Problems and Solutions
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1 Problems ad Solutos Let P be a problem ad S be the set of all solutos to the problem. Deso Problem: Is S empty? Coutg Problem: What s the sze of S? Searh Problem: fd a elemet of S Eumerato Problem: fd S Lous D. Nel 007 COMP 40 Aalyss of Algorthms
2 Problem Populatos ad Searhg Let P be a problem that a be desrbed by a populato of thgs, ad S be the set of all solutos to the problem. Lear Searh: fd those elemets of the populato that satsfy some property. e.g. fd all the studets the lass whose frst ame s Joh Subset Searh: fd those subsets of the populato that satsfy some property. e.g. fd all the subsets of studets the lass who have bee a vehle together at the same tme. Combato Searh: fd those ordergs of the populato that satsfy some property. e.g. fd all the ordergs of studets the lass suh that o two adjaet studets have the same frst ame. Lous D. Nel 007 COMP 40 Aalyss of Algorthms
3 Exhaustve Searhes Problems o populatos of sze a usually be solved wth a exhaustve searh. Lear Searh: fd those elemets of the populato that satsfy some property. e.g. fd all the studets the lass whose frst ame s Joh There are ombatos to searh lear searh Subset Searh: fd those subsets of the populato that satsfy some property. e.g. fd all the subsets of studets the lass who have bee a vehle together at the same tme. There are ombatos to searh Combato Searh: fd those ordergs of the populato that satsfy some property. e.g. fd all the ordergs of studets the lass suh that o two adjaet studets have the same frst ame. There are! ombatos to searh Lous D. Nel 007 COMP 40 Aalyss of Algorthms 3
4 Effey Here are three possble deftos of solvg a problem effetly Solvg the problem wthout u-eessary waste Solvg the problem faster tha a exhaustve searh ould. Solvg the problem polyomal-tme: a amout of tme that a be bouded by some polyomal of the populatos sze. e.g. T the tme to solve a problem of sze s at most p for some polyomal p a a - - a - -. a 0 Sgfae s that ths would be faster tha exhaustve searh of subsets or ordergs! Lous D. Nel 007 COMP 40 Aalyss of Algorthms 4
5 Bg Oh Notato Let T be the tme t taes a algorthm to solve a problem of sze. T s Of f there exsts ostats ad 0 suh that for > 0. T f I other words: f s a upper boud o the tme requred to solve problems of sze greater tha 0. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 5
6 Bg Omega Notato Let T be the tme t taes a algorthm to solve a problem of sze. T s Ωf f there exsts ostats ad 0 suh that for > 0. T f I other words: f s a lower boud o the tme requred to solve problems of sze greater tha 0. Lower bouds are muh harder to prove tha upper bouds Lous D. Nel 007 COMP 40 Aalyss of Algorthms 6
7 Bg Oh Notato -Goals We wat to get the tghtest upper boud we a. We would ot say T s O f we a say, or prove, that T s Olog or eve better T s O. We wat to always gve the smplest boud we a. We would ot say T s O beause s O. Proof: < so < for some, >. hee s O Whe we say T s Of we mea the worst ase tme to solve ay problem of sze, eve though there may be some problems of sze that are easer to solve. Oh-otato gves a pessmst upper boud. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 7
8 Bg Oh Notato -Goals We are usually terested determg f T s oe of the followg. O Olog O Olog O O ostat tme logarthm tme lear tme super lear tme quadrat tme polyomal tme O expoetal tme O! super-expoetal tme Lous D. Nel 007 COMP 40 Aalyss of Algorthms 8
9 Bg Omega Notato -Goals We wat to get the tghtest lower boud we a. We would ot say T s Ω f we a say, or prove, that T s Ωlog or eve better T s Ω. We wat to always gve the smplest boud we a. We would ot say T s Ω beause s Ω. Proof: > so > for some, >. hee s Ω Whe we say T s Ωf we mea that some problems of sze wll requre at least f tme to solve, though there mght be some problems of sze that a be solved more quly. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 9
