6. Suppose lim = constant> 0. Which of the following does not hold?

Transcription

1 CSE 0-00 Nme Test 00 points UTA Stuent ID # Multiple Choie Write your nswer to the LEFT of eh prolem 5 points eh The k lrgest numers in file of n numers n e foun using Θ(k) memory in Θ(n lg k) time using hep insertion sort mergesort PATITION Let f(n) n g(n) e symptotilly positive funtions Whih of the following is true? f(n) = Θ(f(n/)) f(n) = O(g(n)) implies g(n) = Ω(f(n)) f(n) = O(g(n)) implies g(n) = O(f(n)) f(n) + g(n) = Θ(min(f(n),g(n)) Stirling s pproximtion is neee to estlish whih result? H n = ln n + Ο() lg(n!) = Θ(n lg n) n! = n log = log Whih of the following sorts is stle? hepsort insertion quik shell 5 Whih of the following sorts oes not use time in Θ(n lg n) in the verge se? hepsort insertion merge quik f(n) 6 Suppose lim = onstnt> 0 Whih of the following oes not hol? g(n) n f(n) = O(g(n)) f(n) = Ω(g(n)) f(n) = o(g(n)) f(n) = Θ(g(n)) 7 The worst-se time for UILD-MAX-HEAP is: Θ(lg n) Θ(n) Θ(n lg n) Θ(n ) 8 Suppose tht UTA Presient oert Witt woul like ll urrent stuents liste in esening orer se on the numer of A gres on eh stuent s trnsript Theoretilly, the fstest sort for oing this is: ounting merge quik rix Long Answer Use the reursion-tree metho to show tht T(n) = T(n/) + is in Θ(n) 5 points Use the sustitution metho to show tht T(n) = T(n/) + is in Θ(n) 5 points Demonstrte EXTACT-MAX on the following mx-hep y giving the resulting hep 0 points

2 Suppose tht file n file eh ontin set of positive integers in sening orer File is suppose to ontin the union of the sets represente in files n, gin in sening orer Give pseuooe for n lgorithm to verify tht file is orret To re file i, you my simply ll the funtion refile(i), whih will return the next integer from file i If the file hs een exhuste, refile will return - 0 points CSE 0-00 Nme Test 00 points UTA Stuent ID # Multiple Choie Write your nswer to the LEFT of eh prolem 5 points eh The k lrgest numers in file of n numers n e foun Θ(n) verge time using hep insertion sort mergesort PATITION Whih of the following is true? n = Θ(n lg n) n = O(n lg n) n lg n = O(n) n = Ω(n lg lg n) Stirling s pproximtion is neee to estlish whih result? H n = ln n + Ο() lg(n!) = Θ(n lg n) n! = n log = log Whih two sorts hve fetures tht re useful for eveloping externl (eg isk) sorts? hep n shell hep n merge quik n ounting quik n merge 5 Suppose n orere tle with n elements my hve repete keys The worst-se time for etermining the numer of repets, k, for prtiulr key is Θ(k) Θ(log k) Θ(log n) Θ(k log k)

