1. “MM – Median of Medians”
Game:
There are n students in the class. Students are partitioned into n/r groups of size r
(r is odd = 3, 5, or 7).
Each team find its median using just any algorithm. Now we have n/r medians.
Find MM (Median of Medians) using just any algorithms. .
CS331 D. & A. of Algorithms

2. To show why MM ≥ ¼ of n & MM ≤ ¼ of n :
Example: 35 (n) elements are partitioned into 7 (n/r) groups of size 5 (r)
Elements  MM
Non-
decreasing
Order
(all columns)
MM (Median of Medians)
The middle row is in
non-decreasing order
Elements  MM .
CS331 D. & A. of Algorithms

3. To show why MM ≥ ¼ of n & MM ≤ ¼ of n :
Example: 21 (n) elements are partitioned into into 7 (n/r) groups of size 3 (r)
Elements  MM
Non-
decreasing
Order
(all columns)
MM (Median of Medians)
The middle row is in
non-decreasing order
Elements  MM .
CS331 D. & A. of Algorithms

4. To show why MM ≥ ¼ of n & MM ≤ ¼ of n :
Example: 25 (n) elements are partitioned into into 5 (n/r) groups of size 5 (r)
Elements  MM
Non-
decreasing
Order
(all columns)
MM (Median of Medians)
The middle row is in
non-decreasing order
Elements  MM .
CS331 D. & A. of Algorithms

5. Selection Problem using MM (with r = 5)
[ pp , Divide&Conquer class notes ]
Given a list of n elements, find the kth tallest one.
Finding n/r medians takes linear time.
Finding MM from n/5 medians is same as the orginal problem of finding the kth largest element.
MM is at least ≥ and ≤ ¼ of n. Therefore the size of the remaining subproblem is no more than ¾ of n .
CS331 D. & A. of Algorithms