CS 5114: Theory of Algorithms. Sorting. Insertion Sort. Exchange Sorting. Clifford A. Shaffer. Spring 2010

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1 Depatment of Compute Science Viginia Tech Backsbug, Viginia Copyight c 10 by Ciffod A. Shaffe : Theoy of Agoithms Tite page : Theoy of Agoithms Ciffod A. Shaffe Sping 10 Ciffod A. Shaffe Depatment of Compute Science Viginia Tech Backsbug, Viginia Sping 10 Copyight c 10 by Ciffod A. Shaffe : Theoy of Agoithms Sping 10 1 / 31 Soting Each ecod contains a fied caed the key. Linea ode: compaison. Soting Linea ode means: a < b and b < c a < c. Soting Each ecod contains a fied caed the key. Linea ode: compaison. The Soting Pobem Given a sequence of ecods R1, R,..., Rn with key vaues k1, k,..., kn, espectivey, aange the ecods into any ode s such that ecods Rs 1, Rs,..., Rsn have keys obeying the popety ks 1 ks... ksn. Measues of cost: Compaisons Swaps The Soting Pobem Moe simpy, soting means to put keys in ascending ode. Given a sequence of ecods R 1, R,..., R n with key vaues k 1, k,..., k n, espectivey, aange the ecods into any ode s such that ecods R s1, R s,..., R sn have keys obeying the popety k s1 k s... k sn. Measues of cost: Compaisons Swaps : Theoy of Agoithms Sping 10 / 31 Insetion Sot void inssoteem* A, int n) { // Insetion Sot fo int ; i<n; i++) // Inset i th ecod fo int j=i; j>0) && A[j].key<A[j-1].key); j--) swapa, j, j-1); Best Case: Wost Case: Aveage Case: : Theoy of Agoithms Sping 10 3 / 31 Exchange Soting Insetion Sot Best case is 0 swaps, n 1 compaisons. Wost case is n / swaps and compaes. Aveage case is n /4 swaps and compaes. Insetion sot has geat best-case pefomance. Exchange Soting Insetion Sot void inssoteem* A, int n) { // Insetion Sot fo int ; i<n; i++) // Inset i th ecod fo int j=i; j>0) && A[j].key<A[j-1].key); j--) swapa, j, j-1); Best Case: Wost Case: Aveage Case: Exchange Soting Theoem: Any sot esticted to swapping adjacent ecods must be Ωn ) in the wost and aveage cases. Poof: Fo any pemutation P, and any pai of positions i and j, the eative ode of i and j must be wong in eithe P o the invese of P. Thus, the tota numbe of swaps equied by P and the invese of P MUST be i = nn 1). Theoem: Any sot esticted to swapping adjacent ecods must be Ωn ) in the wost and aveage cases. Poof: Fo any pemutation P, and any pai of positions i and j, the eative ode of i and j must be wong in eithe P o the invese of P. Thus, the tota numbe of swaps equied by P and the invese of P MUST be n /4 is the aveage distance fom a ecod to its position in the soted output. i = nn 1). : Theoy of Agoithms Sping 10 4 / 31

2 Initia Pass 1 Swap 1 Pass Swap Pass 3 Swap 3 Revese Swap Pivot = Pivot = Pivot = 57 Pivot = Pivot = Pivot = Fina Soted Aay Pivot = 88 Quicksot Divide and Conque: divide ist into vaues ess than pivot and vaues geate than pivot. void qsoteem* A, int i, int j) { // Quicksot int pivotindex = findpivota, i, j); swapa, pivotindex, j); // Swap to end // k wi be fist position in ight subaay int k = patitiona, i-1, j, A[j].key; swapa, k, j); // Put pivot in pace if k-i) > 1) qsota, i, k-1); // Sot eft if j-k) > 1) qsota, k+1, j); // Sot ight Quicksot Initia ca: qsotaay, 0, n-1); Quicksot Divide and Conque: divide ist into vaues ess than pivot and vaues geate than pivot. void qsoteem* A, int i, int j) { // Quicksot int pivotindex = findpivota, i, j); swapa, pivotindex, j); // Swap to end // k wi be fist position in ight subaay int k = patitiona, i-1, j, A[j].key; swapa, k, j); // Put pivot in pace if k-i) > 1) qsota, i, k-1); // Sot eft if j-k) > 1) qsota, k+1, j); // Sot ight int findpivoteem* A, int i, int j) { etun i+j)/; int findpivoteem* A, int i, int j) { etun i+j)/; : Theoy of Agoithms Sping 10 5 / 31 Quicksot Patition Quicksot Patition Quicksot Patition int patitioneem* A, int, int, int pivot) { do { // Move bounds inwad unti they meet whie A[++].key < pivot); // Move ight whie && A[--].key > pivot));// Left swapa,, ); // Swap out-of-pace vas whie < ); // Stop when they coss swapa,, ); // Revese wasted swap etun ; // Retun fist position in ight The cost fo Patition is Θn). int patitioneem* A, int, int, int pivot) { do { // Move bounds inwad unti they meet whie A[++].key < pivot); // Move ight whie && A[--].key > pivot));// Left swapa,, ); // Swap out-of-pace vas whie < ); // Stop when they coss swapa,, ); // Revese wasted swap etun ; // Retun fist position in ight The cost fo Patition is Θn). : Theoy of Agoithms Sping 10 / 31 Patition Exampe Patition Exampe Patition Exampe Initia Pass Swap Pass Swap Pass Swap Revese Swap : Theoy of Agoithms Sping 10 7 / 31 Quicksot Exampe Quicksot Exampe Quicksot Exampe Pivot = Pivot = Pivot = Pivot = Pivot = 88 Pivot = Pivot = Fina Soted Aay : Theoy of Agoithms Sping 10 8 / 31

