CS4115 Database. 20. What is the time-complexity of squaring a matrix of size n n in Big-Oh notation?

Transcription

1 Sectio 1. Autum 00 Mid-Term 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0; each questio has four possible aswers ad the questios are worth four marks each?. Which of the aswers below is i=0 i?. Which of the aswers below is i= i?. Which of the aswers below is i= i? 1 1) 1 1) 1 1) 5. Which of the aswers below is 5 i=1 i? Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) O ) O 5 ) CS115 Database 1 8. Which of the followig is the correct orderig of the fuctios,, 1000,? 1000 < < < < < 1000 < < < < < 9. Which of the followig is the correct orderig of the fuctios log,!, log log, log? log < log < log log <! log < log log < log <! log log < log < log <! log <! < log log < log 10. If T ) = Of)) the which of the followig is the most precise statemet? T ) = of)) f) = ΩT )) T ) = k f) for some costat k f) = ΘT )) 11. If T ) = Θf)) the which of the followig is the most precise statemet? T ) = Of)) ad T ) = Ωf)) T ) = Of)) but ot T ) = Ωf)) T ) = of)) ad T ) = Ωf)) T ) = of)) but ot T ) = Θf)) 1. If m = Θ) the bucket sort has timecomplexity most precisely) O ) Ω) Ω ) 1. A -time algorithm sortig itegers i the rage [0, 1] is possible by makig passes of radix sort, with buckets of size / / 8 1. The time-complexity of expoetiatio to the power of is most precisely) Olog ) o) Θ) 15. What is the time-complexity of computig i=0 aixi if a dumb expoetiatio algorithm is used? Olog ) O log ) O ) 1 The time-complexity of searchig for a item amogst a array of sorted items is most precisely) Θ log ) olog ) Olog ) Ωlog ) 17. What is the time-complexity of the followig sum = sum + ; O ) Olog ) O log ) 18. What is the time-complexity of the followig for it j = 1; j < ; j = j*) sum = sum + ; O ) Olog ) O log ) 19. What is the time-complexity of the followig for it j = 0; j < ; j++) sum = sum + ; O ) Olog ) O log ) 0. What is the time-complexity of squarig a matrix of size i Big-Oh otatio? O ) O ) O log ) 1. For the AVL Tree show i Figure 1, what type of rotatio would be required i order to isert the value 1? A left rotatio A right rotatio A right-left rotatio A left-right rotatio. For the Biary Search Tree show i Figure 1, what is the iteral path legth of the tree? For the Splay Tree show i Figure 1, how may splay operatios would be required followig a fid5) operatio? 0 1. For the Biary Tree show i Figure 1, which of the followig represets a post-order traversal? 11,1,9,8,10,5,,,,7 7,,10,,5,8,1,,9,11 7,,,,5,10,8,9,1, For the Biary Search Tree show i Figure 1, deletio of ode 7 would result i which of the followig odes becomig the root ode? Sectio. 10 or 10 5 or 8 Autum 00 Fial 1. O the first day of Christmas, O the secod day of Christmas, Two Zetor tractors, ad O the third day of Christmas... How may lies would be i such a poem if it ra for 5 days istead of the usual 1? Which of the aswers below is i= i? 1 1) 1 1) 1 1). Which of the aswers below best approximates Oi 5 ) O ) O ) O 5 ). A -time algorithm sortig itegers i the rage [0, 1] is possible by makig passes of radix sort, with buckets of size / / 8 5. The time-complexity of expoetiatio to the power of is most precisely) Olog ) o) Θ) What is the desired upper limit o the load factor, λ, for ope hashig? λ.0 λ 1.0 λ 0.5 λ How may bits are required per ode to store the height of a ode i a -ode AVL tree? log log log log 8. How may odes are o the bottom layer, h, of a complete biary tree? at least h at most h exactly h oe of the above 9. A iversio i a list of umbers is a occurrece of a pair of umbers ot i order. What is the expected umber of iversios i a list of umbers? ) ) +1) +1) 10. Suppose we exchage elemets A[i] ad A[i+k] of a array, which were origially out of order. How may iversios does this remove from the array, at most? 1 k 1 k Which of the followig sortig algorithms is the odd oe out? Isertio sort Selectio sort Bubble sort Mergesort 1. Although Mergesort has a better worst-case ruig time guaratee tha Quicksort, it is ot the sortig algorithm of choice. This is because It ca be outperformed for small values by, say, isertio sort It has high overheads costs) associated with it It eeds to have all of its iput i primary memory Its performace degrades sigificatly whe sortig o-itegers 1. Embedded i Shellsort is what sortig algorithm? Isertio sort Bucket sort Heapsort Bubblesort 1. For Shellsort, a bad sequece of icremets, hk, is oe where Each of the icremets is a multiple of the previous Each of the icremets is a prime umber The icremets pairwise have o commo factors No set of positios of the array will be i the same subarray too ofte 15. Fidig the media th elemet) of a elemet set ca be doe i average time O log ) O ) 1 A adjacecy array would be a appropriate represetatio of a -vertex graph if the umber of edges was Ω ) o) 17. The ruig time of Depth o a graph, G, is O V E ) Omi V, E )) O V + E ) Omax V, E )) 18. A graph, G, ca be tested for acyclicity i time O V E ) Omi V, E )) O V + E ) Omax V, E )) 19. Every vertex has eve degree is a coditio for a Euleria Circuit. What goes i the blak? ecessary sufficiet both ecessary ad sufficiet either ecessary or sufficiet 0. The Depth -based algorithm for bicoectivity is a example of both ad processig of odes. The blaks are Sectio. iorder, preorder iorder, postorder preorder, postorder iorder, postalorder Autum 01 Mid-Term 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0; each questio has four possible aswers ad the questios are worth 5 marks each? Which of the aswers below is i=0 5 i? 5 5. Which of the aswers below is i= 5 i? Which of the aswers below is +1 i= 5i? 1 5 1) 1 5 1) 5 5 1) 5. Which of the aswers below is log b x? log b log x log x log b log x Which of the aswers below is i=1 i 1)?