10 Mathematal Rules for Bg Oh otato f s Oaf for ay ostat a > 0. If f g ad g s Oh, the f s Oh. If f s Og ad g s Oh, the f s Oh. f g s Omaxf,g. If g s Oh, the fg s Of h. If g s Oh, the fg s Ofh. If f s a polyomal of degree d.e. f a 0 a... a d d, the f s O d. x s Oa for ay fxed x > 0 ad a >. log x s Olog for ay fxed x > 0. logx s O y for ay fxed ostats x > 0 ad y > 0. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 0
11 Useful Log ad Expoetato Idettes log log b b b a / log a a b a b log a log a log log b a log a b b b b log a a a a b b a a b b / b b a log b log log b a b a log a /log b b b Lous D. Nel 007 COMP 40 Aalyss of Algorthms
12 Useful Summato Seres / / / 3 3 / 6 0 a a / a for a > log log log 0 log 0 a / a for 0 < a < 0 Useful Growth Futos: log! s O log ad Ω log Lous D. Nel 007 COMP 40 Aalyss of Algorthms
13 Bubble Sort Java t umbers[] { 43, 5, 67, 3,, 54, 89, 345, 45, 4, 67, 45, 457, 0; //Dsplay the put fort 0; <umbers.legth; System.out.prtumbers[] ", "; System.out.prtl""; //Bubble Sort fort 0; <umbers.legth-; fort j 0; j< umbers.legth--; j fumbers[j] > umbers[j]{ t temp umbers[j]; umbers[j] umbers[j]; umbers[j] temp; //Dsplay the result fort 0; <umbers.legth; System.out.prtumbers[] ", "; System.out.prtl""; 43, 5, 67, 3,, 54, 89, 345, 45, 4, 67, 45, 457, 0, 0, 3, 4, 5,, 43, 45, 45, 54, 67, 67, 89, 345, 457, Press ay ey to otue... Lous D. Nel 007 COMP 40 Aalyss of Algorthms 3
14 4 Aalyss of Algorthms COMP 40 Lous D. Nel 007 Bubble Sort Aalyss 0 0 j T / O s T So For >
15 Bg Oh of a Polyomal Futo Let 3 T a b d For teger values a,b,,d Clam: T s O 3 Proof: 3 T < a b d < a b d a b d 3 3 for > whh s, by defto O 3 Ths a be geeralzed to a polyomal of ay degree Lous D. Nel 007 COMP 40 Aalyss of Algorthms 5
16 publ lass Mystery { Mystery Method Java publ stat vod mysteryt[][][] a{ fort 0; <a.legth-; fort j ; j< a.legth; j fort 0; < j; { a[][j][] *j*; System.out.prtl "," j "," ": " a[][j][]; publ stat vod mastrg args[] { t umbers[][][]; t 0; // s problem sze. 0 s just a example umbers ew t[][][]; mysteryumbers; //ed ma Lous D. Nel 007 COMP 40 Aalyss of Algorthms 6
17 7 Aalyss of Algorthms COMP 40 Lous D. Nel 007 Aalyss of Mystery Java 0 0 j j C T //Based o des java ode 0 j j C T se 0 j j j C j j C T j j [ C T C T [ ] 6 3 [ 3 C T
18 Aalyss of Mystery Java T C [ ] C [ ] C 3 [ ] 3 3 C 3 [ C 3 [ 3 ] whh s 6 O 3 ] Lous D. Nel 007 COMP 40 Aalyss of Algorthms 8
19 Bary Searh Java prvate stat boolea searht[] array, t elemet, t start, t ed{ //searh for elemet sorted array a betwee start dex ad ed dex //bass ase: fstart > ed retur false; fstart ed retur array[start] elemet; //reursve ase t mdpot start ed/; farray[mdpot] elemet retur true; farray[mdpot] < elemet retur searharray, elemet, mdpot, ed; else retur searharray,elemet, start, mdpot-; publ stat boolea searht [] array, t elemet{ //searh for elemet the sorted array a retur searharray, elemet, 0, array.legth-; Lous D. Nel 007 COMP 40 Aalyss of Algorthms 9
20 Aalyss of Bary Searh T T T T 4 Fxed amout of wor plus ew problem half as bg T T T... T log log T log log 0 for > So T s Olog Lous D. Nel 007 COMP 40 Aalyss of Algorthms 0
21 Lear Reurso Java prvate stat t sumt[] array, t start, t ed{ //sum the elemets from start to ed //bass ase: fstart > ed retur 0; //reursve ase retur array[start] sumarray, start, ed; publ stat t sumt [] array{ //sum the elemets of array retur sumarray, 0, array.legth; Lous D. Nel 007 COMP 40 Aalyss of Algorthms