3 f(n) 6 Suppose lim = 0 Whih of the following hols? g(n) n f(n) = O(g(n)) f(n) = Ω(g(n)) f(n) = ω(g(n)) f(n) = Θ(g(n)) 7 The worst-se time for PATITION is: Θ(lg n) Θ(n) Θ(n lg n) Θ(n ) 8 Whih of the following is not true out shellsort on n reors? After h-sorting for the first h vlue, the lrgest key will e lst in the rry Eh group for n h vlue is sorte using insertion sort Eh vlue in the sequene of h vlues is smller thn n The sequene of h vlues ens with Long Answer Use the reursion-tree metho to show tht T(n) = T(n/) + is in Θ(n) 5 points Use the sustitution metho to show tht T(n) = T(n/) + is in Θ(n) 5 points Demonstrte PATITION on the following rry 0 points Suppose tht file n file eh ontin set of positive integers in sening orer File is suppose to ontin the intersetion of the sets represente in files n, gin in sening orer Give pseuooe for n lgorithm to verify tht file is orret To re file i, you my simply ll the funtion refile(i), whih will return the next integer from file i If the file hs een exhuste, refile will return - 0 points CSE 0-00 Nme Test 00 points UTA Stuent ID # Multiple Choie Write your nswer to the LEFT of eh prolem 5 points eh Whih trversl will list the keys in inry serh tree in sening orer? A inorer level-y-level C postorer D preorer Whih t struture opertes in lst-in-first-out fshion? A hshing with hining hshing with open ressing C queue D stk Orere linke lists re useful when: A free storge list is eing use for free noes eletions re to e supporte C hits ominte misses D misses ominte hits Wht t strutures re neee for n infix lultor? A A stk n queue A re-lk tree C Two queues D Two stks 5 Suppose tht you re eiing whether n orere linke list (sening orer) shoul e singly or ouly linke Whih of the following opertions n e one fster if oule linking is use? A Deleting noe Fining the noe with the minimum key C Fining the noe for key D Fining the suessor of noe 6 Suppose tht sequene of n keys will e inserte into n initilly-empty instne of the following t strutures Whih of the following will not tke θ(n ) time for the entire sequene? A liner hsh tle orere linke list C re-lk tree

4 D unlne inry serh tree 7 Cirulr linke lists re osionlly useful euse A some opertions my e one in onstnt time they re n lterntive to re-lk trees C they re useful for implementing irulr queues D they voi mllo()s 8 Qurti proing hs the property tht A Primry lustering is reue Seonry lustering is reue C oth primry lustering n seonry lustering re reue D Neither primry lustering nor seonry lustering is reue Long Answer 0 points eh Give n exmple of inry serh tree whose noes nnot e ssigne olors to mke it legl re-lk tree Suppose you re using oule hshing Wht is the mximum lo ftor tht my e use to ssure tht the expete numer of proes for n unsuessful serh oes not exee? Insert 85 into the following re-lk tree e sure to inite the ses tht re use Insert 55 into the following re-lk tree e sure to inite the ses tht re use Delete 0 from the following re-lk tree e sure to inite the ses tht re use Delete 0 from the following re-lk tree e sure to inite the ses tht re use

5 CSE 0-00 Nme Test 00 points UTA Stuent ID # Multiple Choie Write your nswer to the LEFT of eh prolem 5 points eh Suppose tht you re eiing whether n orere linke list (sening orer) shoul e singly or ouly linke Whih of the following opertions n e one fster if oule linking is use? A Deleting noe Fining the noe with the minimum key C Fining the noe for key D Fining the suessor of noe Suppose tht sequene of n keys will e inserte into n initilly-empty instne of the following t strutures Whih of the following tkes θ(nlgn) time for the entire sequene? A liner hsh tle orere linke list C re-lk tree D unlne inry serh tree Whih trversl will list the keys in inry serh tree in sening orer? A inorer level-y-level C postorer D preorer Whih t struture opertes in first-in-first-out fshion? A hshing with hining hshing with open ressing C queue D stk 5 Orere linke lists re useful when: A free storge list is eing use for free noes eletions re to e supporte C hits ominte misses D misses ominte hits 6 Where is sentinel use with linke list? A nywhere t the eginning C t the en D efore the noe with the smllest key 7 Liner proing hs the property tht A Primry lustering ours frequently Seonry lustering ours frequently C oth primry lustering n seonry lustering our frequently D Neither primry lustering nor seonry lustering our frequently 8 Cirulr linke lists re osionlly useful euse A some opertions my e one in onstnt time they re n lterntive to re-lk trees C they re useful for implementing irulr queues D they voi mllo()s Long Answer 0 points eh Give n exmple of inry serh tree with t lest five noes, ut oes not hve ny re noes