2 7. Which of the aswers below best approximates Oi 5 ) O ) O ) O 5 ) 8. Which of the followig best approximates 1 i= i? 1 log e log e 9. Which of the followig is a correct represetatio of a b ) c? a b+c a cb 10. Usig m buckets, bucket sort has timecomplexity Om ) Om + ) Om ) O m ) 11. Ruig-time Om + ) is equivalet to A) Om) + or, B) Omaxm, )). Which of these possiblities are correct? 1. What is the ruig-time i Big-Oh otatio of the followig chuk of code?: i = -i; O ) O log ) 1. What is the ruig-time i Big-Oh otatio of the followig chuk of code?: for it i = 0; i < *; i++) sum = sum+i; O ) O log ) 5 1. Let T1) = Of)) ad T) = Of)). Give statemets A) T1) T) = ; ad B) T1) + T) = Of)), which of them are true? 15. I a recet court case, a judge cited a city for cotempt ad ordered a fie of for the first day. Each subsequet day, util the city followed the judge s order, the fie was squared that is, the fie progressed,, 1, 5, 5,5,...). What would be the fie o day? Sectio. Autum 01 Fial 1. The umber of odes i a perfect biary tree h 1 h 1 h+1 1. Give the two statemets below, which of them S1: a log c = log c a, for a, c ad iteger; S: logi log i, for i ad iteger. S1 is true, but S is false S1 is false, but S is true. Give the two statemets below, which of them S1: p) = Θq)) q) = Op)); S: f) = Og)) g) = of)). S1 is true, but S is false S1 is false, but S is true. Give the two statemets below, which of them S1: Heapsort s worst-case ruig time is worse tha Quicksort s worst case ruig time; S: Heapsort s average-case ruig time is poorer tha Quicksort s average-case ruig time. S1 is true, but S is false S1 is false, but S is true 5. Fidig the media of a array of elemets has ruig time best described by which of the followig? O log ) Olog ) Ω log ) Whe performig exteral sortig, the data structure used to efficietly do the k-way merge is a liked list a array a biary tree a heap 7. Whe performig exteral sortig, the umber of passes required to sort umbers m at a time usig k tapes is log k m log k m k log m log mk 8. I order to recostruct accurately a biary tree from its ordered traversals all three traversals are ecessary ay two are sufficiet the pre- ad post-order traversals are ecessary some other combiatio might do the job 9. Give the two statemets below, which of them S1: I a closed hash table, a key isertio followed by a deletio of that key returs the hash table exactly to its state prior to the isertio; S: I a closed hash table, a isertio of a o-existig key ca be performed faster tha a deletio of a existig key. S1 is true, but S is false S1 is false, but S is true 10. Give the two statemets below, which of them S1: The step of Heapsort that accouts for its asymptotic ruig time is the buildheap) step; S: O pre-sorted iput, Heapsort s ruig time is. Sectio 5. S1 is true, but S is false S1 is false, but S is true Autum 0 Mid-Term I 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0; each questio has four possible aswers ad the questios are worth 5 marks each? Which of the aswers below is i=0 i?. Which of the aswers below is i= i? 1 1. Which of the aswers below is +1 i= i? 5 1) 1 5 1) 1 5 1) 5. Which of the aswers below is log b x? log b log x log x log b log x Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) Oi ) 8. Which of the followig best approximates i=1 1 i? 1 log e log e 9. Which of the followig is a correct represetatio of a b ) c? a b+c a cb 10. Ruig-time Om + ) is equivalet to A Om) + B Omaxm, )) Which of these possiblities are correct? 11. What is the ruig-time i Big-Oh otatio of the followig chuk of code? i = -i; O ) O log ) 1. What is the ruig-time i Big-Oh otatio of the followig chuk of code? for it i = 0; i < *; i++) sum += sum+i; O ) O log ) 1. Let T1) = Of)) ad T) = Of)). Give statemets A T1) T) = ; B T1) + T) = Of)) which of them 1. I a recet court case, a judge cited a city for cotempt ad ordered a fie of for the first day. Each subsequet day, util the city followed the judge s order, the fie was squared that is, the fie progressed,, 1, 5, 5,5,...). What would be the fie o day? 15. The time-complexity of expoetiatio to the power of is most precisely) Olog ) o) Θ) 1 The time-complexity of searchig for a item amogst a array of sorted items is most precisely) Θ log ) olog ) Olog ) Ωlog ) 17. What is the time-complexity of the followig sum += sum+; O ) Olog ) O log ) 18. What is the time-complexity of the followig for j = 1; j < ; j = j*) sum += ; O ) Olog ) O log ) 19. What is the time-complexity of the followig for j = 0; j < ; j++) sum += ; O ) Olog ) O log ) 0. What is the time-complexity of squarig a matrix of size i Big-Oh otatio? O ) O ) Sectio O log ) Autum 0 Mid-Term II 1. How may bits are required per ode to store the height of a ode i a -ode AVL tree? Olog ) Olog log ) Olog ). Isert the umbers, 7, 1,, 8, 9,, 5 i that order) ito a iitially empty tree; which of the followig represets a preorder traversal of the of the tree?, 1,, 7,, 5, 8, 9, 7, 1,, 8, 9,, 5, 1, 7,,, 8, 5, 9, 1,, 5,, 7, 8, 9. Which of the aswers below is i=0 i? Which of the followig fuctios of grows fastest? f1) = log k, for ay costat k f) = log f) = 1+ɛ/ log, for ay ɛ > 0 f1) f) f) Based o k ad ɛ, it is ot possible to decide 5. What is the ruig-time i Big-Oh otatio of the followig chuk of code? for it j = 0; j < i*i; j++) for it k = 0; k < j; k++) sum++; O ) O ) Oi ) O 5 ) 7 8

