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CMPS 2200 Fall 2017 Sorting Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 11/17/17 1 CMPS 2200 Intro. to Algorithms How fast can we sort? All the sorting algorithms we have seen so far are

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SLIDE 1

11/17/17 CMPS 2200 Intro. to Algorithms 1

CMPS 2200 – Fall 2017

Sorting

Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk

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SLIDE 2

11/17/17 CMPS 2200 Intro. to Algorithms 2

How fast can we sort?

All the sorting algorithms we have seen so far are comparison sorts: only use comparisons to determine the relative order of elements.

  • E.g., insertion sort, merge sort, heapsort.

The best worst-case running time that we’ve seen for comparison sorting is O(nlogn). Is O(nlogn) the best we can do? Decision trees can help us answer this question.

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SLIDE 3

11/17/17 CMPS 2200 Intro. to Algorithms 3

Decision-tree model

A decision tree models the execution of any comparison sorting algorithm:

  • One tree per input size n.
  • The tree contains all possible comparisons (= if-branches)

that could be executed for any input of size n.

  • The tree contains all comparisons along all possible

instruction traces (= control flows) for all inputs of size n.

  • For one input, only one path to a leaf is executed.
  • Running time = length of the path taken.
  • Worst-case running time = height of tree.
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SLIDE 4

11/17/17 CMPS 2200 Intro. to Algorithms 4

Decision-tree for insertion sort:

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1

Each internal node is labeled ai:aj for i, j {1, 2,…, n}.

  • The left subtree shows subsequent comparisons if ai  aj.
  • The right subtree shows subsequent comparisons if ai  aj.

Sort a1, a2, a3

         

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

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SLIDE 5

11/17/17 CMPS 2200 Intro. to Algorithms 5

Decision-tree for insertion sort

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1

Each internal node is labeled ai:aj for i, j {1, 2,…, n}.

  • The left subtree shows subsequent comparisons if ai  aj.
  • The right subtree shows subsequent comparisons if ai  aj.

Sort a1, a2, a3

         

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

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SLIDE 6

11/17/17 CMPS 2200 Intro. to Algorithms 6

Decision-tree for insertion sort

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1

Each internal node is labeled ai:aj for i, j {1, 2,…, n}.

  • The left subtree shows subsequent comparisons if ai  aj.
  • The right subtree shows subsequent comparisons if ai  aj.

Sort a1, a2, a3

        

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

9  4

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SLIDE 7

11/17/17 CMPS 2200 Intro. to Algorithms 7

Decision-tree for insertion sort

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1

Each internal node is labeled ai:aj for i, j {1, 2,…, n}.

  • The left subtree shows subsequent comparisons if ai  aj.
  • The right subtree shows subsequent comparisons if ai  aj.

Sort a1, a2, a3

        

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

9  6

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SLIDE 8

11/17/17 CMPS 2200 Intro. to Algorithms 8

Decision-tree for insertion sort

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1

Each internal node is labeled ai:aj for i, j {1, 2,…, n}.

  • The left subtree shows subsequent comparisons if ai  aj.
  • The right subtree shows subsequent comparisons if ai  aj.

Sort a1, a2, a3

        

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

4  6

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SLIDE 9

11/17/17 CMPS 2200 Intro. to Algorithms 9

Decision-tree for insertion sort

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1

Each internal node is labeled ai:aj for i, j {1, 2,…, n}.

  • The left subtree shows subsequent comparisons if ai  aj.
  • The right subtree shows subsequent comparisons if ai  aj.

Sort a1, a2, a3

         

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

46  9

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SLIDE 10

11/17/17 CMPS 2200 Intro. to Algorithms 10

Decision-tree for insertion sort

a1:a2 a2:a3 a1a2a3 a1:a3 a1a3a2 a3a1a2 a1:a3 a2a1a3 a2:a3 a2a3a1 a3a2a1 Sort a1, a2, a3

         

a1 a2 a3 a1 a2 a3 a2 a1 a3 i j i j i j a2 a1 a3 i j a1 a2 a3 i j insert a3 insert a3 insert a2

46  9 Each leaf contains a permutation , ,…, (n) to indicate that the ordering a(1)  a(2)   a(n) has been established.

