ncue csie wireless communications and networking laboratory chapter 7 search and sort 1

NCUE CSIE Wireless Communications and Networking Laboratory

CHAPTER 7

SEARCH AND SORTSEARCH AND SORT

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Search

Ⅰ. Internal v.s External search

Ⅱ. Static v.s Dynamic search

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Ⅲ. Partial key v.s whole key search

Ⅳ. Actual key v.s Transformation key

Search-Linear SearchBinary SearchFibonacci SearchInterpolation Search

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Search

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If list has n records, with list[i].key referring to the key value for record i, then we can search the list by examining the key values list[0].key,…,list[n-1].key, in that order, until the correct record is located, or we have examined all the records in the list. Since we examine the records in sequence, this searching technique is known as a sequential search.

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1 2 3 4 ... ... n-2 n-1 n

… …

2

1)....321(

n

n

n

Linear Search

Average comparison

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• This search begins by comparing searchnum and list[middle].key where middle .There are three possible outcomes:

searchnum< list[middle].key: In this case, we discard the records between list[middle] and list[n-1], and continue the search with the records between list[0] and list[middle-1].

5

2

)1(

n

Binary Search

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searchnum = list[middle].key:

In this case, the search terminates successfully.

searchnum > list[middle].key: In this case, we discard the records between list[0] and list[middle] and continue the search with the records between list[middle+1] and list[n-1].

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Binary Search

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Procedure BinSearch (f: afile ； var i:integer ； n,k:integer)

Var done : boolean ； l,u,m: integer ；Begin

l:=1 ； u:=n ； i:=0 ； done:=false ； while ((l<=u) and(not done)) do Begin

m:=(l+u) div 2 ； case compare( k , f[m].key) of

〝 > 〞 : l := m+1 {Look in upper half}

〝 = 〞 : Begin

i:=m ； done:= true ； End ； 〝 < 〞 : u:=m-1 ； {Look in lower half}

end ； {of case}

End ； {of while}

End ； {of BinSearch}

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Binary Search

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Number =12

Decision tree is

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Binary Search

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Binary

1.Time Complexity

∵T(n) = T(n/2)+1 ， T(1) = 1

∴T(n) = O( )

2.If n is small Sequential Search

n is large Binary Search

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n2log

Binary Search

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Fibonacci Search F0=0 ， F1 = 1 ， Fi = Fi-1+Fi-2,i 2≧

0,1,1,2,3,5,8,13,21,34,55,…

〝 Fa+m = n+1 〞

(1)n:record number

(2)Fa: the largest Fibonacci number and

it is n+1 )≦ (3)m: 0≧ and it is natural number

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Fibonacci Search

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(1)If n= 33, Fa = ? m = ?

Steps 1.Find the largest Fibonacci number ( ≦n+1≦ 34 )→ F9

Steps 2.m= (n+1)-Fa = 34-34=0

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n 0 1 2 3 4 5 6 7 8 9

Fn 0 1 1 2 3 5 8 13 21 34

Fibonacci Search

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Advantage & Disadvantage:

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Fibonacci Search

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Compare and

f[u].key ， f[l].key is the largest value and the smallest value

Campare (k , f[l].key)

〝 = 〞 : find

〝 < 〞 :u=(l+i)-1

〝 > 〞 :l=(l+i)+1

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)1(].[].[

].[

lukeylfkeyuf

keylfki

keylf ].1[ k

Interpolation Search

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CategoriesCategories:

(1) Internal or External Sorting?

(2) Stable or Unstable Sorting?

(3) Time Complexity

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Sorting

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The step of 26 、 5 、 49 、 13 、 6

Ans: (1)5 、 26 、 49 、 13 、 6

(2)5 、 26 、 49 、 13 、 6

(3)5 、 13 、 26 、 49 、 6

(4)5 、 6 、 13 、 26 、 49

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Insertion Sort

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Algorithm

(1)Insert(r, A[ ],i) Dubroutines

(2)Insort(A[ ],n) main

(1)

Procedure Insert(r , A[ ], i)

Var j : integer ；Begin

j:= i ；While r.key<list[j].key Do Begin

list[j+1]:=list[j] ； j:= j-1 ；End ； List[j+i]:= r ；End ；

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(2)