3 What is the depth of the tree i Figure? 5 7. For the AVL tree show i Figure, i order to rebalace it, the followig rotatios) will have to be doe: A sigle rotatio about ode 7 A sigle rotatio about ode 10 A double rotatio about ode Let S1 = i=1 i ad S = i=1 i). Which oe of the followig statemets is true? Sectio 7. S1 = S for 1 0 oly S1 = S for 1 0 oly S1 = S for oly S1 = S for all Autum 0 Fial 1. The umber of odes i a complete biary tree exactly h 1 exactly h 1 exactly h+1 1. How may odes are o the bottom layer, h, of a perfect biary tree? at least h at most h exactly h oe of the above. Let S1 = i=1 i ad S = i=1 i). Which oe of the followig statemets is true?. O the first day of Christmas, O the secod day of Christmas, Two Zetor tractors, ad O the third day of Christmas... How may lies would be i such a poem if it ra for 5 days istead of the usual 1? What is the time-complexity of the followig for j = 1; j < ; j = j*) sum = sum + ; O ) Olog ) O log ) The worst-case performaces of the heap operatios deletemi) ad isert) are both Olog ). Give the two statemets below, which of them S1 The experimetally foud average case performace of deletemi) is S The experimetally foud average case performace of isert) is S1 is true, but S is false S1 is false, but S is true 7. FIX THIS!!! Give the two statemets below, which of them S1 I a strogly coected graph, every ode coects to every other ode by a edge S If a graph is strogly coected the it caot have a cut vertex articulatio poit) 8. Give the two statemets below, which of them S1 If a -vertex digraph has articulatio poits the the graph must have a cycle S If the Depth-First Tree of a graph G has o back edges the G has o cycles S1 is true, but S is false S1 is false, but S is true 9. Give the two statemets below, which of them S1 Startig from vertex v0 i a graph, the time required by Depth- to fid a path if oe exists) to some vertex v is less tha that required by Breadth- S The space required by Depth- is less tha that required by Breadth- Sectio 8. S1 is true, but S is false S1 is false, but S is true Autum 0 Mid-Term I 1. Which of the followig is a correct represetatio of a b ) c? a b+c a cb. Which of the aswers below is i=0 i?. Which of the aswers below is i= i?. Which of the aswers below is i= i? 1 1) 1 1) 1 1) 5. Which of the aswers below is 5 i=1 i? Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) O ) O 5 ) 8. For the fuctios show i Figure, what best describes the order of the fuctio, g)? g) = oh) g) = Oh) g) = Ωh) g) = Θh) 9. Which of the followig is the correct orderig of the fuctios,, 1000,? 1000 < < < < < 1000 < < < < < 10. Which of the followig is the correct orderig of the fuctios log,!, log log, log? S1 = S for 1 0 oly S1 = S for oly S1 = S for all S1 is true, but S is false S1 is false, but S is true log < log < log log <! log < log log < log <! log log < log < log <! log <! < log log < log Let T1) = Of)) ad T) = Of)). Give statemets A) T1) T) = ; ad B) T1) + T) = Of)), which of them are true? 1. Isert your magic umber, N, ito the Biary Search tree show i Figure. Redraw the tree.. Let S1 = i=1 i ad S = i=1 i). Which oe of the followig statemets is true? S1 = S for 1 0 oly S1 = S for oly S1 = S for all 1. I a recet court case, a judge cited a city for cotempt ad ordered a fie of for the first day. Each subsequet day, util the city followed the judge s order, the fie was squared that is, the fie progressed,, 1, 5, 5,5,...). What would be the fie o day?. For your magic umber, M, evaluate 15 M i i= 5. For your magic umber, M, evaluate 55 i M i i=0. O the first day of Christmas, O the secod day of Christmas, Two Zetor tractors, ad O the third day of Christmas... How may lies would be i such a poem if it ra for 5 days istead of the usual 1? 1. What is the time-complexity of computig i=0 aixi if a dumb expoetiatio algorithm is used? Olog ) O log ) O ) Leave your aswer i the form say) )/ ) What is the time-complexity of the followig Sectio 10. Autum 0 Fial, MC 5. What is the time-complexity of the followig for j = 0; j < ; j++) sum = sum + ; O ) Olog ) O log ) 15. What is the time-complexity of cubig A A A) a matrix of size i Big-Oh otatio? Sectio 9. O ) O ) O 5 ) O ) Autum 0 Mid-Term II. Isert your magic umber, N, ito the AVL tree show i Figure. Redraw the tree.. Delete your magic digit, d, from the Biary Search tree show i Figure. Redraw the resultig tree. Use the machie-readable multiple-choice questio grid that has bee provided to aswer these questios. Please completely mark i black exactly oe circle o the grid for each aswer. A pealty will be charged for wrog aswers. Mark the X bubble for those questios you wish to skip. 1. The umber of odes i a complete biary tree exactly h 1 exactly h 1 exactly h+1 1. How may odes are o the bottom layer, h, of a perfect biary tree? at least h at most h exactly h oe of the above O ) Olog ) O log ) The worst-case performaces of the heap operatios deletemi) ad isert) are both Olog ). Give the two statemets below, which of them S1 The experimetally foud average case performace of deletemi) is S The experimetally foud average case performace of isert) is S1 is true, but S is false S1 is false, but S is true 11 1

