Algorithms and Data Structures

Transcription

1 Recursion Page 1 - Recursion Dr. Fall 2008

2 Recursion Page 2 Outline Introduction Simple Problems Time Complexity and Optimization Advanced Problems

3 Recursion Introduction Page 3 Outline Introduction Simple Problems Time Complexity and Optimization Advanced Problems

4 Recursion Introduction Page 4 Introductory Example Suppose you are responsible for alarming a large group of people in case of an emergency One phone call takes at least 1 minute All phone numbers are publicly known Each individual in the group of size n has a unique identification number i {1,..., n} Your identification number is 1 Everybody is reachable and willing to help How do you organize the alarm most efficiently?

5 Recursion Introduction Page 5 Introductory Example Informal solution: ALARM if the group is empty do nothing else do 1.Divide the group in two parts of equal size (±1) 2.Call a person in the first group a) Inform him/her about the emergency b) Ask him/her to ALARM the rest of the group 3.Call a person in the second group a) Inform him/her about the emergency b) Ask him/her to ALARM the rest of the group

6 Recursion Introduction Page 6 Introductory Example Pseudo-code solution: algorithm alarm(x,y) // x is supposed to alarm x+1 to y print "x is alarmed" if x = y then // no one is left break elseif y x = 1 then // only y is left alarm(y,y) else m (x + y + 1)/2 alarm(x + 1,m) alarm(m + 1,y)

7 Recursion Introduction Page 7 Introductory Example alarm(1,13) 1min. 2min. alarm(3,5) alarm(2,7) alarm(6,7) alarm(9,11) alarm(8,13) 1min. 2min. 1min. 2min. 1min. 2min. 1min. 1min. 2min. alarm(12,13) 1min. max. alarm time alarm(4,4) alarm(5,5) alarm(7,7) alarm(10,10) alarm(11,11) alarm(13,13) Maximal alarm time: t(n) 2( log 2 n 1)min. O(log n) n = 13 t(n) 2 3 = 6min. n = t(n) 2 9 = 18min. n = t(n) 2 19 = 38min.

8 Recursion Introduction Page 8 Recursive Algorithm Recursive Algorithm Algorithm that uses itself as part of the solution Recursive Call A method call in which the method being called is the same as the one making the call Direct Recursion Recursion in which a method directly calls itself Indirect Recursion Recursion in which a chain of two or more method calls returns to the method that originated the chain, e.g. A calls B, B calls C, and C calls A

9 Recursion Introduction Page 9 Example 1: Factorial (Recursion) Compute the factorial of integer n: { 1 n = 1 n! = 1 2 n = n (n 1)! n > 1 algorithm factorial(n) if n = 1 then return 1 // base case of the recursion else return n factorial(n 1) // recursive call

10 Recursion Introduction Page 10 Example 1: Factorial (Recursion) The execution of a recursive algorithm uses a call stack to keep track of previous method call n = 1 factorial(1) Returns 1 n = 2 factorial(2) Returns 2 n = 3 factorial(3) Returns 6 factorial(4) n = 4 Returns 24

11 Recursion Introduction Page 11 Remarks Each recursion must have a base case to stop Each recursive call must reduce the problem size, i.e. leading inescapably to the base case A recursion is called tail-recursive, if the recursive call is always the last executed instruction Each recursive algorithm can be transformed into an iterative algorithm (and vice versa) trivial for tail-recursions easy for so-called augmented recursions not so easy in general (requires a stack to replace the call stack) In a recursive algorithm, the stack is hidden from the user

12 Recursion Introduction Page 12 Example 2: Factorial (Tail-Recursion) The previous factorial algorithm is augmented recursive, i.e. it can be replaced by a tail-recursion algorithm factorial(n) return fact(n,1); algorithm fact(n,result) // auxiliary function if n = 1 then return result; // base case of the recursion else return fact(n 1, n result) // tail recursive call

13 Recursion Introduction Page 13 Example 2: Factorial (Tail-Recursion) n = 1 fact(1,24) Returns 24 fact(2,12) n = 3 n = 2 Returns 24 Tail- Recursion fact(3,4) Returns 24 fact(4,1) n = 4 Returns 24

