Foundations of Data Structures Practical Session #12 Linear Sorting Sorting algorithms criteria 2 Comparison sorting 3 Linear sorting numeric Linear sorting is the problem of sorting a collection of items with numeric keys. The ability to perform arithmetic operations on the keys allows faster algorithms than comparison-based algorithms in many cases. Classical examples include: Counting sort Radix sort Bucket sort 4 Counting sort 5 Counting sort contd Counting-Sort (A, B, k) for i 1 to k C[i] 0 // Calc histogram for j 1 to n C[A[j]] C[A[j]] + 1 // Calc start index (backwards) in output for each key for i 2 to k C[i] C[i] + C[i-1] // Copy to output array for j n downto 1 B[C[A[j]]] A[j] C[A[j]] C[A[j]] 1 return B Example A: C: B: C:0 0 0 Counting sort analysis 7 Radix sort 8 Radix sort analysis For example, sort 7 numbers with 3 digits in decimal base Sorted by 1 st digit Sorted by 2 nd (and 1 st ) digit Sorted! Input array Radix sort contd 10 Bucket sort 11 Bucket sort contd Example 12 A: B: A: Bucket sort analysis 13 Sort algorithms review Stable In Place Extra Space Running time KeysType Worst case Average O(1)O(n 2 )any Insertion sort XO(n)O(nlogn)any Merge sort XO(1)O(nlogn)any Heap sort XO(1)O(n 2 )O(nlogn)anyQuicksort XO(n+k) integers [1..k] Counting sort Depends on the stable sort used O(d(b+n)) d digits in base b Radix sort XO(n)O(n 2 )O(n)[0,1) Bucket sort 14 Question 1 15 Question 1 solution 16 Question 2 17 Question 2 solution 18 Question 2 solution contd 19 Question 2 solution contd 20 Question 3 21 Question 3 solution 22 Question 4 23 Question 4 solution 24 Question 4 solution contd 25