4 7. Give the two statemets below, which of them S1 I a strogly coected graph, every ode coects to every other ode by a edge S If a graph is strogly coected the it caot have a cut vertex articulatio poit) S1 is true, but S is false S1 is false, but S is true 8. Give the two statemets below, which of them S1 If a -vertex digraph has articulatio poits the the graph must have a cycle S If the Depth-First Tree of a graph G has o back edges the G has o cycles S1 is true, but S is false S1 is false, but S is true 9. Give the two statemets below, which of them S1 Startig from vertex v0 i a graph, the time required by Depth- to fid a path if oe exists) to some vertex v is less tha that required by Breadth- S The space required by Depth- is less tha that required by Breadth- Sectio 11. S1 is true, but S is false S1 is false, but S is true Autum 0 Fial, Shorts Please put your aswers to these questios i the aswer book provided to you, labellig your aswers.1,., etc. 1. The uweighted shortest path problem ca be solved i time.. With calls to percolate dow), a heap ca be created i time.. Give the recurrece relatio for the best-case ruig time of QuickSelect), the algorithm 1 for fidig the k th largest elemet i a array:. Recursio is to algorithm implemetatio as is to proof techiques. That is, what is the proof techique aalogue of recursio? 5. Ordiarily the most appropriate way to represet a graph iterally is with ; however, if may queries are of the form Is ode u adjacat to ode v? the the most appropriate represetatio may be Sectio 1. Autum 0, Mid-Term I 1. Which of the aswers below best describes the th Fiboacci umber, F? ) 5 F 7 ) F 7 ) 7 F 5 ) 7 F. Which of the aswers below best describes the th Fiboacci umber, F? F = O ) F = O ) F = Oa ), a > F = Oa ), a < e). Two sowballs rollig dow differet part of the same hill grow at differet rates. Sowball Flakey grows at a rate so that it triples i size every three hours, while sowball Fluffy quadruples i size every four hours. If they both started rollig at the same time ad they were idetical sizes at the outset ad they reach the bottom together, which will be biggest? Flakey Fluffy They will be idetical sizes Depeds o the sow desity o the hill. Two galaxies i the same uiverse expad at differet rates. Galaxy Deuteroium grows at a rate so that it doubles i size every two years, while galaxy Séamus quadruples i size every four years. If they both started expadig at the same time ad they were idetical sizes at the outset which will be bigger at time T = 00, 000 years? Deuteroium Séamus They will be idetical sizes Caot be determied from the iformatio give 5. Cosider the followig algorithm for solvig jigsaw puzzles with pieces: pick a radom piece p from the box let C, the set of pieces i the completed part of the puzzle, start off as p i. for each of the remaiig pieces i the box, r, i tur, see if it fits ito the completed part C by checkig it agaist all exposed pieces of C ii. if it does, place it i its positio iii. if it does t, retur it ad cosider the ext uplaced piece from the box repeat steps i) to iii) while there is somethig i the box O log ) O ) O ) 1 Cosider the followig differet algorithm for solvig jigsaw puzzles with pieces: pick a radom piece p from the box let C, the set of pieces i the completed part of the puzzle start off as p i. omiate a exposed piece of the completed part of the jigsaw; call it c ii. for each of the remaiig pieces i the box, r, i tur, see if it fits ext to c iii. if it does, place it i its positio iv. if it does t, retur it ad cosider the ext uplaced piece from the box repeat steps i) to iv) while there is somethig still left i the box O log ) O ) O ) 7. Give f) = 0, which of the followig is true? f) = O ) ad f) O ) f) O ) ad f) = O ) f) = O ) ad f) = O ) f) O ) ad f) O ) 8. Which of the aswers below is i=1 i? ) i=1 i ) i=1 i ) i=1 i 9. Which of the followig equals log/) + log/)? log log log ) 10. A geometric series is oe that ca be writte i the form i=0 ari, for some costats a ad r; a arithmetic series is oe that ca be writte i the form i=0 a + id. Is the series i=0 i i a) geometric series arithmetic series geometric ad arithmetic series either geometric or arithmetic series 11. Give some costat k, which of the followig statemets log k ) = ad log k = log k ) ad log k = log k ) = ad log k log k ) ad log k 1. Give a matrix, A, of size, what is the time-complexity of calculatig A 5 i Big-Oh otatio? O ) O ) 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0; each questio has three possible aswers ad the questios are worth marks each? 1 1. What is the ruig-time i Big-Oh otatio of the followig chuk of code, where b is some it previously give a value? for it j = i; j < /; j += b) arr[i][j] = 0; O log ) O log b ) O ) What is the time-complexity of the followig for it j = i; j < ; j += /) arr[i][j] = 0; Olog ) O log ) O ) 1 Which of the recurrece relatios below best describes the ruig time of followig fuctio sort)? // retur idex of largest el. it fididlargestcost it a[], it s, it e) it li = s; it l = a[li]; for it i = s+1; i < e; i++) if l < a[i]) li = i; l = a[li]; retur li; void swapit& a, it& b) it t = a; a = b; b = t; void sortcost it a[], it ) for it i = 0; i < ; i ++) it li = fididlargesta, i, ); swapa[i], a[li]); T ) = T k) + Sectio 1. T ) = T k) + c T ) = T k) + c T ) = T 1) + c Autum 0, Mid-Term II 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0; each questio has four possible aswers ad the questios are worth four marks each?. Which of the aswers below is i=0 i?. Which of the aswers below is i= i?. Which of the aswers below is i= i? 1 1) 1 1) 1 1) 5. Which of the aswers below is 5 i=1 i? Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) O ) O 5 ) 8. Which of the followig is the correct orderig of the fuctios,, 1000,? 1000 < < < < < 1000 < < < < < 9. Which of the followig is the correct orderig of the fuctios log,!, log log, log? log < log < log log <! log < log log < log <! log log < log < log <! log <! < log log < log If T ) = Of)) the which of the followig is the most precise statemet? T ) = of)) f) = ΩT )) T ) = k f) for some costat k f) = ΘT )) 11. If you fold a A sheet of paper i half you get a A5 sheet. Similarly, a A sheet is twice the size of a A sheet. This cotiues all the way back to the huge sheet A0. How may A5 sheets fit i a A0 sheet? You re uder pressure. It s your tur to make the Suday breakfast ad you have to get the sausages o the pa i a hurry. If you ve got a lik of sausages to cut with a scissors, what is the fewest o. of cuts eeded to separate them ito idividual sausages, without bedig ay of the sausages? 