Mathematics. Programming

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1 Mathematics for the Digital Age ad Programmig i Pytho >>> Secod Editio: with Pytho 3 Maria Litvi Phillips Academy, Adover, Massachusetts Gary Litvi Skylight Software, Ic. Skylight Publishig Adover, Massachusetts

2 Skylight Publishig 9 Bartlet Street, Suite 70 Adover, MA 080 web: sales@skylit.com support@skylit.com Copyright 200 by Maria Litvi, Gary Litvi, ad Skylight Publishig All rights reserved. No part of this publicatio may be reproduced, stored i a retrieval system, or trasmitted, i ay form or by ay meas, electroic, mechaical, photocopyig, recordig, or otherwise, without the prior writte permissio of the authors ad Skylight Publishig. Library of Cogress Cotrol Number: ISBN (soft cover) ISBN (hard cover) The ames of commercially available software ad products metioed i this book are used for idetificatio purposes oly ad may be trademarks or registered trademarks owed by corporatios ad other commercial etities. Skylight Publishig ad the authors have o affiliatio with ad disclaim ay sposorship or edorsemet by ay of these products maufacturers or trademarks owers Prited i the Uited States of America

3 Recurrece Relatios ad Recursio. Prologue I Chapter we talked about differet ways to defie a fuctio: i words, with a table, with a chart, or with a formula. But we left out oe very importat method: we ca also describe a fuctio, especially a fuctio o positive (or o-egative) itegers, recursively. Here is a example. Suppose you state that a fuctio f with the domai of all positive itegers is defied as follows:, if = f( ) = f( ), if > Copyright by Skylight Publishig This defiitio does ot tell us right away how to fid the value of f ( ). f ( ) is described i terms of f( ) (except for oe simple base case, = ). Ad yet, with some work, we ca fid the value of f ( ) for ay positive. Let s see: we kow that f () =. Next: f(2) = 2 f() = 2 = 2. Next: f(3) = 3 f(2) = 3 2= 6. Next: f(4) = 4 f(3) = 4 6= 24. Ad so o. There is oly oe fuctio that satisfies the above defiitio. I this case you ca easily fid a formula to describe this fuctio: it is our old fried f ( ) =! (-factorial). Recursio is a powerful tool for defiig fuctios. The desigers of computer hardware ad programmig laguages have made sure that recursio is also supported i programmig laguages..2 Recurrece Relatios Recall that a fuctio defied o the set of all positive itegers ca be expressed as a sequece of the fuctio s values: a, a 2,..., a,... To defie such a fuctio recursively, we defie each term of the sequece, except the first (or, perhaps, the first few), through its relatio to the previous term(s). The relatio that coects the 89

4 90 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION -th term to oe or more of the previous terms is ofte the same for each. It is called a recurrece relatio. For example: a = a = a, for ay 2 We get, agai, a =!. This is the same as our recursive defiitio of!, just rewritte i a slightly differet way. It is easy to calculate o a computer the values of a sequece defied through a recurrece relatio. Recall, for example, our calculatio of s( ) = i Chapter 4. We oticed that s (0) = 0 ad s ( ) = s ( ) +, for ay 2. This led to the followig Pytho code: = sum = 0 while <= Max: sum += prit('{0:3d} {:6d}'.format(, sum)) += Copyright by Skylight Publishig Here sum stads for ( ) calculatig successive values of s ( ). s. Iteratios through the while loop are equivalet to Oe of the famous sequeces defied with a recurrece relatio is called Fiboacci umbers: f = ; f2 = f = f + f 2,if > 2. This sequece is amed after Leoardo Fiboacci (the ame meas so of Boaccio) who lived i Pisa i the 3th cetury, but the umbers were kow i Idia over 2000 years ago. Fiboacci came upo the umbers while he was modelig a populatio of rabbits uder simplistic assumptios: we start with oe adult pair; a adult pair produces a pair of baby rabbits each moth; a baby rabbit becomes a adult after oe moth; the rabbits ever die. This model results i the Fiboacci recurrece relatio for the umber of adult pairs after moths (Figure -). The sequece goes like this:,, 2, 3, 5, 8, 3, 2, 34, ad so o. f represets the umber of adult pairs of rabbits at moth. The total umber of pairs of rabbits adults ad

