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Merge and Radix Sorts Data Structures Fall, 2007 13 th

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Merge Sort (1/13) Before looking at the merge sort algorithm to sort n records, let us see how one may merge two sorted lists to get a single sorted list. Before looking at the merge sort algorithm to sort n records, let us see how one may merge two sorted lists to get a single sorted list. Merging Merging The first one, Program 7.7, uses O(n) additional space. The first one, Program 7.7, uses O(n) additional space. It merges the sorted lists (list[i], …, list[m]) and (list[m+1], …, list[n]), into a single sorted list, (sorted[i], …, sorted[n]). It merges the sorted lists (list[i], …, list[m]) and (list[m+1], …, list[n]), into a single sorted list, (sorted[i], …, sorted[n]).

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Merge (using O(n) space) Merge (using O(n) space)

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Merge Sort (3/13) O(1) space merge O(1) space merge Steps in an O(1) space merge when the total number of records, n is a perfect square */ and the number of records in each of the files to be merged is a multiple of n */ Steps in an O(1) space merge when the total number of records, n is a perfect square */ and the number of records in each of the files to be merged is a multiple of n */ Step1: Identify the n records with largest key. This is done by following right to left along the two files to be merged. Step1: Identify the n records with largest key. This is done by following right to left along the two files to be merged. Step2: Exchange the records of the second file that were identified in Step1 with those just to the left of those identified from the first file so that the n record with largest keys form a contiguous block Step2: Exchange the records of the second file that were identified in Step1 with those just to the left of those identified from the first file so that the n record with largest keys form a contiguous block

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Merge Sort (4/13) O(1) space merge (contd) O(1) space merge (contd) Step3: Swap the block of n largest records with the leftmost block (unless it is already the leftmost block). Sort the rightmost block Step3: Swap the block of n largest records with the leftmost block (unless it is already the leftmost block). Sort the rightmost block Step4: Reorder the blocks excluding the block of largest records into nondecreasing order of the last key in the blocks Step4: Reorder the blocks excluding the block of largest records into nondecreasing order of the last key in the blocks

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Merge Sort (5/13) O(1) space merge (contd) O(1) space merge (contd) Step5: Perform as many merge sub steps as needed to merge the n-1 blocks other than the block with the largest keys. Step5: Perform as many merge sub steps as needed to merge the n-1 blocks other than the block with the largest keys. 0 1 2 3 w z u x 4 6 8 a | v y 5 7 9 b | c e g i j k | d f h o p q | l m n r s t 0 1 2 3 4 z u x w 6 8 a | v y 5 7 9 b | c e g i j k | d f h o p q | l m n r s t

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Merge Sort (6/13) 6, 7, 8 are merged Segment one is merged (i.e., 0, 2, 4, 6, 8, a) Change place marker (longest sorted sequence of records) Segment one is merged (i.e., b, c, e, g, i, j, k) Change place marker Segment one is merged (i.e., o, p, q) No other segment. Sort the largest keys. Step6: Sort the block with the largest keys Step6: Sort the block with the largest keys

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When selection sort is used to implement Step 4 each block is regarded as a single record with key equal to that of the last record in the block. The time needed for these is O(n). When selection sort is used to implement Step 4 each block is regarded as a single record with key equal to that of the last record in the block. The time needed for these is O(n). The total time is O(n). The total time is O(n). The additional space used is O(1). The additional space used is O(1). Example: Example: Input list (26, 5, 77, 1, 61, 11, 59, 15, 48, 19) Input list (26, 5, 77, 1, 61, 11, 59, 15, 48, 19)

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selection sort void selectionSort(int numbers[], int array_size) { int i, j; int i, j; int min, temp; int min, temp; for (i = 0; i < array_size-1; i++) { for (i = 0; i < array_size-1; i++) { min = i; min = i; for (j = i+1; j < array_size; j++) { for (j = i+1; j < array_size; j++) { if (numbers[j] < numbers[min]) if (numbers[j] < numbers[min]) min = j; min = j; } temp = numbers[i]; temp = numbers[i]; numbers[i] = numbers[min]; numbers[i] = numbers[min]; numbers[min] = temp; numbers[min] = temp; }} O(n 2 ) O(n 2 )

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Merge Sort (7/13) Iterative merge sort Iterative merge sort 1.We assume that the input sequence has n sorted lists, each of length 1. 2.We merge these lists pairwise to obtain n/2 lists of size 2. 3.We then merge the n/2 lists pairwise, and so on, until a single list remains. Analysis Analysis Total number of passes is the celling of log 2 n Total number of passes is the celling of log 2 n merge two sorted list in linear time: O(n) merge two sorted list in linear time: O(n) The total computing time is O(n log n). The total computing time is O(n log n).