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SLIDE 11

11/17/17 CMPS 2200 Intro. to Algorithms 11

Decision-tree model

A decision tree models the execution of any comparison sorting algorithm:

  • One tree per input size n.
  • The tree contains all possible comparisons (= if-branches)

that could be executed for any input of size n.

  • The tree contains all comparisons along all possible

instruction traces (= control flows) for all inputs of size n.

  • For one input, only one path to a leaf is executed.
  • Running time = length of the path taken.
  • Worst-case running time = height of tree.
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SLIDE 12

11/17/17 CMPS 2200 Intro. to Algorithms 12

Lower bound for comparison sorting

  • Theorem. Any decision tree that can sort n

elements must have height (nlogn).

  • Proof. The tree must contain  n! leaves, since

there are n! possible permutations. For a binary tree of height-h holds that #leaves  2h . Thus, n!  2h.  h  log(n!) (log is mono. increasing)  log ((n/2)n/2) = n/2 log n/2  h  (n log n) .

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SLIDE 13

11/17/17 CMPS 2200 Intro. to Algorithms 13

Lower bound for comparison sorting

  • Corollary. Mergesort is an asymptotically
  • ptimal comparison sorting algorithm.
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SLIDE 14

11/17/17 CMPS 2200 Intro. to Algorithms 14

Sorting in linear time

Counting sort: No comparisons between elements.

  • Input: A[0 . . n-1], where A[ j]{0, 1, 2, …, k-1} .
  • Output: B[0 . . n-1], sorted.
  • Auxiliary storage: C[0 . . k-1] .
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SLIDE 15

11/17/17 CMPS 2200 Intro. to Algorithms 15

Counting sort

for (i = 0; i < k; i++) C[i] = 0 for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == |{key = i}| for (i = 1; i < k; i++) C[i] = C[i] + C[i–1] // C[i] == |{key  i}| for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

1. 2. 3. 4.

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SLIDE 16

11/17/17 CMPS 2200 Intro. to Algorithms 16

Counting-sort example

A: 3 2 3 2 B:

1 2 3 4

C:

1 2 3

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SLIDE 17

11/17/17 CMPS 2200 Intro. to Algorithms 17

Loop 1

A: 3 2 3 2 B: C: for (i = 0; i < k; i++) C[i] = 0

1.

1 2 3 4 1 2 3

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SLIDE 18

11/17/17 CMPS 2200 Intro. to Algorithms 18

Loop 2

A: 3 2 3 2 B: C: 1 for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == |{key = i}|

2.

1 2 3 4 1 2 3

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SLIDE 19

11/17/17 CMPS 2200 Intro. to Algorithms 19

Loop 2

A: 3 2 3 2 B: C: 1 1 for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == |{key = i}|

2.

1 2 3 4 1 2 3

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SLIDE 20

11/17/17 CMPS 2200 Intro. to Algorithms 20

Loop 2

A: 3 2 3 2 B: C: 1 1 1 for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == |{key = i}|

2.

1 2 3 4 1 2 3

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SLIDE 21

11/17/17 CMPS 2200 Intro. to Algorithms 21

Loop 2

A: 3 2 3 2 B: C: 1 1 2 for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == |{key = i}|

2.

1 2 3 4 1 2 3

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SLIDE 22

11/17/17 CMPS 2200 Intro. to Algorithms 22

Loop 2

A: 3 2 3 2 B: C: 1 2 2 for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 // C[i] == |{key = i}|

2.

1 2 3 4 1 2 3

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SLIDE 23

11/17/17 CMPS 2200 Intro. to Algorithms 23

Loop 3

A: 3 2 3 2 B: C: 1 2 2 C': 1 1 2 2 for (i = 1; i < k; i++) C[i] = C[i] + C[i–1] // C[i] == |{key  i}|

3.