5 .2 ~ RECURRENCE RELATIONS 9 babies is also a Fiboacci sequece, but it starts at p = ; p2 = 2, p = p + p 2,if > 2. = = 2 = 3 = 4 = 5... Copyright by Skylight Publishig Figure -. Rabbits populatio growth i the Fiboacci model. adult pair baby pair Fiboacci umbers have may iterestig properties. For example, the sequece of f+ ratios of cosecutive Fiboacci umbers r = coverges to the golde ratio f + 5 τ =.62. Fiboacci umbers pop up i various braches of mathematics 2 ad i the atural world (Figure -2; search the Iteret for fiboacci + ature for other examples).

6 92 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION Figure -2. Broccoli Romaesco. The umbers of clusters i the spirals form Fiboacci sequeces. Photograph by Madelaie Zadik, courtesy Botaic Garde of Smith College. Copyright by Skylight Publishig Exercises. Cosider a fuctio, if = f( ) = 2 f( ) +,if > What is f (4)? 2. A fuctio o positive itegers is described by, if = f( ) = f( ) + 2,if > Redefie it by a simple o-recursive formula. 3. Defie f( ) = 2 recursively, i terms of f( ), for all itegers.

7 .3 ~ RECURSION IN PROGRAMS Write a recursive defiitio of a fuctio f ( ) that has the followig properties: f() = 0, f(2) = 30, ad the values f(), f(2),..., f( ),... form a arithmetic sequece. 5. The same as Questio 4, but for a geometric sequece. 6. Describe the fuctio from Questio with a simple formula, without recursio. 7. Defie recursively, without factorials, the fuctio itegers 3.! f( ) = ( 3)! for 8. Give a =, a = a + for ay 2 o-recursive formula i terms of., describe a with a simple 9. Write a Pytho program that prompts the user to eter a positive iteger ad prits the first Fiboacci umbers. Copyright by Skylight Publishig 0. Fid a sequece that satisfies the Fiboacci equatio a = a + a 2,if > 2 ad, at the same time, is a geometric sequece with a =. How may such sequeces are there?.3 Recursio i Programs Now back to our favorite, s( ) = As we have see, the recursive, if = defiitio of s( ) is s ( ) =. Now suppose, just out of curiosity, s( ) +,if > we implemet this defiitio directly i a Pytho fuctio: def sumton(): if == : retur else: retur sumton(-) +

8 94 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION Will this work? What will sumton(5) retur? As you kow, i a program, a fuctio traslates ito a piece of code that is callable from differet places i the program. The caller passes to the fuctio some argumet values ad a retur address the address of the istructio to which to retur cotrol after the fuctio fiishes its work. So how ca this work whe a piece of code passes cotrol to itself? Wo t the program get cofused? Wo t Pytho choke tryig to swallow its ow tail? Ad yet it works! Try it: >>> sumton(5) 5 Because recursio is such a useful tool i programmig, the developers of Pytho (ad other programmig laguages) have made special efforts to esure it works. There are also CPU istructios that facilitate implemetig recursive calls. Copyright by Skylight Publishig Here is a allegory for what happes. Imagie a fuctio as a mail-order bakery. To place a order you take a empty box, put your writte order ito it ad also put i a label with your address, ad ship it to the bakery. The baker, call him Frak, receives the box, reads the order (the argumets you pass to the fuctio), bakes the cake you ordered, puts it i the box, attaches your label, ad ships the box back to you. Now suppose Frak has received a order ad is workig o it. Meawhile a ew order arrives. Frak immediately drops what he is doig, saves the ufiished cake (the values of the local variables) i the box the order came i, the puts the ew box that has just arrived o top of the oe he was workig o, opes the ew box, ad starts workig o that. Meawhile a ew order arrives. Frak immediately drops what he is doig, saves the ufiished cake (the values of the local variables) i the box the order came i, the puts the ew box that has just arrived o top of the oe he was workig o, opes the ew box, ad starts workig o that. Meawhile... You get the picture. The orders keep comig ad the boxes keep pilig up: ow there is a whole stack of them. Fially, at some poit, Frak gets a break: the orders stop comig. Frak fiishes the top order ad ships it to the customer. He the fiishes the oe that was just below it ad ships that to the customer. Ad so o, util he gets to the bottom of the stack, to the last remaiig order, fiishes that order, ad goes home. You might be woderig: Why is Frak processig orders i this fuy sequece? The order that came i last is completed first. Would t it be more fair to do it i a first-i-first-out maer? I a momet you will see why Frak does it his way.