3 n+1 Cost fo Quicksot Best Case: Aways patition in haf. Wost Case: Bad patition. Aveage Case: fn) = n fi) + fn i 1)) n Optimizations fo Quicksot: Bette pivot. Use bette agoithm fo sma subists. Eiminate ecusion. Best: Don t sot sma ists and just use insetion sot at the end. : Theoy of Agoithms Sping 10 9 / 31 Quicksot Aveage Cost fn) = n n fi) + fn i 1)) n > 1 Since the two haves of the summation ae identica, fn) = n 1 + n fi) n > 1 Mutipying both sides by n yieds nfn) = nn 1) + fi). Cost fo Quicksot Cost fo Quicksot Best Case: Aways patition in haf. Wost Case: Bad patition. Aveage Case: fn) = n n fi) + fn i 1)) Optimizations fo Quicksot: Bette pivot. Use bette agoithm fo sma subists. Eiminate ecusion. Best: Don t sot sma ists and just use insetion sot at the end. Think about when the patition is bad. Note the FindPivot function that we used is petty good, especiay compaed to taking the fist o ast) vaue. Aso, think about the distibution of costs: Line up a the pemuations fom most expensive to cheapest. How many can be expensive? The aea unde this cuve must be ow, since the aveage cost is Θn og n), but some of the vaues cost Θn ). So thee can be VERY few of the expensive ones. This optimization means, fo ist theshod T, that no eement is moe than T positions fom its destination. Thus, insetion sot s best case is neay eaized. Cost is at wost nt. Quicksot Aveage Cost This is a ecuence with fu histoy. Think about what the pieces coespond to. To do Quicksot on an aay of size n, we must: Patation: Cost n Findpivot: Cost c fn) = Quicksot Aveage Cost n n fi) + fn i 1)) n > 1 Since the two haves of the summation ae identica, fn) = n 1 + n fi) n > 1 Mutipying both sides by n yieds nfn) = nn 1) + fi). Do the ecusion: Cost dependent on the pivot s fina position. These pats ae modeed by the equation, incuding the aveage ove a the cases fo position of the pivot. : Theoy of Agoithms Sping / 31 Aveage Cost cont.) Get id of the fu histoy by subtacting nfn) fom )f) Aveage Cost cont.) Aveage Cost cont.) Get id of the fu histoy by subtacting nfn) fom )f) nfn) = nn 1) + fi) )f) = )n + )f) nfn) = n + fn) n fi) )f) = n + n + )fn) n f) = + n + fn). nfn) = nn 1) + fi) )f) = )n + )f) nfn) = n + fn) n fi) )f) = n + n + )fn) f) = n + n + fn). : Theoy of Agoithms Sping / 31 Aveage Cost cont.) Aveage Cost cont.) Aveage Cost cont.) n Note that fo n 1. Expand the ecuence to get: f) + n + fn) = + n + + ) fn 1) = + n n )) fn ) n 1 = + n ) f1)) n Note that fo n 1. n+1 Expand the ecuence to get: f) + n + fn) = + n + + ) fn 1) = + n n n ) f1)) = + n + )) fn ) : Theoy of Agoithms Sping 10 1 / 31