1 log 1 1. Thigs have got costatly worse. The scissors that you were usig is blut ad you ca oly cut a maximum of c sausages at ay oe time with it. How may cuts are ecessary ow? log c c c 1 1. We wat to keep track of the relatioships betwee people, p0, p1,..., p. We create a two-dimesioal array, arr, of bools so that to tell if perso pi is related to pj we check arr[i][j]. There s o eed to check aythig if i = j sice everybody is related to themselves. Sice relatioships are symmetrical, to test if p8 was related to p, we would check arr[][8]. Thus, the etire array is ot eeded. Exactly how may etries of the array are eeded? 1 1) 1

5 15. I the previous questio, suppose the aswer was S). How may differet uique sets of relatios could we have o the people? [Hit: Each uique set of relatios correspods to a uique matrix.] S) S) S) S) S) 1 We wat to fid the largest elemet of a array of umbers, a1,..., a. Assume for hadiess that is a power of.) Our strategy is that we will compare a1 with a, compare a with a, ad, fially, a ad a. That gives us / first roud wiers, f1,..., f, which advace to the ext roud, where they are compared i pairs agai to give secod-roud wiers. This cotiues util there is a wier of the fial, the largest elemet. If this scheme is to work the it will have to perform at least how may comparisos i fidig the champio? Ω1) Ωlog ) Ω 1) Ω) 17. What is the recommeded maximum load factor, λ, for a closed hash table? λ > 0 λ < 0.5 λ < 1 0 < λ < The load factor, λ, of a hash table is defied to be the ratio of etries to table size. Whe does this formula hold? For closed hash tables but ot for ope hash tables For ope hash tables but ot for closed hash tables For both ope ad closed hash tables For either ope or closed hash tables 19. What is the expected time for isertio ito a hash table with load factor, λ, usig liear probig? 1 1 λ) 1 λ) λ) ) 1 λ) ) I a d-ary heap, where will you beig ode i fid your paret? Sectio 1. d i d d i d i d + 1 i+d d Autum 0, Fial 1. I a AVL tree of height h, the umber of probes made i a usuccessful search will be Olog ) Depeds o the way the AVL was first created. Give f) = ad g) =, which oe of the followig statemets is false. f) = Og)) f) = Ωf)) f) = Θf)) f) = og)). How may multiplicatios are used i the calculatio of X, usig the fast expoetiatio algorithm? Olog ) O ) O!). What is the vertex-coectivity of the graph draw i Figure 5? 5 5. How may cut vertices does the graph draw i Figure 5 have? 1 5 O iput of size N, the ruig time of programs A is always less tha 1 N. O iput of size N, the ruig time of program B is always less tha 1 log N ). Is the average ruig time of program A greater tha the average ruig time of program B, for N = 100, 000. Yes, the average ruig time of program A is greater tha the average ruig time of program B No, the average ruig time of program A equals the average ruig time of B, for N = 100, 000 No, the average ruig time of program B is greater tha the average ruig time of program A, for N = 100, 000 There is ot eough iformatio to tell 7. Cosider what happes after is iserted ito the AVL tree draw i Figure The tree becomes ubalaced ad eeds to be re-balaced usig a sigle-rotatio. Followig this siglerotatio what odes), if ay, are the childre of? ode 7 ad ode 8 just ode 7 just ode 8 ode will have o childre 8. Cosider a arbitrary biary tree B where every ode is either full or is a leaf. Let F represet the umber of full odes i B ad let L represet the umber of leaves i B. Recall a full ode is a ode with two childre, ad a leaf is a ode with o childre). Which of the followig statemets is true, o matter what the choice of B? F = L + 1 L = F + 1 L = F F = L 9. For which of the followig sortig algorithms, is it importat that the iput is radom rather tha presorted or i reverse order for example)? quicksort, where the first elemet is chose as the pivot quicksort, where the pivot is chose usig media-of-three partitioig mergesort isertio sort The two formulae we have see regardig the expected umber of probes of hash table operatios usig liear probig are 1 1 λ) ad, 1 operatios are give by them? Sectio λ) ). Which : successful searches; isertios : isertios; successful searches : successful searches; usuccessful searches : usuccessful searches; isertios Autum 0, Fial Shorts) 1. Whe solvig the weighted shortest path a priority queue is used but uses a o-stadard operatio. What is this o-stadard operatio?.. I O-otatio, what is the aalogue of a lieartime algorithm if the iput is a graph G = V, E)?.. The height of a AVL tree is o worse tha times the optimal height.. Give a graph, G = V, E), what is the largest umber of edges exactly a graph ca have i terms of V, the umber of odes?. 5. Ordiarily the most appropriate way to represet a graph iterally is with ; however, if may queries are of the form Is ode u adjacat to ode v? the the most appropriate represetatio may be Sectio 1 Term Sprig 00/007, Mid- 1. Which of the aswers below is i= i? 1 1) 1 1) 1 1). Which of the aswers below best approximates O ) Oi 5 ) O 5 ) O ). A -time algorithm sortig itegers i the rage [0, 1] is possible by makig 8 passes of radix sort, with buckets of size /1 / 1 8. The time-complexity of expoetiatio to the power of is most precisely) Olog ) o) Θ) 5. What is the desired upper limit o the load factor, λ, for ope hashig? λ.0 λ 1.0 λ 0.5 λ 0.0 How may bits are required per ode to store the height of a ode i a -ode AVL tree? log log log log 7. How may odes are o the bottom layer, h, of a complete biary tree? at least h at most h exactly h oe of the above 8. Which of the aswers below is log b x? log x log b log b log x log x 9. Usig m buckets, bucket sort has timecomplexity Om ) Om + ) Om ) O m ) 10. Ruig-time Om + ) is equivalet to A) Om) + or, B) Omaxm, )). Which of these possiblities are correct? 