6 Suppose you re using oule hshing Wht is the mximum lo ftor tht my e use to ssure tht the expete numer of proes for n unsuessful serh oes not exee 0? Insert 95 into the following re-lk tree e sure to inite the ses tht re use Insert 75 into the following re-lk tree e sure to inite the ses tht re use Delete 0 from the following re-lk tree e sure to inite the ses tht re use Delete 0 from the following re-lk tree e sure to inite the ses tht re use

7 CSE 0-00 Nme Test Give oth KMP fil link tles for the given pttern 5 points For the given network, etermine mximum flow n the minimum ut e sure to give eh ugmenting pth, the mount of itionl flow tht it provies, n the resiul grph fter eh ugmenting pth is reore You my hoose the ugmenting pths in ny mnner you wish 5 points s e f t s e f t 5 Augmenting Pth/Flow: Minimum Cut: Use ynmi progrmming to etermine the longest ommon susequene of 0000 n points Demonstrte the Floy-Wrshll lgorithm the following grph In ition to the pth length mtrix, you must give either the suessor mtrix or the preeessor mtrix 0 points Wht re the entries in the hep (for Prim s lgorithm) efore n fter moving the next vertex n ege into the minimum spnning tree? DO NOT COMPLETE THE ENTIE MST!!! Eges lrey in the MST re the thik ones Eges urrently not in the MST re the nrrow ones You o not nee to show the inry tree for the hep orering 0 points

8 A F 7 E H 5 M C I L G 6 6 Complete the following instne of the optiml mtrix multiplition orering prolem, inluing the tree showing the optiml orering 0 points p[0]=5 p[]= p[]=5 p[]= p[]= ???? J Suppose tht n ugmenting pth for the mximum flow prolem is to mximize the mount of itionl flow tht the AP provies Desrie greey lgorithm tht solves this prolem 0 points 8 Demonstrte the tehnique tht uses two epth-first serhes to fin the strongly-onnete omponents of the following grph 0 points G C 9 8 D K F E A H D 9 For eh of the previous eight prolems, give the worst-se time for the relevnt lgorithm 0 points (KMP) (Mx flow) (LCS) (Floy-Wrshll) 5 (MST) 6 (Optiml mtrix multiplition) 7 (Greey mximum AP) 8 (SCCs) CSE 0-00 Nme Test Give oth KMP fil link tles for the given pttern 5 points

9 For the given network, etermine mximum flow n the minimum ut e sure to give eh ugmenting pth, the mount of itionl flow tht it provies, n the resiul grph fter eh ugmenting pth is reore You my hoose the ugmenting pths in ny mnner you wish 5 points s e f t s e f t 5 Augmenting Pth/Flow: Minimum Cut: Use ynmi progrmming to etermine the longest ommon susequene of n 0 points Demonstrte the Floy-Wrshll lgorithm on the following grph In ition to the pth length mtrix, you must give either the suessor mtrix or the preeessor mtrix 0 points Wht re the entries in the hep (for Prim s lgorithm) efore n fter moving the next vertex n ege into the minimum spnning tree? DO NOT COMPLETE THE ENTIE MST!!! Eges lrey in the MST re the thik ones Eges urrently not in the MST re the nrrow ones You o not nee to show the inry tree for the hep orering 0 points

10 A F 7 E H 5 M C I 7 J 9 0 L G 8 D 6 6 Demonstrte the tehnique tht uses epth-first serh to topologilly sort the verties in the following grph 0 points G C 9 K F E A H D 7 Disuss the ifferene etween the frtionl n 0/ knpsk prolems 0 points 8 Give n optiml Huffmn oe tree for the provie symols n proilities In ition, ompute the expete numer of its per symol 0 points A 5 C 05 D E F G 9 For eh of the previous eight prolems, give the worst-se time for the relevnt lgorithm 0 points (KMP) (Mx flow) (LCS) (Floy-Wrshll) 5 (MST) 6 (Topologil sort) 7 (Knpsk) 8 (Huffmn oe tree onstrution)