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" # $ %&')*+,-./01246789:;<=>5?3MABCD EFGHIJKLPb$f۰b$h~" O>S0R۰b$U9`dE^Jh۰b$u0ȒC'{۰b$!pAu[kJ۰b$ک)4"[3 ̜۰b$~߹!,۰b$8:g *b3d۰b$w.]ͺU#awY۰b$® &LηB۰b$!#qf۰b$J08+TM۰b$3_fz!g 0vc۰b$R)qTɟ:Ao.۰b$;H te}HE۰b$)-D)W1\۰c $ n3f3@8d >$ʚ;+4ʚ;g4UdUd@t@ppp@ <4!d!d,gX<4dddd,gXg4`d`d4@@8p@ pp>___PPT9 h___PPT2001D<4X?-"2003 W L Truppel bCS 14: Intro. Data Structures & Algs. " Lecture 5O =%Intro to Data Structures & AlgorithmsWagner Truppel Lecturer, Dept. of Computer Science & Engineering UC Riverside wagner@cs.ucr.edu http://www.cs.ucr.edu/~wagner http://www.cs.ucr.edu/cs14zO  O`a~Today s TopicsSorting (I) The best sorting algorithm The worst sorting algorithm The rest& BubbleSort SelectionSort InsertionSort MergeSort: B1 B1N1Formal conceptsAscending (non-decreasing) for all i in [0, length-2]: a[i] <= a[i+1] 2, 7, 9, 9, 14, 27, 31, 31, 56 Descending (non-increasing) for all i in [0, length-2]: a[i] >= a[i+1] 56, 31, 31, 27, 14, 9, 9, 7, 2ZZZ ZZZZZ      Sorting (I)Sorting requires additional information Comparison/equality operator (==) alone isn t enough That is, sorting requires support for an additional operation: an order operator (<, >, <=, or >=) So sorting applies only to those ADTs which also satisfy the interface of the ADT Ordered ADT Ordered defines one or more operators which lets us order elements bFW T=|The best sorting alg$ The human sorting 12, 4, 7, 3, 9 We  magically sort this array by moving each element to its correct place in a single step: 3, 4, 7, 9, 12 Time complexity: roughly constant time Problems with this approach: We can do it by eye; the computer can t do it at all It seems we have some global view of the array; the computer only sees things locally Even we cannot do it for large arrays\Z{ZDZZ{DThe worst sorting alg$ Suppose we have a function bool isAscending(Array<T> & a) which returns true if the array is in ascending order First, how much time does that function take to execute ? It should take O(n) time Here s an implementation bool isAscending(Array<T> & a) { for (int i = 0; i < a.length() - 1; i++) { if (a[i] > a[i + 1]) return false; } return true; } ZUZlZZZ& I>kThe worst sorting alg$ Another question: how many different possibilities (permutations) are there for an array of n elements ? Example: n = 3, a = { 3, 2, 7 } { 2, 3, 7 }, { 2, 7, 3 } { 3, 2, 7 }, { 3, 7, 2 } { 7, 2, 3 }, { 7, 3, 2 } For n = 3 we have 6 permutations In general, there are n! permutations [ n factorial = n! = n x (n-1) x & x 2 x 1 ]:Z.Z\The worst sorting alg$ So here s a sorting algorithm: while there are permutations produce the next permutation as an array if isAscending(permutation) Break How much time does it take ? There are n! permutations, each taking O(n) time, so the total time complexity is O(n x n!) - exponential ! Worse than O(n!), which is really bad ! The computer could do it, but would take a loooooooooooong timeZZEZ ZZZE  { .,h Sorting extremes!So we have two extremes The best sorting algorithm Roughly O(1), which is awesome But the computer cannot do it The worst sorting algorithm Worse than O(n!), which is horrible The computer could do it, but it would take too long All other sorting algorithms lie in between these extremesZZ>ZZZZ;Z     6  ;BubbleSort ('Nave')  Suppose we want to sort (ascending) the