22 Aalyss of Lear Reurso T T T T Fxed amout of wor plus ew problem oe smaller T T 3 T... T 0 T 0 Bass ase s problem of sze 0 for > So T s O Lous D. Nel 007 COMP 40 Aalyss of Algorthms
23 Paldrome Test publ stat boolea spaldromestrg s { //aswer whether Strg s s a paldrome //.e. reads the same bawards ad forwards: "level", "v" //BASIS CASES f s.legth < retur true; //RECURSIVE STEP f s.harat0 s.harats.legth - { retur spaldromes.substrg, s.legth - ; else retur false; publ stat vod mastrg args[] { Strg s "v"; Strg s "toyota"; System.out.prtls " s paldrome: " spaldromes; System.out.prtls " s paldrome: " spaldromes; //ed ma Lous D. Nel 007 COMP 40 Aalyss of Algorthms 3
24 4 Aalyss of Algorthms COMP 40 Lous D. Nel 007 Aalyss of spaldrome T T T T 0, ; O s T So / / / T 4 4 T T / / / 4 4
25 A Better Paldrome Test prvate stat boolea spaldromestrg s, t start, t ed { //aswer whether Strg s s a paldrome //.e. read the same bawards ad forwards: "level", "v" //BASIS CASES f start > ed retur true; //RECURSIVE STEP f s.haratstart s.harated { retur spaldromes, start, ed-; else retur false; publ stat boolea spaldromestrg s{ retur spaldromes, 0, s.legth-; Lous D. Nel 007 COMP 40 Aalyss of Algorthms 5
26 Aalyss of a better spaldrome T T ; T, T 0 T 4 T 6 T / So T s O Exerse: Show ths geeralzes for ay stuato where a problem s made smaller by a fxed amout. That s, for ay ostat ad, the reurree relato T T s O Lous D. Nel 007 COMP 40 Aalyss of Algorthms 6
27 Solvg Two Half Problems prvate stat t sumt[] array, t start, t ed{ //sum the elemets from start to ed //bass ase: fstart ed retur array[start]; fstart > ed retur 0; //reursve ase t mdpot started/; retur sumarray, start, mdpot sumarray, mdpot, ed; publ stat t sumt [] array{ retur sumarray, 0, array.legth-; Lous D. Nel 007 COMP 40 Aalyss of Algorthms 7
28 Aalyss of Lear Reurso T T / ; T 0 T T T 4 T 8... T.... log T log 0 log log So T s O Exerse: Show ths geeralzes for ay stuato where you solve d problems of sze /d Lous D. Nel 007 COMP 40 Aalyss of Algorthms 8
29 Fboa Numbers Java Fb Fb Fb ; Fb0 Fb publ stat log fboat { //aswer the 'th fboa umber //BASIS CASES f<0{ System.out.prtl"ERROR:; System.ext-; f 0 retur ; f retur ; //RECURSIVE STEP retur fboa- fboa-; Lous D. Nel 007 COMP 40 Aalyss of Algorthms 9
30 Aalyss of Fboa T T T ; T 0 T T T 3 T T T 3 T 3 T 4 T 3 4 3T 3 T T 4 T 5 T 4 7 5T 4 3T 5 What s the patter?? Lous D. Nel 007 COMP 40 Aalyss of Algorthms 30
31 Alteratve Aalyss of Fboa T T T ; T 0 T T T T T 4 T T T 6 T T 3 / T 0 0 / T?? T C So s T s Ω ad O? Lous D. Nel 007 COMP 40 Aalyss of Algorthms 3
32 Faster Fboa publ lass LogDuple { //objet represetg a par of log s publ log value, value; publ LogDuplelog l, log l{ value l; value l; prvate stat LogDuple fastfboalog { //BASIS CASES f 0 retur ew LogDuple,0; f retur ew LogDuple,; f < - {System.out.prtl"ERROR "; System.ext0; //RECURSIVE STEP LogDuple prevous fastfboa-; retur ew LogDupleprevous.value prevous.value, prevous.value; publ stat log fboat { LogDuple result fastfboa; retur result.value; Lous D. Nel 007 COMP 40 Aalyss of Algorthms 3
33 Aalyss of Faster Fboa T T ; T, T 0 T... T T So T s O You deftely do ot wat to ome up wth a expoetal tme algorthm to solve a problem that a be solved lear tme!! Lous D. Nel 007 COMP 40 Aalyss of Algorthms 33
34 Dffult Problems Suppose your boss ass you to wrte a effet algorthm to solve the Travelg Salesma problem. Travelg Salesma Problem: INSTANCE: A fte set of tes, a postve teger dstae betwee eah par of tes. OBJECTIVE: Fd a tour of all the tes suh that the dstae travelled s mmum. More formally, Istae: C {,,..., dstaes d, j > 0, j... Objetve: fd a orderg:,,..., > that mmzes d, d < Lous D. Nel 007 COMP 40 Aalyss of Algorthms 34
