Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1)
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1 Merge Sort Merge Sort versio Outlie d Redig Divide-d-coquer prdigm ( Merge-sort ( Algorithm Mergig two sorted sequeces Merge-sort tree Executio exmple Alysis Geeric mergig d set opertios ( Summry of sortig lgorithms ( Merge Sort versio Divide-d-Coquer Divide-d coquer desig prdigm: Divide: divide the iput dt S i two (or more disjoit subsets S 1 d S 2 Recur: solve the subproblems ssocited with S 1 d S 2 Coquer: combie the solutios for S 1 d S 2 ito solutio for S Bse cse: directly solve d do ot divide for smll subproblem sizes (typiclly 0 or 1. Merge-sort is sortig lgorithm bsed o divide-d-coquer Lie hep-sort O( ruig time Ulie hep-sort No uxiliry priority queue Accesses dt sequetilly (suitble to sort dt o dis Merge Sort versio 1.3 3
2 Merge-Sort Merge-sort o iput sequece S with elemets cosists of three steps: Divide: prtitio S ito two sequeces S 1 d S 2 of bout /2 elemets ech Recur: recursively sort S 1 d S 2 Coquer: merge S 1 d S 2 ito uique sorted sequece Algorithm mergesort(s Iput sequece S with elemets Output sequece S sorted if > 1 (S 1, S 2 prtitio(s, /2 S 1 mergesort(s 1 S 2 mergesort(s 2 S merge(s 1, S 2 retur S Merge Sort versio Prtitioig Sequece The divide step of merge-sort cosists of prtitioig iput sequece S Use doubly lied list with hed d til poiter The ll sequece ADT opertios te O(1 time. With totl elemets, prtitio tes O( time. Algorithm prtitio(s, Iput sequece S, with items;, prtitio size Output prtitio of S ito S 1 of size d S 2 of size - S 1 empty sequece S 2 empty sequece pos S.first( for i 1 to do S 1.isertLst(pos.elemet( pos S.fter(pos for i to do S 2.isertLst(pos.elemet( pos S.fter(pos retur (S 1,S 2 Merge Sort versio Mergig Two Sorted Sequeces The coquer step of merge-sort cosists of mergig two sorted sequeces A d B Use doubly lied list with hed d til poiter The ll sequece ADT opertios te O(1 time. With totl elemets, merge tes O( time. Algorithm merge(a, B Iput sequece A d B, both sorted, with totl items combied Output sorted sequece of A B S empty sequece while A.isEmpty( B.isEmpty( if A.first(.elemet( < B.first(.elemet( S.isertLst(A.remove(A.first( else S.isertLst(B.remove(B.first( while A.isEmpty( S.isertLst(A.remove(A.first( while B.isEmpty( S.isertLst(B.remove(B.first( retur S Merge Sort versio 1.3 6
3 Merge-Sort Tree A executio of merge-sort is depicted by biry tree ech ode represets recursive cll of merge-sort d stores usorted sequece before the executio d its prtitio sorted sequece t the ed of the executio the root is the iitil cll the leves re clls o subsequeces of size 0 or Merge Sort versio Executio Exmple Prtitio Merge Sort versio Executio Exmple (cot. Recursive cll, prtitio Merge Sort versio 1.3 9
4 Executio Exmple (cot. Recursive cll, prtitio Merge Sort versio Executio Exmple (cot. Recursive cll, bse cse Merge Sort versio Executio Exmple (cot. Recursive cll, bse cse Merge Sort versio
5 Executio Exmple (cot. Merge Merge Sort versio Executio Exmple (cot. Recursive cll,, bse cse, merge Merge Sort versio Executio Exmple (cot. Merge Merge Sort versio
6 Executio Exmple (cot. Recursive cll,, merge, merge Merge Sort versio Executio Exmple (cot. Merge Merge Sort versio Merge-Sort Alysis Use recurrece equtio. Algorithm mergesort(s Iput sequece S with elemets Output sequece S sorted if > 1 (S 1, S 2 prtitio(s, /2 S 1 mergesort(s 1 S 2 mergesort(s 2 S merge(s 1, S 2 retur S Merge Sort versio
