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1 Recurrece reltio & Recursio 9 Chpter 3 Recurrece reltio & Recursio Recurrece reltio : A recurrece is equtio or iequlity tht describes fuctio i term of itself with its smller iputs. T T The problem of size is divided ito subproblems ech of size / d the cost of dividig + combiig the solutio is / T T the cost of dividig + combiig. There re THREE phses : () Divide brek complex problem () Coquer solve recursively. (3) Combie Combied solutios of ll sub problems. Exmples: T If 4T If Or, i geerl, T T f b SP P SP where d b I the bove recurrece the whole problem of iput size is divided ito subproblems of b of dividigs d combiig the solutio is f(). Methods to solve recurrece reltio : () Bckwrd substitutio () Recurrece tree method (3) Mster method (4) Akr Bzzi method. Bckwrd substitutio methods: I this method we reduce vlue of i bckwrd wy utill the size of problem becomes. Let us cosider the followig recurrece. T T We c further expd it s follows: T SP size d the cost

2 30 3 T : : T if T Recurrece Tree Method : Three step process : () Costruct the recurrece tree () Add the cost t ech levels of tree (3) Add the cost of ll levels to get overll cost. e.g. RR give : T T T I bove rr. T T This is free from I, so we ll tret this s root of tree. root Level 0 Recurrece reltio & Recursio T() = T(/) T(/) Level Childre ccordig to the umber of sub-problems here it is sub problem. If we dd both childre d root, we will get the origil R.R. Now we hve to grow this tree. So, we will clculte vlue of child lef d grow it dowwrd. Or, T T u T T cost of dividig 4 Now, this is root c 3 it is free from T. / T(/ ) T(/ ) Now ttch this prtil tree to bove prts tree. T(/) T(/) T(/ ) T(/ ) T(/ ) T(/ ) Similrly : T T 3

3 Recurrece reltio & Recursio 3 / T(/ 3 ) T(/ 3 ) Now combie this prtil tree ito previous. / / / / / / (/ 3 ) (/ 3 ) (/ 3 ) (/ 3 ) (/ 3 ) (/ 3 ) (/ 3 ) (/ 3 )... T() T() T() T() T() T() T() T() T() T() Now we crete recurrece tree we will proceed to ext step : Step - : Add cost of ech level. / / / / / / T() T() T() T() T() T() T() T() Step - 3 : Add ll cost cos... times T t Now, how my times we will dd deped o HEIGHT OF TREE At ech level is divided ito prts i.e i level i At lst level tht is i vlue i Put o both sides we get i i i i

4 3 i = height of the tree =. so, totl cost will be T... times O Mster Method : Mster method is used for the followig recurreces of the form. T T f b the. Cse : If Recurrece reltio & Recursio where d b re costts d f() is symptoticlly positive fuctio f O b for some costt 0 the T b Cse : If f b the T b Cse 3: If f b for some costt 0, the T f Exmples: () T T 9 3 here = 9, b = 3 d f() =. 9 b 3 sice Where 9 f O 3 we c pply cse of mster method d clculte tht T. (b) T T 3 here = b = 3 f ; b 3/ 0 Cse of mster method pplies d thus the solutio to the recurrece reltio is T (c) T 3T 4 We hve 3, b 4; f 3 b O Sice f Where 0. cse 3 pplies s f b c f 4 4 There for by cse of mster theorem: 3 T for 3 c 4 3 4

5 Recurrece reltio & Recursio (d) T T Here =, b = d f b s mster theorem. (e) f is ot polyomilly lrger th. Therefore it does ot follow y cse of T T Sol. Let m d igorig the sig. m m T T m Now let T m S S m S m m the here d f b m m b m m m f m so by cse II of mster method. S m m m m T S m m Akr bzzi Method : If the recurrece hs form With T T f i b Now chgig bck. i i 0 k i 3 k... p p p p p i bi b b b3 bk where, p = row., b i p T du u f u p f f u We will cosider the equtio d fid vlue of P put it i formul, perform itegrtio d the get the vlue. e.g. 3 T T T 4 4 T T T f b b 4 b 3 b 4 i 33

6 34 b p p b 3 p 4 4 p p p 3 p 4 4 u u T du du u u u T e.g. T T T T let T x T f f f,,, 3, 5, 8,... x x x x x x x x x x x x x x 0 Recurrece reltio & Recursio x b b 4c 5 x Solutio is T c c c c T Cocept of Recurrsio: Recursio mes self-referece. A recursive fuctio is fuctio which clls itself durig its executio. It must hve followig properties. () There must be certi criteri, clled bse criteri for which fuctio does ot cll itself. (b) Ech time the fuctio cll itself (directly or idirectly), it must be closer to the bse criteri i.e. it uses the rgumets smller th the oe it ws give. A recursive fuctio with these properties is sid to be well defied. Exmples: () Fctoril fuctio:! d 0!= if 0! if