14 Recursion Introduction Page 14 Example 3: Factorial (Iteration) The tail-recursive factorial algorithm can be replaced by a simple iterative loop algorithm factorial(n) result 1 while n > 0 do result result n n n 1 return result

15 Recursion Introduction Page 15 Recursive vs. Iterative Algorithms Recursive algorithms are often less efficient than their iterative counterparts The overhead of managing the call stack is non-negligible The space complexity is often O(n) or O(log n) instead of O(1) Risk to get "stack overflow" errors Some (functional) programming languages such as Scheme, Lisp, or Haskell detect tail-recursions automatically Nevertheless, recursion has been used to solve some of the biggest and hardest algorithmic problems in computer science!!!

16 Recursion Simple Problems Page 16 Outline Introduction Simple Problems Time Complexity and Optimization Advanced Problems

17 Recursion Simple Problems Page 17 Problem 1: Fibonacci Compute the Fibonacci number 0 if n = 0 F (n) = 1 if n = 1 F (n 1) + F (n 2) if n > 1 Input Integer n 0 Ouput F (n) Examples fibonacci(10) 55 fibonacci(21) 10946

18 Recursion Simple Problems Page 18 Problem 2: Integer Printing Print integers relative to various bases Input Integer n 0, base b 2 Ouput Screen output of [d k d 1 d 0 ] b for n = d k b k + + d 1 b + d 0 Examples printinteger(61, 2) printinteger(61, 16) 3D Auxiliary Function printdigit(d)

19 Recursion Simple Problems Page 19 Problem 3: Palindrome Detect whether a sequence is a palindrome Input Sequence s Ouput true or false Examples palindrome([amanaplanacanalpanama]) true palindrome([recursioniscomplicated]) false Auxiliary Functions size(), first(), last(), removefirst(), removelast()

20 Recursion Simple Problems Page 20 Problem 4: Prefix Determine whether a sequence is the prefix of another sequence Input Sequences s 1 and s 2 Ouput true or false Examples prefix([recursive], [recursivecall]) true prefix([call], [recursivecall]) false Auxiliary Functions isempty(), first(), removefirst()

21 Recursion Simple Problems Page 21 Problem 5: Minimum/Maximum Find both the minimum and the maximum of an sequence of numbers. Input Sequence s of numbers Ouput Pair (min, max) with min = min{s} and max = max{s} Example minmax([6, 1, 2, 3, 12, 4, 3, 8, 0]) ( 4, 12) Auxiliary Functions first(), isempty(), removefirst(), makepair(first, second), firstelt(), secondelt()

22 Recursion Simple Problems Page 22 Problem 6: Line Plotter Plot a line between two points p 1 = (x 1, y 1 ) and p 2 = (x 2, y 2 ) Input Points x 1, y 1, x 2, y 2 Ouput Plot the connecting line on the screen Example plotline(8, 5, 21, 11) see next slide Auxiliary Functions plotpixel(x, y)

23 Recursion Simple Problems Page 23 Problem 6: Line Plotter (cont.) y x

24 Recursion Simple Problems Page 24 Problem 7: Koch Fractal Plot a Koch fractal of maximal depth d between two points d=0 middle d=1 left right d=2

25 Recursion Simple Problems Page 25 Problem 7: Koch Fractal (cont.) Input Integer d, points p 1 = (x 1, y 1 ) and p 2 = (x 2, y 2 ) Ouput Plot the Koch fractal on the screen Example kochfractal(6, p 1, p 2 ) see previous slide Auxiliary Functions plotline(p 1, p 2 ), left(p 1, p 2 ), middle(p 1, p 2 ), right(p 1, p 2 )

26 Recursion Simple Problems Page 26 Problem 8: Towers of Hanoi Move all the blocks from the first to the third tower, one after another, and without ever putting a block on a smaller one Tower 1 Tower 2 Tower 3

27 Recursion Simple Problems Page 27 Problem 8: Towers of Hanoi (cont.) Input Integer n (number of blocks) Ouput Print sequence of instructions on screen Example hanoi(3) 1 3, 1 2, 3 2, 1 3, 2 1, 2 3, 1 3 Auxiliary Functions makestack(), push(e), pop()

