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Bead-Sort - A natural algorithm for sorting

Bead-Sort - A natural algorithm for sorting. Beads sort themselves out. Bead-Stack machine The “tilt” operation. 90 0. Bead-Sort: illustration 1. 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.

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Bead-Sort - A natural algorithm for sorting

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  1. Bead-Sort - A natural algorithm for sorting

  2. Beads sort themselves out

  3. Bead-Stack machine The “tilt” operation 900

  4. Bead-Sort: illustration 1 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

  5. Bead-Sort: illustration 2 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

  6. Bead-Sort: illustration 3 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

  7. Bead-Sort: illustration 4 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

  8. Bead-Sort: illustration 5 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

  9. 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.

  10. 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)

  11. 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

  12. 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

  13. Parallel implementation 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

  14. Parallel simulation 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

  15. Bead-Stack machine 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.

  16. Bead-Stack machine The “tilt” operation 900

  17. 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

  18. Bead-Stack machine 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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