11/28/18 FIBONACCI NUMBERS GOLDEN RATIO, RECURRENCES. Announcements. Announcements. Announcements

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1 Fiboncci (Leonrdo Pisno) 0-0? Sttue in Pis Itly FIBONACCI NUERS GOLDEN RATIO, RECURRENCES Lecture CS0 Fll 08 Announcements A: NO LATE DAYS. No need to put in time nd comments. We hve to grde quickly. No regrde requests for A. Grde bsed only on your score on bunch of sewer systems. Plese check submission guidelines crefully. Every mistke you mke in submitting A slows down grding of A nd consequent dely of publishing tenttive course grdes. Announcements Announcements Finl is optionl! As soon s we grde A nd get it into the CMS, we determine tenttive course grdes. You will complete ssignment Accept course grde? on the CMS by Wednesdy night. If you ccept it, tht IS your grde. It won t chnge. Don t ccept it? Tke finl. Cn lower nd well s rise grde. More pst finls re now on Exms pge of course website. Not ll nswers yet. Course evlution: Completing it is prt of your course ssignment. Worth % of grde. Must be completed by Sturdy night. DEC We then get file tht sys who completed the evlution. We do not see your evlutions until fter we submit grdes to to the Cornell system. We never see nmes ssocited with evlutions. Fiboncci function fib(0) 0 fib() fib(n) fib(n-) + fib(n-) for n 0,,,,,, 8,,, In his book in 0 titled Liber Abci Hs nothing to do with the fmous pinist Liberci But sequence described much erlier in Indi: Virhṅk Gopl before Hemcndr bout 0 The so-clled Fiboncci numbers in ncient nd medievl Indi. Prmnd Singh, 98 pdf on course website Fiboncci function (yer 0) fib(0) 0 fib() fib(n) fib(n-) + fib(n-) for n /** Return fib(n). Precondition: n 0.*/ public sttic int f(int n) { if ( n < ) return n; return f(n-) + f(n-); 0,,,,,, 8,,,, We ll see tht this is lousy wy to compute f(n)

2 Golden rtio Φ ( + )/.8098 Φ ( + )/ Divide line into two prts: Cll long prt nd short prt b 0,,,,,, 8,,,, fib(n) / fib(n-) is close to Φ. So Φ * fib(n-) is close to fib(n) Use formul to clculte fib(n) from fib(n-) b ( + b) / / b Solution is the golden rtio, Φ In fct, /b 8/. /8. /. /.9 /. limit f(n)/fib(n-) Φ n -> See webpge: Golden rtio nd Fiboncci numbers: inextricbly linked Golden rtio Φ ( + )/ Fiboncci, golden rtio, golden ngle 0 Find the golden rtio when we divide line into two prts nd b such tht ( + b) / / b Φ Golden rectngle limit f(n)/fib(n-) golden rtio n -> /b 8/. /8. /. /.9 /. 0,,,,,, 8,,,, b 0/ golden ngle For successive Fiboncci numbers, b, /b is close to Φ but not quite it Φ. 0,,,,,, 8,,,,, Fiboncci function (yer 0) The Prthenon Downloded from wikipedi Golden rectngle Fiboncci tiling Fiboncci spirl 0,,,,,, 8,,,

3 fiboncci nd bees Drwing golden rectngle with ruler nd compss 0 Mle bee hs only mother Femle bee hs mother nd fther The number of ncestors t ny level is Fibonnci number b golden rectngle How to drw golden rectngle 8 : mle bee, : femle bee Fiboncci in Pscl s Tringle hypotenuse: (* + (½)(½)) (/) Suppose you re plnt You wnt to grow your leves so tht they ll get good mount of sunlight. You decide to grow them t successive ngles of 80 degrees Pretty stupid plnt! The two bottom leves get VERY little sunlight! p[i][j] is the number of wys i elements cn be chosen from set of size j Suppose you re plnt Fiboncci in nture 8 You wnt to grow your leves so tht they ll get good mount of sunlight. 90 degrees, mybe? Where does the fifth lef go? The rtichoke uses the Fiboncci pttern to spirl the sprouts of its flowers. 0/(golden rtio).9 The rtichoke sprouts its lefs t constnt mount of rottion:. degrees (in other words the distnce between one lef nd the next is. degrees). Recll: golden ngle topones.weebly.com//post/0/0/the-rtichoke-nd-fiboncci.html