7 Merge-Sort Alysis Use recurrece equtio. T(0 = T(1 = 2 T( = c + T(/2 + T(/2 + c = 2c + 2T(/2 c is costt. Algorithm mergesort(s Iput sequece S with elemets Output sequece S sorted if > 1 (S 1, S 2 prtitio(s, /2 S 1 mergesort(s 1 S 2 mergesort(s 2 S merge(s 1, S 2 retur S Merge Sort versio Alysis of Merge-Sort The height h of the merge-sort tree t ech recursive cll we divide i hlf the sequece, The overll mout or wor doe t the odes of depth i we prtitio d merge 2 i sequeces of size /2 i we me 2 i+1 recursive clls Thus, the totl ruig time of merge-sort is bout 2c, or O( depth 0 #clls 1 size cost 2c 1 2 /2 2c i 2 i /2 i 2c Merge Sort versio The Recursio Tree For solvig divide-d-coquer recurrece reltios: b T ( 2T ( / 2 + b if < 2 if 2 depth 0 T s 1 size cost b 1 2 /2 b i 2 i /2 i b Totl cost = b + b (lst level plus ll previous levels Merge Sort versio
8 Summry of Sortig Algorithms Algorithm selectio-sort isertio-sort hep-sort merge-sort Time O( 2 O( 2 O( O( Notes slow i-plce for smll dt sets (< 1K slow i-plce for smll dt sets (< 1K fst i-plce for lrge dt sets (1K 1M fst sequetil dt ccess for huge dt sets (> 1M Merge Sort versio Divide-d-Coquer Merge Sort versio Divide-d-Coquer Alysis c be doe usig recurrece equtios Wht would recurrece equtio loo lie for this tree? Merge Sort versio
9 Recurrece Equtio Alysis The coquer step of merge-sort cosists of mergig two sorted sequeces, ech with /2 elemets d implemeted by mes of doubly lied list, tes t most b steps, for some costt b. The bsis cse ( < 2 tes 2 steps. Therefore, if we let T( deote the ruig time of merge-sort: 2 T ( 2T ( / 2 + b if < 2 if 2 We c therefore lyze the ruig time of merge-sort by fidig closed form solutio to the bove equtio. Tht is, solutio tht hs T( oly o the left-hd side. Merge Sort versio Itertive Substitutio I the itertive substitutio, or plug-d-chug, techique, we itertively pply the recurrece equtio to itself d see if we c fid ptter: T ( = 2T ( / 2 + b 2 = 2(2T ( / 2 + b( / 2 + b 2 2 = 2 T ( / 2 + 2b 3 3 = 2 T ( / 2 + 3b 4 4 = 2 T ( / 2 + 4b =... i i = 2 T ( / 2 + ib Note tht bse, T(1=2, cse occurs whe /2 i =1. (Or i =. So, T( = 2 + b Thus, T(. Merge Sort versio Solvig recurrece equtios Recurrece Trees (lredy show Itertive Substitutio (lredy show Guess-d-Test Method (i boo Mster Method (ext does ot pply to ll recurrece equtios! Merge Sort versio
10 Mster Method My divide-d-coquer recurrece equtios hve the form: if < d T ( T ( / b + f ( if d The Mster Theorem: Note:, re costts you pic. 1. if f (, the T ( f ( f ( +, the T (, the T ( f (, provided f ( / b ( Merge Sort versio Mster Method, Exmple 1 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: T ( = 4T ( / 2 + Solutio: b =2, so cse 1 sys T( 2. Merge Sort versio Mster Method, Exmple 2 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: T ( = 2T ( / 2 + Solutio: b =1, so cse 2 sys T( 2. Merge Sort versio
11 Mster Method, Exmple 3 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: T ( = T ( / 3 + Solutio: b =0, so cse 3 sys T(. Merge Sort versio Mster Method, Exmple 4 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: 2 T ( = 8T ( / 2 + Solutio: b =3, so cse 1 sys T( 3. Merge Sort versio Mster Method, Exmple 5 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: 3 T ( = 9T ( / 3 + Solutio: b =2, so cse 3 sys T( 3. Merge Sort versio
12 Mster Method, Exmple 6 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: (biry serch T ( = T ( / 2 Solutio: b =0, so cse 2 sys T(. Merge Sort versio Mster Method, Exmple 7 T ( T ( / b + f ( The Mster Theorem: 1. if f (, the T ( f ( f ( if < d if d, the T ( +, the T ( f (, provided f ( / b ( Exmple: T ( = 2T ( / 2 + (hep costructio Solutio: b =1, so cse 1 sys T(. Merge Sort versio Itertive Proof of the Mster Theorem Usig itertive substitutio, let us see if we c fid ptter: T ( = T ( / b + f ( 2 = ( T ( / b + f ( / b + b 2 2 = T ( / b + f ( / b + f ( = T ( / b + f ( / b + f ( / b + f ( =... = T (1 + 1 i b 1 b i i = T(1 + f ( / b i= 0 We the distiguish the three cses s The first term is domit ( ( i= 0 i f ( / b Ech prt of the summtio is eqully domit The summtio is geometric series Merge Sort versio
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