CPS 616 W2017 MIDTERM SOLUTIONS 1

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1 CPS 616 W2017 MIDTERM SOLUTIONS 1 PART 1 20 MARKS - MULTIPLE CHOICE Instructions Plas ntr your answrs on t bubbl st wit your nam unlss you ar writin tis xam at t Tst Cntr, in wic cas you sould just circl your answr or ac qustion in tis booklt. Us pncil only or bubbl sts.. YOU CAN GUESS. MARKS WILL NOT BE DEDUCTED FOR FALSE ANSWERS. Qustions 1. (2 marks) Is tis rap: A. Dirctd and cyclical B. Dirctd and acyclical C. Undirctd and cyclical D. Undirctd and acyclical 2. (2 marks) Wic on o ts squncs o vrtics is not a topoloical sort?: A. a b c d B. C. a c b d D. Non o t abov (i.. ty ar all topoloical sorts) 3. (2 marks) Wic on o ts squncs o vrtics was nratd rom a BFS (Bradt First Sarc) travrsal o t rap on t lt? Assum tis ar brokn by t alpabtical ordr o t vrtics. A. c d a b B. a b c d C. b a c d D. Non o t abov (i.. non was nratd rom a BFS travrsal) 4. (2 marks) Wic on o ts squncs o vrtics was not nratd rom a DFS (Dpt First Sarc) travrsal o t rap on t lt? Assum tis ar brokn by t alpabtical ordr o t vrtics. A. b c d a B. a b c d C. b a c d D. Non o t abov (i.. ty wr all nratd rom a DFS travrsal)

2 CPS 616 W2017 MIDTERM SOLUTIONS 2 5. (3 marks) Wic is t most dsirabl complxity or an aloritm (i.. wic o ts complxitis provids t most op tat t aloritm will b ast)? A. O(n) B. θ(n) C. Ω(lo n) D. o(n) 6. (3 marks) Wat is t xact cost o tis aloritm as a unction o n: myloops (n) or (i=0; i<=n; i++) or (j=n; j>i, j--) // t basic opration is r A. n(n+1)/2 B. n(n-1)/2 C. (n+1) 2 /2 D. (n+1)(n-1)/2 7. (3 marks) Wic o t tr claims (A, B, C) about t two cod ramnts undrnat is als: // Framnt 1 rsult1 = 0 or i=1 to n do or j=1 to n do or k=1 to n do i i<j rsult1 = rsult1++; // Framnt 2 rsult2 = 0 or i=1 to n do or j=1 to n do i i<j or k=1 to n do rsult2 = rsult2++; A. T valu o rsult1 at t nd o t xcution o ramnt1 is t sam as t valu o rsult2 at t nd o t xcution o ramnt 2. B. Framnts 1 and 2 av t sam xact xcution cost C. Framnts 1 and 2 av t sam asymptotic xcution cost 8. (3 marks) Wic o ts squncs is not a Gray cod: A. 111,101, 100, 000, 001, 011, 010, 110 B. abc, bac, bca, cba, cab, acb C. 00,01,02,12,11,10,20,21,22 D. Non o t abov: ty ar all Gray cods

3 CPS 616 W2017 MIDTERM SOLUTIONS 3 PART 2 SHORT ANSWERS 30 MARKS - PLEASE WRITE YOUR ANSWERS DIRECTLY IN THIS EXAM 9. (30 marks) Hr is a dcras by on aloritm to solv t knapsack problm // Dscription o itms is stord in arrays o n lmnts. Intr wits[n] // wits[i] = wit o itm i Intr valus[n] // valus[i] = vau o itm i // St Knapsack (Intr capacity, Intr n) // capacity is t knapsack capacity // n = numbr o itms to it in t knapsack, n 1 // Tis unction rturns a st containin t most valuabl combination o itms // wic can it in t knapsack St Knapsack (Intr capacity, Intr n) { I n=1 I wits[1]<= capacity rturn {1 Els rturn { // I itm n dos not it, lav it out Intr witn = wits[n] I witn>capacity rturn Knapsack(capacity,n-1) // Bst valu wn itm n is includd in knapsack St includd = {n Knapsack(capacity-witn, n-1) // Bst valu wn itm n is not includd in knapsack St xcludd = Knapsack(capacity,n-1) I Wort(xcludd) >= Wort(includd) Els rturn includd // Intr Wort(StoItms) // StoItms is a st o itms // Tis unction rturns t wort o a st o itms Intr Wort(StoItms) { Intr rsult = 0 For itm in StoItms do Rsult += valus[itm] Rturn rsult For t analysis tat ollows, you will b assumin tat t basic opration is an array rrnc wits[i] or valus[i], i.. you will b askd to count ow many tims an itm is toucd. a) (2 marks) In t cod abov, idntiy clarly t locations o t basic opration (or xampl you can circl tm, or ilit tm, or undrlin tm, tc.) b) (1 mark) Wat is t siz o tis aloritm? (i.. wat variabls will t cost o tis aloritm b a unction o?) n