7 Recurrece reltio & Recursio (b) Computig fctoril by fuctio (Recursive) it fctoril (it ) { If ( = = 0) //Bse criteri. retur ; else retur *fctoril ( ); Iductive step How it executes if = 4. 4! = 4.3!. 3! = 3.! 3.! =.! 4.! =.0! 5. 0!= (Termitio) 6.!=. = 7.! = = 8. 3! = 3 = ! = 4 6 = 4 Fibocci Numbers: A fibocci sequece f 0, f, f, f 3... f is defied s 0,,,, 3, 5, 8, 3,... Tht is f 0 = 0, f =, f = d the subsequet terms re sum of precedig terms. f f f if if Computig th fibocci umber by recurrece. It fib (it ) { If // Bse criteri. retur ; else retur fib ( ) + fib ( ); (Iductive step) fib (5) will be computed s follows: fib(5) 35 fib(4) fib(3) fib(3) fib() fib() fib() fib() fib() fib() fib(0) fib() fib(0) fib() fib(0)

8 36 TOWER OF HANOI PROBLEMS Recurrece reltio & Recursio Iitil cofigurtio of TOH problem is : source peg PegA PegB PegC Source Peg Auxillry Peg Destitio Peg helpig uxiliry peg Fil cofigurtio of TOH problem : destitio peg e.g. For Oe disk : Iitil Stge : 3 p move : A C Fil stge : Whe = umber of movemet = For two disk : 0 p Iitil Stge : Fil stte : Moves : (i) A B (ii) A C (iii) B C So, whe = Number of movemet (3) Gol: All discs re set to Peg C from PegA by movemets of disc i other pegs. (i) Oly oe disc movemet is llowed t time.

9 Recurrece reltio & Recursio (ii) A lrger disc cot be plced over smller disc. Number of movemet of disc i tower of Hoi problem is. T = totl umebr of disc movemet Number of disc = i THP T T T T 37 Recursio or Itertio : A recursive fuctio works through the process of cllig itself util coditio is met. Itertio uses loopig cotrol structure (while, do while, for) to repet sectio of code util coditio is met.recursio is best whe you hve lot of vribles to keep trck of with ech loop, d it s usig divide d coquer type of lgorithm. Recursio helps to trim dow the frme of referece (the portio of the problem you re ow iterested i), very esily.whe we hve few vribles to keep trck of, or the lgorithm is t some type of divide d coquer, the itertive loopig is better.itertio is slightly fster, but tht should ot be the criteri used to choose it. The differece is very smll, d the clrity d shortess of the recursive code, c be quite helpful for debuggig. With itertio it is the sme vrible whose vlue is costtly chged, d you c ccess its vlue t ech stge. With recursio, differet locl vrible (by the sme me) is creted with ech cll. SOLVED PROBLEMS. Let T() be fuctio defied by recurrece [GATE - 005] T T () T (b) T (c) T Sol. T T f So, T Correct optio is () b f b (d) Noe of these. The recurrece reltio [GATE - 996] Sol. () () T T 4 T 3 (b) (c) 3 4 (d) Noe of these 3. For k for some k 0 the recurrece reltio As. T T T () (b) () (c) (d)

10 38 Recurrece reltio & Recursio 4. The ruig time of lgorithm is represeted by the followig recurrece reltio [GATE 009] 3 T T c, otherwise 3 which of the followig represets the complexity of the bove recurrece reltio. Sol. () (b) 3 T T c, otherwise 3 Solve usig mster theorem. T T 3 b 3 f So, b Correct optio is () (c) (d) T T f b 5. The recurrece reltio tht rises with the complexity of biry serch is: [JNUEE-008] Sol. () T T T T k (b) (c) T T (d) T T k For biry serch if we hve umber of elemets. So, first it sorts the rry i scedig or descedig order the divide the rry i hlf of its umber d repet it till we fid out specified umber. T T k Correct optio is (b) 6. Cosider the followig c-code. it j, ; j = ; while (j < ) j = j*; The umbers of comprisio mde i the executio of the loop for y > 0 is [GATE 007] Sol. it j, ; () (b) (c) (d) r j = ; while (j < ) j = j*; For =, J = J =