array {5,4,1,2,7} BubbleSort selects two adjacent elements at a time and swaps them if they re out of sequence This is done for every sequential pair (inner loop) And then repeated from the start again (outer loop)nZ  | * ,: yABubbleSort ('Nave')  5Outer loop: 0Inner loop: 0 Select the pair {5,4}N6Z BubbleSort ('Nave')  dOuter loop: 0Inner loop: 0 Select the pair {5,4} Its elements are out of sequence, so swap themNeZ /BubbleSort ('Nave')  5Outer loop: 0Inner loop: 1 Select the pair {5,1}N6Z BubbleSort ('Nave')  dOuter loop: 0Inner loop: 1 Select the pair {5,1} Its elements are out of sequence, so swap themXeZ .BubbleSort ('Nave')  5Outer loop: 0Inner loop: 2 Select the pair {5,2}N6Z BubbleSort ('Nave')  dOuter loop: 0Inner loop: 2 Select the pair {5,2} Its elements are out of sequence, so swap themXeZ .BubbleSort ('Nave')  5Outer loop: 0Inner loop: 3 Select the pair {5,7}X6Z  BubbleSort ('Nave')  Outer loop: 0Inner loop: 3 Select the pair {5,7} Its elements are not out of sequence, so don t swap themXoZ 8!BubbleSort ('Nave')   The element 7 is now in its final position and won t be ever moved again Outer loop: 1Inner loop: 0 Start over by selecting the pair {4,1}dZ I'IG"BubbleSort ('Nave')  ~The element 7 is now in its final position and won t be ever moved again Outer loop: 1Inner loop: 0 Start over by selecting the pair {4,1} Its elements are out of sequence, so swap themZ I!.#BubbleSort ('Nave')  The element 7 is now in its final position and won t be ever moved again Outer loop: 1Inner loop: 1 Select the pair {4,2}dZ I$BubbleSort ('Nave')  \The element 7 is now in its final position and won t be ever moved again Outer loop: 1Inner loop: 1 Select the pair {4,2} Its elements are out of sequence, so swap themnZ I.%BubbleSort ('Nave')  The element 7 is now in its final position and won t be ever moved again Outer loop: 1Inner loop: 2 Select the pair {4,5}nZ I&BubbleSort ('Nave')  pThe element 7 is now in its final position and won t be ever moved again Outer loop: 1Inner loop: 2 Select the pair {4,5} Its elements are not out of sequence, so don t swap themZ I 'BubbleSort ('Nave')  *The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Inner loop: 0 Start over by selecting the pair {1,2}Z G&(BubbleSort ('Nave')  The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Inner loop: 0 Start over by selecting the pair {1,2} Its elements are not out of sequence, so don t swap themZ G! )BubbleSort ('Nave')  The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Inner loop: 1 Select the pair {2,4}zZ G*BubbleSort ('Nave')  zThe elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Inner loop: 1 Select the pair {2,4} Its elements are not out of sequence, so don t swap themZ G +BubbleSort ('Nave')  2The elements 7, 5, and 4 are now in their final positions and won t be ever again Outer loop: 3Inner loop: 0 Start over by selecting the pair {1,2}Z G',BubbleSort ('Nave')  The elements 7, 5, and 4 are now in their final positions and won t be ever again Outer loop: 3Inner loop: 0 Select the pair {1,2} Its elements are not out of sequence, so don t swap themZ G -BubbleSort ('Nave')  \The elements 7, 5, 4, and 2 are now in their final positions and won t be ever again Outer loop: 4Inner loop: 0 There are no more pairs to select, so the algorithm endsZ G9.BubbleSort ('Nave')  The elements 7, 5, 4, and 2 are now in their final positions and won t be ever again Outer loop: 4Inner loop: 0 There are no more pairs to select, so the algorithm ends Note that the algorithm does not end when the array is  obviously sorted ZJ GW*BubbleSort ('Nave')  *int n = a.length(); for (int i = 0; i < n - 1; i++) // outer loop for (int j = 0; j < n - 1 - i; j++) // inner loop if (a[j+1] < a[j]) Swap(a[j+1], a[j]); It s very slow& O(n2) in the worst case With an optimization, it s possible to get O(n) in the best case [ array already sorted ] The optimization is: stop when no swaps have occurred for an entire pass (an entire iteration of the outer variable i)BZ2ZZ ZZB2     > 6*K/BubbleSort ('Smart')  2The smart version of BubbleSort starts the same way as the nave version, so I ll skip the steps from slides 10 to 20 - this is slide 21 The element 7 is now in its final position and won t be ever moved again Outer loop: 1Swapped: YesInner loop: 1 Select the pair {4,2}Z  UI 0BubbleSort ('Smart')  vThe element 7 is now in its final position and won t be ever moved again Outer loop: 1Swapped: YesInner loop: 1 Select the pair {4,2} Its elements are out of sequence, so swap themzZ I /1BubbleSort ('Smart')  The element 7 is now in its final position and won t be ever moved again Outer loop: 1Swapped: YesInner loop: 2 Select the pair {4,5}zZ I 2BubbleSort ('Smart')  The element 7 is now in its final position and won t be ever moved again Outer loop: 1Swapped: YesInner loop: 2 Select the pair {4,5} Its elements are not out of sequence, so don t swap themZ I  3BubbleSort ('Smart')  The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Swapped: NoInner loop: 0 Start over by selecting the pair {1,2} Starting over means to start a new iteration of the outer loop (on the i variable) If we detect no swaps through an entire iteration of that variable, we can stopEZ G &5j4BubbleSort ('Smart')  The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Swapped: NoInner loop: 0 Start over by selecting the pair {1,2} Its elements are not out of sequence, so don t swap themZ G ( 5BubbleSort ('Smart')   The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Swapped: NoInner loop: 1 Select the pair {2,4}Z G 6BubbleSort ('Smart')  The elements 7 and 5 are now in their final positions and won t be ever again Outer loop: 2Swapped: NoInner loop: 1 Select the pair {2,4} Its elements are not out of sequence, so don t swap themZ G  7BubbleSort ('Smart')  The elements 7, 5, and 4 are now in their final positions and won t be ever again Outer loop: 2Swapped: NoInner loop: 1 The nave version would continue by starting over and selecting the pair {1,2} but& Z G U8BubbleSort ('Smart')  |The elements 7, 5, and 4 are now in their       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~final positions and won t be ever again Outer loop: 2Swapped: NoInner loop: 1 & the smart version realizes that an entire iteration of the outer loop went by without any swaps, so it stops The smart version is stil O(n2) in the worst case But it s O(n) in the best caseN?Z G >1     7I SelectionSort  Similar to BubbleSort but instead of bubbling the largest element towards the end of the current sub-array, we find that largest element and just move it to its final position It s also slow& O(n2) in the worst case, but typically much better than BubbleSort Two variations: Select the minimum Select the maximum No conceptual differences between themZ'Z'Z  Z=   !    '6  ^:SelectionSort (Min)Suppose again that we want to sort (ascending) the array {5,4,1,2,7} The Min version of SelectionSort finds the minimum element of the current sub-array and swaps it with the first element of the current sub-array Then it decreases the size of the current sub-array by 1 and starts again $    8lWFSelectionSort (Min)The current sub-array is the entire initial array, so its size is 5. The algorithm searches for the min throughout the entire current sub-array, which takes O(n) time. The alg finds that the minimum element is 1.B %<SelectionSort (Min) The current sub-array is the entire initial array, so its size is 5. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 1. The alg swaps 1 with 5, since 5 is the first element of the current sub-array. B ",)G=SelectionSort (Min)The current sub-array has now size 4. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 2.# v%>SelectionSort (Min)The current sub-array has now size 4. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 2. The alg swaps 2 with 4, since 4 is the first element of the current sub-array.# #,v)G?SelectionSort (Min)The current sub-array has now size 3. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 4.# v%@SelectionSort (Min)The current sub-array has now size 3. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 4. The alg swaps 4 with 5, since 5 is the first element of the current sub-array.# #,v)GASelectionSort (Min)The current sub-array has now size 2. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 5.# v%BSelectionSort (Min)The current sub-array has now size 2. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 5. The alg swaps 5 with 5, since 5 is the first element of the current sub-array.# #,v)GCSelectionSort (Min)The current sub-array has now size 1. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 7.# v%DSelectionSort (Min)DThe current sub-array has now size 1. The algorithm searches for the min throughout the entire current sub-array. The alg finds that the minimum element is 7. The alg swaps 7 with 7, since 7 is the first element of the current sub-array. The size of the current sub-array would now be decreased to 0, so the algorithm stops.EZ# z,v)GSelectionSort (Max) int csas = a.length(); // current sub-array size while (csas > 0 ) { int idx_of_max = 0; for (int i = 0; i < csas; i++) { if (a[i] > a[idx_of_max]) idx_of_max = i; } swap(a[idx_of_max], a[csas-1]); csas--; }6Z 0     InsertionSort  InsertionSort divides the array in two sections: the sorted region and the unsorted region At every step, the algorithm selects the first element of the unsorted region and inserts it in the correct position inside the sorted region Thus, in every step, the sorted region grows in size by 1 while the unsorted region shrinks in size by 1 It s also slow& O(n2) in the worst case.|1   !    