9 .3 ~ RECURSION IN PROGRAMS 95 Oe day, Frak received a order for a multi-layered weddig cake with four layers. The layers i such a cake are basically the same, oly each layer is smaller i diameter tha the oe beeath it. Frak started workig o the bottom layer, but bega to worry that he d get cofused i all the layers. Luckily, a fried of his (a computer programmer) came up with this fabulous idea: why does t Frak sed a order to himself for a multi-layered cake that cosists of the three top layers, ad the, whe the three-layer cake is ready, just put it o top of the layer he is workig o? Frak does t have to ship his order very far, of course, but he still follows his usual procedure: takes a empty box, writes out a order for a three-layer cake, writes a label with the retur address (his ow) ad seds i the order (by puttig the box o the top of the stack). The as usual, he opes the box ad sees: aha, a order for a three-layer cake! So he writes himself a order for a two-layer cake. You get the picture. Fially Frak receives (his ow) order for a sigle-layer cake. He fiishes it quickly ad ships it back (to himself, of course). He receives it, uses it to complete the two-layer cake, ad ships that back (to himself). Receives it, completes the three-layer cake, ships back, receives that, completes the four-layer cake, ad fially ships it to the customer. Due to his LIFO (last-i-first-out) method, Frak utagles the whole layered busiess without ay cofusio. Copyright by Skylight Publishig This seems like a lot of shippig back ad forth, but it is easy for a computer. Pytho allocates a special chuk of memory, called a stack, for recursive fuctio calls. The CPU has the push istructio that puts a data item oto the top of the stack. The complemetary pop istructio removes ad returs the top item. I Pytho, Frak s activity may be represeted as somethig like this (assumig the + operator is properly defied for cakes): def bakesiglelayeredcake(d): 'Bakes a simple cake of diameter d' retur Cake(d) def bakemultilayeredcake(): 'Bakes a cake with layers' d = 3 * # the diameter of the bottom layer i iches cake = bakesiglelayeredcake(d) # Base case: == -- do othig # Recursive case: if > : cake += bakemultilayeredcake( - ) retur cake For a recursive fuctio to work, it must have a simple base case, where recursive calls are ot eeded.

10 96 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION I the above code, == is the base case. Whe a fuctio is called recursively, it must be called for smaller ad smaller tasks, which evetually coverge to the base case (or oe of the base cases). Each time we call the fuctio bakemultilayeredcake(-) recursively, we call it with a argumet that is smaller by tha the previous time. Recursio has its cost: first, you eed to have eough space o the stack ad, secod, the code speds extra time pushig ad poppig data ad callig the same fuctio. Iteratios are ofte more efficiet. But some applicatios, especially those dealig with ested structures or brachig processes, require a stack ayway, ad recursio allows you to write very cocise ad clear code. Example Write a recursive fuctio that calculates 0 for a o-egative iteger. Copyright by Skylight Publishig Solutio def pow0(): if == 0: # base case retur else: retur pow0(-) * 0 Example 2 Write a recursive fuctio that returs the sum of the digits i a o-egative iteger. Solutio def sumdigits(): s = % 0 # Base case: < 0 -- do othig # Recursive case: if >= 0: s += sumdigits( // 0) retur s

11 .3 ~ RECURSION IN PROGRAMS 97 Example 3 Write a recursive fuctio that takes a strig ad returs a reversed strig. Solutio def reverse(s): if le(s) > : s = reverse(s[:]) + s[0] retur s The base case here is implicit: whe the strig is empty or cosists of oly oe character, there is othig to do. A very popular example of recursio i computer programmig books ad tutorials is the Towers of Haoi puzzle. We have three pegs ad disks of icreasig size. Iitially all the disks are stacked i a tower o oe of the pegs (Figure -3). The objective is to move the tower to aother peg, oe disk at a time, without ever placig a bigger disk o top of a smaller oe. Copyright by Skylight Publishig Figure -3. The Towers of Haoi puzzle The key to the solutio is to otice that i order to move the whole tower, we first eed to move a smaller tower of - disks to the spare peg, the move the bottom disk to the destiatio peg, the move the tower of - disks from the spare peg to the destiatio peg (Figure -4).

