Lecture 21

‫‪Data Structures‬‬ ‫‪Notes for Lecture 21‬‬ ‫‪Introduction of Searching‬‬ ‫‪By‬‬ ‫‪Samaher Hussein Ali‬‬ ‫‪2007-2008‬‬

‫‪Searching‬‬ ‫اﻟﺒﺤﺚ هﻮ ﻋﻤﻠﻴﺔ اﻟﺘﻔﻴﺶ ﻋﻦ ﻋﻨﺼﺮ ﻣﻌﻴﻦ ﻓﻲ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﺒﻴﺎﻧﺎت أوهﻮ ﻋﻤﻠﻴﺔ اﻳﺠﺎد ﻋﻨﺼﺮ ﻣﻌﻴﻦ ﻓﻲ ﻣﺠﻤﻮﻋﺔ ﻣﻦ اﻟﺒﻴﺎﻧﺎت ﻓﺎذا آﺎن‬ ‫ذﻟﻚ اﻟﻌﻨﺼﺮ ﻣﻮﺟﻮد ﺗﻜﻮن ﻋﻤﻠﻴﺔ اﻟﺒﺤﺚ )اﻳﺠﺎﺑﻴﺔ( ﻓﻌﺎﻟﺔ وهﺬا ﻳﺤﺪث اذا آﺎﻧﺖ ﻗﺎﺋﻤﺔ اﻟﺒﺤﺚ ﻣﺮﺗﺒﺔ ﺑﺎﻻﻋﺘﻤﺎد ﻋﻠﻰ ﺣﻘﻞ ﻣﻌﻴﻦ ﺑﻴﻨﻤﺎ ﺗﻜﻮن‬ ‫ﻋﻤﻠﻴﺔ اﻟﺒﺤﺚ ﻏﻴﺮ ﻓﻌﺎﻟﺔ ﻋﻨﺪ ﻋﺪم وﺟﻮد اﻟﻌﻨﺼﺮ ﻓﻲ ﻗﺎﺋﻤﺔ اﻟﺒﺤﺚ ‪.‬‬

‫*اﻧﻮاع ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ‬ ‫وﻳﻮﺟﺪ ﻋﺪة ﺗﺼﻨﻴﻔﺎت ﻟﺨﻮارزﻣﻴﺎت اﻟﺒﺤﺚ ﻣﻨﻬﺎ‪-:‬‬ ‫‪.1‬ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ اﻟﺘﺴﻠﺴﻠﻲ‬

‫‪Sequential Search Algorithm‬‬

‫‪ .2‬ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ اﻟﺜﻨﺎﺋﻲ‬

‫‪Binary Search Algorithm‬‬

‫‪ .3‬ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ اﻟﻜﺘﻠﻲ‬

‫‪Block Search Algorithm‬‬

‫‪ .4‬ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ اﻟﻤﺤﺮم‬ ‫‪ .5‬ﺑﺤﺚ اﻟﺸﺠﺮة اﻟﺜﻨﺎﺋﻴﺔ‬

‫‪Tabu Search Algorithm‬‬

‫*اﺳﺲ ﻗﻴﺎس آﻔﺎءة ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ‬ ‫‪ .1‬ﻣﻘﺪار اﻟﺨﺰن اﻟﻼزم‬ ‫‪ .2‬ﻣﻘﺪار اﻟﺘﻌﻘﻴﺪات ﻓﻲ اﻟﺨﻮارزﻣﻴﺔ‬ ‫‪ .3‬ﻋﺪد اﻟﻨﻘﻼت او اﻟﺘﺤﺮﻳﻜﺎت او اﻟﻤﻘﺎرﻧﺎت ﻓﻲ اﻟﺨﻮارزﻣﻴﺔ‬ ‫‪ .4‬ﻣﻌﺪل ﻃﻮل اﻟﺒﺤﺚ‬

