ECE4050 Data Structures and Algorithms. Lecture 6: Searching

Transcription

1 ECE4050 Data Structures ad Algorithms Lecture 6: Searchig 1

2 Search Give: Distict keys k 1, k 2,, k ad collectio L of records of the form (k 1, I 1 ), (k 2, I 2 ),, (k, I ) where I j is the iformatio associated with key k j for 1 <= j <=. Search Problem: For key value K, locate the record (k j, I j ) i L such that k j = K. Searchig is a systematic method for locatig the record(s) with key value k j = K. 2

3 Successful vs. Usuccessful A successful search is oe i which a record with key k j = K is foud. A usuccessful search is oe i which o record with k j = K is foud (ad presumably o such record exists). 3

4 Approaches to Search 1. Sequetial ad list methods (lists, tables, arrays). 2. Direct access by key value (hashig) 3. Tree idexig methods. 4

5 Average Cost for Sequetial Search How may comparisos does sequetial search do o average? We must kow the probability of occurrece for each possible iput. Must K be i L? 5

6 Average Cost (cot) Let k i = i+1 be the umber of comparisos whe X = L[i]. Let k = be the umber of comparisos whe X is ot i L. Let p i be the probability that X = L[i]. Let p be the probability that X is ot i L[i] for ay i. 6

7 Geeralizig Average Cost What happes to the equatio if we assume all p i 's are equal (except p )? Depedig o the value of p, (+1)/2 < T() <. 7

8 Searchig Ordered Arrays Chage the model: Assume that the elemets are i ascedig order. Is liear search still optimal? Why ot? Optimizatio: Use liear search, but test if the elemet is greater tha K. Why? Observatio: If we look at L[5] ad fid that K is bigger, the we rule out L[1] to L[4] as well. More is Better: If K > L[], the we kow i oe test that K is ot i L. What is wrog here? 8

9 Jump Search What is the right amout to jump? Algorithm: Check every k'th elemet (L[k], L[2k],...). If K is greater, the go o. If K is less, the use liear search o the k elemets. This is called Jump Search. 9

10 10 Aalysis of Jump Search If mk <= < (m+1)k, the the total cost is at most m + k 1 3-way comparisos. What should k be? 1 mi 1 k k k 1 1 ), ( T k k k m k

11 Jump Search Aalysis (cot) Take the derivative ad solve for T'(x) = 0 to fid the miimum. This is a miimum whe k What is the worst case cost? Roughly 2 11

12 Lessos We wat to balace the work doe while selectig a sublist with the work doe while searchig aother sublist. I geeral, make sub-problems of equal effort. This is a example of divide ad coquer. What if we exted this to three levels?. biary search 12

13 Lists Ordered by Frequecy Order lists by (expected) frequecy of occurrece. Perform sequetial search Cost to access first record: 1 Cost to access secod record: 2 Expected search cost: C 1p1 2p2... p. 13

14 Examples(1) (1) All records have equal frequecy. C i / ( 1) / 2 i1 I the more geeral case we must cosider the probability (labeled p ) that the search key does ot match ay oe i the list. 14

15 (2) Geometric Frequecy Examples(2) p i { 1/ 1/ 2 2 i 1 if if 1 i i 1 C ( i / 2 i ) 2. i1 15

16 Zipf Distributios Applicatios: Distributio for frequecy of word usage i atural laguages. Distributio for populatios of cities, etc. 80/20 rule: C i1 i / i Η 80% of accesses are to 20% of the records. For distributios followig 80/20 rule, / H C / log e. 16

17 Self-Orgaizig Lists Self-orgaizig lists modify the order of records withi the list based o the actual patter of record accesses. Self-orgaizig lists use a heuristic for decidig how to reorder the list. These heuristics are similar to the rules for maagig buffer pools. 17

18 Heuristics Order by actual historical frequecy of access. (Similar to LFU buffer pool replacemet strategy.) Move-to-Frot: Whe a record is foud, move it to the frot of the list. Traspose: Whe a record is foud, swap it with the record ahead of it. 18

19 Text Compressio Example Applicatio: Text Compressio ad trasmissio. Keep a table of words already see, orgaized via Moveto-Frot heuristic. If a word ot yet see, sed the word. Otherwise, sed (curret) idex i the table. The car o the left hit the car I left. The car o 3 left hit 3 5 I 5. This is similar i spirit to Ziv-Lempel codig. 19

20 Searchig i Sets For dese sets (small rage, high percetage of elemets i set). Ca use logical bit operators. Example: To fid all primes that are odd umbers, compute: & Documet processig: Sigature files: a list of keywords, each associated with a bit vector idicatig which documets cotai the keyword. 20

21 Hashig Hash: compute the positio, rather tha search for the positio, of a key. Hash fuctio: a fuctio used to compute a record s positio Hash table: the data structure i which records are placed ad retrieved o positios calculated by hash fuctios. A positio i the table is called a slot, umbered from 0 to M-1. I a hashig system, ay key value K ad some hash fuctio h, h(k) is a slot i the table such that 0 <= h(k) < M, ad we have the record of key K stored at positio h(k). 21

22 Hashig Key rage is usually much larger tha slot cout (M). Multiple keys ca be mapped to the same slot. Give a hash fuctio h ad two keys k1 ad k2, if h(k1) = h(k2) the we say that k1 ad k2 have a collisio at slot h(k1) (or h(k2)) uder hash fuctio h. a collisio resolutio mechaism is required. 22

23 Hash Fuctios If we kow othig about keys distributio, a hash fuctio should evely distribute keys across the hash table. Example hash fuctios: (1) (2) mid-square: square a key ad takes middle digits (3) What are good hash fuctios whe key is a character strig? 23

24 (3) What are good hash fuctios whe key is a character strig? Aother versio: 24

25 Ope Hashig The collisios are stored outside of the table. What should be cosidered to decide the table size? 25

26 Closed Hashig All records are stored directly withi the hash table. Two versio of bucket hashig: home positio of k is h(k) Hash to the bucket ad start searchig from the first slot of bucket Hash to the home positio ad start searchig from the positio i the bucket 26