Median Finding and Quick Sort

1. Median Finding and Quick Sort Suvarna Angal

2. Project Requirements • Implement the median-finding algorithms – Random and Linear Median Finding Algorithms. • The user is able to select the “k”, i.e., the rank of the number desired as output (k = n/2 is the median). • The user is also able to select groups of 3 or 5 in the linear-time median finding algorithm.

3. Project Requirements • The user can compare the performance of Random and Linear Median Finding Algorithms. • Implement quick sort using both algorithms and compare the performances of these different versions.

4. Randomized Median Finding • QSel(S,k) • m = a random element of S is the pivot. • S1 = all numbers in S < m • S2 = all numbers in S > m • if |S1| >= k return Qsel(S1,k); else if |S| - |S2| >= k return m; else return Qsel(S2,k-|S|+|S2|);

5. Median Finding • The difference with this approach is about the pivot selection. • To find the pivot element, we divide S into n/5 groups of 5 elements each. • Each group is then sorted and its median is selected. • Then invoke a recursive call to the same function to find median of medians.

6. Median Finding • Then this median of medians becomes the pivot element for the QSel function. • This will find the k-th smallest element in a sequence S of n elements in worst-case time O(n).

7. Implementation • The project is implemented in java. • The User Interface is done using Java Swing- MedianClient.java • This takes a list of numbers, k and number of elements in a group as input. • 4 classes – MedianQuickSort.java, MedianRandomQuickSort.java, Median.java, and MedianRandom.java are written.

8. User Interface

9. Analysis • Used simple counter to measure performance. • Checked performance for varying input sizes like 100, 1000 and 10000 for all 4 algortihms. • Also changed the group size 3 or 5 for the Linear Median Finding Algorithm.

10. Thank You! Demo.