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Merge Sort (8/13) merge_pass merge_pass Invokes merge (Program 7.7) to merge the sorted sublists Invokes merge (Program 7.7) to merge the sorted sublists Perform one pass of the merge sort. It merges adjancent pairs of subfiles from list into sorted. Perform one pass of the merge sort. It merges adjancent pairs of subfiles from list into sorted. the number of elements in the list the length of the subfile [0][1][2][3][4][5][6][7][8][9] length=2 n=10 i=0 0 13 4 list sorted 457 8

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merge_sort: Perform a merge sort on the file merge_sort: Perform a merge sort on the file [0][1][2][3][4][5][6] [7][8][9] length=1 list extra n=10 2 list 4 extra 8 list 16

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Merge Sort (10/13) Recursive merge sort concept Recursive merge sort concept

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Merge Sort (10/13) Recursive merge sort concept Recursive merge sort concept

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Merge Sort (10/13) Recursive merge sort concept Recursive merge sort concept

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Merge Sort (10/13) Recursive merge sort concept Recursive merge sort concept

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Merge Sort (10/13) Recursive merge sort concept Recursive merge sort concept

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listmerge: listmerge: Takes two sorted chains and returns an integer that points to the start of the sorted list Takes two sorted chains and returns an integer that points to the start of the sorted list The link field in each record is initially set to -1 Since the elements were numbered from 0 to n-1, we use list[n] to store the start pointer

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rmerge: sort the list, list[lower], …, list[upper]. The link field in each record is initially set to -1 rmerge: sort the list, list[lower], …, list[upper]. The link field in each record is initially set to -1 start = rmerge(list, 0, n-1); [0][1][2][3][4][5][6] [7][8][9] lower= upper= middle= 0 9 4 4 2 2 1 1 0 01 0 1 1 0 2 1 2 2 0 4 2 3 3 3 4 3 43 4 0 3 0 2 9 4 5 7 7 6 6 5 5 5 6 6 6 5 7 6 7 7 5 9 7 8 8 8 8 9 9 9 8 5 8 5 7 0 5 0 4 = 0 list

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Merge Sort (13/13) Variation: Natural merge sort : Variation: Natural merge sort : We can modify merge_sort to take into account the prevailing order within the input list. We can modify merge_sort to take into account the prevailing order within the input list. In this implementation we make an initial pass over the data to determine the sequences of records that are in order. In this implementation we make an initial pass over the data to determine the sequences of records that are in order. The merge sort then uses these initially ordered sublists for the remainder of the passes. The merge sort then uses these initially ordered sublists for the remainder of the passes.

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Heap Sort (1/3) The challenges of merge sort The challenges of merge sort The merge sort requires additional storage proportional to the number of records in the file being sorted. The merge sort requires additional storage proportional to the number of records in the file being sorted. By using the O(1) space merge algorithm, the space requirements can be reduced to O(1), but significantly slower than the original one. By using the O(1) space merge algorithm, the space requirements can be reduced to O(1), but significantly slower than the original one. Heap sort Heap sort Require only a fixed amount of additional storage Require only a fixed amount of additional storage Slightly slower than merge sort using O(n) additional space Slightly slower than merge sort using O(n) additional space Faster than merge sort using O(1) additional space. Faster than merge sort using O(1) additional space. The worst case and average computing time is O(n log n), same as merge sort The worst case and average computing time is O(n log n), same as merge sort Unstable Unstable

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adjust adjust adjust the binary tree to establish the heap adjust the binary tree to establish the heap /* compare root and max. root */ /* move to parent */ [1] [2][3] [4][5][6][7] [8][9][10] 26 577 1611159 154819 rootkey = root = 1 n = 10 26 child =23 77 67 59 14 26

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Heap Sort (3/3) [1] [2][3] [4][5][6][7] [8][9][10] 26 577 1611159 154819 heapsort heapsort n = 10 i =54 48 1 32 61 19 5 1 77 59 26 9 5 77 61 48 15 5 8 1 61 59 26 1 7 5 59 48 19 5 6 1 48 26 11 1 5 1 26 19 15 1 4 5 19 15 5 3 1 11 1 2 1 5 1 1 1 5 ascending order (max heap) bottom-up top-down

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Radix Sort (1/8) We considers the problem of sorting records that have several keys We considers the problem of sorting records that have several keys These keys are labeled K 0 (most significant key), K 1, …, K r-1 (least significant key). These keys are labeled K 0 (most significant key), K 1, …, K r-1 (least significant key). Let K i j denote key K j of record R i. Let K i j denote key K j of record R i. A list of records R 0, …, R n-1, is lexically sorted with respect to the keys K 0, K 1, …, K r-1 iff (K i 0, K i 1, …, K i r-1 ) (K 0 i+1, K 1 i+1, …, K r-1 i+1 ), 0 i < n-1 A list of records R 0, …, R n-1, is lexically sorted with respect to the keys K 0, K 1, …, K r-1 iff (K i 0, K i 1, …, K i r-1 ) (K 0 i+1, K 1 i+1, …, K r-1 i+1 ), 0 i < n-1

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Radix Sort (2/8) Example Example sorting a deck of cards on two keys, suit and face value, in which the keys have the ordering relation: K 0 [Suit]: < < < K 1 [Face value]: 2 < 3 < 4 < … < 10 < J < Q < K < A sorting a deck of cards on two keys, suit and face value, in which the keys have the ordering relation: K 0 [Suit]: < < < K 1 [Face value]: 2 < 3 < 4 < … < 10 < J < Q < K < A Thus, a sorted deck of cards has the ordering: 2, …, A, …, 2, …, A Thus, a sorted deck of cards has the ordering: 2, …, A, …, 2, …, A Two approaches to sort: Two approaches to sort: 1. MSD (Most Significant Digit) first: 1. MSD (Most Significant Digit) first: sort on K 0, then K 1,... 2. LSD (Least Significant Digit) first: 2. LSD (Least Significant Digit) first: sort on K r-1, then K r-2,...

