Discrrete mathematics for computer science recurrence warmup

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.

Analysis

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•
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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)

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Linear solution!

2
Because n + .9n + .9 n + … < 10n
So T(n) = O(n)


FINIS