1 2 3 4 1 2 3

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SLIDE 24

11/17/17 CMPS 2200 Intro. to Algorithms 24

Loop 3

A: 3 2 3 2 B: C: 1 2 2 C': 1 1 3 2 for (i = 1; i < k; i++) C[i] = C[i] + C[i–1] // C[i] == |{key  i}|

3.

1 2 3 4 1 2 3

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SLIDE 25

11/17/17 CMPS 2200 Intro. to Algorithms 25

Loop 3

A: 3 2 3 2 B: C: 1 2 2 C': 1 1 3 5 for (i = 1; i < k; i++) C[i] = C[i] + C[i–1] // C[i] == |{key  i}|

3.

1 2 3 4 1 2 3

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SLIDE 26

11/17/17 CMPS 2200 Intro. to Algorithms 26

Loop 4

A: 3 2 3 2 B: 2 C: 1 1 3 5 C': 1 1 3 5 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 27

11/17/17 CMPS 2200 Intro. to Algorithms 27

Loop 4

A: 3 2 3 2 B: 2 C: 1 1 3 5 C': 1 1 2 5 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 28

11/17/17 CMPS 2200 Intro. to Algorithms 28

Loop 4

A: 3 2 3 2 B: 2 3 C: 1 1 2 5 C': 1 1 2 5 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 29

11/17/17 CMPS 2200 Intro. to Algorithms 29

Loop 4

A: 3 2 3 2 B: 2 3 C: 1 1 2 5 C': 1 1 2 4 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 30

11/17/17 CMPS 2200 Intro. to Algorithms 30

Loop 4

A: 3 2 3 2 B: 2 2 3 C: 1 1 2 4 C': 1 1 2 4 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 31

11/17/17 CMPS 2200 Intro. to Algorithms 31

Loop 4

A: 3 2 3 2 B: 2 2 3 C: 1 1 2 4 C': 1 1 1 4 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 32

11/17/17 CMPS 2200 Intro. to Algorithms 32

Loop 4

A: 3 2 3 2 B: 2 2 3 C: 1 1 1 4 C': 1 1 1 4 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 33

11/17/17 CMPS 2200 Intro. to Algorithms 33

Loop 4

A: 3 2 3 2 B: 2 2 3 C: 1 1 1 4 C': 1 1 4 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 34

11/17/17 CMPS 2200 Intro. to Algorithms 34

Loop 4

A: 3 2 3 2 B: 2 2 3 3 C: 1 1 4 C': 1 1 4 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 35

11/17/17 CMPS 2200 Intro. to Algorithms 35

Loop 4

A: 3 2 3 2 B: 2 2 3 3 C: 1 1 4 C': 1 1 3 for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

4.

1 2 3 4 1 2 3

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SLIDE 36

11/17/17 CMPS 2200 Intro. to Algorithms 36

Analysis

for (i = 0; i < k; i++) C[i] = 0

(n) (k) (n) (k)

for (j = 0; i < n; j++) C[A[ j]] = C[A[ j]] + 1 for (i = 1; i < k; i++) C[i] = C[i] + C[i–1] for (j = n-1; i ≥ 0; j--) B[C[A[ j]]-1] = A[ j] C[A[ j]] = C[A[ j]] – 1

(n + k)

1. 2. 3. 4.

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SLIDE 37

11/17/17 CMPS 2200 Intro. to Algorithms 37

Running time

If k = O(n), then counting sort takes (n) time.

  • But, sorting takes (n log n) time!
  • Where’s the fallacy?

Answer:

  • Comparison sorting takes (n log n) time.
  • Counting sort is not a comparison sort.
  • In fact, not a single comparison between

elements occurs!

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SLIDE 38

11/17/17 CMPS 2200 Intro. to Algorithms 38

Stable sorting

Counting sort is a stable sort: it preserves the input order among equal elements. A: 3 2 3 2 B: 2 2 3 3 Exercise: What other sorts have this property?

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SLIDE 39

11/17/17 CMPS 2200 Intro. to Algorithms 39

Radix sort

  • Origin: Herman Hollerith’s card-sorting

machine for the 1890 U.S. Census. (See Appendix .)