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Radix Sort (3/8) MSD first 1. 1.MSD sort first, e.g., bin sort, four bins 2. 2.LSD sort second 2, …, A, …, 2, …, A Result: 2, …, A, …, 2, …, A

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Radix Sort (4/8) LSD first 1. 1.LSD sort first, e.g., face sort, 13 bins 2, 3, 4, …, 10, J, Q, K, A 2. 2.MSD sort second (may not needed, we can just classify these 13 piles into 4 separated piles by considering them from face 2 to face A) Simpler than the MSD one because we do not have to sort the subpiles independently Result: 2, …, A, …, 2, …, A 2, …, A

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Radix Sort (5/8) We also can use an LSD or MSD sort when we have only one logical key, if we interpret this key as a composite of several keys. We also can use an LSD or MSD sort when we have only one logical key, if we interpret this key as a composite of several keys. Example: Example: integer: the digit in the far right position is the least significant and the most significant for the far left position integer: the digit in the far right position is the least significant and the most significant for the far left position range: range: 0 K 999 using LSD or MSD sort for three keys (K 0, K 1, K 2 ) since an LSD sort does not require the maintainence of independent subpiles, it is easier to implement MSDLSD 0-9

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Radix Sort (6/8) radix sort radix sort decompose the sort key into digits using a radix r. decompose the sort key into digits using a radix r. Ex: When r =10, we get the common base 10 or decimal decomposition of the key Ex: When r =10, we get the common base 10 or decimal decomposition of the key LSD radix r sort LSD radix r sort The records, R 0, …,R n-1 The records, R 0, …,R n-1 Keys: d-tuples (x 0, x 1, …, x d-1 ) and that 0 x i < r. Keys: d-tuples (x 0, x 1, …, x d-1 ) and that 0 x i < r. Each record has a link field, and that the input list is stored as a dynamically linked list. Each record has a link field, and that the input list is stored as a dynamically linked list. We implement the bins as queues We implement the bins as queues front[i], 0 i < r, pointing to the first record in bin i front[i], 0 i < r, pointing to the first record in bin i rear[i], 0 i < r, pointing to the last record in bin i rear[i], 0 i < r, pointing to the last record in bin i #define MAX_DIGIT 3 /* 0 to 999 */ #define RADIX_SIZE 10 typedef struct list_node *list_pointer; typedef struct list_node { int key[MAX_DIGIT]; list_pointer link;};

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LSD Radix Sort LSD Radix Sort Time complexity: O(MAX_DIGIT(RADIX_SIZE+n)) MAX_DIGIT passes O(RADIX_SIZE) O(n) RADIX_SIZE = 10 MAX_DIGIT = 3 f[9] f[8] f[7] f[6] f[5] f[4] f[3] f[2] f[1] f[0] 271 NULL 93 33 NULL 984 NULL 55 306 NULL 208 NULL 179 859 9 NULL r[9] r[8] r[7] r[6] r[5] r[4] r[3] r[2] r[1] r[0] Initial input: 17920830693859 98455927133 Chain after first pass, i=2: 271933398455 3062081798599

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Radix Sort (8/8) Simulation of radix_sort Simulation of radix_sort f[9] f[8] f[7] f[6] f[5] f[4] f[3] f[2] f[1] f[0] 271 NULL 93 33 NULL 984 NULL 55 306 208 NULL 179 859 9 NULL r[9] r[8] r[7] r[6] r[5] r[4] r[3] r[2] r[1] r[0] f[9] f[8] f[7] f[6] f[5] f[4] f[3] f[2] f[1] f[0] 271 NULL 93 33 984 NULL 55 306 NULL 208 179 859 9 NULL r[9] r[8] r[7] r[6] r[5] r[4] r[3] r[2] r[1] r[0] NULL Chain after second pass, i=1: 306208933 55859271 17998493 Chain after third pass, i=0: 9335593 179208271 306859984

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Summary of Internal Sorting (1/2) Insertion Sort Insertion Sort Works well when the list is already partially ordered Works well when the list is already partially ordered The best sorting method for small n The best sorting method for small n Merge Sort Merge Sort The best/worst case (O(nlogn)) The best/worst case (O(nlogn)) Require more storage than a heap sort Require more storage than a heap sort Slightly more overhead than quick sort Slightly more overhead than quick sort Quick Sort Quick Sort The best average behavior The best average behavior The worst complexity in worst case (O(n 2 )) The worst complexity in worst case (O(n 2 )) Radix Sort Radix Sort Depend on the size of the keys and the choice of the radix Depend on the size of the keys and the choice of the radix

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Summary of Internal Sorting (2/2) Analysis of the average running times Analysis of the average running times

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