Final Exam : Solutions

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1 Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b ihr n or n +. A w ar auming ha n = k, hi man h mdian i ihr (i) k or (ii) k. In h ormr ca vry numbr bwn 0 and k mu b prn in h li; hu h miing numbr i largr han k. W hn rcur on h numbr in h li largr han h mdian o which hr ar k. In h lar ca vry numbr bwn k + and n = k mu b prn in h li; hu h miing numbr i mallr han k. W hn rcur on h numbr in h li mallr han h mdian, o which hr ar k. In h ba ca n = hr ar wo poibl numbr ha can appar bu only on do. Thu w ind h miing numbr in O() im. (b) Th running im i T (n) = T ( n ) + O(n) wih ba ca T (c) = O(), or c. Nx l vriy ha h addiiv rm i indd O(n). Obrv ha w can ind h mdian in O(n) im by h mhod n in cla. W can hn pli h li in wo group, on coniing o numbr mallr han h mdian and h ohr coniing o numbr largr han h mdian in O(n) im. Finally w chck whhr h mdian i h mallr or largr o h wo poibilii (.g. n or n + in h ir p) in O() im; hi anwr drmin which ubproblm o nx conidr. So w hav a =, b = and d =. Thu, a < b d and w ar in Ca i) o h Mar Thorm. Thror h running im i T (n) = O(n d ) = O(n)

2 . Daa Srucur. (a) W add ach nw lmn a a la a h nd o h hap, and u h hapiy-up opraion o ror h hap ordr.

3 (b) So w iniially plac all h ky in a compl binary r. Clarly h hap propry i no aiid. W amp o ix hi rom h lav up. Th ubr rood a h lav aiy h hap propry a hy coni o a ingl nod ach. So w may bgin by looking a h ubr rood a parn o h lav, namly h nod wih ky 7 and. Th ubr rood a 7 aii h hap propry o w do no chang i. Th ubr rood a do no aiy h hap propry. So w apply hapiy-down o ix hi. So w wap wih i mall child. Finally w conidr h ubr rood a h nod wih ky 6, ha i h whol r. Thi r do no aiy h hap propry. So w apply hapiy-down o ix hi. So w wap 6 wih i mall child, and hn wap 6 wih i nw mall child. Th hap propry now hold or h nir r. No: hi giv a dirn hap han h on ound in (a). Morovr, a n in cla, h run im i O(n) rahr han O(n log n).

4 (c) Aum h li ar rom mall o larg. Sor h mall im in ach li a ky in a hap o iz k. I ak O(k) im o build h hap. W hn xrac h minimum rom h hap. Thi i h minimum o all n im. I hi numbr cam rom li l hn w inr ino h hap h nx numbr in li l. Each o h opraion ak O(log k) im, a h hap conain k nod. W rpa hi n im o cra h nir ord li. So h runim i O(k + n log k) = O(n log k). (Th corrcn o hi algorihm i vidn.)

5 . Grdy Algorihm (a) Th rvr-dl algorihm conidr h dg in dcraing ordr o co and oupu {d9, d8, d7, d6, d, d, k, k, d0, k9, d8, k7, k6, k, k} Hr d man ha h dg i dld and k man ha h dg i kp. For xampl, dg 9 i hn rmovd a i i in a cycl; dg 8 i rmovd a i i par o a cycl in G {9}, c. Th inal r i hown abov. (b) Th mo xpniv dg in a cycl δ(s) i no in vry minimum panning r. (I h dg co ar diinc hn i no in h uniqu MST.) (c) Th nw dg = (i, j) cra a cycl C = Pij whn addd o T, whr Pij i h uniqu pah bwn i and j in T. L b h mo xpniv dg on Pij. I c c hn i h mo xpniv dg in h cycl C. Thu, by b), i i no in h MST. So T i ill h MST. I c < c hn i h mo xpniv dg in C and T + i h nw MST. Sinc T ha xacly n dg (and n vric) i ak O(n) im o ind Pij and h mo xpniv dg on Pij. So w can ind h nw MST in im O(n).

