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Insertion Sort and Radix Sort Algorithms - Prof. Hanan Samet, Study notes of Data Structures and Algorithms

University of Maryland Data Structures and Algorithms

Prof. Hanan Samet

An overview of two sorting algorithms: insertion sort and radix sort. Insertion sort is a simple sorting algorithm that builds a sorted list one item at a time. Radix sort is a non-comparison-based sorting algorithm that sorts data based on the value of their digits or bits. The document also discusses the properties and advantages of each algorithm.

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These notes may not be reproduced by any means (mechanical or elec-

tronic or any other) without the express written permission of Hanan Samet

SORTING TECHNIQUES

Hanan Samet

Computer Science Department and

Center for Automation Research and

Institute for Advanced Computer Studies

University of Maryland

College Park, Maryland 20742

e-mail: [email protected]

Discover Study notes of Data Structures and Algorithms University of Maryland

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st

These notes may not be reproduced by any means (mechanical or elec- tronic or any other) without the express written permission of Hanan Samet

SORTING TECHNIQUES

Hanan Samet

Computer Science Department and Center for Automation Research and Institute for Advanced Computer Studies University of Maryland College Park, Maryland 20742 e-mail: [email protected]

st

ISSUES

Nature of storage media
- fast memory
- secondary storage
Random versus sequential access of the data
Space
- can we use space in excess of the space occupied by the data?
Stability
- is the relative order of duplicate occurrences of a data value preserved?
- useful as adds a dimension of priority (e.g., time stamps in a scheduling application) for free

(^1) st SELECTION SORT b

Find the smallest element and output it 113 400 818 612 411 113 113 113 113 113

(^2) st r

(^1) st SELECTION SORT b

Find the smallest element and output it 113 400 818 612 411 113 113 113 113 113

(^2) st r

(^3) st z

(^1) st SELECTION SORT b

Find the smallest element and output it 113 400 818 612 411 113 113 113 113 113

(^2) st r

(^3) st z

(^4) st g

(^5) st v

(^1) st SELECTION SORT b

Find the smallest element and output it 113 400 818 612 411 113 113 113 113 113

(^2) st r

(^3) st z

(^4) st g

(^5) st v

(^6) st b

Step 1 causes no change as 113 is the smallest element

Step 2 causes 311 in position 6 to be exchanged with 400 in position 2 resulting in the first instance of 400 becoming the second instance

In-place variant can be made stable:
1. use linked lists

• onlyn ·(n –1)/2 comparisons

exchange adjacent elements

• O (n 2 ) time as alwaysn ·(n –1) comparisons

Needs extra space for output
In-place variant is not stable
Ex: 1 113

1 2

st INSERTION SORT

Values on the input list are taken in sequence and put in their proper position in the output list
Previously moved values may have to be moved again
Makes only one pass on the input list at cost of move operations

1 b

st INSERTION SORT

Values on the input list are taken in sequence and put in their proper position in the output list
Previously moved values may have to be moved again
Makes only one pass on the input list at cost of move operations

1 b

(^2) st r

st INSERTION SORT

Values on the input list are taken in sequence and put in their proper position in the output list
Previously moved values may have to be moved again
Makes only one pass on the input list at cost of move operations

1 b

(^2) st r

(^3) st z

(^4) st g

st INSERTION SORT

Values on the input list are taken in sequence and put in their proper position in the output list
Previously moved values may have to be moved again
Makes only one pass on the input list at cost of move operations

1 b

(^2) st r

(^3) st z

(^4) st g

(^5) st v

st

MECHANICS OF INSERTION

• Assume insertingx, thei th^ element

Must search for position to make the insertion
Two methods:
1. search in increasing order
  - Ex: insert 411 in [113 400 612 818]
2. search in decreasing order
  - Ex: insert 411 in [113 400 612 818]

st

SEARCH IN INCREASING ORDER

Ex: insert 411 in [113 400 612 818]

• Initialize the content of the output positioni to ∞ so that

we don’t have to test if we are at end of current list

• Start atj=1 and find first positionj wherex belongs — i.e.,

x < out[j ]

• Move all elements at positionsk (k ≥ j) one to right (i.e., to

k+1)

• j comparisons andi – j+1 moves

• Total ofi+1 operations for stepi

• Total of (n+1)(n+2)/2–1 =n(n+3)/2 operations

independent of whether the file is initially sorted, not sorted, or sorted in reverse order

st

SORTING BY DISTRIBUTION COUNTING

Insertion sort must perform move operations because when a value is inserted in the output list, we don’t know its final position
If we know the distribution of the various values, then we can calculate their final position in advance
If range of values is finite, use an extra array of counters
Algorithm:
1. count frequency of each value
2. calculate the destination address for each value by making a pass over the array of counters in reverse order and reuse the counter field
3. place the values in the right place in the output list

• O (m +n ) time forn values in the range 0 …m –1 and stable

Not in-place because on the final step we can’t tell if a value is in its proper location as values are not inserted in the output list in sorted order
Ex:

input 0 1 2 3 4 5 6 7 list: 3a 0a 8a 2a 1a 1b 2b 0b

counter: 0 1 2 3 4 5 6 7 8 9 value: 2 2 2 1 0 0 0 0 1 0 position: 0 2 4 6 7 7 7 7 7 8

output 0 1 2 3 4 5 6 7 list: 0a 0b 1a 1b 2a 2b 3a 8a

st SHELLSORT

Drawback of insertion and bubble sort is that move operations only involve adjacent elements
If largest element is initially at the start, then by the end of

the sort it will have been the subject ofn move or

exchange operations

Can reduce number of move operations by allowing values to make larger jumps
Sort successively larger sublists
1. e.g., sort every fourth element
  - elements 1, 5, 9, 13, 17 form one sorted list
  - elements 2, 6, 10, 14, 18 form a second sorted list
  - elements 3, 7, 11, 15, 19 form a third sorted list
  - elements 4, 8, 12, 16, 20 form a fourth sorted list
2. apply same technique with successively shorter increments
3. final step uses an increment of size 1
Usually implement by using insertion sort (that searches for position to make the insertion in decreasing order) at each step
Ex: with increment sequence of 4, 2, 1

1 b

start 113 400 818 612 411 311 412 420

Insertion Sort and Radix Sort Algorithms - Prof. Hanan Samet, Study notes of Data Structures and Algorithms

Related documents

Partial preview of the text

Download Insertion Sort and Radix Sort Algorithms - Prof. Hanan Samet and more Study notes Data Structures and Algorithms in PDF only on Docsity!

SORTING TECHNIQUES

• onlyn ·(n –1)/2 comparisons

• O (n 2 ) time as alwaysn ·(n –1) comparisons

• Assume insertingx, thei th^ element

• Initialize the content of the output positioni to ∞ so that

• Start atj=1 and find first positionj wherex belongs — i.e.,

x < out[j ]

• Move all elements at positionsk (k ≥ j) one to right (i.e., to

k+1)

• j comparisons andi – j+1 moves

• Total ofi+1 operations for stepi

• Total of (n+1)(n+2)/2–1 =n(n+3)/2 operations

• O (m +n ) time forn values in the range 0 …m –1 and stable

the sort it will have been the subject ofn move or