mK InsertionSort bInitially, the unsorted region is the entire array, {5,4,1,2,7}, while the sorted region is empty.ncZ   L InsertionSort Initially, the unsorted region is the entire array, {5,4,1,2,7}, while the sorted region is empty. Select the first element of the unsorted region, that is, 5.Z    M InsertionSort Initially, the unsorted region is the entire array, {5,4,1,2,7}, while the sorted region is empty. Select the first element of the unsorted region, that is, 5. Insert it at the proper position inside the sorted region.Z    ;N InsertionSort The unsorted region is now {4,1,2,7}, while the sorted region is {5}. Select the first element of the unsorted region, that is, 4.Z    O InsertionSort The unsorted region is now {4,1,2,7}, while the sorted region is {5}. Select the first element of the unsorted region, that is, 4. Insert it at the proper position inside the sorted region.Z    ;P InsertionSort The unsorted region is now {1,2,7}, while the sorted region is {4,5}. Select the first element of the unsorted region, that is, 1.Z   Q InsertionSort The unsorted region is now {1,2,7}, while the sorted region is {4,5}. Select the first element of the unsorted region, that is, 1. Insert it at the proper position inside the sorted region.Z   ;R InsertionSort The unsorted region is now {2,7}, while the sorted region is {1,4,5}. Select the first element of the unsorted region, that is, 2. ZZ   S InsertionSort The unsorted region is now {2,7}, while the sorted region is {1,4,5}. Select the first element of the unsorted region, that is, 2. Insert it at the proper position inside the sorted region.Z   ;T InsertionSort The unsorted region is now {7}, while the sorted region is {1,2,4,5}. Select the first element of the unsorted region, that is, 7.Z    U InsertionSort The unsorted region is now {7}, while the sorted region is {1,2,4,5}. Select the first element of the unsorted region, that is, 7. Insert it at the proper position inside the sorted region.Z    ;V InsertionSort The unsorted region is now empty, while the sorted region is {1,2,4,5,7}. Since the unsorted region is empty, the algorithm stops.zZ   :J InsertionSort // n = size of the array // uns_1st= 1st index of unsorted region // ins_idx = index of where to insert // uns_next = next item (of some type T) // in the unsorted region for (int uns_1st = 1; uns_1st < n; uns_1st++) { T uns_next = a[uns_1st]; int ins_idx = uns_1st; for (; ins_idx > 0 && a[ins_idx - 1] > uns_next; ins_idx--) { a[ins_idx] = a[ins_idx - 1]; } a[ins_idx] = uns_next; }*&          MergeSort 6MergeSort is a recursive divide-and-conquer algorithm The algorithm divides the array in half (the divide step) sorts each half separately (the recursion step) then merges the two sorted parts together (the conquer step) It s always O[n log(n)]. Fast! It requires additional storage. Not so good. Merging the two sorted halves is the complicated part of the algorithm. I ll let you read this in the textbook.DDZZZZZ*ZZZ   ( 1           H T-?WSorting (I) summary* we ll look at the last three algorithms later in the course. ** QuickSort s worst-case is very rare and the algorithm is much faster in practice than the others. *** RadixSort is not like the other sorting algorithms, so it s a bit unfair to compare it to them.Z> Y W,B Y WXSorting (I) summaryWhy is sorting so important? It s essential for any kind of fast search on large databases It s often the first step in more complicated algorithms, for example, in computational geometry and scene rendering For java animations of all of these sorting algorithms, point your web browser to http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/sorting.html ZZZR=* R=u#`/PsxHH@Rg(HH(dh   ` www3ff` 3fff` ___>?" dF@,?n<d@uA @ " d`  n?" dd@   @@``PR   @ ` `P0p>> -"2003 W L Truppel bCS 14: Intro. Data Structures & Algs. " Lecture 5 j( = 2  N`U0?"