Discrrete mathematics for computer science recurrence warmup

Finding the Median Without SortingGeneralize to finding the k’th largest element of a list: FindL, k Median ::= FindL, |L|/2... Assume distinct elements Divide L into blocks of 5 and fin

Recurrence Warmup

Finding the Median Without Sorting

Generalize to finding the k’th largest element of a list: Find(L, k) Median ::= Find(L, |L|/2)

Find(L, k)

Let |L| = n Assume distinct elements

Divide L into blocks of 5 and find the medians (third of five

elements) of the blocks:

O(n) time.

Recursively find the median of the medians, M

M < half the medians, and each median < 2 of the 5 elements of its block

So those medians are < 2/10 of the elements of L

So M < 3/10 of the elements of L

Likewise M > 3/10 the elements of L

Find(L, k)

Use M to split L into two sublists: elements < M and elements > M

On the basis of the size of these lists, figure out which part the k’th element of L belongs to

Recursively find the corresponding element within that sublist,

which is of size at most 7n/10

• T(1) = 1

• T(n) ≤ T(n/5) + T(7n/10)

• (Time to find median of medians plus time to select from elements that have not been excluded)

• Linear solution!

Because n + 9n + 92n + … < 10n

So T(n) = O(n)

FINIS