  • Digit-by-digit sort.
  • Hollerith’s original (bad) idea: sort on

most-significant digit first (left to right).

  • Good idea: Sort on least-significant digit

first (right to left) with an auxiliary stable sorting algorithm (like counting sort). 3 2 9 4 5 7 6 5 7 8 3 9 4 3 6 7 2 0 3 5 5

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SLIDE 40

11/17/17 CMPS 2200 Intro. to Algorithms 40

Operation of radix sort

3 2 9 4 5 7 6 5 7 8 3 9 4 3 6 7 2 0 3 5 5 7 2 0 3 5 5 4 3 6 4 5 7 6 5 7 3 2 9 8 3 9 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9

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SLIDE 41

11/17/17 CMPS 2200 Intro. to Algorithms 41

  • Sort on digit t

Correctness of radix sort

Induction on digit position

  • Assume that the numbers

are sorted by their low-order t – 1 digits. 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9

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SLIDE 42

11/17/17 CMPS 2200 Intro. to Algorithms 42

  • Sort on digit t

Correctness of radix sort

Induction on digit position

  • Assume that the numbers

are sorted by their low-order t – 1 digits. 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9

  • Two numbers that differ in

digit t are correctly sorted.

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SLIDE 43

11/17/17 CMPS 2200 Intro. to Algorithms 43

  • Sort on digit t

Correctness of radix sort

Induction on digit position

  • Assume that the numbers

are sorted by their low-order t – 1 digits. 7 2 0 3 2 9 4 3 6 8 3 9 3 5 5 4 5 7 6 5 7 3 2 9 3 5 5 4 3 6 4 5 7 6 5 7 7 2 0 8 3 9

  • Two numbers that differ in

digit t are correctly sorted.

  • Two numbers equal in digit t

are put in the same order as the input  correct order.

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SLIDE 44

11/17/17 CMPS 2200 Intro. to Algorithms 44

Analysis of radix sort

  • Sort n computer words of b bits each.
  • View each word as having b/r base-2r digits.

Example: 32-bit word (b=32)

  • r = 1: 32 base-2 digits

 b/r = 32 passes of counting sort on base-2 digits

231 2423222120 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

  • r = 4: 32/4 base-24 digits (hexadecimal numbers)

(24)3 (24)2 (24)1 (24)0

0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

(24)7 (24)6 (24)5(24)4 163 162 161 160

0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

167 166 165 164

 b/r = 8 passes of counting sort on base-24 digits

163 162 161 160

2 8 13=D 3 12=C 6 7 5

167 166 165 164

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SLIDE 45

11/17/17 CMPS 2200 Intro. to Algorithms 45

Analysis of radix sort (cont.)

Example: 32-bit word (b=32)

  • r = 8: 32/8 base-28 digits

(28)3 (28)2 (28)1 (28)0

0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

2563 2562 2561 2560

0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

 b/r = 4 passes of counting sort on base-28 digits

2563 2562 2561 2560

40 211 198 117

  • r = 16: 32/16 base-216 digits

(216)1 (216)0

0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

655361 655360

0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1

 b/r = 2 passes of counting sort on base-216 digits

655361 655360

10451 50805

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SLIDE 46

11/17/17 CMPS 2200 Intro. to Algorithms 46

Analysis of radix sort

  • Sort n computer words of b bits each.
  • View each word as having b/r base-2r digits.
  • Assume counting sort is the auxiliary stable sort.
  • Make b/r passes of counting sort on base-2r digits

How many passes should we make?

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SLIDE 47

11/17/17 CMPS 2200 Intro. to Algorithms 47

Analysis (continued)

Recall: Counting sort takes (n + k) time to sort n numbers in the range from 0 to k – 1.

  • If each b-bit word is broken into r-bit pieces,

each pass of counting sort takes (n + 2r) time.

  • Since there are b/r passes, we have

 

       

r

n r b b n T 2 ) , ( .

  • Choose r to minimize T(n,b):

Increasing r means fewer passes, but as r >> log n, the time grows exponentially.

