Jonathan Turner Exam 2-10/28/03
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1 CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm v only i v h lry lo wo hilrn. How o hi hng lr h lmm hown low (hi lmm i rom h nlyi o h running im o Fioni hp)? Explin your nwr. Lmm. L x ny no in n Fhp. L y,...,y r h hilrn o x, in orr o im in whih hy wr link o x (rli o l). Thn, rnk(y i ) i or ll i. Th inquliy in h lmm om rnk(y i ) i. Sin y i h h m rnk x whn i m hil o x n x mu hv h l i hilrn h im, y i mu hv h rnk o l i whn i m hil o x. Sin i ill i hil o x, i n hv lo mo wo hilrn in h im, o i rnk mu l i. L S k h mll poil numr o nn h no o rnk k h, in our moii vrion o Fioni hp. Giv ruriv lowr oun on S k. Th i, giv n inquliy o h orm S k (S,S,..., S k ) whr i om union o h S i or i<k. Clrly S =, S = n S =. For k>, w n u h moii lmm o onlu h S + S + S + + S. No h h irn wn h oun or S k n or S k i S k. k k U hi o giv lowr oun on h mll numr o nn h no wih rnk n hv. From h ov, w hv S +S =, S +S =, S +S =9, S 9+S.
2 . ( poin) Conir n xuion o h rhir hor ph lgorihm on h grph hown low, uming h h our vrx i. Wh vri r nn in p o h lgorihm? Wh vri r nn in p? P? P? (Rll h p n r h our vrx i nn h ir im n or j>, p j n whn ll h vri on h quu h n o h prviou p hv n nn). Vri n in p, vri,,, n in p, vri,,,, n in p, vri,,, n in p. Eg wih lngh How mny im i h g o h orm ( i, j ) xmin? im h. Explin how o gnrliz hi xmpl o how h hr r grph on n vri or whih h rhir nning lgorihm k Ω(n ) im. Th xmpl n gnrliz y xning h hin l o k vri n inring h iz o h ompl ipri ugrph righ o k vri. Th k vri o h orm i will h hv n g o vrx, wih ring lngh, in h ov xmpl. Thi will u h g in h ipri ugrph o xmin k im. Sin h grph h k vri n hr r k g in h ipri ugrph, h ol numr o im n g i xmin will Ω(k )=Ω(n ).
3 . ( poin) Th igur low how grph in whih w hv u h g lngh rnormion ri in h no o limin ngiv g lngh. Th in u o ompu h rnormion r hown long wih h rnorm g lngh. Wh i h originl lngh o h ph? Th lngh in h rnorm grph i. To hi, w n ur o g h originl lngh, o h originl lngh w. Wh i h hor ph rom o? Wh i i lngh (uing h originl g lngh)? Th hor ph i. I lngh in h rnorm grph i, o i originl lngh i.
4 . ( poin) L G ir grph in whih h g h om poiiv piy (or h g wigh) n l ign inion vrx. L T rvr olnk ph r wih inion. Th i, T i pnning r o G ir owr in whih ll ph hv h lrg olnk piy o ny ph joining h m wo npoin (rll h h olnk piy o ph i h piy o i mll piy g). L p(u) o h ir vrx (r u) on h r ph rom u o. L onb(u) h ir vrx (r u) on on ph rom u o. Mor prily, i v i onb(u), hn v p(u), (u,v) i n g n hr i no ohr uh vrx w p(u), or whih h olnk ph rom u o h r wih w i h lrgr piy hn h olnk ph rom u o h r wih v. Fill in h oy o h C++ progrm hown low o ompu onb(u) or ll vri u. Your progrm houl run in O(m) im, whr m i h numr o g in G. voi onb(wigrph G, vrx p[], in [], vrx []) { // Rurn h on nx hop vrx o u in [u] or // ll vri u. p[u] i h prn o u in h hor // ph r n [u] i h piy o h ph // rom u o h inion. Th inion i vrx n. vrx u, v; g ; in p; or (u = ; u!= G.n; u++) { [u] = Null; p = ; or ( = G.irou(u);!=Null; = G.nxou(u,)){ v = G.h(); i (v!= p[u] && min(g.w(),[v]) > p) { [u] = v; p = min(g.w(),[v]); } } } } Dri in wor, how you oul gnrliz hi, o llow on o ily rmin h hir ph o, h ourh,. Wh i h running im or hi lgorihm? W oul o hi y oring h jny li, o h or vrx u, h g h r h olnk piy ph om ir, hn h g h r h on, hn h g h r h hir, n o orh. In gnrl, n g (u,v) will pr n g (u,w) i min(p(u,v),(v))>min(p(u,w),(w)). Uing n O(log n) lgorihm o or h o h jny li giv running im o O(m log n).
5 . ( poin) An inn o h low grph ruur wih our n ink i hown low. Th low hown i vli low irin irou 9 9 g nx l p low hl hnx h 9 Wh h mgniu o h low, hown? Fill in h g o h riul grph low, inluing hir pii. Show how h low grph ruur hng whn w ur h ph,,,,. You my ju mrk h hng on h igur ov. irin irou 9 9 nx l g p low hl hnx h 9
6 . ( poin) Th igur low how n inn o h mx low prolm wih low prou y h piy ling vrion o h ugmning ph lgorihm n inrmi g in h ompuion., piy, low,,,,,,,,,,, Drw h l riul grph, R or hi low ( =). Drw R h r o h nx ling ph ( =).
7 . ( poin) Th igur low how n inrmi g in h xuion o h miil ph lgorihm wih ynmi r. Moiy h low vlu on h grph o h hy rl h urrn low (inluing h low ror impliily in h ynmi r ruur)., 9, h,,,,,,,,,,,, g,,,,,, i H H h H H g i Explin wh hppn in h nx p o h lgorihm. Vrx i rll o, h g nring in h ynmi r ruur r rmov n h low o rom g o n h low o rom o r ror in h low grph ruur.
8 . ( poin) Th igur low how n inrmi g in h xuion o h gnrl prlowpuh lgorihm. In h wor, how mny mor p r n or h lgorihm rmin? (Aum h whnvr hr i hoi o puh low o h l or h righ, h lgorihm puh low o h righ.) piy, low,,,,,,,, in ll, x W woul ir rll o n n on uni o low k o, hn rll o n n low o, hn rll o n n low k o hn, hn rll o n n low o, hn rll o n n low o, hn rll o n n low k o, hn, hn rll o n n low o. So h ol numr o p i. Wh woul h nw in ll, i you rompu hm hi poin? (Ju how h in ll on h igur.) Th ll or, n woul, n rpivly. How mny mor p r n i you run h lgorihm rom hi, uing h moii in ll?
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