‫‪Average Search Length‬‬

‫وﻳﻤﻜﻦ ﺣﺴﺎﺑﻪ ﺣﺴﺐ اﻟﻘﺎﻧﻮن اﻟﺘﺎﻟﻲ‪-:‬‬ ‫‪n‬‬

‫‪Pi‬‬

‫‪i‬‬

‫‪C‬‬

‫∑‬

‫= ‪L‬‬

‫‪i=1‬‬

‫ﺣﻴﺚ ‪-:‬‬ ‫‪ L‬ﻣﻌﺪل ﻃﻮل اﻟﺒﺤﺚ‬ ‫‪ Ci‬ﻋﺪد اﻟﻤﻘﺎرﻧﺎت اﻟﻤﻄﻠﻮﺑﺔ ﻟﻠﻮﺻﻮل اﻟﻰ اﻟﻌﻨﺼﺮ‬ ‫‪ P‬اﺣﺘﻤﺎﻟﻴﺔ اﻟﻮﺻﻮل اﻟﻰ اﻟﻌﻨﺼﺮ‬

‫‪Sequential Search Algorithm‬‬ ‫ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ اﻟﺘﺴﻠﺴﻠﻲ‬

‫ﺧﻮارزﻣﻴﺎت اﻟﺒﺤﺚ اﻟﺘﺴﻠﺴﻠﻲ وﺗﺘﻀﻤﻦ ﻋﻤﻠﻴﺔ اﻟﺒﺤﺚ ﻋﻦ ﻋﻨﺼﺮ ﻣﻌﻴﻦ ﻓﻲ ﻗﺎﺋﻤﺔ ﻣﻦ اﻟﻌﻨﺎﺻﺮ ﻣﻦ ﺧ ﻼل ﻣ ﺴﺢ ﺟﻤﻴ ﻊ ﻋﻨﺎﺻ ﺮ اﻟﻘﺎﺋﻤ ﺔ ﻣ ﻦ‬ ‫ﺑﺪاﻳﺘﻬﺎ وﺑﺎﻟﺘﺴﻠﺴﻞ ﻟﺤﻴﻦ اﻟﻮﺻﻮل اﻟﻰ اﻟﻌﻨﺼﺮ اﻟﻤﻄﻠﻮب ﻓﻲ ﺣﺎﻟﺔ وﺟﻮدﻩ او اﻟﻮﺻﻮل اﻟﻰ ﻧﻬﺎﻳﺔ اﻟﻘﺎﺋﻤﺔ ﻋﻨﺪﻣﺎ ﻳﻜﻮن اﻟﻌﻨﺼﺮ ﻏﻴﺮ ﻣﻮﺟﻮد‪-:‬‬ ‫‪1‬‬

‫ﻟﺬا ﻓﺎن ﻣﻌﺪل ﻋﺪد اﻟﻤﻘﺎرﻧﺎت هﻮ )‬

‫‪n‬‬ ‫‪2‬‬

‫( ‪ .‬ووﻗﺖ ﺗﻨﻔﻴﺬ اﻟﺨﻮارزﻣﻴﺔ هﻮ ) )‪.( Ο(n‬‬

‫ﻳﺴﺘﺨﺪم اﻟﺒﺤﺚ اﻟﺘﺴﻠﺴﻠﻲ ﻣﻊ اﻟﺒﻴﺎﻧﺎت اﻟﺘﻲ ﺗﺤﺘﺎج اﻟﻰ ﻣﻌﺎﻟﺠﺔ دورﻳﺔ ﻣﺜﻞ ﻧﻈﺎم اﻟﺮواﺗﺐ‪.‬‬ ‫‪Algorithm of sequential search‬‬

‫‪(Given an unordered vector k consisting of n+1 elements, (n>=1) this algorithm search the vector for a‬‬ ‫)‪particular elements having the value x‬‬ ‫]‪1. [ initialize search‬‬ ‫‪I=1‬‬ ‫‪K(n+1)=x‬‬ ‫] ‪2. [search the vector‬‬ ‫‪Repeat‬‬ ‫‪While k(i)<>x‬‬ ‫‪I=i+1‬‬ ‫]‪3. [successful search‬‬ ‫‪If i=n+1 then‬‬ ‫)"‪Msgbox("unsuccessful search‬‬ ‫‪Else‬‬ ‫)"‪Msgbox("successful search‬‬ ‫ﻣﻼﺣﻈﺔ‪-:‬‬ ‫ﻋﺪد اﻟﻤﻘﺎرﻧﺎت = ﻋﺪد اﻟﻌﻨﺎﺻﺮ اﻟﻤﻘﺮءة او اﻟﻤﺤﺴﻮﺑﺔ أي‬

