. On the other hand here is overlapping problem. Same thing have to calculate again and again.

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3.3 Dynamic Programming

Now we will solve this problem by using dynamic programming.

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1 2 6

9 2 5 4

7 7 4

3 2

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11 5 5

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11

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Figure 6: Multistage graph.

Cost (i,j) = cost of edge i,j

Cost (i,j)= cost of shortest path from node j in stage i to the next stage

Cost (i,j)= min (cost (j,l) + cost (i+l,l))

We are going to solve this by backward method.

Cost of (3, 6) thats mean third stage and node 6. Third stage

Cost (3, 6) = min(c (6, 9) +c (4,9)

c (6,10)+c (4,10)) = 7

Cost (3, 7) = min (c (7, 9) + c (4, 9)

c (7,10) +c (4,10)) = 5

Cost (3, 8) = min(c(8,10) +c(4,10)

c (8,11) + c(4,11) = 7

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7

5

7

Figure: Third stage minimum weight

Second Stage:

Cost (2, 2) = min (4+7, 2+5, 1+7) = 7

Cost (2, 3) = min (2+7, 7+5) = 9

Cost (2, 4) = min (11+7) = 18

Cost (2, 5) = min (11+5, 11+7) = 16

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18

16

Figure: Second stage minimum path

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First Stage:

Cost (1, 1) = min (9+7, 7+9, 3+18, 2+16) = 16

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9

9

7

3 18

2 16

Figure: Shortest path

The shortest path is 16. We solved this problem by backward method. There are no

overlapping and unnecessary delays. We found two shortest path in our multistage graph.

Every stage we calculated one time and saved the result in the memory. So, next time

when we need the result then we called it.

Conclusion:

Finally, we can say that dynamic programming is very effective to solve any problem.

To design anything we should always consider the time and cost. Dynamic programming

concept can give that solution.

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References

1. Wikipedia: http://en.wikipedia.org/wiki/Dynamic_programming#History

2. MIT:https://www.youtube.com/watch?v=OQ5jsbhAv_M

3. Shortest path: https://www.youtube.com/watch?v=m5Y-4TsXsJ0

4. https://www.youtube.com/watch?v=WN3Rb9wVYDY