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90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Data Structures.

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Presentation on theme: "90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Data Structures."— Presentation transcript:

1 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Data Structures and Algorithms for Information Processing Lecture 8: Recursion

2 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Recursion We’ve seen several examples of the use of recursion We’ll take a closer look at recursion as a style of programming Lends itself to analytical methods; proving program properties

3 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Verifying Program Properties How can we be sure that a program is correct? –Debug –Test cases –Make sure output matches another program –... None of these give absolute assurance

4 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Imperative Programming The “usual” style in Java, using commands Programs are written by create data (“state”) and storing it in variables Flow of control insures that correct sequence of assignments is executed

5 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Applicative Programming No references to other objects No side effects (assignments, output...) Some advantages: –Functions only return values –No need for loops –Easier to prove properties A different programming style, and a different way to think about programming

6 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Recursion l General pattern: recursive_fn(params) { if (…) return some_value; else... recursive_fn(new_params)... } l A recursive function call is made somewhere in the body of the function

7 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Tail Recursion l General pattern: tail_recursive_fn(params) { if (…) return some_value; else return tail_recursive_fn(new_params) } Tail recursive: the function does no work after the recursive call

8 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Tail Recursion l “Usual” recursive factorial // Precondition: n >= 0 static int fact1(int n) { if (n==0) return 1; else return n*fact1(n-1); }

9 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Tail Recursion Tail recursive factorial static int fact2(int n) { // Invariant: n >= 0 return fact2_aux(n, 1); } static int fact2_aux(int n, int accum) { if (n==0) return accum; else return fact2_aux(n-1, n*accum); }

10 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Execution Trace fact1(5) 5*fact1(4) 5*4*fact1(3) 5*4*3*fact1(2) 5*4*3*2*fact1(1) 5*4*3*2*1*fact1(0) 5*4*3*2*1*1 => 120

11 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Execution Trace fact2(5) fact2_aux(5,1) fact2_aux(4,5) fact2_aux(3,20) fact2_aux(2,60) fact2_aux(1,120) fact2_aux(0,120) => 120

12 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Example of Non-Tail Recursion // Precondition: y > 0 static int mult (int x, int y) { if (y==1) return x; else return x + mult(x, y-1); } Addition operation carried out after the recursive call

13 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Tail Recursion (cont) Tail recursive functions can be more efficient Often easier to prove properties of tail recursive functions –correctness, termination, cost –technique: induction

14 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Example: fact2 Want to prove using induction that fact2(n) => n! We do this by proving an appropriate property of the auxiliary function: fact2_aux(n, p) => n! * p

15 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Example: fact2 (cont) Base case: n=0 –for all p fact2_aux(0, p) => p = 0! * p Inductive step: n > 0: –Assume true for n-1 –For all p fact2_aux(n-1, p) =>(n-1)! * p

16 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Example: fact2 (cont) Inductive step: n > 0: –Assume true for n-1 –For all p fact2_aux(n-1, p) =>(n-1)! * p –So: fact2_aux(n, p) => fact2_aux(n-1, n*p) => (n-1)! * (n*p) = (n*(n-1)!)*p = n!*p

17 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Example: fact2 (cont) Proving termination by induction: –Base case: fact2_aux(0, p) => return p –Inductive case: terminates if operator * terminates

18 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Proving Properties of Programs that use Lists List concatenation: A^B 1010 7 null hea d 1515 null 5 hea d

19 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Append Input: Two lists x and y Output: x ^ y List append (List x, List y) { if (x == null) return y; else return new List(x.value(), append(x.next(), y)); }

20 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Append (cont) Want to prove: – Terminates – Returns x^y – Cost is O(n)

21 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Termination Base case: x.length ()=0 –append (x,y) => append ( null, y) => y Inductive case: x.length ()=k-1>0 –append(x’,y) terminates for x’ with x’.length() < k –append(x.next(),y) terminates –constructor terminates

22 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Correctness of append Base case: x.length ()=0 –append (x,y) => append ( null, y) = empty list ^ y = x ^ y Inductive case: x.length ()=k-1>0 –assumption: append(x.next (), y) => (list referred to by x.next) ^ y –new List(x.value(), append(x.next(), y)) => x ^ y

23 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Cost of append Let Cost[n,m] be the number of operations for append(x,y) with x.length ()=n and y.length () = m Cost[0,m] = A (constant) For n>0, Cost[n,m] = Cost[n-1,m] + B Thus, Cost[n,m] = A + nB = O(n)

24 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Reversing Lists List reversal: Rev[x] 1010 7 null hea d 1515 7 10 null hea d 1515

25 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Reversing Lists: first implementation List reverse1 (List l) { if (l == null) return null; else { return append(reverse(l.next()), new List(l.value())); } }

26 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Cost of reverse1 Cost[n] is cost of reversing list with length n Cost[0] = A (constant) Cost[n] = B + Cost[n-1] + (n-1)C Solving the recurrence gives Cost[n] = O(n^2)

27 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Tail recursive reverse We can easily get an O(n) implementation using tail recursion: List rev2_aux(List x, List y) { if (x==null) return y; else return rev2_aux(x.next(), new List(x.value(), y)) } List reverse2(x) { return rev2_aux(x, null); }

28 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Cost of rev2_aux Let Cost[n,m] be the number of operations for rev2_aux(x,y) with x.length ()=n and y.length () = m Cost[0,m] = A (constant) n>0, Cost[n,m] = Cost[n-1,m+1] + B Thus, Cost[n,m] = A + nB = O(n)

29 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Fibonacci Numbers Want fib(0)=1, fib(1)=1, fib(n) = fib(n-1) + fib(n-2) if n>1 Simple recursive implementation: // Precondition: n>=0 i nt fib(int n) { if (n < 2) return 1; else return fib(n-1)+fib(n-2); }

30 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Fibonacci Numbers Cost is the same as the Fibonacci numbers themselves! fib(n) rises exponentially with n: –fib(n) > (1.618)^n / 2

31 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Imperative version int i=1; int a=1, b=1; while (i<n) { int c = a+b; // fib(i+1) a = b; b = c; i++; }

32 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Recursive Version Define an auxiliary function fib_aux Use two accumulator variables, one set to fib(i-1), the other to fib(i)

33 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Recursive Version int fib_aux (int n, int i, int x, int y) { if (i==n) return y; else return fib_aux(n, i+1, y, x+y); } int fib(int n) { if (n==0) return 1; else return fib_aux(n, 1, 1, 1); }

34 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Backtracking Search General pattern: Test if current position satisfies goal If not, mark current position as visited and make a recursive call to search procedure on neighboring points Exhaustive search, terminates as soon as goal is found

35 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Backtracking search: simple example The “bear game”: Start with initial number of bears Need to reach goal number within a certain number of step s Possible moves: –Add incr number of bears –Divide current number in half

36 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Backtracking search: simple example Public static boolean bears (int initial, int goal, int incr, int n) { if (initial == goal) return true; else if (n==0) return false; else if (bears(initial+incr, goal, incr, n-1)) return true; else if (initial % 2 == 0) return bears(initial/2, goal, incr, n-1); else return false; }

37 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Backtracking search: simple example Why does this program terminate? What if we remove the restriction on the number of steps?

38 90-723: Data Structures and Algorithms for Information Processing Copyright © 1999, Carnegie Mellon. All Rights Reserved. Lecture 8: Recursion Benefits of Recursion Recursion can often be implemented efficiently Requires less work (programming and computation) Tail recursive versions require less stack space The code is typically much cleaner than imperative versions

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