4 Aveage Cost cont.) Aveage Cost cont.) Aveage Cost cont.) f) 1 + n + + n n + 3 ) 1 = 1 + n + ) + 1 n )) = +n + )Hn+1 1) = Θn og n). f) 1 + n + + n + = 1 + n + ) + n + = + n + )H n+1 1) n 3 + ) n )) H n+1 = Θog n) = Θn og n). : Theoy of Agoithms Sping 10 / 31 Megesot List megesotlist inist) { if inist.ength) <= 1) etun inist;; List 1 = haf of the items fom inist; List = othe haf of the items fom inist; etun megemegesot1), megesot)); Megesot Megesot List megesotlist inist) { if inist.ength) <= 1) etun inist;; List 1 = haf of the items fom inist; List = othe haf of the items fom inist; etun megemegesot1), megesot)); : Theoy of Agoithms Sping 10 / 31 Megesot Impementation 1) Megesot Impementation 1) Megesot Impementation 1) Megesot is ticky to impement. void megesoteem* A, Eem* temp, int eft, int ight) { int mid = eft+ight)/; if eft == ight) etun; // List of one megesota, temp, eft, mid); // Sot haf megesota, temp, mid+1, ight);// Sot haf fo int i=eft; i<=ight; i++) // Copy to temp temp[i] = A[i]; Megesot is ticky to impement. This impementation equies a second aay. void megesoteem* A, Eem* temp, int eft, int ight) { int mid = eft+ight)/; if eft == ight) etun; // List of one megesota, temp, eft, mid); // Sot haf megesota, temp, mid+1, ight);// Sot haf fo int i=eft; i<=ight; i++) // Copy to temp temp[i] = A[i]; : Theoy of Agoithms Sping 10 / 31 Megesot Impementation ) Megesot Impementation ) Megesot Impementation ) // Do the mege opeation back to aay int i1 = eft; int i = mid + 1; fo int cu=eft; cu<=ight; cu++) { if i1 == mid+1) // Left ist exhausted A[cu] = temp[i++]; ese if i > ight) // Right ist exhausted A[cu] = temp[i1++]; ese if temp[i1].key < temp[i].key) A[cu] = temp[i1++]; ese A[cu] = temp[i++]; Megesot cost: Megesot is good fo soting inked ists. // Do the mege opeation back to aay int i1 = eft; int i = mid + 1; fo int cu=eft; cu<=ight; cu++) { if i1 == mid+1) // Left ist exhausted A[cu] = temp[i++]; ese if i > ight) // Right ist exhausted A[cu] = temp[i1++]; ese if temp[i1].key < temp[i].key) A[cu] = temp[i1++]; ese A[cu] = temp[i++]; Megesot cost: Megesot is good fo soting inked ists. : Theoy of Agoithms Sping 10 1 / 31 Megesot cost: Θn og n) Linked ists: Send ecods to atenating inked ists, megesot each, then mege.

5 a) b) og n Heap: Compete binay tee with the Heap Popety: Min-heap: a vaues ess than chid vaues. Max-heap: a vaues geate than chid vaues. The vaues in a heap ae patiay odeed. Heap epesentation: nomay the aay based compete binay tee epesentation. Heap: Compete binay tee with the Heap Popety: Min-heap: a vaues ess than chid vaues. Max-heap: a vaues geate than chid vaues. The vaues in a heap ae patiay odeed. Heap epesentation: nomay the aay based compete binay tee epesentation. : Theoy of Agoithms Sping 10 / 31 Buiding the Heap Buiding the Heap Buiding the Heap a) equies exchanges 4-), 4-1), -1), 5-), 5-4), -3), -5), 7-5), 7-). b) equies exchanges 5-), 7-3), 7-1), -1) a) This is a Max Heap How to get a good numbe of exchanges? By induction. Heapify the oot s subtees, then push the oot to the coect eve b) a) equies exchanges 4-), 4-1), -1), 5-), 5-4), -3), -5), 7-5), 7-). b) equies exchanges 5-), 7-3), 7-1), -1). : Theoy of Agoithms Sping / 31 void heap::siftdownint pos) { // Sift ELEM down assetpos >= 0) && pos < n)); whie!isleafpos)) { int j = eftchidpos); if j<n-1)) && Heap[j].key < Heap[j+1].key)) j++; // j now index of chid with > vaue if Heap[pos].key >= Heap[j].key) etun; swapheap, pos, j); pos = j; // Move down void heap::siftdownint pos) { // Sift ELEM down assetpos >= 0) && pos < n)); whie!isleafpos)) { int j = eftchidpos); if j<n-1)) && Heap[j].key < Heap[j+1].key)) j++; // j now index of chid with > vaue if Heap[pos].key >= Heap[j].key) etun; swapheap, pos, j); pos = j; // Move down : Theoy of Agoithms Sping / 31 BuidHeap Fo fast heap constuction: Wok fom high end of aay to ow end. Ca siftdown fo each item. Don t need to ca siftdown on eaf nodes. void heap::buidheap) // Heapify contents { fo int i=n/-1; i>=0; i--) siftdowni); Cost fo heap constuction: BuidHeap Fo fast heap constuction: BuidHeap Wok fom high end of aay to ow end. Ca siftdown fo each item. Don t need to ca siftdown on eaf nodes. void heap::buidheap) // Heapify contents { fo int i=n/-1; i>=0; i--) siftdowni); Cost fo heap constuction: i 1) n n. i i 1) is numbe of steps down, n/ i is numbe of nodes at that eve. The intuition fo why this cost is Θn) is impotant. Fundamentay, the issue is that neay a nodes in a tee ae cose to the bottom, and we ae wost case) pushing a nodes down to the bottom. So most nodes have nowhee to go, eading to ow cost. og n i 1) n n. i : Theoy of Agoithms Sping 10 / 31