‫‪= i‬‬ ‫‪i * Pi‬‬

‫‪n‬‬

‫∑‬

‫‪i‬‬

‫‪C‬‬

‫= ‪L‬‬

‫‪i=1‬‬

‫ﻣﺜﺎل ‪ -:‬اﻓﺮض ان اﺣﺘﻤﺎﻟﻴﺔ اﻟﻮﺻﻮل اﻟﻰ آﻞ ﻋﻨﺼﺮ ﻏﻴﺮ ﻣﺘﺴﺎوﻳﺔ ﻓﻲ ﻗﺎﺋﻤﺔ ﺗﺤﺘﻮي ﻋﻠﻰ ﺧﻤﺴﺔ ﻗﻴﻮد اوﺟﺪ ﻣﻌﺪل ﻃﻮل اﻟﺒﺤﺚ ﺑﺎﺳﺘﺨﺪام‬ ‫اﻟﺒﺤﺚ اﻟﺘﺴﻠﺴﻠﻲ اذا آﺎﻧﺖ اﺣﺘﻤﺎﻟﻴﺔ اﻟﻮﺻﻮل ﻟﻜﻞ ﻋﻨﺼﺮ هﻲ‪-:‬‬ ‫‪0.4‬‬

‫‪P1‬‬

‫‪0.3‬‬

‫‪P2‬‬

‫‪0.2‬‬

‫‪P3‬‬

‫‪0.07‬‬

‫‪P4‬‬

‫‪0.03‬‬

‫‪P5‬‬

‫‪i * Pi‬‬

‫‪n‬‬

‫∑‬

‫= ‪L‬‬

‫‪i=1‬‬

‫‪L=1*0.4+2*0.3+3*0.2+4*0.07+5*0.03 = 2.03‬‬ ‫‪2‬‬

‫‪Binary Search Algorithm‬‬ ‫ﺧﻮارزﻣﻴﺔ اﻟﺒﺤﺚ اﻟﺜﻨﺎﺋﻲ‬ ‫ﻓﻜﺮﺗﻬﺎ‪-:‬‬

‫ان هﺬﻩ اﻟﺨﻮارزﻣﻴﺔ ﺗﻔﺘﺮض اﻟﺒﺤﺚ ﻋﻦ ﻋﻨﺼﺮ ﻓﻲ ﻗﺎﺋﻤﺔ ﻣﺮﺗﺒﺔ ﺣﺴﺐ ﺗﺮﺗﻴﺐ ﻣﻌﻴﻦ وﺗﺘﻀﻤﻦ اﻟﺨﻄﻮات اﻟﺘﺎﻟﻴﺔ ‪-:‬‬ ‫‪ .1‬ﻣﻘﺎرﻧﺔ اﻟﻌﻨﺼﺮ اﻟﻤﺮاد اﻟﺒﺤﺚ ﻋﻨﻪ وﻟﻴﻜﻦ )‪ (X‬ﻣﻊ اﻟﻌﻨﺼﺮ اﻟﺬي ﻳﻘﻊ ﻣﻨﺘﺼﻒ اﻟﻘﺎﺋﻤﺔ‬ ‫‪ .2‬اذا آﺎن اﻟﻌﻨ ﺼﺮ اﻟﻤﻄﻠ ﻮب )‪ (X‬اﻗ ﻞ ﻣ ﻦ ﻗﻴﻤ ﺔ اﻟﻌﻨ ﺼﺮ اﻟ ﺬي ﻳﻘ ﻊ ﻣﻨﺘ ﺼﻒ اﻟﻘﺎﺋﻤ ﺔ اذن ﺳﻴﻨﺤ ﺼﺮ اﻟﺒﺤ ﺚ ﻓ ﻲ اﻟﺠ ﺰء اﻟ ﺬي ﻳ ﻀﻢ‬ ‫اﻟﻌﻨﺎﺻﺮ اﻻﺻﻐﺮ ﻣﻦ ﻋﻨﺼﺮ اﻟﻤﻨﺘﺼﻒ )ﺟﻬﺔ اﻟﻴﺴﺎر(‬ ‫‪ .3‬اذا آﺎن اﻟﻌﻨﺼﺮ اﻟﻤﻄﻠﻮب )‪ (X‬اآﺒ ﺮ ﻣ ﻦ ﻗﻴﻤ ﺔ اﻟﻌﻨ ﺼﺮ اﻟ ﺬي ﻳﻘ ﻊ ﻣﻨﺘ ﺼﻒ اﻟﻘﺎﺋﻤ ﺔ اذن ﺳﻴﻨﺤ ﺼﺮ اﻟﺒﺤ ﺚ ﻓ ﻲ اﻟﺠ ﺰء اﻟ ﺬي ﻳ ﻀﻢ‬ ‫اﻟﻌﻨﺎﺻﺮ اﻻآﺒﺮ ﻣﻦ ﻋﻨﺼﺮ اﻟﻤﻨﺘﺼﻒ )ﺟﻬﺔ اﻟﻴﻤﻦ(‪ .‬وﺗﺴﺘﻤﺮ اﻟﻌﻤﻠﻴﺔ ﺣﺘﻰ ﻳﺘﻢ اﻟﻌﺜﻮر ﻋﻠﻰ اﻟﻌﻨﺼﺮوذﻟﻚ ﺑﺘﻜﺮار اﻟﺨﻄﻮﺗﻴﻦ ‪2‬و‪3‬‬ ‫‪ .4‬اذا آﺎن اﻟﻌﻨﺼﺮ اﻟﻤﻄﻠﻮب )‪ (X‬ﻣﺴﺎوﻳًﺎ ﺑﺎﻟﻘﻴﻤﺔ اﻟﻰ اﻟﻌﻨﺼﺮ ﻓﻲ ﻣﻨﺘﺼﻒ اﻟﻘﺎﺋﻤﺔ اذن ﺳﺘﻨﺘﻬﻲ ﻋﻤﻠﻴﺔ اﻟﺒﺤﺚ ﻣﺒﺎﺷﺮﺗًﺎ‪.‬‬ ‫‪Algorithm of binary search‬‬

