Sorting in Linear Time
How fast can we sort?
Decision-tree example


Decision-tree example
Decision-tree example
Decision-tree example
Decision-tree example
Decision-tree example
Lower Bound for decision-tree
sorting
Lower bound for comparison
sorting
Sorting in linear time
Counting sort
COUNTING-SORT(A,B,k)
// Initialize aux. store
// Count how many times occurred.
// Cumulative sums of aux. store
// ???
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Counting-sort example
Analysis
Running Time
Stable sorting
Counting sort array size
limitation problem
• If we need sort 32 bits integers, size of
count array 232 = 4GB !
• Unsorted array size is usually smaller than
this (1KB << 4GB)
• Solution... Use smaller parts like digits and
sort these parts particularly : Radix sort
Radix sort
Operation of radix sort
Analysis of radix sort
• Given n d-digit number in which each digit can take on up
to k-possible values, the running time of RADIX-SORT
sorts these numbers is
(d*(n+k))
Bucket Sort
BUCKET - SORT(A)
1 n  lengthA
2 for i  1 to n
3
do insert Ai  into list B nAi   
4 for i  0 to n  1
5
do sort list Bi  with insertion sort
6 concatanete the lists B0, B1,..., Bn  1 together in order
Bucket Sort
A
B
1
.78
0
2
.17
1
.12
.17
3
.39
2
.21
.23
4
.26
3
5
.72
4
6
.94
5
7
.21
6
.68
8
.12
7
.72
9
.23
8
10
.68
9
.94
/
.26
/
.78
/
/
/
Bucket Sort
Running time:
n 1
T (n)  (n)   O(n )
i 0
2
i
Order Statistics