Quicksort. Part 1: Understanding Quicksort

Transcription

1 Qucksort Part 1: Understandng Qucksort

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3 Qucksort A practcal algorthm The hdden constants are small (hdden by Bg-O) Succnct algorthm The runnng tme = O(n lg n) Superor to other O(n lg n) n some respects Why are we coverng Qucksort? Neat algorthm Addtonal practce wth dvde-and-conquer Our frst stochastc algorthm

4 Qucksort Input : an array of n elements n any order Output : a reorderng of the nput array such that the elements are n non-decreasng order Key dea of Qucksort: partton the array around a pvot element

5 Key concept of Qucksort Pck an element and call t the pvot Partton (rearrange) the elements so that: Everythng to the left of the pvot s less than the pvot Everythng to the rght of the pvot s greater than the pvot Let s gnore tes for now Ths s a partal sortng nto buckets What can you tell me about the pvot? Pvot s now n the correct spot (we ve made progress!) What would be the runnng tme of callng partton on every element?

6 Parttonng

7 Parttonng < P P > P

8 Pvot around hello [ hello, are, you, how, today, dong ]

9 Qucksort (NOT IN-PLACE PARTITIONING) QUICKSORT (A) f A.length <= 1 then return A pvot_ndex = CHOOSE-PIVOT (A) // many ways to do ths left, rght = PARTITION (A, pvot_ndex) left = QUICKSORT (left) rght = QUICKSORT (rght) return left ++ A[pvot_ndex] ++ rght

10 Parttonng the Easy Way How would you partton? (how dd we perform a merge?)

11 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array

12 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array 8

13 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array 2 8

14 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array 2 5 8

15 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array

16 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array

17 Parttonng the Easy Way Coped elements to a new array How would you partton? (how dd we perform a merge?) Orgnal array New array Ths would be smlar to merge sort. Lots of memory allocatons.

18 Parttonng the Easy Way Nothng nherently wrong wth ths approach n theory But can we do the same thng wthout the extra memory? Note: mplementng merge sort n-place s possble You can do so wth an teratve (stack based) approach

19 Parttonng In-Place For now, assume that the pvot s n the frst spot of a subarray (we can swap the pvot wth the frst spot f needed) Idea: gradually buld up a subarray that s correctly parttoned by scannng through the array P < P > P Un-parttoned

20 Parttonng In-Place P < P > P Un-parttoned

21 P < P > P Un-parttoned Un-parttoned

22 P < P > P Un-parttoned Un-parttoned

23 P < P > P Un-parttoned swap Un-parttoned

24 P < P > P Un-parttoned Un-parttoned

25 P < P > P Un-parttoned Un-parttoned

26 P < P > P Un-parttoned swap Un-parttoned

27 P < P > P Un-parttoned Un-parttoned

28 P < P > P Un-parttoned

29 P < P > P Un-parttoned swap

30 P < P > P Un-parttoned

31 Wrte the partton functon PARTITION (A, left, rght) // nput s the subarray A[left..rght] pvot = A[left] = left + 1 for = (left + 1)..rght // nclusve f A[] < pvot swap A[] and A[] += 1 // else do nothng swap A[left] and A[-1] return -1 1.O(n) where n = r - l In-place no extra memory

32 Wrte the partton functon PARTITION (A, left, rght) // nput s the subarray A[left..rght] pvot = A[left] = left + 1 for = (left + 1)..rght // nclusve f A[] < pvot swap A[] and A[] += 1 // else do nothng swap A[left] and A[-1] return -1 What would you wrte as your loop nvarant for ths functon?

33 Qucksort QUICKSORT (A, left, rght) f left >= rght return A // Optonally swap pvot nto poston pvot_ndex = PARTITION (A, left, rght) QUICKSORT (A, left, pvot_ndex - 1) QUICKSORT (A, pvot_ndex + 1, rght) PARTITION (A, left, rght) // nput s A[left..rght] nclusve pvot = A[left] = left + 1 for = (left + 1)..rght // nclusve f A[] < pvot swap A[] and A[] += 1 // else do nothng swap A[left] and A[-1] return -1 // return ndex of pvot