Recursion. recursion. (now rare or obsolete) A backward movement, return. The Shorter Oxford English Dictionary

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1 Recursion recursion. (now rare or obsolete) A backward movement, return. The Shorter Oxford English Dictionary A recursive routine is one that calls itself. Familiar examples of recursion: recurrence relations recursive structures COMP1120: DA Lecture 6: Recursion

2 Recurrence Relations factorial f (0) = 1 f (n) = f (n 1) n 1,1,2,6,24,120,... triangular numbers Fibonacci F(1) = 1 F(2) = 1 F(n) = F(n 1) + F(n 2) 1,1,2,3,5,8,13,... t(1) = 1 t(n) = t(n 1) + n 1,3,6,10,15,... COMP1120: DA Lecture 6: Recursion

3 Some familiar recursive structures: Recursive Structures Stephen = Elise families Freda Elvis = Buffy Alan Bob = Anne natural numbers Charles = Diana Bill = Hillary Kate Greg Megan fractals Winston George Zain = Carol Mick Freya Pat COMP1120: DA Lecture 6: Recursion

4 Recursive Routines Sum of first n numbers (triangular numbers): sum(n: INTEGER): INTEGER is local i : INTEGER from Result := 0 i := 1 until i > n loop Result := Result + i i := i + 1 sum(n: INTEGER): INTEGER is if n = 1 then Result := 1 else Result := sum(n 1) + n COMP1120: DA Lecture 6: Recursion

5 Recursive Routines (continued) Factorial of n: fact (n: INTEGER):INTEGER is local i : INTEGER from i := 1 Result := 1 until i > n loop Result := Result i i := i + 1 fact (n: INTEGER):INTEGER is if n = 0 then Result := 1 else Result := fact (n 1) n COMP1120: DA Lecture 6: Recursion

6 Routine Execution a is b(3,7) b(x,y:integer) is c(x) c(y:integer) is local x:integer x := 2 y call c: x=6 c(3) y=3 b: x=3 b(3,7) y=7 a: Execution Stack COMP1120: DA Lecture 6: Recursion

7 Factorial Execution fact (n: INTEGER):INTEGER is if n = 0 then Result := 1 else Result := fact (n 1) n call fact(0) fact(1) fact: n=0 Result=1 fact: n=1 Result=1 fact: return fact(2) n=2 Result=2 fact: 2 io. put integer( fact (3)) n=3 Result=6 COMP1120: DA Lecture 6: Recursion

8 Application of Recursion Recursion is just another tool in your toolbox right times to use it when it s simpler to code when it s simpler to understand when it fits the problem wrong times to use it when it s more complex to code when it esn t really apply COMP1120: DA Lecture 6: Recursion

9 What Does This Code Do? f (n: INTEGER): INTEGER is if n = 1 then Result := 99 elseif n.even then Result := f (n //2) else Result := f(3 n+1) Does it terminate? (This is the Lothar Collatz problem) COMP1120: DA Lecture 6: Recursion

10 Linear Search Revisited search(x: BOOK; c: ARRAY[BOOK]; i: INTEGER) is if i > c.upper then found := False elseif c.item( i ) = x then found := True index := i else search(x, c, i+1)... search(b, library, library.lower) COMP1120: DA Lecture 6: Recursion

11 Binary Search Revisited search(x: T; c : ARRAY[T]; lo, hi : INTEGER) is local mid: INTEGER if lo + 1 >= hi then found := c. valid index (lo ) and then c.item(lo) = x index := lo else mid := ( lo+hi ) // 2 if c.item(mid) <= x then search(x, c, mid, hi) else search(x, c, lo, mid)... search(t, ts, ts.lower, ts.upper+1) COMP1120: DA Lecture 6: Recursion

12 Reasoning With Recursive Routines Iterative: invariant variant termination condition is the invariant maintained? Recursive: postcondition variant (on each call) base case(s) it s induction es the termination condition and invariant imply postcondition? COMP1120: DA Lecture 6: Recursion

13 Reasoning With the Factorial Function Induction: general case: base: es P(b) hold? step: es P(k) imply P(k + 1) For factorial: base: es fact(0) = 0! hold? step: es fact(k) = k! imply f act(k + 1) = (k + 1)!? variant: n fact (n: INTEGER):INTEGER is if n = 0 then Result := 1 else Result := fact (n 1) n ensure Result = n!... io. put integer( fact (3)) COMP1120: DA Lecture 6: Recursion

14 Family Reunion Invitations reunion(p: PERSON) is reunion for nuclear family invite (p) if p.has partner then invite (p.partner) if p.has children then reunion(p.son) reunion(p.daughter) COMP1120: DA Lecture 6: Recursion

15 Family Reunion (continued) reunion(mary) numbered invitations for a nuclear family 2 1 John = Mary 5 3 Jim = Kate John = Hong Glynis = Brian Geoff Sandy = Tom Mary = Paul Sam Helen Michael Mary COMP1120: DA Lecture 6: Recursion

16 Just For Fun If you re interested... Type in the program on slide 9. Try it out on a few examples. Perhaps even call it with every possible natural number. (at least those that Eiffel can represent) What have you noticed? What if you change the function f(3 n+1) to be f(3 n 1)? What if you change 99 to another value? COMP1120: DA Lecture 6: Recursion