Selection --Medians and Order Statistics (Chap. 9)

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Selection --Medians and Order Statistics (Chap. 9)

• The ith order statistic of n elements Sa1,
  a2,, an ith smallest elements
• Also called selection problem
• Minimum and maximum
• Median, lower median, upper median
• Selection in expected/average linear time
• Selection in worst-case linear time

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O(nlg n) Algorithm

• Suppose n elements are sorted by an O(nlg n)
  algorithm, e.g., MERGE-SORT
• Minimum the first element
• Maximum the last element
• The ith order statistic the ith element.
• Median
• If n is odd, then ((n1)/2)th element.
• If n is even,
• then (?(n1)/2?)th element, lower median
• then (?(n1)/2?)th element, upper median
• All selections can be done in O(1), so total
  O(nlg n).
• Can we do better?

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Selection in Expected Linear Time O(n)

• Select ith element
• A divide-and-conquer algorithm RANDOMIZED-SELECT
• Similar to quicksort, partition the input array
  recursively
• Unlike quicksort, which works on both sides of
  the partition, just work on one side of the
  partition.
• Called prune-and-search, prune one side, just
  search the other side).
• (Please review or read quicksort in chapter 7.)

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RANDOMIZED-SELECT(A,p,r,i)

1. if pr then return Ap
2. q?RANDOMIZED-PARTITION(A,p,r)
3. //the q holds for Ap,q-1?Aq ?Aq1,r
4. k ?q-p1
5. if ik then return Aq
6. else if iltk
7. then return RANDOMIZED-SELECT(A,p
   ,q-1,i)
8. else return RANDOMIZED-SELECT(A,
   q1,r,i-k)

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Analysis of RANDOMIZED-SELECT

• Worst-case running time ?(n2), why???

it may be unlucky and always partition into Aq,
an empty side and a side with remaining
elements. So every partitioning of m elements
will take ?(m) time, and mn,n-1,,2. Thus
total is ?(n) ?(n-1) ?(2) ? (n(n1)/2-1)
?(n2). Moreover, no particular input elicits the
worst-case behavior, Because of randomness.
But in average, it is good.
By using probabilistic analysis/random variable,
it can be proven that the expected running time
is O(n). (ref. to page 187).
Can we do better, such that O(n) in worst case??
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Selection in worst case linear time O(n)

• Select the ith smallest element of Sa1, a2,,
  an
• Use so called prune-and-search technique
• Let x? S, and partition S into three subsets
• S1aj aj ltx, S2aj aj x, S3aj aj gtx
  
• If S1 gti, search ith smallest element in S1
  recursively, (prune S2 and S3 away)
• Else If S1 S2 gti, then return x (the ith
  smallest element)
• Else search (i-( S1 S2 ))th in S3
  recursively, (prune S1 and S2 away)
• The question is how to select x such that S1 and
  S3 are nearly equal.?

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The Way to Select x
At least (3n/10)-6 elements ltx
Divide elements into ?n/5? groups of 5 elements
each. Find the median of each group Find the
median of the medians
At least (3n/10)-6 elements gtx
Because each of 1/2 ?n/5?-2 groups contributes 3
elements which are ? x
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SELECT ith Element in n Elements)

• Divide n elements into ?n/5? groups of 5
  elements.
• Find the median of each group.
• Use SELECT recursively to find the median x of
  the above ?n/5? medians.
• Partition n elements around x into S1, S2 , and
  S3.
• If S1gti, search ith smallest element in S1
  recursively,
• Else If S1S2gti, then return x (the
  ith smallest element)
• Else search (i-(S1S2))th in S3
  recursively,

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Analysis of SELECT (cont.)

• Steps 1,2,4 take O(n),
• Step 3 takes T(?n/5?).
• Let us see step 5
• At least half of medians in step 2 are ? x, thus
  at least 1/2 ?n/5?-2 groups contribute 3 elements
  which are ? x. i.e, 3(?1/2 ?n/5? ? -2) ?
  (3n/10)-6.
• Similarly, the number of elements ? x is also at
  least (3n/10)-6.
• Thus, S1 is at most (7n/10)6, similarly for
  S3.
• Thus SELECT in step 5 is called recursively on at
  most (7n/10)6 elements.
• Recurrence is
• T(n) O(1)
  if nlt some value (i.e. 140)
• T(?n/5?)T(7n/106)O(n) if n ?the
  value (i.e, 140)

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Solve recurrence by substitution

• Suppose T(n) ? cn, for some c.
• T(n) ? c ?n/5? c(7n/106) an
• ? cn/5 c 7/10cn6c an
• 9/10cnan7c
• cn(-cn/10an7c)
• Which is at most cn if -cn/10an7clt0.
• i.e., c ?10a(n/(n-70)) when ngt70.
• So select n140, and then c ?20a.
• Note n may not be 140, any integer gt70 is OK.

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Summary

• Bucket sort, counting sort, radix sort
• Their running times,
• Modifications
• The ith order statistic of n elements Sa1,
  a2,, an ith smallest elements
• Minimum and maximum.
• Median, lower median, upper median
• Selection in expected/average linear time
• Worst case running time
• Prune-and-search
• Selection in worst-case linear time
• Why group size 5?