University of Waterloo CS240 Winter 2018 Assignment 4 Due Date: Wednesday, Mar. 14th, at 5pm

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Alexander Owen
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1 Univesit of Wateloo CS Winte Assinment Due Date: Wednesda, Ma. th, at pm vesion: -- : Please ead the uidelines on sumissions: ~cs//uidelines.pdf. This assinment contains itten questions and a poammin question. Sumit ou itten solutions electonicall as a PDF ith the file named ap.pdf usin MakUs. We ill also accept individual question files named aq.pdf, aq.pdf,..., aq.pdf if ou ish to sumit questions as ou complete them. Fo Question, sumit ou files quadteecompession.h and quadteecompession.cpp ased on the skeleton files povided on the e pae. Polem [++ maks] a) Give the contents of the hash tale that esults hen ou inset items ith the kes I, E, A, T, P, R, S, L, Y, G in that ode into an initiall empt tale of size M =. Use doule hashin, ith the hash functions iven elo: c A E G I L P R S T Y ASCII(c) h (c) = ASCII(c) mod h (c)=(( ASCII(c)) mod )+ 9 Fo full cedit it suffices to ive the final coect anse, ut e ecommend shoin some intemediate aas to have a chance of pat-maks in case of eos. ) Find one chaacte c {A,..., Z} that is diffeent fom all the ones in pat (a) such that insetin c into ou esult fom pat (a) ith the same hash-function ould lead to failue to inset (i.e., to e-hashin). Biefl justif ou anse. Hint: Make use of the fact that the tale-size M is not ell-chosen fo doule-hashin. c) Suppose ou have an implementation of doule-hashin (not ith the aove hashin functions). Unfotunatel thee is a u in ou doule-hashin code such that one o oth of the hash functions alas etun the same value x >. Descie hat happens in each of these situations: (i) h has the u, (ii) h has the u, (iii) oth h and h have us.

2 You should comment on hat the esultin poe sequence ill e and ho lon ou ould expect that an Inset ill take. (You need not pove the un-time of Inset fomall, ut compae its ehaviou to hashin methods that e have seen in class.) You ma assume that M and the hashin-functions ae suitale fo doule-hashin. In paticula the load facto α is kept small (α < ), M is a pime, h (k) {,..., M }, and h and h hash unifoml if the don t have us. Polem [++ maks] a) Give the contents of the hash tale that esults hen ou inset items ith the kes I, E, A, T, P, R, S, L, Y, G in that ode into an initiall empt tale of size M =, usin cuckoo hashin and the folloin hash functions (h is the same as in the pevious question): c A E G I L P R S T Y ASCII(c) h (c) = ASCII(c) mod h (c) = ( ASCII(c)) mod 7 Fo full cedit it suffices to ive the final coect anse, ut e ecommend shoin some intemediate aas to have a chance of pat-maks in case of eos. ) Find one chaacte c {A,..., Z} that is diffeent fom all the ones in pat (a) such that insetin c into ou esult fom pat (a) ith the same hash-function ould lead to failue to inset (i.e., to e-hashin). (The chaacte need not e the same one as in Q().) Biefl justif ou anse. c) Suppose that ou find that ou hash tale contains the folloin items: 7 9 S I L E R G Y A T P You kno that this hash tale as otained insetin the kes ith the aove hash functions h and h. You also kno that no othe ke as eve inseted, and no ke as eve deleted. But ou do not kno in hich ode the insetions happened. Give thee distinct easons (involvin diffeent kes) h this hash tale could not possil have een otained via cuckoo hashin, no matte hat the insetion ode as. Polem [++ maks] a) Conside the set of points in Fiue. Da the coespondin kd-tee, folloin the conventions fom the couse slides, and statin splittin alon a suitale x-coodinate. You should sho oth the splittin lines that ae inseted into the pictue and the coespondin tee.

3 9 7 p p p p p 9 p p p p p p p A p p 7 p p 7 9 Fiue : A set of points, and the que-ectanle A. ) Pefom a ane-que fo the ectanle A indicated in Fiue, hich is ectanle [.,.7] [, ] (folloin the pseudo-code on Module slide ). You should sho ho the alothm opeates indicatin (in the tee that ou ceated in pat (a)) all nodes v such that kdtee-raneseach(v, A) as called on v. (We call such a node a a node.) c) Define a point set P fo n = k as follos: P = {p,..., p n }, and fo l =,..., n p l has -coodinate l If l = i n + j, then p l has x-coodinate j n + i. (Fiue ives the point-set fo k =.) Sho that thee exists a que ectanle A such that no point of P lies in A, ut thee ae Ω( n) a nodes duin the ane-que fo A. (Motivation: Slide of module aued that the nume of a nodes is O( n), so the pupose of this assinment is to aue that this ound is tiht. You do not need to kno the poof of the O( n) ound to do this assinment.) Polem [7 maks] We ae iven a set W of n of indos on the compute sceen S (i.e., axis-paallel ectanles in the plane). A ne indo pops up and e ish to find all indos in W that ae completel coveed the ne indo. Give an aloithm that finds all these indos in O((lo n) c + s) time, hee s is the nume of indos that ae found, and c is a constant. The modules have not een finalized et and slide-numes ma chane ±.