‫‪(Given a vector k, consisting of n elements in ascending order this algorithm searches the structure for a‬‬ ‫‪given element whose value is given by x. the variables low, middle, and high denote the lower, middle‬‬ ‫)‪and upper limits of the search interval respectively‬‬ ‫]‪1. [ initialize search‬‬ ‫‪Low =1‬‬ ‫‪High =n‬‬ ‫]‪2. [perform search‬‬ ‫‪Repeat thru step4‬‬ ‫‪While low<=high‬‬ ‫]‪3. [obtain index of mid point of interval‬‬ ‫‪Middle=(low + high)/2‬‬ ‫]‪4. [compare‬‬ ‫‪If xk(middle) then‬‬ ‫‪Low=middle+1‬‬ ‫‪Else‬‬ ‫)"‪Msgbox("successful‬‬ ‫]‪5. [unsuccessful‬‬ ‫)"‪Msgbox("unsuccessful‬‬ ‫‪3‬‬

‫ ﻓﻲ اﻟﻘﺎﺋﻤﺔ اﻟﺘﺎﻟﻴﺔ‬45 ‫ اﺑﺤﺚ ﻋﻦ اﻟﻌﻨﺼﺮ‬-: ‫ﻣﺜﺎل‬ 1

17

45

52

63

81

92

100

-:‫اﻟﺤﻞ‬ Low=1 High=8 Middle=(1+8)/2=4 K(middle)=k(4)=52 X=52 52>45 The element in smallest element list

1

17

45

Low=1 High=3 Middle=(1+3)/2=2 K(middle)=k(2)=17 X=17 17<45 Go to right 45=45 "the element is found and the search is successful"

4