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Three Classic Computer Science Problems, and the

Algorithms to Solve Them

- Matthew Hyde
- PhD Student in Computer Science
- ASAP Research Group

Outline

- 1) The 1 Dimensional Bin Packing
- 2) The Knapsack Problem
- 3) The Travelling Salesman Problem
- A hard real life problem made up of two of these

Outline

- 1) The 1 Dimensional Bin Packing
- 2) The Knapsack Problem
- 3) The Travelling Salesman Problem
- A hard real life problem made up of two of these

1 Dimensional Bin Packing

- You have a number of pieces of different sizes
- The pieces can be put into containers, called

bins - The bins have a certain capacity
- You must pack the pieces into the smallest amount

of bins possible

1 Dimensional Bin Packing

106

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37

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Capacity 524

Practical Applications

- Lengths of material in factories
- Cast metal comes in predefined stock lengths
- You have to cut the stock lengths to get the

lengths you want - Any waste at the end of a stock length is a cost

to be avoided - A 2 Dimensional example
- wrapping paper for presents comes in predefined

sizes - Any unused areas are waste and will mean you have

to buy more sheets

Practical Applications

- TV advert scheduling
- 5 minute break
- Adverts are all different lengths
- Fit as many adverts in as possible

Why do you need algorithms?

- For this problem you can solve a small example

with a pen and paper just trying all the

combinations of pieces - Like going through combinations on a combination

lock - What about if there were 100 pieces
- Or 100,000 pieces?

Algorithm First-Fit

- Order the pieces from largest to smallest
- Put each piece in the first bin that it will fit

in

Algorithm First-Fit

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Capacity 524

Algorithm First-Fit

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487

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Algorithm First-Fit

524 524 524 524 524 524

524

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Algorithm First-Fit

46

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442 504 504 504 506 487

487

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Algorithm First-Fit

516 516 516 516 516 517

516 9

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Algorithm Best-Fit

- Order the pieces from largest to smallest
- Put the piece in the bin with the smallest gap,

out of the bins that it fits in

Algorithm Best-Fit

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Capacity 524

Algorithm Best-Fit

Still fits into 7 bins!

10

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442 504 504 504 506 487

487

442 504 516 516 518 524

524

500 524 516 516 518 524

524

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Outline

- 1) The 1 Dimensional Bin Packing
- 2) The Knapsack Problem
- 3) The Travelling Salesman Problem
- A hard real life problem made up of two of these

The Knapsack Problem

- The Burglars dilemma!
- If a burglar breaks into a house, they cannot

take everything

- Some things are more valuable
- Their bag can only hold a certain amount

The Knapsack Problem

- One container
- Each piece has a weight
- Each piece has a value
- Maximise the value of the items in the bag

Algorithm Highest-Value-First

- For each piece, calculate its value per kg
- Order the pieces by value per kg
- Put the pieces in one by one
- If the piece doesnt fit, go to the next one

until you reach the end

Algorithm Highest-Value-First

A Problem For You

Weight / Value

50 / 100

50 / 25

30/80

30/80

30/80

5/15

5/15

5/15

15/40

5/15

5/15

15/40

15/40

15/40

The Knapsack

Capacity 120

Value 315

Optimal Answer

Weight / Value

50 / 100

50 / 25

5/15

5/15

15/40

15/40

15/40

The Knapsack

5/15

30/80

5/15

30/80

30/80

15/40

5/15

Capacity 120

Value 325

Outline

- 1) The 1 Dimensional Bin Packing
- 2) The Knapsack Problem
- 3) The Travelling Salesman Problem
- A hard real life problem made up of two of these

Travelling Salesman problem

- What is the shortest route visiting each city

exactly once and then returns to the starting

city?

Travelling monkeys

- Monkeys can solve the travelling salesman problem

to a certain extent - Apparently they consider three points into the

future at least

from Nature. 29 May 1997 Vol 387 pp437-532

Applications

- Satellite Navigation in a car finds the shortest

path to your destination, and is constantly

updating - School busses, picking up from many stops in an

area - Parcelforce, FedEx, DHL delivery companies. One

driver will pick up many parcels in an area

Algorithms

- Nearest neighbour algorithm
- Greedy, doesnt think ahead
- Insertion algorithm
- Starts with a rough outline and fills in the gaps
- http//www-e.uni-magdeburg.de/mertens/TSP/node2.ht

ml

Outline

- 1) The 1 Dimensional Bin Packing
- 2) The Knapsack Problem
- 3) The Travelling Salesman Problem
- A hard real life problem made up of two of these

Difficult real world example

- Superstore delivery - 2 problems combined

Superstore delivery

- Pack the lorries efficiently - 3D bin packing
- Using less lorries saves money

Superstore delivery

- Decide the route the lorry will take
- But more difficult than the TSP because of

limited fuel and space so it has to make multiple

trips from the depot

Superstore delivery

- The houses you decide to visit determines which

goods you need to fit into the lorry - If some of the goods fit together very well, then

you may decide to alter the planned routes to

take advantage - So the two problems are interrelated in the

superstore delivery problem

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