Procedure Insort(Var list : afile ； n:Integer) ；Var j: Integer ；Begin

list[0].key:= -∞ ； for j:= 2 to n do

insert (list[j], list, j-1) ；End ；

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Insertion Sort

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Ⅰ. Time complexity :

worst case & average case = O( )

best case= O(n) n-1 Comparative

Ⅱ. Insertion Sort is stable

e.g ……5… 5+ …

after:……5…… 5+ …

Ⅲ. Space Complexity = O(1)

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2n

Insertion Sort

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64 25 12 22 11

-> 11 25 12 22 64

-> 11 12 25 22 64

-> 11 12 22 25 64 -> 11 12 22 25 64

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Selection Sort

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Procedure SelectSort(R,n)Begin For i:= 1 to n-1 do Begin m := i ； For j = i+1 to n do If kj< km then m := j ； End ； {of For loop} If i < > m then Begin Swap (Ri,Rm) ； End ；End ； {of SelectSort}

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Selection Sort

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Ⅰ. Time complexity:

Best 、 Average 、 Worst case=O(n2)

Ⅱ. Space complexity : O(1)

Ⅲ. unstable sort

e.g.

pass1:

pass2:

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2.5.5

5.5.2

5.5.2

Selection Sort

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Bubble sort, is a simple sorting algorithm that works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted.

e.g. Step of 26 、 5 、 47 、 19 、 6

pass1 :5 、 26 、 19 、 6 、 47

pass2 :5 、 19 、 26 、 6 、 47

pass3 :5 、 6 、 19 、 26 、 47

pass4 :5 、 6 、 19 、 26 、 4722

Bubble Sort

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Procedure BubbleSort(R, n)Begin for i=1 to (n-1) do begin f = 0; for j=1 to (n-1) do if R[j+1].key<R[j].key then [swap(Rj, Rj+1); f=1; ] if f=0 then exit // No swap: exit// end;End; {of BubbleSort}

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Bubble Sort

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Ⅰ. Time complexity :

Best case : O(n)

Worst case : O(n2)

Average case : O(n2)

Ⅱ . Bubble Sort is a stable sort

Ⅲ. Space complexity : O(1)

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Bubble Sort

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Given: (R0, R1, R2, R3, ……….., Rn-3, Rn-2 Rn-1)

After first pass: R1, ..…, RS(i)-1, R0, RS(i), RS(i)+1, …, RS(n-1)

Pivot key

i j

two partitions

Quick Sort

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R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

26 5 37 1 61 11 59 15 48 19

11 5 19 1 15 26 59 61 48 37

1 5 11 19 15 26 59 61 48 37

1 5 11 15 19 26 59 61 48 37

1 5 11 15 19 26 48 37 59 61

1 5 11 15 19 26 37 48 59 61

1 5 11 15 19 26 37 48 59 61

1 5 11 15 19 26 37 48 59 61

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Quick Sort

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void procedure QuickSort(list, m, n) if (m<n) then i=m, j=n+1, p.k.=list[m].key Repeat repeat i=i+1 until list[i].key ≥p.k. repeat j=j-1 until list[i].key ≤p.k. if (i<j) then swap(list[i], list[j]) Until i ≥ j swap(list[m], list[j]) QuickSort(list, m, j-1) QuickSort(list, j+1, n)End ； {of if}End ； {of Qsort}

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Quick Sort

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Ⅰ. Time complexity:

Best case

T(n)= c*n + 2T( )

= 4*T ( )+2cn

= n*T ( )+

= n+ cnlogn

= O(nlogn)28

2

n

4

n

n

n cnnlog

Quick Sort

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Worst caseT(n) cn + T(n-1)≦ ≦ cn + T( cn + T(n-2)) ≦ 2cn + T(n-2) ≦ 3cn + T(n-3) . . ≦ (n-1)cn + T(1) = O( )

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2n

Quick Sort

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Average case-> Time complexity is O(nlogn)

Ⅱ. Quick Sort is unstable

Ⅲ. Space complexity: O(logn)~O(n)

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Quick Sort

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It merges the sorted lists(list[i],…,list[m]) and (list[m+1],…,list[n]), into a single sorted list,(sorted[i],…,sorted[n]).