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SLIDE 48

11/17/17 CMPS 2200 Intro. to Algorithms 48

Choosing r

 

       

r

n r b b n T 2 ) , ( Minimize T(n,b) : Observe that we don’t want 2r > n, and there’s no harm asymptotically in choosing r as large as possible subject to this constraint. > Choosing r = log n implies T(n,b) = (bn/log n).

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SLIDE 49

11/17/17 CMPS 2200 Intro. to Algorithms 49

Radix Sort with optimized r

  • Example:

For numbers in the range from 0 to nd – 1, we have b = d log n  radix sort runs in (dn) time.

  • Notice that counting sort runs in O(n+k) time,

where all numbers are in the range 1 through k.

  • Assume counting sort is the auxiliary stable sort.
  • Sort n computer words of b bits each.

The runtime of radix sort is: T(n,b) = (bn/log n).

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SLIDE 50

11/17/17 CMPS 2200 Intro. to Algorithms 50

Conclusions

Example (32-bit numbers):

  • At most 3 passes when sorting  2000 numbers.
  • Merge sort and quicksort do at least log2000

= 11 passes. In practice, radix sort is fast for large inputs, as well as simple to code and maintain. Downside: Unlike quicksort, radix sort displays little locality of reference, and thus a well-tuned quicksort fares better on modern processors, which feature steep memory hierarchies.

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SLIDE 51

11/17/17 CMPS 2200 Intro. to Algorithms 51

Appendix: Punched-card technology

  • Herman Hollerith (1860-1929)
  • Punched cards
  • Hollerith’s tabulating system
  • Operation of the sorter
  • Origin of radix sort
  • “Modern” IBM card

Return to last slide viewed.

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SLIDE 52

11/17/17 CMPS 2200 Intro. to Algorithms 52

Herman Hollerith (1860-1929)

  • The 1880 U.S. Census took almost

10 years to process.

  • While a lecturer at MIT, Hollerith

prototyped punched-card technology.

  • His machines, including a “card sorter,” allowed

the 1890 census total to be reported in 6 weeks.

  • He founded the Tabulating Machine Company in

1911, which merged with other companies in 1924 to form International Business Machines.

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SLIDE 53

11/17/17 CMPS 2200 Intro. to Algorithms 53

Punched cards

  • Punched card = data record.
  • Hole = value.
  • Algorithm = machine + human operator.

Replica of punch card from the 1900 U.S. census. [Howells 2000]

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SLIDE 54

11/17/17 CMPS 2200 Intro. to Algorithms 54

Hollerith’s tabulating system

  • Pantograph card

punch

  • Hand-press reader
  • Dial counters
  • Sorting box

Figure from [Howells 2000].

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SLIDE 55

11/17/17 CMPS 2200 Intro. to Algorithms 55

Operation of the sorter

  • An operator inserts a card into

the press.

  • Pins on the press reach through

the punched holes to make electrical contact with mercury- filled cups beneath the card.

  • Whenever a particular digit

value is punched, the lid of the corresponding sorting bin lifts.

  • The operator deposits the card

into the bin and closes the lid.

  • When all cards have been processed, the front panel is opened, and

the cards are collected in order, yielding one pass of a stable sort.

Hollerith Tabulator, Pantograph, Press, and Sorter

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SLIDE 56

11/17/17 CMPS 2200 Intro. to Algorithms 56

Origin of radix sort

Hollerith’s original 1889 patent alludes to a most- significant-digit-first radix sort:

“The most complicated combinations can readily be counted with comparatively few counters or relays by first assorting the cards according to the first items entering into the combinations, then reassorting each group according to the second item entering into the combination, and so on, and finally counting on a few counters the last item of the combination for each group of cards.”

Least-significant-digit-first radix sort seems to be a folk invention originated by machine operators.

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SLIDE 57

11/17/17 CMPS 2200 Intro. to Algorithms 57

“Modern” IBM card

So, that’s why text windows have 80 columns!

Produced by the WWW Virtual Punch- Card Server.

  • One character per column.