35 Two dfferet deftos of effey What does effet mea? Pratal: A effet algorthm s oe that does ot use muh more tme ad memory tha s eessary. If your algorthm rug tme T s Of ad the problem s Ωf we a all the algorthm optmal. Theoretal: A effet algorthm s oe that a solve a problem of sze polyomal tme. T s Of for some polyomal f For example: 3 T a b That s, t does ot tae expoetal tme, or worse to solve the problem d Lous D. Nel 007 COMP 40 Aalyss of Algorthms 35
36 The Problems All Admt a Expoetal Tme Soluto You a solve the travelg salesma problem, or ay other problem you eouter ths ourse, expoetal, or worse, tme as follows. Geerate all possble ordergs of the tes. Add the dstaes for eah possble tour ad remember the best oe. Ths algorthm would be O! whh s worse tha expoetal tme ad so s ot theoretally effet. Suppose you aot ome up wth a better algorthm, what are you gog to tell your boss? Lous D. Nel 007 COMP 40 Aalyss of Algorthms 36
37 What to tell your boss? Opto I foud a effet algorthm Yeah! Opto Prove the problem s tratable o effet algorthm a exst. That s, the problem has a expoetal tme lower boud. Opto 3 Say to your boss: I guess I m just to dumb. Opto 4 Gve evdee that the problem mght be tratable. Show that f you ould solve ths problem effetly, you ould the solve other problems that people have thus far foud very dffult. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 37
38 Travelg Salesma Problem To date obody has ome up wth a effet algorthm to solve the travelg salesma problem. To date obody has bee able to prove that the problem s tratable has a expoetal lower boud They have gve evdee that the problem mght be hard by mappg t to the famous hamltoa yle problem whh s osdered to be hard. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 38
39 Hamltoa Cyle Problem Istae: Gve a Graph GV,E wth set of vertes V ad set of edges E. Questo: Do the edges of the graph allow a tour of the graph? NO YES Lous D. Nel 007 COMP 40 Aalyss of Algorthms 39
40 Hamltoa Cyle Problem Formal Istae: G V, E V { v, v,..., v, E { e, e Questo: Is there a orderg,..., e m < v v,..., v, > suh that v, v E, for... ad v, v E Lous D. Nel 007 COMP 40 Aalyss of Algorthms 40
41 Clam: TSP s at least has hard at HC Clam: Travelg salesma problem s at least as hard as the Hamlto Cyle Problem. Proof: We wll show that f you ould somehow solve the Travelg Salesma problem effetly, the you would also be able to solve the Hamlto Cyle Problem effetly Proof Idea: Buld a stae of the Travelg Salesma problem out of a Hamlto Cyle problem ad show that the aswer to the Travelg Salesma problem the provdes a aswer for the Hamlto Cyle problem But: we must show that the mappg s effet we get to use at most a polyomal amout to tme to do the mappg of the problem ad the aswer. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 4
42 A Travelg Salesma Algm. ould solve Hamlto Cyle HC Istae: Is there a orderg suh that G V, E V { v, v,..., v, E { e, e < v v,..., v, >,..., e m v, v E, for... ad v, v E New TS Problem C { v, v,..., v d v, v f j v, v j E ad d v, v j f v, v j E Clam: If the mmum legth tour foud for the TS problem s <, the G has a Hamlto Cyle, othewse t does ot. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 4
43 A Travelg Salesma Algm. ould solve Hamlto Cyle Justfato: The oly way for a tour to have a legth of s f t uses oly dstaes that orrespod to edges of the graph G. Otherwse the legth of the tour wll be greater tha. If G has o Hamltoa yle there a be o tour of legth. Importae: If you foud a effet algorthm to solve the Travelg Salesma problem, you ould use t to solve the Hamlto Cyle Problem. So the Travelg Salesma problem s at least as hard as the Hamlto Cyle Problem whh may beleve s hard. Lous D. Nel 007 COMP 40 Aalyss of Algorthms 43
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