12 98 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION Start: Move a smaller tower (recursio) : Move oe disk: Move a smaller tower (recursio) : Figure -4. A recursive solutio to Towers of Haoi Copyright by Skylight Publishig Example 4 Write a recursive fuctio that solves the Towers of Haoi puzzle for disks. Solutio def movetower(disks, frompeg, topeg): sparepeg = 6 - frompeg - topeg if Disks > : movetower(disks -, frompeg, sparepeg) prit('from ' + str(frompeg) + ' to ' + str(topeg)) # This code does ot model disk movemets # -- it just prits out each move if Disks > : movetower(disks -, sparepeg, topeg) If we wat to move a tower of, say, seve disks from peg to peg 2, a iitial call to movetower would be movetower(7,, 2).

13 .3 ~ RECURSION IN PROGRAMS 99 Note that i the above example the fuctio does ot retur a value (more precisely, it returs Noe), so it is a recursive procedure. Programmig this task without recursio would be rather difficult. Exercises. Write ad test a recursive fuctio that returs!. Do ot use ay loops. 2. I Sectio 5.3 Questio 7, you created a fuctio ittobi() that takes a o-egative iteger ad returs a strig of its biary digits. For example, ittobi(5) returs '0'. Rewrite this fuctio recursively, without ay loops. Hit: implemet two base cases: == 0 ad == : they cover the situatios whe has oly oe biary digit. 3. Write ad test a fuctio def pritdigits(): Copyright by Skylight Publishig that displays a triagle with rows, made of digits. For example, pritdigits(5) should display The fuctio s code uder the def lie should be three lies. Hit: prit *str() prits times. 4. The same as Questio 3, but ivert the triagle, so that pritdigits(5) prits Explai the statemet sparepeg = 6 - frompeg - topeg i Example 4.

14 200 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION 6. Idetify the base case i the movetower fuctio i Example Without ruig the code i Example 4, determie the output whe movetower(3,, 2) is called. 8. The Towers of Haoi puzzle was iveted by the Frech mathematicia Edouard Lucas i 883. The leged that accompaied the puzzle stated that i Beares, Idia, there was a temple with a dome that marked the ceter of the world. The Hidu priests i the temple moved golde disks betwee three diamod eedles. God placed 64 gold disks o oe eedle at the time of creatio ad the uiverse will come to a ed whe the priests have moved all 64 disks to aother eedle. What is the total umber of moves eeded to move a tower of 2 disks? 3 disks? 4 disks? disks? Assumig oe move per secod, estimate the lifespa of the uiverse..4 Mathematical Iductio Copyright by Skylight Publishig Let d be the umber of moves required to move a tower of disks i the Towers of Haoi puzzle. It is pretty obvious that, for >, d = d + + d = 2d +, because to move a tower of disks, we eed to first move a tower of - disks, the oe disk, the agai a tower of - disks. If you fiished the last exercise, you probably guessed correctly that d = 2. How ca we prove this fact more rigorously? I questios of this kid we ca use the method of proof called mathematical iductio. Suppose you have two sequeces, { a } ad { b } a. = b (base case).. Suppose that: 2. For ay > you have maaged to establish the followig fact: if a = b the a = b. Is it true the that a = b for all? Let s see: we kow that a = b. We kow that if a = b the a2 = b2. So we must have a2 = b2. Next step: we kow that a 2 = b 2. We kow that if a 2 = b 2 the a3 = b3. So we must have a3 = b3. Next step:... ad so o. We have to coclude that a = b for all. I the future, we do ot have to repeat this chai of