11. Give f) = ad g) =, which oe of the followig statemets is false. f) = Og)) f) = Ωf)) f) = Θf)) f) = og)) 1. How may multiplicatios are used i the calculatio of X, usig the fast expoetiatio algorithm? O!) O ) Olog ) 1. The load factor, λ, of a hash table is defied to be the ratio of etries to table size. Whe does this formula hold? For closed hash tables but ot for ope hash tables For ope hash tables but ot for closed hash tables For both ope ad closed hash tables For either ope or closed hash tables 1. What is the expected time for isertio ito a hash table with load factor, λ, usig liear probig? 1 1 λ) 1 λ) λ) ) 1 λ) ) 15. We wat to keep track of the relatioships betwee people, p0, p1,..., p. We create a two-dimesioal array, arr, of bools so that to tell if perso pi is related to pj we check arr[i][j]. There s o eed to check aythig if i = j sice everybody is related to themselves. Sice relatioships are symmetrical, to test if p8 was related to p, we would check arr[][8]. Thus, the etire array is ot eeded. Exactly how may etries of the array are eeded? 1 1) 1 1 I the previous questio, suppose the aswer was S). How may differet uique sets of relatios could we have o the people? [Hit: Each uique set of relatios correspods to a uique matrix.] S) S) S) S) S) 17. We wat to fid the largest elemet of a array of umbers, a1,..., a. Assume for hadiess that is a power of.) Our strategy is that we will compare a1 with a, compare a with a, ad, fially, a ad a. That gives us / first roud wiers, f1,..., f, which advace to the ext roud, where they are compared i pairs agai to give secod-roud wiers. This cotiues util there is a wier of the fial, the largest elemet. If this scheme is to work the it will have to perform at least how may comparisos i fidig the champio? Ω1) Ωlog ) Ω 1) Ω) 18. I a AVL tree of height h, the umber of probes made i a successful search will be most accurately) Olog ) Sectio 17. Depeds o the way the AVL was created Sprit , Fial 1. The umber of odes i a complete biary tree exactly h 1 exactly h 1 exactly h+1 1. How may odes are o the bottom layer, h, of a perfect biary tree? at least h at most h exactly h oe of the above. Let S1 = i=1 i ad S = i=1 i). Which oe of the followig statemets is true? S1 = S for 1 0 oly S1 = S for oly S1 = S for all. O the first day of Christmas, O the secod day of Christmas, Two Zetor tractors, ad O the third day of Christmas... How may lies would be i such a poem if it ra for 5 days istead of the usual 1? What is the time-complexity of the followig O ) Olog ) O log ) 19 0

6 The worst-case performaces of the heap operatios deletemi) ad isert) are both Term Sectio 18. Olog ). Give the two statemets below, which of them S1 The experimetally foud average case performace of deletemi) is S The experimetally foud average case performace of isert) is S1 is true, but S is false S1 is false, but S is true 7. Give the two statemets below, which of them S1 I a strogly coected graph, every ode coects to every other ode by a edge S If a graph is strogly coected the it caot have a cut vertex articulatio poit) S1 is true, but S is false S1 is false, but S is true 8. Give the two statemets below, which of them S1 If a -vertex digraph has articulatio poits the the graph must have a cycle S If the Depth-First Tree of a graph G has o back edges the G has o cycles S1 is true, but S is false S1 is false, but S is true 9. Give the two statemets below, which of them S1 Startig from vertex v0 i a graph, the time required by Depth- to fid a path if oe exists) to some vertex v is less tha that required by Breadth- S The space required by Depth- is less tha that required by Breadth- S1 is true, but S is false S1 is false, but S is true 1 Sprig 010/011, Mid- 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0; each questio has four possible aswers ad the questios are worth four marks each?. Which of the aswers below is i=0 i?. Which of the aswers below is i= i?. Which of the aswers below is i= i? 1 1) 1 1) 1 1) 5. Which of the aswers below is 5 i=1 i? Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) O ) O 5 ) 8. Which of the followig is the correct orderig of the fuctios,, 1000,? 1000 < < < < < 1000 < < < < < 9. Which of the followig is the correct orderig of the fuctios log,!, log log, log? log < log < log log <! log < log log < log <! log log < log < log <! log <! < log log < log 10. If T ) = Of)) the which of the followig is the most precise statemet? T ) = of)) f) = ΩT )) T ) = k f) for some costat k f) = ΘT )) 11. If T ) = Θf)) the which of the followig is the most precise statemet? T ) = Of)) ad T ) = Ωf)) T ) = Of)) but ot T ) = Ωf)) T ) = of)) ad T ) = Ωf)) T ) = of)) but ot T ) = Θf)) 1. If m = Θ) the bucket sort has timecomplexity most precisely) O ) Ω) Ω ) 1. A -time algorithm sortig itegers i the rage [0, 1] is possible by makig passes of radix sort, with buckets of size / / 8 1. The time-complexity of expoetiatio to the power of is most precisely) Olog ) o) Θ) 15. What is the time-complexity of computig i=0 aixi if a aïve expoetiatio algorithm is used? Olog ) O log ) O ) 1 The time-complexity of searchig for a item amogst a array of sorted items is most precisely) Θ log ) olog ) Olog ) Ωlog ) 17. What is the time-complexity of the followig sum = sum + ; O ) Olog ) O log ) 18. What is the time-complexity of the followig for it j = 0; j < ; j = j*) sum = sum + ; O ) Olog ) O log ) 19. What is the time-complexity of the followig for it j = 0; j < ; j++) sum = sum + ; O ) Olog ) O log ) 0. What is the time-complexity of squarig a matrix of size i Big-Oh otatio? O ) O ) O log ) 1. For the AVL Tree show i Figure 1, what type of rotatio would be required i order to isert the value 1? A left rotatio A right rotatio A right-left rotatio A left-right rotatio. For the Biary Tree show i Figure 1, which of the followig represets a post-order traversal? 