CSE 2320 Notes 8: Sorting. (Last updated 10/3/18 7:16 PM) Idea: Take an unsorted (sub)array and partition into two subarrays such that.

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1 CSE Notes 8: Sortig (Last updated //8 7:6 PM) CLRS 7.-7., 9., A. QUICKSORT Cocepts Idea: Take a usorted (sub)array ad partitio ito two subarrays such that p q r x y z x y y z Pivot Customarily, the last subarray elemet (subscript r) is used as the pivot value. After partitioig, each of the two subarrays, p... q ad q +... r, are sorted recursively. Subscript q is retured from PARTITION (aside: some versios do t place pivot i its fial positio). Like MERGESORT, QUICKSORT is a divide-ad-coquer techique: MERGESORT QUICKSORT Divide Trivial PARTITION (i-place) Subproblems Sort Two Parts Sort Two Parts Combie MERGE Trivial (ot i-place) Bottom-up Yes No possible?

2 Versio : PARTITION (i Θ( ) time, see ) A B x # z * y Pivot Already kow to have x y Already kow to have y < z Utouched Termiatio y < *: Move B over * y: Swap # & * Move B over Move A over A ad B ca be at the the same positio... A # B y Swap # & y to place y i its fial positio. it ewpartitio(it arr[],it p,it r) // From CLRS, d ed. { it x,i,j,temp; x=arr[r]; i=p-; for (j=p;j<r;j++) if (arr[j]<=x) { i++; temp=arr[i]; arr[i]=arr[j]; arr[j]=temp; } temp=arr[i+]; arr[i+]=arr[r]; arr[r]=temp; retur i+; }

3 Example: AB A 6 B A 6 B A 6 7 B A 7 6 B A B A B A B 5 4 A B 5 4 A B 5 4 < 5 > Versio (Aside: Sedgewick, similar to CLRS, p. 85): Poiters move toward each other (also i Θ( ) time, see ) A B x # * z y Pivot Already kow to have x y Already kow to have y z Utouched a b # < y: Move A right y < *: Move B left c Swap # ad * (uless A ad B have collided) Termiatio A B * # y Swap # & y to place y i its fial positio.

4 it partitio(item *a,it ell,it r) { // From Sedgewick, but more complicated sice poiters move // towards each other. // Elemets before i are <= pivot. // Elemets after j are >= pivot. it i = ell-, j = r; Item v = a[r]; 4 pritf("iput\"); dump(arr,ell,r); for (;;) { // Sice pivot is the right ed, this while has a setiel. // Stops at ay elemet >= pivot while (less(a[++i], v)) ; // Stops at ay elemet <= pivot (but ot the pivot) or at the left ed while (less(v, a[--j])) if (j == ell) break; if (i >= j) break; // Do't eed to swap exch(a[i], a[j]); } exch(a[i], a[r]); // Place pivot at fial positio for sort retur i; } Examples: A B Left positioed A B 5 Right positioed A B 5 After swap A B 5 Left cotiues A B 5 Left positioed A B 6 5 Right positioed A B 6 5 After swap A B 6 5 Left cotiues A B 6 5 Left positioed A B Right cotiues A 8 4 B Right positioed A 4 8 B After swap 4 A 8 B Left positioed 4 AB Poiters collided 4 < 5> Pivot positioed

5 A B Left positioed A B 5 Right positioed A B 5 After swap 4 A B 5 Left positioed 4 A B 9 5 Right positioed 4 A B 9 5 After swap 4 A B 9 5 Left positioed 4 A B Right positioed 4 A B After swap 4 A B Left positioed 4 A 6 5 B Right positioed 4 A 5 6 B After swap 4 A 5 6 B Left positioed 4 A 5 B Poiters collided 4 < 5> Pivot positioed 5 QUICKSORT Aalysis [Aside: also applies to the biary search trees of Notes ] Worst Case Pivot is smallest or largest key i subarray every time. (Icludes ascedig or descedig order.) Let T be the umber of comparisos. T( ) = T( ) + = i = Θ i= Best Case Pivot ( media ) always eds up i the middle. T( ) = T( ) + (Similar to mergesort.) Expected Case Assume all! permutatios are equally likely to occur. Likewise, each elemet is equally likely to occur as the pivot (each of the elemets will be the pivot i ( )! cases). E the expected umber of comparisos. E =. is E( ) = + E i i= Show Ο( log ). Suppose E i + c E = + c = + c ( + E( i) ) = + cili for i <. E i i= ili -+ c xl xdx [Boud above by itegral] i= x l x x [From 4 l = + cl c + c cl for c =6 i i 5 4

6 Other issues: Ubalaced partitioig also leads to worst-case stack depth i Θ( ). Small subfiles - use simpler sort o each subfile or delay util quicksort fiishes. Pivot selectio - radom, media-of-three Subfile with all keys equal for versio ad partitioig? 8.B. SELECTION AND RANKING USING QUICKSORT PARTITIONING IDEAS Fidig kth largest (or smallest) elemet i uordered table of elemets (Aside: If k is small, e.g. Ο log, use a heap.) Sort everythig? Use PARTITION several times. Always throw away the subarray that caot iclude the target. 5 (quickselectio) (quicklargest) Θ worst case (e.g. iput ordered) expected. Let E( k,) represet the expected umber of comparisos to fid the Θ kth largest i a set of umbers. (Assume all! permutatios are equally likely.) Suppose = 7 ad k =. After 6 comparisos to place a pivot, the 7 possible pivot positios require differet umbers of additioal comparisos: E,6 E,5 E,4 4 E, 5 6 E,5 7 E,6