15 .4 ~ MATHEMATICAL INDUCTION 20 reasoig steps every time we just say our coclusio is true by mathematical iductio. Example Prove that the umber of moves i the Towers of Haoi puzzle for disks d = 2. Solutio d satisfies the relatioship d =, d = 2d + for > We wat to show that d = 2 for all. Copyright by Skylight Publishig. First let s establish the base case, =. d = ad 2 =, so OK. d = Now let s proceed with the iductio step. Assume that d = 2. The d = 2d + = 2(2 ) + = 2 2+ = 2. So, for ay >, we were able to show that if d = 2 the 2 d =. By mathematical iductio, d = 2 for all, Q.E.D. The method of mathematical iductio applies to ay sequece of propositios { P }. Suppose we kow that P is true; suppose we ca show for ay > that if P is true the P is true. The, by mathematical iductio, P is true for all. (I the above example, P is the propositio that d = 2.)

16 202 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION Sometimes it is ecessary to use a more geeral versio of mathematical iductio, with several base cases ad a more geeral iductio step. Suppose that:. P,..., P k are true (base case(s)). 2. For ay > k we ca prove that if P, P2,..., P are all true, the P is true. The, by mathematical iductio, P is true for all. Example 2 Prove that the -th Fiboacci umber f Solutio. We ca establish two base cases: Copyright by Skylight Publishig f = = = f = = Now the iductio step. Assume that for ay m< particular, for > 2, f ad f f m 3 2. The m 2. I f = f + f 2 + = + = > = So the propositio is true for the two base cases, ad if it is true for ad 2 the it is also true for. By math iductio, it is true for all, Q.E.D.

17 .4 ~ MATHEMATICAL INDUCTION 203 If you fiished Questio 9 from Sectio.2, you probably tried to prit the first 00 Fiboacci umbers ad got f 00 = The above result explais why Fiboacci umbers grow so fast. Exercises 2. Prove usig mathematical iductio that (2 ) =. 2. Cosider the followig fuctio: def tagle(s): = le(s) if < 2: retur '' # empty strig else: retur tagle(s[:]) + s[:-] + tagle(s[0:-]) Show that tagle('abcde') returs a strig that cotais either 'a' or 'e'. Copyright by Skylight Publishig 3. Show that for a strig s of legth, tagle(s), defied i Questio 2, returs a strig of legth 2 L =. Hit: first obtai a recurrece relatio for L, the prove it usig mathematical iductio. 4. Suppose you have straight lies i a plae, such that o two lies are parallel to each other ad o three lies go through the same poit. Show that the umber of regios ito which these lies cut the plae depeds oly o, but ot o a particular cofiguratio of the lies. Fid a formula for that umber ad prove it usig mathematical iductio. Hit: whe you add a lie, the existig lies cut it ito pieces. Each of these pieces cuts a regio ito two, addig a ew regio. 5. Cosider the sequece a0 = 2, a = 2, a = 2a + 3a 2 for 2. Write a Pytho program that prits out the first terms of this sequece. Examie the patter ad come up with a o-recursive formula for a, the prove that your guess is correct usig mathematical iductio. Hit: compare a to 3.

18 204 CHAPTER ~ RECURRENCE RELATIONS AND RECURSION 6. Cosider a recursive versio of the fuctio that returs the -th Fiboacci umber: def fiboacci(): if == or == 2: retur else: retur fiboacci(-) + fiboacci(-2) Test it with = 6 ad = 20. Now test it with = 00. The computer wo t retur the result right away. Perhaps the recursive versio takes a little loger... Prove by mathematical iductio that the total umber of calls to fiboacci, icludig the origial call ad all the recursive calls combied, is ot less tha the -th Fiboacci umber f. We have show above that 2 3 f. Assumig that your computer ca execute oe trillio calls per 2 secod, estimate how log you will have to wait for fiboacci(00) (i years). Press Ctrl-C to abort the program. As we said earlier, sometimes recursio ca be costly. Copyright by Skylight Publishig Terms itroduced i this chapter: Recurrece relatio Fiboacci umbers Golde ratio Recursio Recursive fuctio Recursive procedure Base case Mathematical iductio.5 Review Pytho feature itroduced i this chapter: Recursive fuctios