11 Recurrece reltio & Recursio For =, J = For J =, = 3, J = For J =, J = ; out of loop = 4, J = for J = For J =, J = 4; out of loop Similrly for = 00 It will execute for steps. So, optio () is true. 7. How to fid the complexity of progrm Sol. Complexity is bsic rough estimte complexity is deped o umber of loops for (i =0; i < ; i++) for (j = 0; j < ; j++) 8. Fid out the complexity of progrm. i = ; while (i > 0) { i = i/; Sol. i i i i i i i 9. Fid solutio to the followig recurrece equtio Sol. By usig substitutio method we get followig series: T k k k So put k the we get T O() 0 0. The recurrece reltio As. Sol. 39 T() T, T() [GATE-987 : Mrks] T() = T() = 3T(/4) + Hs the solutio T() equl to () O() (b) O( ) (c) O( 3/4 ) (d) oe of these [GATE-996 : Mrks] () Usig Mster Theorem 4 3 We get

12 40. Let T() be the fuctio defied by T() =, sttemets is true? Recurrece reltio & Recursio T() T for. Which of the followig () T() O( ) (b) T() = O() (c) T() = O( ) (d) oe of these [GATE-997 : Mrks] (b) As. Sol. Use Mster Theorem.. I the followig C fuctios, let m. As. Sol. it gcd (, m) { if (%m = = 0 retur m; = %m; retur gcd (m, ); How my recursive clls re mde by this fuctio? () ( ) (b) () (c) ( ) (d) [GATE-007 : Mrks] () Let T(m, ) be the totl umber of steps. So T(m, 0) = 0, T(m, ) = T(, m mod ) o verge T T(k, ) ; T (T T T ) 0 k 0 T S S (S0S S ) S (S0 S S ) ((S ) S ) S So T ( ) 0() T ( ) 3. Which oe of the followig is the recurrece equtio for the worst cse time complexity of the Quicksort lgorithm for sortig () umbers? I the recurrece equtios give i the optio below, c is costt. As. Sol. () T() T(/) c (b) T() T( ) T() c (c) T() T( ) c (d) T() T(/) c [GATE-05 (Set-) : Mrk] (b) I worst cse the pivot elemet positio my be either first or lst. I tht cse the elemets re lwys divided i : ( ) proportio. The recurrece reltio for such proportiol divisio would be T() T() T( ) O() O( ) () T() T (/) c O( ) (b) T() T( ) T() c O( ) (c) T() T ( ) c O( ) (d) T() T(/) c O()

13 Recurrece reltio & Recursio 4 PRACTICE SET. Cosider the followig recursive fuctio fu(x, y). Wht is the vlue of fu(4, 3) it fu(it x, it y) { if(x == 0) retur y; retur fu(x -, x + y); () 3 (b) (c) 9 (d) 5. Wht does the followig fuctio do? it fu(it x, it y) { if(y == 0) retur 0; retur(x + fu(x, y-)); () x+y (b) x*y (c) x+x*y (d) x^y 3. Output of followig progrm? #iclude<stdio.h> void prit(it ) { if( > 4000) retur; pritf( %d, ); prit(*); pritf( %d, ); it mi() { prit(000); getchr(); retur 0; () (b) (c) (d) Cosider the sme recursive C fuctio tht tkes two rgumets usiged it foo(usiged it, usiged it r) { if( > 0) retur(%r + foo (/r, r )); else retur 0; () 9 (b) 8 (c) 5 (d) 5. Cosider the followig fuctio double f(double x) { if(bs(x*x - 3) < 0.0) retur x; else retur f(x/ +.5/x); Give vlue q (to decimls) such tht f(q) will retur q:. ().73 (b).4 (c) 4. (d) 3.4

14 4 6. T() = T() = T(-) +, evlutes to () + (c) + (b) (d) Recurrece reltio & Recursio 7. Cosider the followig three clims. ( + k) m = Θ( m ), where k d m re costts. + = O( ) 3. + = O( ) Which of these clims re correct? () d (b) d 3 (c) d 3 (d), d 3 8. Wht is the worst cse time complexity of followig implemettio of subset sum problem. // Returs true if there is subset of set[] with sum equl to give sum bool SubsetSum(it set[], it, it sum) { // Bse Cses if(sum == 0) retur true; if( == 0 && sum!= 0) retur flse; // If lst elemet is greter th sum, the igore it if(set[-] > sum) retur issubsetsum(set, -, sum); /* else, check if sum c be obtied by y of the followig () icludig the lst elemet (b) excludig the lst elemet */ retur issubsetsum(set, -, sum) issubsetsum(set, -, sum-set[-]); () O( * ^) (b) O(^) (c) O(^ * ^) (d) O(^) ANSWER KEY. (). (b) 3. (b) 4. (d) 5. () 6. () 7. () 8. (d)