11,1,9,8,10,5,,,,7 7,,10,,5,8,1,,9,11 7,,,,5,10,8,9,1,11.. For the Biary Search Tree show i Figure 1, deletio of ode 7 would result i which of the followig odes becomig the root ode? 10 or 10 5 or 8. Two sowballs rollig dow differet parts of the same hill grow at differet rates. Sowball Flakey grows at a rate so that it triples i size every three hours, while sowball Fluffy quadruples i size every four hours. If they both started rollig at the same time ad they were idetical sizes at the outset ad they reach the bottom together, which will be biggest? The depth of sow is the same all over the hill.) Flakey Fluffy They will be idetical sizes Depeds o the sow desity o the hill 5. Two galaxies i the same uiverse expad at differet rates. Galaxy Deuteroium grows at a rate so that it doubles i size every two years, while galaxy Séamus quadruples i size every four years. If they both started expadig at the same time ad they were idetical sizes at the outset which will be bigger at time T = 00, 000 years? Sectio 19. Deuteroium Séamus They will be idetical sizes Caot be determied from the iformatio give Sprig 010/011, Fial Use the machie-readable multiple-choice questio grid that has bee provided to aswer these questios. Please completely mark i black exactly oe circle o the grid for each aswer. A pealty will be charged for wrog aswers. Mark the X bubble for those questios you wish to skip. 1. What is the time-complexity of the followig for j = 1; j < ; j = j*) sum = sum + ; O ) O log ) Olog ). The worst-case performaces of the heap operatios deletemi) ad isert) are both Olog ). Give the two statemets below, which of them S1 The experimetally foud average case performace of deletemi) is S The experimetally foud average case performace of isert) is S1 is true, but S is false S1 is false, but S is true. The umber of odes i a complete biary tree exactly h 1 exactly h 1 exactly h+1 1. How may odes are o the bottom layer, h, of a perfect biary tree? exactly h at least h at most h oe of the above 5. Let S1 = i=1 i ad S = i=1 i). Which oe of the followig statemets is true? S1 = S for 1 0 oly S1 = S for oly S1 = S for all O the first day of Christmas, O the secod day of Christmas, Two Zetor tractors, ad O the third day of Christmas... How may lies would be i such a poem if it ra for 5 days istead of the usual 1? Give the two statemets below, which of them S1 I a strogly coected graph, every ode coects to every other ode by a directed edge S If a graph is strogly coected the it caot have a cut vertex articulatio poit) S1 is true, but S is false S1 is false, but S is true 8. Give the two statemets below, which of them S1 If a -vertex digraph has articulatio poits the the graph must have a cycle S If the Depth-First Tree of a graph G has o back edges the G has o cycles S1 is true, but S is false S1 is false, but S is true 9. Give the two statemets below, which of them S1 Startig from vertex v0 i a graph, the time required by Depth- to fid a path if oe exists) to some vertex v is less tha that required by Breadth- S The space required by Depth- is less tha that required by Breadth- Sectio 0. Term S1 is true, but S is false S1 is false, but S is true Sprig 011/01, Mid- 1. I a multiple choice exam, if egative markig is beig used, what should be the pealty for wrog aswers so that somebody who guesses throughout eds up with 0? Each questio has four possible aswers ad the questios are worth 5 marks each Which of the aswers below is i=0 i?. Which of the aswers below is i= i? 1 1

7 . Which of the aswers below is +1 i= i? 5 1) 1 5 1) 1 5 1) 5. Which of the aswers below is log b x? log b log x log x log b log x Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) Oi ) 8. Which of the followig best approximates i=1 1 i? 1 log e log e 9. Which of the followig is a correct represetatio of a b ) c? a b+c a cb 10. Ruig-time Om + ) is equivalet to A Om) + B Omaxm, )) Which of these possiblities are correct? 11. What is the ruig-time i Big-Oh otatio of the followig chuk of code? i = -i; O ) O log ) 1. What is the ruig-time i Big-Oh otatio of the followig chuk of code? for it i = 0; i < *; i++) sum += sum+i; O ) O log ) 1. Let T1) = Of)) ad T) = Of)). Give statemets A T1) T) = ; B T1) + T) = Of)) which of them 1. I a recet court case, a judge cited a city for cotempt ad ordered a fie of for the first day. Each subsequet day, util the city followed the judge s order, the fie was squared that is, the fie progressed,, 1, 5, 5,5,...). What would be the fie o day? 15. The time-complexity of expoetiatio to the power of is most precisely) 1 The time-complexity of searchig for a item amogst a array of sorted items is most precisely) Θ log ) olog ) Olog ) Ωlog ) 17. What is the time-complexity of the followig sum += sum+; O ) Olog ) O log ) 18. What is the time-complexity of the followig for j = 1; j < ; j = j*) sum += ; O ) Olog ) O log ) 19. What is the time-complexity of the followig for j = 0; j < ; j++) sum += ; O ) Olog ) O log ) 0. What is the time-complexity of squarig a matrix of size i Big-Oh otatio? 