Merge sort:•Iterative•Recursive

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Merge Sort

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e.g. 26 、 5 、 77 、 1 、 61 、 11 、 59 、 15 、 48 、 19

[26] [5] [77] [1] [61] [11] [59] [15] [48] [19]

[ 5 、 26 ] [ 1 、 77 ] [ 11 、 61 ] [ 15 、 59 ] [19 、 48 ]

[ 1 、 5 、 26 、 77 ] [11 、 15 、 59 、 61] [19 、 48]

[1 、 5 、 11 、 15 、 19 、 26 、 48 、 59 、 61 、 77]

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Iterative

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Ⅰ. Time complexity:

Best ， Worst ， Average case is O(nlogn)

Ⅱ . stable

Ⅲ. Require O(n) Space

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)(nT1n 1, if

1 ,)2

(2 nifcnn

T

Iterative

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26 、 5 、 77 、 1 、 61 、 11 、 59 、 15 、 48、 19

[5 、 26] [77] [1 、 61] [11 、 59] [15] [19 、 48]

[5 、 26 、 77] [1 、 61] [11 、 15 、 59] [19 、 48]

[1 、 5 、 26 、 61 、 77] [11 、 15 、 19 、 48、 59]

[1 、 5 、 11 、 15 、 19 、 26 、 48 、 59 、 61、 77]

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Recursive

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Procedure rMergeSort (Var x:afile ； l, u:Integer ； Var p:Integer )

Begin

If l u Then p:=l≧ else Begin

mid := (1+u) div 2 ； rMergeSort(x,l,mid,q) ； //left

rMergeSort(x,mid+1,u,r) ； //right

ListMerge(x,q,r,p) ； //merge

End ； {of if}

End ； {of rMergeSort}

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Recursive

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Heap sort begins by building a heap out of the data set, and then removing the largest item and placing it at the end of the partially sorted array.

After removing the largest item, it reconstructs the heap, removes the largest remaining item, and places it in the next open position from the end of the partially sorted array. This is repeated until there are no items left in the heap and the sorted array is full.

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Heap Sort

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Example: 35 、 21 、 37 、 15 、 52 、 15+ 、 5 、 40

Heap Sort

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Heap Sort

1. 2.

output 52 output 40

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3. 4.

output 37 output 35

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Heap Sort

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5. 6.

output 21 output 15

40

Heap Sort

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7. 8.

output 15+ output 5

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Heap Sort

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Ⅰ. Time complexity: Best, Average, Worst case is O(nlogn)

Ⅱ. Unstable method

Ⅲ. Space complexity: O(1)

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Heap Sort

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• LSD radix sort(1)If r is base -> r buckets(2)If m digit -> m steps

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Radix Sort

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179 、 208 、 306 、 93 、 859 、 984 、 55 、 9 、 271 、 33 Radix sort

Pass 1.

(Single-digit)

Merge:271 、 93 、 33 、 984 、 55 、 306 、 208 、 179 、 859 、 9

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0 1 2 3 4 5 6 7 8 9

9

33 859

271 93 984 55 306 208 179

Radix Sort

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Pass 2. (tens'digit)

Merge:306 、 208 、 9 、 33 、 55 、 859 、 271 、 179 、 984 、 93

Pass 3. (hundreds'digit)

Merge:9 、 33 、 55 、 93 、 179 、 208 、 271 、 306 、 859 、 984

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0 1 2 3 4 5 6 7 8 9

93

55

33 271

9 179 208 306 859 984

0 1 2 3 4 5 6 7 8 9

9

208 859 179

306 33 55 271 984 93

Radix Sort

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Ⅰ. Time complexity :

d is the largest number of keys, n=data, r = base

Ⅱ. Space complexity :

→ 〝 Bucket size 〞 = r * bucket*n

Ⅲ. Stable

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))(( rndO

)( nrO

Radix Sort

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Summary

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Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Freed〝 Fundamentals of Data Structures in C 〞 , W. H. Freeman & Co Ltd, 1992.

Ellis Horowitz, Sartaj Sahni, and Dinesh Mehta〝 Fundamentals of Data Structures in C++ 〞 Silicon Pr, 2006

Richard F.Gilberg, Behrouz A. Forouzan, 〝 Data Structures: A Pseudocode Approach with C 〞 , SBaker & Taylor Books, 2004

Fred Buckley, and Marty Lewinter 〝 A Friendly Introduction to Graph Theory 〞 Prentice Hall, 2002

〝資料結構 - 使用 C 語言〞蘇維雅譯，松崗， 2004 〝資料結構 - 使用 C 語言 〞 蔡明志編著，全華， 2004 〝資料結構 ( 含精選試題 ) 〞洪逸編著，鼎茂， 2005

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Reference