! :    NT0gֳgֳ ?"P  T Click to edit Master title style! !<  HV0gֳgֳ ?"  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www3ff   zr` ( 00// r  S @q0  D $  C A figA21.pdf 00187D56Earth B9B583FF:`@p,  Tqgֳgֳ ?p$ DPH___PPT2001$ΐ H  0޽h ? www3ff   zrp (  r  S q0  D $  C A figA22.pdf 00187D56Earth B9B583FF:`@p,  T`qgֳgֳ ?p$ DPH___PPT2001$ΐ H  0޽h ? www3ff(   LP( | Lr L S @q0  D  L S q0<$ D H L 0޽h ? www3ff   (  r  S Or0  D $  C A figA11.pdf 00187D56Earth B9B583FF:`@p,  TpOrgֳgֳ ?p$ DPH___PPT2001$ΐ   HOrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   ( p r  S Rr0  D $  C A  figA12.pdf 00187D56Earth B9B583FF:`@p,  TQrgֳgֳ ?p$ DPH___PPT2001$ΐ   H0Prgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   (  r  S Sr0  D $  C A  figA13.pdf 00187D56Earth B9B583FF:`@p,  T0Srgֳgֳ ?p$ DPH___PPT2001$ΐ   HRrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   ( 4kh4 r  S Ur0  D $  C A  figA14.pdf 00187D56Earth B9B583FF:`@p,  TpUrgֳgֳ ?p$ DPH___PPT2001$ΐ   HUrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   (  r  S Xr0  D $  C A  figA15.pdf 00187D56Earth B9B583FF:`@p,  TWrgֳgֳ ?p$ DPH___PPT2001$ΐ   HPWrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   (  r  S r0  D $  C A  figA16.pdf 00187D56Earth B9B583FF:`@p,  T0rgֳgֳ ?p$ DPH___PPT2001$ΐ   HPrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   (  r  S r0  D $  C A figA17.pdf 00187D56Earth B9B583FF:`@p,  TPrgֳgֳ ?p$ DPH___PPT2001$ΐ   Hrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   (  r  S Pr0  D $  C A figA18.pdf 00187D56Earth B9B583FF:`@p,  Trgֳgֳ ?p$ DPH___PPT2001$ΐ   Hrgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff   (  r  S r0  D $  C A figA19.pdf 00187D56Earth B9B583FF:`@p,  Trgֳgֳ ?p$ DPH___PPT2001$ΐ   H0rgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff    ( D8bbpb`b r  S q0  D ,  Tqgֳgֳ ?p$ DPH___PPT2001$ΐ $  C A figA23.pdf 00187D56Earth B9B583FF:`@p  H@esgֳgֳ ? `$PH___PPT2001$ΐ nSwapped is a boolean flag to indicate whether there has been a swap in the current iteration of the outer loopXoZ] ZH  0޽h ? www3ff(   0pP( 00// pr p S @hs0  D  p S gs<$ D H p 0޽h ? www3ff   zr@ ( 00// r  S js0  D ,  Tjsgֳgֳ ?p$ DPH___PPT2001$ΐ $  C A figB00.pdf 00187D56Earth B9B583FF:8PH  0޽h ? www3ff   zrPd ( 00// dr d S hs0  D $ d C A" figB01.pdf 00187D56Earth B9B583FF:8P, d Tisgֳgֳ ?p$ DPH___PPT2001$ΐ H d 0޽h ? www3ff   zr`< ( 00// <r < S ls0  D $  < C A! figB02.pdf 00187D56Earth B9B583FF:8P, < T`lsgֳgֳ ?p$ DPH___PPT2001$ΐ H < 0޽h ? www3ff   zrp@ ( 00// @r @ S  ms0  D $ @ C A  figB03.pdf 00187D56Earth B9B583FF:8P, @ T jsgֳgֳ ?p$ DPH___PPT2001$ΐ H @ 0޽h ? www3ff   zrD ( 00// Dr D S ms0  D $ D C A figB04.pdf 00187D56Earth B9B583FF:8P, D T@nsgֳgֳ ?p$ DPH___PPT2001$ΐ H D 0޽h ? www3ff   zrH ( 00// Hr H S g0  D $ H C A figB05.pdf 00187D56Earth B9B583FF:8P, H Thgֳgֳ ?p$ DPH___PPT2001$ΐ H H 0޽h ? www3ff   zrL ( 00// Lr L S k0  D $ L C A figB06.pdf 00187D56Earth B9B583FF:8P, L Tjgֳgֳ ?p$ DPH___PPT2001$ΐ H L 0޽h ? www3ff   zrP ( 00// Pr P S @m0  D $ P C A figB07.pdf 00187D56Earth B9B583FF:8P, P Tlgֳgֳ ?p$ DPH___PPT2001$ΐ H P 0޽h ? www3ff   zrT ( 00// Tr T S `n0  D $ T C A figB08.pdf 00187D56Earth B9B583FF:8P, T Tngֳgֳ ?p$ DPH___PPT2001$ΐ H T 0޽h ? www3ff   zrX (   Xr X S o0  D $ X C A figB09.pdf 00187D56Earth B9B583FF:8P, X Togֳgֳ ?p$ DPH___PPT2001$ΐ H X 0޽h ? www3ff   zr\ ( 4mlTl l \r \ S q0  D $ \ C A figB10.pdf 00187D56Earth B9B583FF:8P, \ T`qgֳgֳ ?p$ DPH___PPT2001$ΐ H \ 0޽h ? www3ff(   hP( 00// hr h S r0  D  h S  r0<$ D H h 0޽h ? www3ff(   @P(  @r @ S :0  D  @ S  :<$ D H @ 0޽h ? www3ff   zrx ( 00// xr x S :0  D , x T:gֳgֳ ?p$ DPH___PPT2001$ΐ $ x C A figC00.pdf 00187D56Earth B9B583FF:`@pH x 0޽h ? www3ff   zr | (  |r | S :0  D , | T:gֳgֳ ?p$ DPH___PPT2001$ΐ $ | C A# figC01.pdf 00187D56Earth B9B583FF:`@pH | 0޽h ? www3ff   zr0 (  r  S :0  D ,  T@:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A$ figC02.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr@ (  r  S  :0  D ,  T:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A% figC03.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zrP (  r  S @:0  D ,  T:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A& figC04.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr` ( 00// r  S @:0  D ,  T:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A' figC05.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zrp (  r  S :0  D ,  T`:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A( figC06.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr (  r  S :0  D ,  T :gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A) figC07.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr (  r  S :0  D ,  T@:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A* figC08.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr ( 00// r  S :0  D ,  T:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A+ figC09.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr ( <7rd=rCrIr r  S :0  D ,  T`:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A, figC10.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff   zr ( {rrhr r  S  :0  D ,  T:gֳgֳ ?p$ DPH___PPT2001$ΐ $  C A- figC11.pdf 00187D56Earth B9B583FF:`@pH  0޽h ? www3ff(   tP( 00// tr t S r0  D  t S  l<$ D H t 0޽h ? www3ff(   HP( 00// Hr H S P0  D  H S p<$ D H H 0޽h ? www3ff   kc0)O( 00// r  S ]<0  D mr 0  O#"2&:9:9bbb:x @  = 6d<?"0   ZO[n]$ @` ; 6e<?" 0  NO[n] @` 9 6pf<?"   RadixSort***(    @` 6 6@<?"0   a O[n Log(n)]$   @` 4 6<?"  0  iO[n2]2  @` 2 6`<?"   QuickSort**(    @` / 6<?"0   a O[n Log(n)]$   @` - 6<?" 0  U O[n Log(n)]   @` + 6 <?"    } HeapSort*(   @`  6<?"0  a O[n Log(n)]$   @`  6<?" 0  U O[n Log(n)]   @`  6<?"  u MergeSort(    @`  6<?"0 iO[n2]2  @`  6<?" 0 iO[n2]2  @`  6`=?"  y InsertionSort(   @`   6=?"0" iO[n2]2  @`   6=?" "0 iO[n2]2  @`   6@=?""  y SelectionSort(   @`   6=?"0)" iO[n2]2  @`   6 =?" )0" iO[n2]2  @`  6=?") " v BubbleSort(    @`  6=?"00) f Average case(   @`  6 !=?" 00) d Worst case(   @`  6"=?"0 ) T  @`nB  0o ?"00hB  s *1 ?"))hB  s *1 ?"""hB  s *1 ?"hB  s *1 ?"nB  0o ?"  nB  0o ?"0 hB  s *1 ?" 0 hB  s *1 ?"000 nB  0o ?"0 hB , s *1 ?"  hB 3 s *1 ?"  hB : s *1 ?"   M T@"=gֳgֳ ? 