1. For the Biary Tree show i Figure 1, which of the followig represets a post-order traversal? 11,1,9,8,10,5,,,,7 7,,10,,5,8,1,,9,11 7,,,,5,10,8,9,1,11.. For the Biary Search Tree show i Figure 1, deletio of ode 7 would result i which of the followig odes becomig the root ode? 10 or 10 5 or 8. For the Biary Search Tree show i Figure 1, deletio of ode 7 would result i the iorder traversal beig which of the followig?, 10,, 5, 8, 1,, 9, 11 10,, 1, 8, 5,, 11, 9,,, 10,, 5, 8, 1,, 9, 11,,, 5, 8, 9, 10, 11, 1. Which is larger, 18 9 or 9 18? They are both the same We did t do a problem like this before 5. How may odes are cotaied i a perfect terary -way) tree of height k levels 0... k)? k k k+1 How may odes are cotaied i the lowest layer of a perfect terary -way) tree of height k levels 0... k)? Olog ) o) Θ) O ) O ) O log ) k k k k How may odes are cotaied i layers 10 to of a perfect terary -way) tree of height 0 levels )? The best aswer below is: All three of the followig aswers ) 8. The Fiboacci umbers obey the recurrece relatio F = F + F,, F0 = F1 = 1 Which oe of the followig statemets is false? F+1 = F + F, 1 F+ = F+1 + F, 0 F = 1 + i=0 Fi, F =, 0 I the followig two questios BST<Comparable> is a Biary Search Tree C++ class ito which you may isert objects that are comparable, that is, it is possible to decide amogst two which is larger ad smaller. Each ode of this tree will be a class, BNode, with a poiters to the left ad right subtrees, ->left ad ->right, respectively. 9. What does the fuctio what) below, or, more accurately, the member fuctio BST<Comparable>::what)), perform? BNode<Comparable> * BST<Comparable>::what BNode<Comparable> *t ) if t == NULL ) retur NULL; if t->left == NULL ) retur t; else retur what t->left ); Cout the umber of NULL odes i the Fid the leftmost ode i the tree rooted at t Fid a ode smaller tha t Fid a ode larger tha t 7 0. What does the member) fuctio whatever) below, perform? The fuctio max) returs the larger of two umbers. it BST<Comparable>::whatEver BNode<Comparable> *t ) if t == NULL ) retur 0; it l = whatevert->left); it r = whatevert->right); it m = maxl, r); // larger of two its retur m+1; Sectio 1. Cout the umber of NULL odes i the Cout the umber of o-null odes i the Compute the height of the tree rooted at t Compute the age of the tree rooted at t Sprig 011/01, Fial 1. Which of the aswers below is i=0 i?. Which of the aswers below is i= i? 1 1. Which of the aswers below is +1 i= i? 5 1) 1 5 1) 1 5 1). Which of the aswers below is log b x? log b log x log x log b log x 5. Which of the aswers below is i=1 i 1)? Which of the aswers below best approximates Oi 5 ) O ) Oi ) 7. Which of the followig best approximates i=1 1 i? 1 log e log e 8. Ruig-time Om + ) is equivalet to A Om) + B Omaxm, )) Which of these possiblities are correct? 9. What is the ruig-time i Big-Oh otatio of the followig chuk of code? i = -i; O ) O log ) 10. Let T1) = Of)) ad T) = Of)). Give statemets A T1) T) = ; B T1) + T) = Of)) which of them The time-complexity of searchig for a item amogst a array of sorted items is most precisely) Θ log ) olog ) Olog ) Ωlog ) 1. What is the time-complexity of the followig for j = 1; j < ; j = j*) sum += ; O ) Olog ) O log ) 1. Which is larger, 18 9 or 9 18? They are both the same I the limit they will differ oly slightly 1. How may odes are cotaied i a perfect terary -way) tree of height k levels 0... k)? k k k How may odes are cotaied i the lowest layer of a perfect terary -way) tree of height k levels 0... k)? k k k k+1 1 How may odes are cotaied i layers 10 to of a perfect terary -way) tree of height 0 levels )? The best aswer below is: All three of the followig aswers ) I the followig three questios BST<Comparable> is a Biary Search Tree C++ class ito which you may

8 isert objects that are comparable, that is, it is possible to decide amogst two which is larger ad smaller. Each ode of this tree will be a class, BNode, with a poiters to the left ad right subtrees, ->left ad ->right, respectively. 17. What does the fuctio what) below, or, more accurately, the member fuctio BST<Comparable>::what)), perform? BNode<Comparable> * BST<Comparable>::what BNode<Comparable> *t ) if t == NULL ) retur NULL; if t->left == NULL ) retur t; else retur what t->left ); Cout the umber of NULL odes i the Fid the leftmost ode i the tree rooted at t Fid a ode smaller tha t Fid a ode larger tha t 18. What does the member) fuctio whatever) below, perform? The fuctio max) returs the larger of two umbers. it BST<Comparable>::whatEver BNode<Comparable> *t ) if t == NULL ) retur 0; 19. What does the member) fuctio whatnow) below, perform? it BST<Comparable>::whatNow BNode<Comparable> *t ) if t == NULL ) retur 0; retur whatnowt->left) + whatnowt->right) + 1; Cout the umber of NULL odes i the Cout the umber of o-null odes i the Compute the height of the tree rooted at t Compute the age of the tree rooted at t 0. The Fiboacci umbers obey the recurrece relatio F = F + F,, F0 = F1 = 1 Which oe of the followig statemets is false? F+1 = F + F, 1 F+ = F+1 + F, 0 F = 1 + i=0 Fi, F =, 0 Figure 1: A Biary Search / AVL / Splay Tree Figure : Some fuctios it l = whatevert->left); it r = whatevert->right); it m = maxl, r); // larger of two its retur m+1; Cout the umber of NULL odes i the Cout the umber of o-null odes i the Compute the height of the tree rooted at t Compute the age of the tree rooted at t 9 0 Figure : A Biary Search / AVL Tree Figure : A Biary Search Tree A -regular graph. A graph with cutvertices. Figure 5: Some example graphs. Figure : A AVL Tree. 1