Chapter Chapter 99..

Medians and Medians and

Order StatisticsOrder Statistics

Medians and Order Medians and Order StatisticsStatistics

The ith order statistic of a set of n elements is the

ith smallest element. Example: minimum and

maximum

Median: lower median and upper medianMedian: lower median and upper median

Selection problem: selecting the ith order statistic

from a set of n distinct numbers

Selection problem can be solved in O(nlgn)—

faster algorithms?

Average Case- O(n)

Worst case -O(n)

Selection ProblemSelection Problem

Maximum and MinimumMaximum and Minimum

Simultaneously Maximum Simultaneously Maximum and Minimumand Minimum

Selection in Expected Selection in Expected Linear TimeLinear Time

Once position of pivot in A is fixed, it will not changed anymore

AnalysisAnalysis

Worst-case running time: ΘΘ(n2)– We could be extremely unlucky and always recurse on a

sub-array that is only 1 element smaller than the

previous sub-arrayprevious sub-array

Expected running time: O(n)– Randomized-select works well on average

– It is randomized, no particular input bring out the worst-case behavior

The running time of Randomized-select is a random variable T(n), we will obtain an upper bound on E[T(n)]

Expected Running Time(1)Expected Running Time(1)

Including pivot A[q]

With equal probability for all possible k

Expected Running Time(2)Expected Running Time(2)

1

1 / , 1n

k k

i

EX n X

=

= =∑

Expected Running Time(3)Expected Running Time(3)

Expected Running Time(4)Expected Running Time(4)

hypothesis

>0?

Linear Expected Running timeLinear Expected Running time

Selection in worstSelection in worst--case linear timecase linear time

Select algorithm: its running time is O(n) in Select algorithm: its running time is O(n) in

the the worst caseworst case

It finds the desired element by recursively It finds the desired element by recursively

partitioning the input arraypartitioning the input arraypartitioning the input arraypartitioning the input array

The The main ideamain idea behind the algorithm, behind the algorithm,

however, is to however, is to guaranteeguarantee a a good splitgood split

Select use deterministic PARTITION Select use deterministic PARTITION

algorithm with algorithm with pivotpivot as inputas input

SELECT (1)SELECT (1)

Median of median

SELECT (2)SELECT (2)

Analysis (1)Analysis (1)

Analysis (2): #nodes greater Analysis (2): #nodes greater than xthan x

Both elements greater

than and less than x are

all at least 3n/10-6

Analysis (3)Analysis (3)

No matter recursively call on which sides of x, at most 7n/10+6 elements

RecurrenceRecurrence

Constants 140?Constants 140?

ConclusionConclusion

Class ExerciseClass Exercise

Exercise 9.1Exercise 9.1--2, 9.22, 9.2--2, 9.32, 9.3--55