4 Fomall, each indo is descied via a -tuple (x l, x,, t ) ith x l < x and < t, hich coesponds to the ectanle [x l, x ] [, t ]. If the ne indo is W = (x l, x,, t), ou que should etun in O((lo n) c ) time all those indos W = (x l, x,, t ) in W that satisf [x l, x ] [x l, x ] and [, t ] [, t]. To make this possile, ou ill need to assume that W is stoed in a suitale data stuctue. Descie hat ou ae usin. This data stuctue should take at most O(n lo c n) space, and one should e ale to uild it in O(n lo c n) time. Polem [+(+) maks] One of the applications of quadtees is compession of pictues. The pictue is ecusivel divided in quadants until the entie quadant is of the same colou. a) (Witten) Usin this ule, ive the quad-tee that epesents the eion [, ) [, ) of the fiue iven elo. To explain the dain conventions, e ive ou elo the quad-tee fo the eion [, ) [, ) of the fiue. (You ma omit laels NE, NW, etc., especiall nea leaves.) An unlaelled pixel is hite ( ). [,) [,) l l ) (Poammin) l l l [,) [,) NE NW [,) [,) [,) [,) Wite a poam that convets a iven (lack-and-hite) pictue into a quadtee that epesents it as descied in pat (a), and vice vesa. (Details ae elo.) c) (Bonus) Comment on the effectiveness of this method of compessin pictues. To suppot ou auments, povide expeimental esults (usin ou on implementation) that compae A and T fo vaious pictues. Comin up ith methods of eneatin pictues and hat values of n to test ae pat of the question.

5 Details: On the e pae, ou ill find a stu-file quadteecompession.h and quadteecompession. Implement stin * aatotee(int n, int * A). Hee n is a poe of, aa A has size n and each ent is o. The function etuns a efeence to a stin of a quad-tee that epesents the same pictue. Implement int * teetoaa(int n, stin * T). Hee n is a poe of, T is the stin of a quad-tee that stoes a pictue of size n n, hee all leaves ae o. The function etuns an aa that epesents the same pictue. A -aa A[..n ] stoes a lack-and-hite pictue as follos: Fo x, < n, ent A[indexOf(n, x, )] stoes the colo of the pixel hose ottom-left cone is (x, ), Hee indexof is a function povided in the stu. We use the convention that epesents hite and epesents lack. A stin T epesents a quadtee havin a chaacte fo each node in pe-ode. A chaacte i means an intenal node, a chaacte means a leaf that is lack and a chaacte means a leaf that is hite. Stin T lists the nodes in pe-ode, i.e., it fist has the chaacte of the oot, then ecusivel the chaactes of the NE-child, the chaactes of the NW-child, the chaactes of the -child, and the chaactes of the -child. Fo example, if n =, then T = (i,, i, i,,,,,,,,, ) epesents the folloin quad-tee (and pictue): [, ) [, ) 7 NE NW lack [, ) [, ) lack hite lack hite [, ) [, ) lack hite hite lack hite With the aove conventions and fo fixed n, one can aue that an aa A detemines a unique stin T and vice vesa (ou need not pove this). Fiuin out ho to do the convesion eteen them is pat of ou assinment. The stus contain #ifndef SCRIPTS and #endif at vaious places. Do not emove these. Anthin that is suounded #ifndef SCRIPTS and #endif ill not e tanslated ou testin scipts, and ou can add testin-outines ithin them. The stu-file contains some pintin outines that ou ma use, ut do not have to use. You can add othe functions as ou see fit. You should not use (o need) an liaies eond the ones povided in the stu-files. 7

EE 168 Handout #32 Introduction to Digital Image Processing March 7, 2012 HOMEWORK 7 SOLUTIONS

EE 168 Handout #32 Introduction to Digital Image Processing March 7, 2012 HOMEWORK 7 SOLUTIONS EE 168 Handout #32 Intoduction to Diital Imae Pocessin Mach 7, 2012 HOMEWORK 7 SOLUTIONS Polem 1: Colo Wheels We can epesent an N x N colo imae y a thee-dimensional aay such that the fist two dimensions