4 9 Blooms: strobe-nimted sculptures 0 Uses of Fiboncci sequence in CS Fiboncci serch Fiboncci hep dt strcture Fiboncci cubes: grphs used for interconnecting prllel nd distributed systems Fiboncci serch of sorted b[0..n-] Fiboncci serch history binry serch: cut in hlf t ech step 0 e n 0 e e e e e (n-0)/ e (e-0)/ 8 89 Fibonncci serch: (n ) cut by Fiboncci numbers 0 e 0 e e e e e e 0 + Dvid Ferguson. Fibonccin serching. Communictions of the ACM, () 90: 8 Wiki: Fiboncci serch divides the rry into two prts tht hve sizes tht re consecutive Fiboncci numbers. On verge, this leds to bout % more comprisons to be executed, but only one ddition nd subtrction is needed to clculte the indices of the ccessed rry elements, while clssicl binry serch needs bit-shift, division or multipliction. If the dt is stored on mgnetic tpe where seek time depends on the current hed position, trdeoff between longer seek time nd more comprisons my led to serch lgorithm tht is skewed similrly to Fiboncci serch. Fiboncci serch LOUSY WAY TO COMPUTE: O(^n) Dvid Ferguson. Fibonccin serching. This flowchrt is how Ferguson describes the lgorithm in this -pge pper. There is some English verbige but no code. Only high-level lnguge vilble t the time: Fortrn. /** Return fib(n). Precondition: n 0.*/ public sttic int f(int n) { if ( n < ) return n; Clcultes f() 8 times! return f(n-) + f(n-); Wht is complexity of f(n)?

5 Recursion for fib: f(n) f(n-) + f(n-) Recursion for fib: f(n) f(n-) + f(n-) T(0) T(n): Time to clculte f(n) T() Just recursive function T(n) + T(n-) + T(n-) recurrence reltion We cn prove tht T(n) is O( n ) It s proof by induction. Proof by induction is not covered in this course. But we cn give you n ide bout why T(n) is O( n ) T(n) < c* n for n > N T(0) T() T(n) + T(n-) + T(n-) T(0) * 0 T() * T(n) < c* n for n > N T() + T() + T(0) + * + * 0 <rithmetic> * () <rithmetic> * Recursion for fib: f(n) f(n-) + f(n-) Recursion for fib: f(n) f(n-) + f(n-) 8 T(0) T() T(n) T(n-) + T(n-) T(0) * 0 T() * T() * T(n) < c* n for n > N T() + T() + T() + * + * <rithmetic> * () <rithmetic> * T(0) T() T(n) T(n-) + T(n-) T(0) * 0 T() * T() * T() * T(n) < c* n for n > N T() + T() + T() + * + * <rithmetic> * () <rithmetic> * Recursion for fib: f(n) f(n-) + f(n-) Recursion for fib: f(n) f(n-) + f(n-) 9 0 T(0) T() T(n) T(n-) + T(n-) T(0) * 0 T() * T() * T() * T() * * WE CAN GO ON FOREVER LIKE THIS T(n) < c* n for n > N T() + T() + T() + * + * <rithmetic> * () <rithmetic> T(0) T() T(n) T(n-) + T(n-) T(0) * 0 T() * T() * T() * T() * T(n) < c* n for n > N T(k) + T(k-) + T(k-) + * k- + * k- <rithmetic> * ( + k- + k- ) <rithmetic> * k

6 Cching As vlues of f(n) re clculted, sve them in n ArryList. Cll it cche. When sked to clculte f(n) see if it is in the cche. If yes, just return the cched vlue. If no, clculte f(n), dd it to the cche, nd return it. Must be done in such wy tht if f(n) is bout to be cched, f(0), f(), f(n-) re lredy cched. Cching /** For 0 n < cche.size, fib(n) is cche[n] * If fibcched(k) hs been clled, its result in in cche[k] */ public sttic ArryList<Integer> cche new ArryList<>(); /** Return fiboncci(n). Pre: n > 0. Use the cche. */ public sttic int fibcched(int n) { if (n < cche.size()) return cche.get(n); if (n 0) { cche.dd(0); return 0; if (n ) { cche.dd(); return ; int ns fibcched(n-) + fibcched(n-); cche.dd(ns); return ns; Liner lgorithm to clculte fib(n) Logrithmic lgorithm! /** Return fib(n), for n > 0. */ public sttic int f(int n) { if (n < ) return ; int p 0; int c ; int i ; // invrint: p fib(i-) nd c fib(i-) while (i < n) { int fibi c + p; p c; c fibi; i i+; return c + p; f 0 0 f f n+ f n+ + f n 0 0 k 0 f n f n+ 0 f n 0 f n+ f n+ f n+ f n+k f n+k+ f n f n+ f n+ f n+ f n+ f n+ Logrithmic lgorithm! Another log lgorithm! f 0 0 f f n+ f n+ + f n 0 k f n f n+ f n+k f n+k+ Define φ ( + ) / φ ( - ) / The golden rtio gin. 0 k f 0 f f k f k+ You know logrithmic lgorithm for exponentition recursive nd itertive versions Gries nd Levin Computing Fiboncci number in log time. IPL (October 980), 8-9. Prove by induction on n tht fn (φ n - φ n ) /