This lecture and the next. Why Sorting? Sorting Algorithms so far. Why Sorting? (2) Selection Sort. Heap Sort. Heapsort

Transcription

1 Ths lecture ad the ext Heapsort Heap data structure ad prorty queue ADT Qucksort a popular algorthm, very fast o average Why Sortg? Whe doubt, sort oe of the prcples of algorthm desg. Sortg used as a subroute may of the algorthms: Searchg databases: we ca do bary search o sorted data A large umber of computer graphcs ad computatoal geometry problems Closest par, elemet uqueess Why Sortg? () A large umber of sortg algorthms are developed represetg dfferet algorthm desg techques. A lower boud for sortg Ω( log ) s used to prove lower bouds of other problems Sortg Algorthms so far Iserto sort, selecto sort Worst-case rug tme Θ( ); -place Merge sort Worst-case rug tme Θ( log ), but requres addtoal memory Θ(); (WHY?) 3 4 Selecto Sort Selecto-Sort(A[..]): For For dowto A: A: Fd Fd the the largest elemet amog amog A[..] B: B: Exchage t t wth wth A[] A[] A takes Θ() ad B takes Θ(): Θ( ) total Idea for mprovemet: use a data structure, to do both A ad B O(lg ) tme, balacg the work, achevg a better trade-off, ad a total rug tme O( log ) Heap Sort Bary heap data structure A array Ca be vewed as a early complete bary tree All levels, except the lowest oe are completely flled The key root s greater or equal tha all ts chldre, ad the left ad rght subtrees are aga bary heaps Two attrbutes legth[a] heap-sze[a]

2 CHOROCHRONOS Mdter Revew Heap Sort (3) Heap Sort (4) Paret () retur / Left () retur Rght () retur + Heap property: A[Paret()] A[] Level: I a bary represetato, a multplcato/dvso by two s left/rght shft Addg ca be doe by addg the lowest bt 0 7 Heapfy Notce the mplct tree lks; chldre of ode are ad + Why s ths useful? Heapfy () s dex to the array A Bary trees rooted at Left() ad Rght() are heaps But, A[] mght be smaller tha ts chldre, thus volatg the heap property The method Heapfy makes A a heap oce more by movg A[] dow the heap utl the heap property s satsfed aga 9 Heapfy Example The rug tme of Heapfy o a subtree of sze rooted at ode s determg the relatoshp betwee elemets: Θ() plus the tme to ru Heapfy o a subtree rooted at oe of the chldre of, where /3 s the worst-case sze of ths subtree. T () T ( / 3) + Θ() T () O(log ) Alteratvely 0 Heapfy: Rug Tme Tmos Sells Rug tme o a ode of heght h: O(h)

3 CHOROCHRONOS Mdter Revew Buldg a Heap Buldg a Heap Covert a array A[...], where legth[a], to a heap Notce that the elemets the subarray A[( / + )...] are already -elemet heaps to beg wth! 3 Buldg a Heap: Aalyss Buldg a Heap: Aalyss () Correctess: ducto o, all trees rooted at m > are heaps Rug tme: calls to Heapfy O(lg ) Deftos heght of ode: logest path from ode to leaf heght of tree: heght of root O( lg ) Good eough for a O( lg ) boud o Heapsort, but sometmes we buld heaps for other reasos, would be ce to have a tght boud k T () O 4 Ituto: for most of the tme Heapfy works o smaller tha elemet heaps lg O ( + ) sce O( ) lg / ( / ) Buldg a Heap: Aalyss (3) tme to Heapfy O(heght of subtree rooted at ) assume k (a complete bary tree k lg ) Heap Sort How? By usg the followg "trck" f x < //dfferetate x x //multply by x ( x ) x O() 0 x //plug x / / 4 The total rug tme of heap sort s O( lg ) + Buld-Heap(A) tme, whch s O() Therefore Buld-Heap tme s O() Tmos Sells x ( x ) 7 3

