More on Sorting: Quick Sort and Heap Sort

Transcription

1 More on Sortng: Quck Sort and Heap Sort Antono Carzanga Faculty of Informatcs Unversty of Lugano October 12, 2007 c 2006 Antono Carzanga 1 Another dvde-and-conuer sortng algorthm The heap Heap sort Outlne c 2006 Antono Carzanga 2

2 Remember k-smallest Selecton Problem: fnd the k-smallest element of a seuence A a value v A such that A k elements of A are greater or eual to v Idea: splt A n three parts based on a chosen value v A A L contans the set of elements that are less than v A v contans the set of elements that are eual to v A R contans the set of elements that are greater then v E.g., A = 2, 36, 5, 21, 8, 13, 11, 20, 5, 4, 1 we pck a splttng value, say v = 5 A L = 2, 4, 1 A v = 5, 5 A R = 36, 21, 8, 13, 11, 20 Can we use the same dea for sortng A? c 2006 Antono Carzanga 3 Problem: sortng Another Strategy for Sortng Idea: rearrange the seuence A[1...N] n three parts based on a chosen pvot value v A A[1... 1] contan elements that are less than or eual to v A[] = v A[ N] contan elements that are greater than v v = 8 = A[1... 1] A[ N] c 2006 Antono Carzanga 4

3 Another Dvde-and-Conuer for Sortng Dvde: partton A n A[1... 1] and A[ N] such that 1 < < j N A[] A[] A[j] Conuer: sort A[1... 1] and A[ N] Combne: nothng to do here notce the dfference wth MERGESORT QUICKSORT(A, begn, end) 1 f begn < end 2 then PARTITION(A, begn, end) 3 QUICKSORT(A, begn, 1) 4 QUICKSORT(A, + 1, end) c 2006 Antono Carzanga 5 Start wth = 1.e., assume all elements are greater than the pvot Scan the array left-to-rght, startng at poston 2 Partton If an element A[] s less than or eual to pvot, then swap t wth the current poston and shft to the rght Invarant begn k < A[k] v k end A[k] > v v A[end] c 2006 Antono Carzanga 6

4 Complete QUICKSORT Algorthm PARTITION(A, begn, end) 1 begn 2 v A[end] 3 for begn +1 to end 1 4 do f A[] v 5 then swap(a[], A[]) swap(a[end], A[]) 8 return QUICKSORT(A, begn, end) 1 f begn < end 2 then PARTITION(A, begn, end) 3 QUICKSORT(A, begn, 1) 4 QUICKSORT(A, + 1, end) c 2006 Antono Carzanga 7 Complexty of PARTITION PARTITION(A, begn, end) 1 begn 2 v A[end] 3 for begn +1 to end 1 4 do f A[] v 5 then swap(a[], A[]) swap(a[end], A[]) 8 return T(N) = Θ(N) c 2006 Antono Carzanga 8

5 Complexty of QUICKSORT QUICKSORT(A, begn, end) 1 f begn < end 2 then PARTITION(A, begn, end) 3 QUICKSORT(A, begn, 1) 4 QUICKSORT(A, + 1, end) Worst case = begn or = end the partton transforms P of sze N n P of sze N 1 T(N) = T(N 1) + Θ(N) T(N) = Θ(N 2 ) c 2006 Antono Carzanga 9 Complexty of QUICKSORT (2) QUICKSORT(A, begn, end) 1 f begn < end 2 then PARTITION(A, begn, end) 3 QUICKSORT(A, begn, 1) 4 QUICKSORT(A, + 1, end) Best case = N/2 the partton transforms P of sze N nto two problems P of sze N/2 and N/2 1, respectvely T(N) = 2T(N/2) + Θ(N) T(N) = Θ(N log N) c 2006 Antono Carzanga 10

6 Bnary Heap Our frst real data structure Interface HEAP-INSERT(H, key) nserts key n the heap HEAP-EXTRACT-MAX(H, ) extracts the maxmum key heap-sze(h) s the number of keys n H Two knds of bnary heaps max-heaps mn-heaps Useful applcatons sortng prorty ueue c 2006 Antono Carzanga 11 Conceptually a full bnary tree Bnary Heap: Structure Implemented as an array c 2006 Antono Carzanga 12

7 Bnary Heap: Propertes PARENT() return /2 LEFT() return 2 RIGHT() return Max-heap property: for all > 1, A[PARENT()] A[] c 2006 Antono Carzanga 13 Example Max-heap property: for all > 1, A[PARENT()] A[] E.g., Where s the max element? How can we mplement HEAP-EXTRACT-MAX? c 2006 Antono Carzanga 14

8 Heap-Extract-Max HEAP-EXTRACT-MAX procedure extract the max key rearrange the heap to mantan the max-heap property Now we have two subtrees n whch the max-heap property holds c 2006 Antono Carzanga 15 Max-Heapfy MAX-HEAPIFY(A, ) procedure assume: the left subtree of node s a heap assume: the rght subtree of node s a heap goal: rearrange the heap to mantan the max-heap property c 2006 Antono Carzanga 16

9 Max-Heapfy MAX-HEAPIFY(A, ) 1 l LEFT() 2 r RIGHT() 3 f l heap-sze(a) A[l] > A[] 4 then largest l 5 else largest 6 f r heap-sze(a) A[r] > A[largest] 7 then largest r 8 f largest = 9 then swap(a[], A[largest]) 10 MAX-HEAPIFY(A, largest) Complexty of MAX-HEAPIFY? The heght of the tree! T(N) = Θ(log N) c 2006 Antono Carzanga 17 BUILD-MAX-HEAP(A) 1 heap-sze(a) length(a) 2 for length(a)/2 downto 1 3 do MAX-HEAPIFY(A, ) Buldng a Heap heapfy ponts leafs length(a) = c 2006 Antono Carzanga 18

10 Heap Sort Idea: we can use a heap to sort an array HEAPSORT(A) 1 BUILD-MAX-HEAP(A) 2 for length(a) downto 1 3 do swap(a[], A[1]) 4 heap-sze(a) heap-sze(a) 1 5 MAX-HEAPIFY(A, 1) What s the complexty of HEAPSORT? T(N) = Θ(N log N) Benefts n-place sortng; worst-case s Θ(N log N) c 2006 Antono Carzanga 19 Summary of Sortng Algorthms Algorthm Complexty In place? worst average best INSERTION-SORT Θ(N 2 ) Θ(N 2 ) Θ(N) yes SELECTION-SORT Θ(N 2 ) Θ(N 2 ) Θ(N 2 ) yes BUBBLE-SORT Θ(N 2 ) Θ(N 2 ) Θ(N 2 ) yes MERGE-SORT Θ(N log N) Θ(N log N) Θ(N log N) no QUICK-SORT Θ(N 2 ) Θ(N log N) Θ(N log N) yes HEAP-SORT Θ(N log N) Θ(N log N) Θ(N log N) yes c 2006 Antono Carzanga 20