7 Suppose = 8 ad k = 6. After 7 comparisos to place a pivot, the 8 possible pivot positios require differet umbers of additioal comparisos: 7 E 6,7 E 6,6 4 E, 5 E,4 6 E,5 7 E 4,6 8 E 5,7 Observatio: Fidig the media is slightly more difficult tha all other cases. E( k,) = + k E i, k + i i= + E k,i i=k Show Ο(). Usig substitutio method, suppose E( i, j) cj for j <. k c( k + i) + ci i= i=k = + c k ( k + i) + c i i= i=k = + c k ( k) + c k i + c i i= i= i=k = + c ( k )( k) + c i = + c ( k )( k) + c ( ) i= + c c ( ) k = + maximizes c k E( k,) + = + c 4 c + 4 c + c c c for c 4 ( k) = + c 4 c + c 4 = c c c 4 c 8.C. LOWER BOUNDS ON SORTING Sice a lower boud o a problem is to apply to a umber of algorithms, it is ecessary to have a model that captures the essetial features of those algorithms. It is possible for algorithms to exist that do ot follow the model.

8 Example Decisio Tree 8 < go left. > go right. A:A A:A A:A A/A/A A:A A/A/A A:A A/A/A A/A/A A/A/A A/A/A Decisio Tree Model for Sortig Two keys may be compared i Θ( ) time. The time for other processig is proportioal to the umber of comparisos. All! possible iput permutatios must be successfully sorted. (Leaves are labeled to show how iput array has bee rearraged.) A tree with outcomes as leaves ad decisios as iteral odes may be costructed for a algorithm ad a specific value of. Worst-case comparisos? Expected comparisos? What is the miimum height of a decisio tree for sortig keys? Sice there must be! leaves, the the height is Ω( lg(! )) = Ω( lg). [Notes.D] Other Examples of Lower Bouds (aside) Biary search o ordered table leaves for the outcomes. Ω( lg) lower boud. (Searchig uordered table? Use adversary istead.) Problem: Give decisio tree model for mergig two ordered tables with elemets each.. Number of outcomes is based o: a. elemets i output table. b. elemets of output table will receive elemets of first table i their origial order (but possibly separated by elemets from secod table).

9 . Number of leaves = umber of outcomes = 9 = ( )! 4 6! =!!!! ( 5! ) ( 5! ) =!!! = i + i= i i= i i =. Height of tree is bouded below by lg(umber of leaves) = lg lg = lg. 8.D. STABLE SORTING (review) A sort is stable if two elemets with equal keys maitai their origial (iput) order i the output. Practical sigificace is for situatios with a compoud key:. Each time a user logs ito a computer, a record is created with user ame, date, ad time.. Oce a year, each user receives a chroological report listig their log-is.. If a stable sort is available, the the sort for () ca use just the user ame as the sort key. Which sorts ca be coded aturally to achieve stability? Selectio Isertio Merge Heap Quick How ca a ustable sort be forced to behave like a stable sort? 8.E. LINEAR TIME SORTING If the rage of keys is limited, the sortig by direct key comparisos might ot be the fastest method. Coutig Sort Sort records with keys i rage... k.. Clear cout table oe couter for each value i rage. Θ( k) for (i=; i<k; i++) cout[i]=;. Pass through iput table add to appropriate couter for each key. Θ( ) for (i=; i<; i++) cout[ip[i]]++;

10 . Determie first slot that will receive a record for each rage value. Θ( k) slot[]=; for (i=; i<k; i++) slot[i]=slot[i-]+cout[i-]; 4. Copy each record to output, icremet idex i table from (). Θ( ) for (i=; i<; i++) out[slot[ip[i]]++]=ip[i]; Overall, takes time Θ( k + ) which will be Θ( ) if k is bouded (LSD first) Radix Sort Example: Sortig records whose keys are 9-digit Social Security Numbers.. Treat each SSN as three digit umber ABC where each digit is i the rage c=ss%; b=(ss/)%; a=ss/;. Use coutig sort to sort all records o C.. Use coutig sort to sort all records o B. (Must be doe i stable fashio.) 4. Use coutig sort to sort all records o A. (Must be doe i stable fashio.)

11 A B C A B C A B C Time is Θ d k + umber of records. ( ) where d is the umber of digits (), k is the size of the radix (), ad is the Aside: I your favorite programmig laguage, give geeral code for isolatig a eeded digit from a key. Aside: d ad k deped o each other ragesize = k d k = d ragesize Icoveiet to compare asymptotically with key-compariso based sorts. If the radix is biary, code similar to PARTITION may be used istead of coutig sort. Test Questio: A billio umbers i the rage... 9,999,999 are to be sorted by LSD radix sort. How much faster will this be doe if a decimal radix is used rather tha a biary radix? Show your work.