4 CHOROCHRONOS Mdter Revew Heap Sort Heap Sort: Summary Heap sort uses a heap data structure to mprove selecto sort ad make the rug tme asymptotcally optmal Rug tme s O( log ) lke merge sort, but ulke selecto, serto, or bubble sorts Sorts place lke serto, selecto or bubble sorts, but ulke merge sort Prorty Queues Prorty Queues () A prorty queue s a ADT(abstract data type) for matag a set S of elemets, each wth a assocated value called key A PQ supports the followg operatos Isert(S,x) sert elemet x set S (S S {x}) Maxmum(S) returs the elemet of S wth the largest key Extract-Max(S) returs ad removes the elemet of S wth the largest key Applcatos: A Heap ca be used to mplemet a PQ Iserto of a ew elemet Tmos Sells Prorty Queues (4) Removal of max takes costat tme o top of Heapfy Θ(lg ) ob schedulg shared computg resources (Ux) Evet smulato As a buldg block for other algorthms Prorty Queues (3) 0 3 elarge the PQ ad propagate the ew elemet from last place up the PQ tree s of heght lg, rug tme: Θ(lg ) 4 4

5 Prorty Queues () Quck Sort Characterstcs sorts almost "place,".e., does ot requre a addtoal array, lke serto sort Dvde-ad-coquer, lke merge sort very practcal, average sort performace O( log ) (wth small costat factors), but worst case O( ) Quck Sort the Prcple Parttog To uderstad quck-sort, let s look at a hgh-level descrpto of the algorthm A dvde-ad-coquer algorthm Dvde: partto array to subarrays such that elemets the lower part < elemets the hgher part Coquer: recursvely sort the subarrays Combe: trval sce sortg s doe place Lear tme parttog procedure Partto(A,p,r) 0 x A[r] 0 p r+ X0 04 whle TRUE 0 repeat utl A[] x 07 repeat + 0 utl A[] x 0 09 f < 0 the exchage A[] A[] else retur Quck Sort Algorthm Ital call Qucksort(A,, legth[a]) Qucksort(A,p,r) 0 f p<r 0 the q Partto(A,p,r) 03 Qucksort(A,p,q) 04 Qucksort(A,q+,r) Aalyss of Qucksort Assume that all put elemets are dstct The rug tme depeds o the dstrbuto of splts 9 30

6 Best Case If we are lucky, Partto splts the array evely T ( ) T ( /) +Θ( ) Usg the meda as a pvot The recurrece the prevous slde works out, BUT Q: Ca we fd the meda lear-tme? A: YES! But we eed to wat utl we get to Chapter Worst Case Worst Case () What s the worst case? Oe sde of the parto has oly oe elemet T ( ) T() + T ( ) +Θ( ) T ( ) +Θ( ) Θ( k) k Θ( k) Θ k ( ) Worst Case (3) Whe does the worst case appear? put s sorted put reverse sorted Same recurrece for the worst case of serto sort However, sorted put yelds the best case for serto sort! Aalyss of Qucksort Suppose the splt s /0 : 9/0 T ( ) T ( /0) + T(9 /0) +Θ ( ) Θ( log )! 3 3

7 A Average Case Scearo A Average Case Scearo () Suppose, we alterate lucky ad ulucky cases to get a average behavor - (-)/ (-)/ Θ() L( ) U( /) +Θ( ) lucky U ( ) L ( ) +Θ( ) ulucky we cosequetly get L( ) ( L( / ) +Θ ( /)) +Θ( ) L ( / ) +Θ( ) Θ( log ) (-)/+ Θ( ) (-)/ How ca we make sure that we are usually lucky? Partto aroud the mddle (/th) elemet? Partto aroud a radom elemet (works well practce) Radomzed algorthm rug tme s depedet of the put orderg o specfc put trggers worst-case behavor the worst-case s oly determed by the output of the radom-umber geerator 37 3 Radomzed Qucksort Assume all elemets are dstct Partto aroud a radom elemet Cosequetly, all splts (:-, :-,..., - :) are equally lkely wth probablty / Radomzato s a geeral tool to mprove algorthms wth bad worst-case but good average-case complexty Next Week Q: Ca we beat the Ω( log ) lower boud for sortg? A: I geeral o, but some specal cases YES! Ch 7: Sortg lear tme 39 40