Bead-Sort- A natural algorithm for sorting

Beads sort themselves out

900

Bead-Stack machineThe “tilt” operation

Sorting {3, 1, 2}

2. Drop 1 bead

3. Drop 2 beads

1. Drop 3 beads(Remember, always from left-to-right)

12

3

Bead-Sort: illustration 1

Sorting {2, 3, 1}

2. Drop 3 beads

3. Drop 1 beads

1. Drop 2 beads(Remember, always from left-to-right)

1

2

3

Bead-Sort: illustration 2

Sorting {1, 3, 2}

2. Drop 3 beads

3. Drop 2 beads

1. Drop 1 bead(Remember, always from left-to-right)

1

2

3

Bead-Sort: illustration 3

Sorting {1, 3, 2, 4}

2. Drop 3 beads

3. Drop 2 beads

1. Drop 1 bead(Remember, always from left-to-right)

4. Drop 4 beads

3

4

1

2

Bead-Sort: illustration 4

Sorting {2, 4, 3, 2}

2. Drop 4 beads

3. Drop 3 beads

1. Drop 2 beads(Remember, always from left-to-right)

4. Drop 2 beads

4

2

2

3

Bead-Sort: illustration 5

Simulating Bead-Sort with a program

• 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.

1

3

0

0

Lev

el_C

ount

Rod_Count

2 1 1

Bead-Sort: Proof of correctnessMathematical induction on number of rows of beadsClaims:

(i) Rows of beads represent the same set of positive integers before and after dropping down.

(ii) After dropping, each row has beads less than (or equal to) that on the row directly below it.

1 row (k = 1)

2 rows (k = 2)

(i)

(ii)

(iii)

x

x + i x i

xx + i

x

i

Bead-Sort: Proof of correctness

k+1th row

k rows

the smallest amongst k rows

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

Parallel implementation of Bead-SortUsing a digital circuit

flip-flops

presence of bead - 1

absence of bead - 0

c1

c2

cn

1 1 1

1 1

1 1 1

0

0

0

0 0

0

0

0

0

0

00

c1

c2

cn

1 1 1

1 1

1 1

1

0

0

0

0 0

0

0

0

0

00

0Data entry register 1 1 1

0ci

t

ci-1

Cit+1

Input from corresponding cell in data entry register

Calculating currentResistor chain 1:I1 = V1/R = 3/(1+2+3) = 1/2 AResistor chain 2:I2 = V2/R = 2/(1+2+3) = 1/3 AResistor chain 3:I3 = V3/R = 1/(1+2+3) = 1/6 A

1 1 0 (no.2)

1 0 0 (no. 1)

1 1 1 (no. 3)

3 2 1

‘Trim’ Voltage :

Trim( v ) =

1 if v >= 0.5

0 otherwise

V2V2V1V1 V3V3

Trim

Trim

Trim

Trim

Trim

Trim

Trim

Trim

Trim

vv11

vv22

vv33

1

2

3

1

2

3

1

2

3

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

1 1 0

(strings of 1’s similar to balls)DATA ENTRY

1 1 01 1 02 1 02 1 03 2 13 2 1

1/2 A 1/3 A 1/6 A

0.5 V

1 V

1.5 V

0.3 V

0.7 V

1 V

0.2 V

0.3 V

0.5 V

1 V

2 V

3 V

sorted data

Parallel implementation Using an analog circuit

000

initial configuration

{2,1,3}

1 1 1

1

11

0 0

0

final configuration

{1,2,3}

000

1

1

1

1

11

0

00

0

0 0

0

1

0 1

0

0

0 0

1

1

0 1

1

0

1 0

0

1

1 0

0

0

1 1

1

1

1 1

1

1 2 3 4 5 6 7 8

CA rules for Bead-Sort simulation

Parallel simulationUsing a cellular automaton (CA)

Bead-Stack machineLoading 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.

900

Bead-Stack machineThe “tilt” operation

Bead-Stack machineReading the output

only label ‘3’ is seen

11 22 33

11 22 33 44

11 22

11

labels stuck on the sides (of the beads)

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

Bead-Stack machineComplexity of inputting, sorting and outputting a listInputting 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).