Introduction to Algorithms (Counting Sort, Radix Sort, Lower Bounds for Sorting)

Comparison model

• all input items are black boxes (ADTS)
• only operations allowed are comparisons (<,<=,>,>=)
• time cost = # comparisons

Decision tree

any comparison algorithm can be viewed as a tree of all possible comps & outcomes

•  internal node = binary decision
• leaf = output
• root-to-leaf path = algorithm execution
• path length = running time
• the height of tree = worst-case running time

 Lower bound

Claim:

• n preprocessed items
• finding a given item among them in comparison model requires  in worst-case
• sorting n items requires                   

Linear time sorting(Integer sorting) 

• assume n keys sorting are integers ∈ {0, 1, 2, 3, ... , k-1} (&each fits in a word)
• can do a lot more than comparisons
• can sort in O(n) time

Counting Sort

L = array of k empty lists lists

for j in range n:
    L[key(A[j])].append(A[j])

output = []

for i in range k: 
    output.extend(L[i])

Radix Sort

• imagine each integer in base b
• 
• sort by least significant digit
• ...
• sort by most signigicant digit
• use counting sort for digit sort

minimized when b = n