4 CPS 616 W2017 MIDTERM SOLUTIONS 4 c) (2 marks) Wat is t xact cost o t unction Wort? i.. ow many tims will its basic opration b xcutd wn it is calld wit t paramtr StoItms? Numbr o lmnts in t paramtr StoItms. d) (4 marks) Wat is t bst cas cost o t Knapsack aloritm and in wic situation dos it appn? T bst cas appns wn non o t itms it in t knapsack. As a rsult ac call to Knapsack or n>1 calls itsl rcursivly only onc (on t 7 t lin o Knapsack), and Wort is also nvr calld. T cost is xactly n bcaus wits[i] is calld xactly onc or ac o t n lmnts. ) (3 marks) In wic situation dos t worst cas cost occur? (i.. in wic situation dos tis aloritm do t most work, maximizin bot t work o Knapsack and Wort?) T worst cas occurs wn all t lmnts can it totr in t knapsack. Eac call to t Knapsack unction will mak t two rcursiv calls to crat includd and xcludd. Furtrmor t Wort unction will always b calld wit t maximum possibl numbr o lmnts in StoItms. ) (5 marks) Din rcursivly t worst cas cost Kn o t Knapsack unction or n itms. Rmmbr tat you nd to provid bot t bas cas and t rcurrnc rlation. Also do not ort to includ t cost o t unction Wort in your cost. Justiy your answr (i.. xplain wat ac componnt o t ormula rprsnts) K1 = 1 (wits[1] on lin 3 o Knapsack) Kn = 1 (wits[n] on lin 6 o Knapsack) + Kn-1 (rcursiv call or includd) + Kn-1 (rcursiv call or xcludd) + n-1 (Wort(xcludd) + n (Wort(includd)) = 2Kn-1 + 2n

5 CPS 616 W2017 MIDTERM SOLUTIONS 5 ) (8 marks) Mak a w modiications to tis aloritm to intrat t unctionality o t Wort unction dirctly into t Knapsack unction in an icint way tat minimizs t numbr o rrncs to t valus array. You do not nd to worry about prcis syntax as lon as your psudocod rsmbls actual cod in a rasonabl way. You can us pairs in your psudocod, i.. data structurs o t orm (a,b). You would rr to t irst lmnt a as (a,b)[1] and t scond lmnt b as (a,b)[2] Pair Knapsack (Intr capacity, Intr n) { I n=1 I wits[1]<= capacity rturn ( {1, valus[1]) Els rturn ( {, 0 ) Intr witn = wits[n] I witn>capacity rturn Knapsack(capacity,n-1) St includd = {n Knapsack(capacity-witn, n-1) Pair willinclud = Knapsack(capacity-witn, n-1) Pair includd = ( {n willinclud[1], valus[n] + willinclud[2]) Pair xcludd = Knapsack(capacity,n-1) I Wort(xcludd) >= Wort(includd) I xcludd[2] >= includd[2] Els rturn includd ) (5 marks) Din rcursivly t worst cas cost In o your improvd Knapsack unction or n itms. Rmmbr tat you nd to provid bot t bas cas and t rcurrnc rlation Justiy your answr (i.. xplain wat ac componnt o t ormula rprsnts) I1 = 2 (wits[1] and valus[1] on lin 3 o Knapsack) In = 1 (wits[n] on lin 6 o Knapsack) + In-1 (rcursiv call or includd) + 1 (valus[n] in calculation o includd) + In-1 (rcursiv call or xcludd) = 2In-1 + 2