In-Class Exercise. Hashing Used in: Hashing Algorithm

Transcription

1 In-Class Exercise Hashing Used in: Encryptin fr authenticatin Hash a digital signature, get the value assciated with the digital signature,and bth are sent separately t receiver. The receiver then uses the same hash functin n the signature, gets the value assciated with that signature, cmpares the messages. If the same - authenticatin Database access Names->phne numbers Authr->articles (r bks) Tpic->articles usernames->passwrds Scial Security Number->everything abut yu Zip cdes->regins Translatin Anagrams (hw?) Idea: Hashing is designed slely fr finding keys quickly. With hashing, the keys are nt put in any rder. We cannt find the smallest, the largest, the median, etc withut cnverting frm a hash set t a different data structure (r ding a lt f lping). If we hash crrectly, we can, hwever, find whether an item is in a database very quickly. Usually a key is assciated with a value r set f values. In ther wrds, with ur student example, the key was the student s id, but the values include the student s first and last name, their address, their year, their majr, their gpa, etc. Nne f the values have t be unique within the database (2 students can even have the same name, let alne the same majr), but the key we use fr finding this infrmatin must be unique. There are 3 parts t dealing with hashing: Cming up with a gd hashing functin Must be quickly cmputable Must lead t indices that are evenly distributed thrughut the array Must be cnsistent (always cmpute t the same index) A gd hash functin can depend n the type f data yu re dealing with Cming up with a gd array size Prime numbers are best Yu want an array that has abut 2x the number f keys S a prime number that is greater than 2x the number f keys is a gd number Dealing with the inevitable cllisins Chaining Make yur hash array be a linked list Insert every cllisin int the linked list Wrst case: every key ends up in the same index Linked list is n in length fr n keys REALLY bad hash functin 1

2 Linear prbing When yu have a secnd key hashing t the same index as a first key, keep prbing at the next index(es) in the array until yu find an empty spt Insert at that index S if a secnd key hashes t index 32 and there s already a value there, lk at index 33, then 34, then 35, etc. When yu get t the end f the array, lp back t index 0 Watch ut fr infinite lps! (when will that happen?) Causes clustering When a bunch f keys are clustered tgether in ne area in the array This slws the whle finding f the key quickly prcess dwn The mre clustering, the mre there s a tendency t cluster S we prbably want t try t avid this Quadratic prbing An attempt t avid clustering After a cllisin, instead f lking at the next index (h(k) + 1), lk at indices quadratically (h(k) + 1 2, then 2 2, then 3 2, etc, s lk at h(k)+1, then +4, then +9, etc. Again, lp back t 0 when get t end f array Helps mst when a lt f keys cluster t the same area Desn t help when a lt keys hash t the same index And still end up having areas f clustering Pseud-randm prbing An alternative t quadratic prbing Idea: can t randmly chse next index after a cllisin culd never find the key again But we culd generate a list f randm numbers and use that list t determine the next index E.g., {0, 8, 3, 4, 7, 2, 9, 6, 1, 5} index = h(k) + randmarr[0] If cllisin, index = h(k) + randmarr[1], Then h(k)+ randmarr[2], etc. VERY quick t calculate! Maybe a bit better than quadratic prbing when it cmes t clustering, but yu pretty much have the same prblems verall Duble hashing Have 2 different hashing functins. (there are MANY hashing functins, and yu can always make yur wn) If the first hash functin n a key gives an index that is already taken, try the secnd hashing functin n the key. The secnd hashing functin can be smething simple like h(k) ct*(k md (m)) where ct is the number f times yu ve had a cllisin with k and m is a prime number less than the size f the array (7 is nice). K is the numeric equivalent f the key (s in a string, it might just be adding up the asci values f the letters, r even a=1, b=2, etc.) We have t add 1 because if k md m is 0, we culd end up in an infinite lp E.g., arraysize = 11, m = 7 h2(k) = i+(k md(m-1)) h2(k) = i+(k md(m) h0(55) = 55%11 = 0 h0(66) = 66%11 = 0 X 2

3 H2((66) =(1+k%(M))) = 1 + (66%7) = 4 P2(66) = 0 + 1*4 = 4%11 = 4 h0(11) = 11%11 = 0 X H2(11) = 1+k%(M))) = 1 + (11%7) =5 P2(11) = 0 + 1*5 = 5%11 = 5 h0(88) = 88%11 = 0X H2(88) = 1+k%(M))) = 1 + (88%7) =5 P2(88) = 0 + 1*5 = 5%11 = 5X P3(88) = 0 + 2*5 = 10%11 = 10 Duble hashing helps with the prblem f many keys hashing t the same first value in thery the 2 hashing functins shuld be different enugh that the chances f bth hashing t the same index with the same key shuld be small Back t array size: Gal find things quickly (in 0(1) if pssible) REALLY large array sizes few cllisins (get clse t 0(1)) But a lt f extra space Smaller array sizes Many mre cllisins But less extra space Ideal abut 2x number f keys (up t next prime number) Lad factr: What happens if number f keys is changing? Lad factr = number f keys/ array size Want this t be less than 70% If the lad factr gets ver 70%, we want t: Create a new array duble in size f ld array, then up t next prime number Rehash all the keys in the ld array int the new array Equally, if the lad factr gets belw 25%, Halve the array size, up t the next prime Re-hash all the keys in the ld array Rehashing 0(n) - nt gd Dn t want t d this t ften Hashing Prblems: 1. Currently yu ve gt an array f size 23 and 17 keys. What shuld yu d? 2. With chaining, what wuld be the wrst case fr finding a particular key, and when wuld that ccur? 3

4 3. With linear prbing, why d we want t put as a cnditin, while (ct < arraysize) { 4. What is the best case fr finding a particular key in the hash table using duble hashing? 5. Why is deleting keys a prblem using linear/quadratic/pseud-randm prbing? 6. With a perfect hashing functin, and a set T cnsisting f {t1,t2,t3, tn} that has already been hashed int a hash array, hw lng wuld it take t tell whether set S cnsisting f {s1,s2,s3, sm} is a subset f set T? 7. Hash the fllwing wrds int the hash map belw using chaining, assuming each character has the crrespnding numerical value, and the hash functin is: wrd[0]+wrd[1]+wrd[2]+ wrd[wrdlength-1]%arraysize a b c d e f g h i j k l m n p q r s t u v w x y z Wrds t hash: cat, dg, bird, ant, bug, bat, fx 4

5 8. Hash the fllwing wrds int the hash map belw using linear prbing, assuming the same hash functin as abve (same list f wrds t hash): Wrds t hash: cat, dg, bird, ant, bug, bat, fx 9. Hash the fllwing wrds int the hash map belw using duble hashing, assuming the same hash functin as abve (same list f wrds t hash) fr the first hash, and h(k) ct*(h(k)%7) as the secnd hashing functin: Wrds t hash: cat, dg, bird, ant, bug, bat, fx If Finished, Start Lab 5 5