6 . Dynamic Programming. (a) W will hav a ubproblm or ach pair i j coniing o valu in h inrval {v i, v i+,..., v j, v j }. L Π(i, j) b h maximum guarand proi h ir playr o mov can mak in hi ubproblm. Thn w obain h rcurrnc: Π(i, j) = max ( v i Π(i +, j), v j Π(i, j ) ) wih ba ca Π(i, i) = v i or all i n. To hi i corrc noic ha h ir movr can lc ihr v i or v j. Bu ar doing o h i hn h cond-movr in h ruling ubgam. Thu rom hn on h will pay h ohr playr h rcuriv oluion o Π(i +, j) or Π(i, j ), rpcivly. (b) Thr ar n ubproblm and ach rquir ha w look up h oluion o ub-ubproblm. Thu, h rcurrnc can b olvd in O(n ) im. 6

7 . Maximum Flow. (a) Th algorihm ak p. Th ridual graph or ach o h our p ar hown blow. In p (i) h ridual graph G i ju G il. A hor augmning pah i hn {, a, d, }. W choo P = {, a, d, } and augmn i by i bolnck valu b(p ) = min(,, ) =. Th ruling low and h nw ridual graph ar hown in (ii). Th hor augmning pah i P = {, a,,, } wih b(p ) = min(,,, ) =. Th ruling low and h nw ridual graph ar hown in (iii). Th hor augmning pah i P = {, b, c,,, } wih b(p ) = min(,,,, ) =. Th ruling low and h nw ridual graph ar hown in (iv). Th hor augmning pah i P = {, b, c, d, a,, } wih b(p ) = min(,,,,,, ) =. Th ruling low i hown in (v) bu now hr ar no pah in h ridual graph o w op. Th low w hav ound ha valu. (i) (0,) a (0,) (0,) (0,) b (0,) (0,) c (0,) d (0,) (0,) (0,) (0,) a d b c (ii) (,) a (,) (iii) (0,) (,) (0,) a (,) b (0,) (0,) (0,) (,) b (0,) c (0,) (0,) c (0,) d (,) d(0,) (,) (0,) (0,) (0,) (,) (,) a d a d b c b c (iv) (,) a (,) (,) (,) b (0,) (,) c (,) d (,) (0,) (,) (,) a d b c 7

8 (v) (,) a (0,) (,) (,) b (,) (,) c (,) d (,) (0,) (,) (,) a d b c (b) Conidr h ridual graph in p (v). W g uck bcau hr ar dircd pah rom only o {b, c, d}. So h minimum cu i δ + (S) whr S = {, b, c, d}. Thi cu conain h arc {(, a), (c, ), (d, )}. Th arc hav capacii,,, rpcivly. So h minimum cu ha valu. Thu, by h maxlow-mincu horm hi cu i a criica ha h low w ound in (a) i maximum. (c) Whn h Ford-Fulkron algorihm g uck hn l T b h o vric in h ridual graph ha hav dircd pah o. Thn S = V T ha h dird propry. 8

9 6. Sarching or Rar Elmn L z = x b h mall numbr in h array. W wan o know i z i rar. A h array i ord, i hr ar k copi o Z hn hy mu b in poiion o k o h array. So w can aily drmin k in O(log n) im by binary arch. I k n 00 hn z i rar and w op. Ohrwi k > n and z 00 i no rar. W hn conidr h ub-array coniing o numbr {x k +, x x +,..., x n }. Bcau z wa no rar hi li ha a mo n n = 99 n numbr. W l z = x k + b h mall numbr in h array and w whhr z i rar by h mhod abov. I i i rar w ar don. n I i i no rar, hn w rcu on a li ha ha a la wr numbr. 00 Thror w can rcur a mo 00 im bor h li i mpy. Thu our running im i O(00 log n) = O(log n), a dird. Clarly, hi algorihm ihr ind a rar lmn or hr ar no rar lmn. 9

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