0<$ D H  0޽h ? www3ff(   pP( p r  S )0  D   S 0+`p<$ D H  0޽h ? www3ff tlPH(  HR H 3 <   4r H # p'< @  4  H H 0޽h ? ̙33dxp^RЀ3ÿ lHbP  @AL G@;b `B&V(xbs""DocumentSummaryInformation8 Current User,. Oh+'0X6 px  'GeneraleneeneWagner Truppelo176Microsoft PowerPointrP@J'@@S@@S@ʥ  G25PICT5* HH HH  EnnnnnnnnnnnnnnnngggmqPظgggmqPnb,Jn]ngdn:,,/9]ngdnZ1";[ngdn]nq2! ;q2! ;nZ1n8 nT Qn HnT Qn YUnT Qn8 nqݯ4 wׯ2Үw2lw4ޮηqݯ4 wׯ2Үw2lw4ޮηnZ1jUd_KbnbUmX^[SmneMVmn j3?dnn]K_nnn^ inm[K[nj3?dnjROcn27Xn_ /j3?daUeNgYZnnYZnnkSPjnbUmYZnnYZnaUeNgn`KZnmYOfn qB Ђ'9F|2 u4x't,'0+-+'0+-t,e  qB Ђ'9F|2 u4x't,'0+-+'0+-t,e n Z1]Z8h k?gn]Dj5nn 2Ul4]D[TnW8[nn]D2NnnndS8hnnn2NC,ng  Hnq0>Hk }P2m-)j9u+tj'0+-m;r>'0+-tj ?šq0>Hk }P2m-)j9u+tj'0+-m;r>'0+-tj ?šnZ1]iJ:idm MXZa@.n l<Jj6Nnm n BQnknS6nn9MnQ, nBQnjA".nhI(nBQn2Znnnnnmnnnnnn2ZnU& >g,[nqyD5 4 L2KްaB\n0d=t)*-@D)*-tNP$qyD5 4 L2KްaB\n0d=t)*-@D)*-tNP$nZ1]7n`4nnn nnEFnjn BQn!hnK;nn `L*V nBQn(@Z/n0+ fnnkBQn2annnn ninTXnnnnn n2an`"EYnd?"mnqyL9*b ܔZ02Vd4ߍf;>q7tH-hOLH-t(&s#؛ A"qyL9*b ܔZ02Vd4ߍf;>q7tH-hOLH-t(&s#؛ A"nZ1]7n`4n!Mk nnaI*=fnn~ pjkgo g sjkgo g?9fnR..Ung,0mn97n e,Mnn]nn97n~LY0' /)LY0' /)@Gmna!&dn@Nnn _9n8 nnn~,1P-0'4/)ܼ11P-0'4/)ܼA%]nM>MPn "n-nngMZfUhnneMVmnnX^[SjUabUmgMXnn_WjPWncLancL_n~+ P{N0'; 9'}/f /]aT + P{N0'; 9'}/f /]aTCGnk"knX5"cnnnc Tn? gn \68h =n,(Xn* $n~E=jح0'2Hk}>/3fԷaﴔJ=jح0'2Hk}>/3fԷaﴔD1hnT cin=.]K8nnn@LeTZa@.n MX^6idmXin,Ej[O8h*gN@jn~$;C"V=0'EĐ)'}D//%f&!pXC*;C"V=0'EĐ)'}D//%f&!pXCE(ancD%n=j B? mnnmn8T>Nnm n nn^6nnn nnn,cnnb0n&Jn~ 70'Ufġ 45}D/)4$f).pg;70'Ufġ 45}D/)4$f).pg;F$ZnDLn7DnF7Nnn%gn,TEFnjn nn^6nnnnnn,hnnd0nlK, Xn ~EsH>?O0'N *b9}L/)4$f).p^ɾEsH>?O0'N *b9}L/)4$f).p^ɾG!#Wn^O2dn>ahhW"nnC"7TaI*N.Emnnnnnnnnnnn~zV?Vnnnnnnnnnnnnnnnn~XBCgnnnnnnnnnnnnn@~?LYDNnnnn"~ZF`nn%~ 0[HMlnn"~]L^nns~ ^NPknnnnnnnnn~u^O]nnnnnnnnnnnnnnnnn܁~ `RSlnnnnnnnnnnnnnnnnnnnnnnnnn~q}aT`nnnnnnnnnnnnnnnnnH~bVlnnnn"~wcYcnn%~#e\]mnn"~f^fnnF~.f_amnnnnn~hbjnnnnnnnnnnnn~9Miehnnnnnnnnnnnnnnnnnnnnnnnn~jglnnnnnnnnnnnnnnnnn4~arkiknnn$~ "lkmnn~mnn ~7MoqnnM~psnnnnnnnj~tqurnnnnnnnnnnz~6rwvnnnnnnnnnnnm~  syqnn nnnnnnnnn"~i{t{unn$~1u}|nn"~ āwӀsnn"~k ~xӂynn"~!:yӄnn"~ˁ!ҁz҇snn"~{"{҈{nn"~0"L|Ҋnn#~#}ьqnnn#~#~юznnn#~X$pёnnn#~$4ѓonn ~́%ՁЕunn4~%Зnnn~M&hИnnnnnnnnnnnnn~&3Кonnnnnnnnnnnnnnnnnnnn~'Ϝvnnnnnnnnnnnnnn4~'Ϟnnn(~Yˀ'sϟnn~ (CϢn~(Σsn~)Υn^~y)Φnnnnnnnnف~G *f Ψnnnnnnnnnnnnnnn nnnnnnnnn~  *=  Ωnn nnnnnnnnnnn nnnnnnnnnnnnnnnnف~ * ͫtnn nnnnnnnnnnnnnnnnnnnn7~+ͭnnn(~+ͮnn~W+tͯn~,,Qͱn~,1Ͳon~؄,߄̴xn~-̴n~-̶n~b-~̷n~?.b̸n~.D̹n~.˺pn~ԅ.܅˻znv~/˼nnnnnnnnnnnn~/˽nnnnnnnnnnnnnnnnnnnnn~| / ˾nnnnnnnnnnnnn nnnnnnnnnnnnn~_/}˿nnnnnnnnnnnnnnnn+~B0fnn(~'0Pnn~0=n~0on~0wn~͆0׆~n~1Ćn~1n~1n~1n~s1n~c1n~S1tn~F2kn~<2cn~32[n܁~ )22222222 2S ɺnnnnnnnnnnnnnnnnn#~ 2222222222222Lɽnnnnnnnnnnnnnnnnnnnnnnnnnn~ 22222 2D nnnnnnnnnnnnnnnnnnn7~222?nn~ 2=ņn~ 2:Ɔn~27Ɇn~25Ɇn~2ˆn ՜.+,D՜.+,T    'On-screen Showsn-sIr MTimes New RomanArial WingdingsBlank Presentation&Intro to Data Structures & AlgorithmsTodays TopicsFormal concepts Sorting (I)The best sorting algThe worst sorting algThe worst sorting algThe worst sorting algSorting extremesBubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Nave')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')BubbleSort ('Smart')SelectionSortSelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Min)SelectionSort (Max)InsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSortInsertionSort MergeSortSorting (I) summarySorting (I) summary  Fonts UsedDesign Template Slide TitlesI@ 8@ _PID_HLINKS'Amailto:Wagner@cs.ucr.eduhttp://www.cs.ucr.edu/~wagnerhttp://www.cs.ucr.edu/cs12>http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/sorting.html&_g$CTWagner Truppel