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Bead-Sort - A natural algorithm for sortingPowerPoint Presentation

Bead-Sort - A natural algorithm for sorting

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- A natural algorithm for sorting

1

2

3

Sorting {3, 1, 2}

1. Drop 3 beads

(Remember, always from left-to-right)

2. Drop 1 bead

3. Drop 2 beads

1

2

3

Sorting {2, 3, 1}

1. Drop 2 beads

(Remember, always from left-to-right)

2. Drop 3 beads

3. Drop 1 beads

1

2

3

Sorting {1, 3, 2}

1. Drop 1 bead

(Remember, always from left-to-right)

2. Drop 3 beads

3. Drop 2 beads

1

2

3

4

Sorting {1, 3, 2, 4}

1. Drop 1 bead

(Remember, always from left-to-right)

2. Drop 3 beads

3. Drop 2 beads

4. Drop 4 beads

2

2

3

4

Sorting {2, 4, 3, 2}

1. Drop 2 beads

(Remember, always from left-to-right)

2. Drop 4 beads

3. Drop 3 beads

4. Drop 2 beads

Simulating Bead-Sort with a program

0

0

Level_Count

1

3

2

1

1

Rod_Count

- Two linear arrays used.
- Rod_Count keeps track of number
- of beads in each rod.
- Level_Count records number of
- beads in each level.
- Finally, Level_Count will contain
- sorted data.

Bead-Sort: Proof of correctness

x + i

x

x

i

1 row (k = 1)

(i)

i

(iii)

x

x

x + i

(ii)

Mathematical induction on number of rows of beads

- Claims:
- Rows of beads represent the same set of positive integers before and after dropping down.
- After dropping, each row has beads less than (or equal to) that on the row directly below it.

2 rows (k = 2)

Bead-Sort: Proof of correctness

the smallest amongst k+1 rows has already emerged on top

k+1th row

k rows

the smallest amongst k rows

Parallel implementation of Bead-Sort

Using a digital circuit

flip-flops

absence of bead - 0

presence of bead - 1

cn

cn

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

0

Data entry register

c2

c2

0

0

0

Input from corresponding cell in

data entry register

1

1

c1

c1

0

0

1

1

1

cit

Cit+1

0

1

1

ci-1

1

1

1

1

1

1

Using an analog circuit

1/2 A

1/3 A

1/6 A

sorted data

Trim

Trim

Trim

v1

1 V

1

1

1

0.5 V

0.3 V

0.2 V

3 2 1

Trim

Trim

Trim

2

2

v2

2 V

V1

V2

V3

2

1 V

0.7 V

0.3 V

Trim

Trim

Trim

v3

3 V

3

3

3

1.5 V

1 V

0.5 V

‘Trim’ Voltage :

Trim( v ) =

1 if v >= 0.5

0 otherwise

1 1 1 (no. 3)

1 0 0 (no. 1)

1 1 0 (no.2)

Calculating current

Resistor chain 1:

I1 = V1/R = 3/(1+2+3) = 1/2 A

Resistor chain 2:

I2 = V2/R = 2/(1+2+3) = 1/3 A

Resistor chain 3:

I3 = V3/R = 1/(1+2+3) = 1/6 A

1 1 0

2 1 0

3 2 1

1

1

0

DATA ENTRY

(strings of 1’s similar to balls)

Increase voltage by 1 unit every time a ‘1’ is sent

Using a cellular automaton (CA)

0

0

0

0

0

0

1

1

1

1

0

0

1

0

0

1

1

0

initial configuration

{2,1,3}

final configuration

{1,2,3}

1

1

0

1

1

1

0

1

0

1

0

1

0

1

0

0

0

1

1

0

1

0

0

0

0

1

1

1

1

1

0

0

0

0

1

1

1

1

1

2

3

4

5

6

7

8

CA rules for Bead-Sort simulation

Loading the input (beads)

1

2

3

4

Pushing button-1 will make one bead to drop down, pushing button-2 will make two beads to drop down (in parallel), and so on.

Beads rolled down along rod-1 will have the label “1”, those rolled down along rod-2 will have “2”, and so on.

1

1

2

1

2

3

1

2

3

4

only label ‘3’ is seen

labels stuck on the sides (of the beads)

Bead-Stack machine

Reading the output

Bead-Stack machine those rolled down along rod-2 will have “2”, and so on.

Complexity of inputting, sorting and outputting a list

Inputting a list:Using the gadget that loads a row-of-beads, we need N (loading) operations to input a list of size N (into the frame of the Bead-Stack machine).

Sorting a list:We tilt the frame 90o (a single operation, from the Bead-Stack machine’s perspective). The time taken by the falling beads to settle down is sqrt(2h/g), where h is the height of the rods and g, the acceleration due to gravity. If we fix the height of rods to be the same as the size (N) of the list, then the time taken is given by sqrt(2N/g), i.e. O(sqrt(N)).

Outputting a list:Nothing needs to be done to explicitly output rows of beads; the user could (literally)read the output (row by row).

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