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6.006 Introduction to Algorithms
Spring 2008
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Lecture 11
Sorting IV: Stable Sorting, Radix Sort
6.006 Spring 2008
Lecture 11: Sorting IV: Stable Sorting, Radix Sort
Lecture Overview
• Stable Sorting
• Radix Sort
• Quick Sort ← not officially a part of 6.006
• Sorting Races
Stable Sorting
Preserves input order among equal elements
4’ 1
3* 4 3
1 3* 3 4’ 4
counting sort is stable
merge sort is stable
Figure 1: Stability
Selection Sort and Heap: Find maximum element and put it at end of array (swap with
element at end of array) NOT STABLE!
2a
2b
←
3
2b 2a 3
define
2a <2b
Figure 2: Selection Sort Instability
Radix Sort
• Herman Hollerith card-sorting machine for 1890 census.
• Digit by Digit sort by mechanical machine
1. Examine given column of each card in a deck
2. Distribute the card into one of 10 bins
1
Lecture 11
Sorting IV: Stable Sorting, Radix Sort
6.006 Spring 2008
3. Gather cards bin by bin, so cards with first place punched are on top of cards
with second place punched, etc.
80 cols
.
.
.
10
.
places .
Figure 3: Punch Card
MSB vs. LSB?
Sort on most significant digit first or least significant digit first?
MSB strategy: Cards in 9 of 10 bins must be put aside, leading to a large number of
intermediate piles
LSB strategy: Can gather sorted cards in bins appropriately to create a deck!
Example
3
4
6
8
4
7
3
2
5
5
3
3
2
5
9
7
7
9
6
0
5
7
3
4
4
6
3
8
2
5
3
5
5
2
3
0
5
6
7
7
9
9
7
3
4
8
3
4
6
2
2
3
3
5
5
5
0
9
6
9
5
7
7
3
3
4
4
6
7
8
2
5
3
5
5
2
3
9
5
6
7
7
0
9
Digit sort needs to be stable, else will get wrong result!
Figure 4: Example of Radix Sort
2
Lecture 11
Sorting IV: Stable Sorting, Radix Sort
6.006 Spring 2008
Analysis
Assume counting sort is auxiliary stable sort. Θ(n + k) complexity.
Suppose we have n words of b bits each.
One pass of counting sort
b passes of counting sort
b
passes
r
Θ(n + 2b )
Θ(b(n + 2)) = Θ(nb)
b
Θ( (n + 2r )) minimized when r = lg n
r
Θ(
bn
)
lg n
Quick Sort
This section is for “enrichment” only.
Divide: Partition the array into two. Sub-arrays around a pivot x such that elements in
lower sub array ≤ x ≤ elements in upper sub array. ← Linear Time
pivot
≤x
x
≥x
Figure 5: Pivot Definition
Conquer: Recursively sort the two sub arrays
Combine: Trivial
If we choose a pivot such that two sub arrays are roughly equal:
T (n) = 2T (n/2) + Θ(n) =⇒ T (n) = Θ(n lg n)
If one array is much bigger:
T (n) = T (n − 1) + Θ(n) =⇒ T (n) = Θ(n2 )
Average case Θ(n lg n) assuming input array is randomized!
3
Lecture 11
Sorting IV: Stable Sorting, Radix Sort
6.006 Spring 2008
Sorting Races
Click here for a reference on this.
Bubble Sort: Repeatedly step through the list to be sorted. Compare 2 items, swap if they
are in the wrong order. Continue through list, until no swaps. Repeat pass through
list until no swaps. Θ(n2 )
Shell Sort: Improves insertion sort by comparing elements separated by gaps
4
Θ(nlg2 n)