28 Recursion Time Complexity and Optimization Page 28 Outline Introduction Simple Problems Time Complexity and Optimization Advanced Problems

29 Recursion Time Complexity and Optimization Page 29 Running Time of Recursions In principle, recursive versions of algorithms can be analyzed like non-recursive algorithms Count primitive operations Express the running time as a function T (n) Classify T (n) using the big-oh notation The problem with recursive algorithms is that T (n) refers to itself Recurrence equation (difficult to solve in general) { 2, for n = 1 Example: T (n) = T (n 1) + 5, for n > 1 for factorial(n), which leads to T (n) = 5(n 1) + 2

30 Recursion Time Complexity and Optimization Page 30 Big-Oh for Recurrence Equations Recurrence equations are often in one of the two following forms: Reducing the problem size by k: (Rules R1 R2) { c if n = d T (n) = q T (n k) + f (n) if n > d Example: c = 2, q = k = d = 1, f (n) = 5 for factorial(n) Dividing the problem size by r: (Rules D1 D4) { c if n = d T (n) = q T (n/r) + f (n) if n > d (see Master-Theorem in [Goodrich et al., 2001], page 268)

31 Recursion Time Complexity and Optimization Page 31 Complexity Rules Overview f (n) O(1) O(log n) O(n) O(n 2 ) R1 q = 1 k 1 n n log n n 2 n 3 R2 q 2 k 1 q n log n q n n q n n 2 qn D1 q = 1 r 2 log n log n n n 2 D2 q 2 r = q n n n log n n 2 D3 q 2 r > q n log r q n log r q n n 2 D4 q 2 r < q n log r q n log r q n log r q n 2

32 Recursion Time Complexity and Optimization Page 32 Memoization A common technique to speed up recursive algorithms is called memoization A memoized function stores (caches) results of previous calls When the function is later called with the same parameters, it returns the stored result rather than recalculating it Memoized functions are not allowed to have side-effects no screen output no global variable assignments no change of object attributes Memoization is usually implemented with hash tables Difficult space/time trade-off

33 Recursion Time Complexity and Optimization Page 33 Memoization: Example fibonacci(5) memoizedfibonacci(5) Complexity is reduced from O(2 n ) to O(n)

34 Recursion Advanced Problems Page 34 Outline Introduction Simple Problems Time Complexity and Optimization Advanced Problems

35 Recursion Advanced Problems Page 35 Problem 9: Flood Fill In a black-and-white bitmap, fill the neighborhood of a given point with black. Consider the bitmap as a 2-dimensional array A[i, j] with A[i, j] {white, black}

36 Recursion Advanced Problems Page 36 Problem 9: Flood Fill (cont.) Input Bitmap A[i, j], integers x, y Ouput Modified bitmap A[i, j] Example floodfill(a, 3, 6) see previous slide Auxiliary Functions arrayxsize(a), arrayysize(a)

37 Recursion Advanced Problems Page 37 Problem 10: Ruler Draw a ruler with n different (sub-) units. Input Integer n Ouput Draw ruler on screen Example ruler(5) Auxiliary Functions drawline(n) draws a line of size n from left to right

38 Recursion Advanced Problems Page 38 Problem 11: Powerset Compute the set of all subsets (= powerset) of a given set Input Set s Ouput Powerset P = 2 s Example powerset({1, 2, 3}) {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} Auxiliary Functions first(), isempty(), removefirst(), makeset(e), union(s)

39 Recursion Advanced Problems Page 39 Problem 12: Cartesian Product Compute the Cartesian product of a given set of sets Input Sequence s = [s 1 s n ] of sets s i Ouput Cartesian product s 1 s n (a set of sequences) Example product([{1, 2}{a, b}{x, y}]) {[1ax], [1ay],..., [2by]} Auxiliary Functions first(), isempty(), removefirst(), insertfirst(e), makeset(e), union(s)

40 Recursion Advanced Problems Page 40 Problem 13: Permutations Compute the set of all permutations of an ordered set Input Set/sequence s Ouput Set of all permutations of s Example permute([1, 2, 3]) {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]} Auxiliary Functions first(), isempty(), removefirst(), insertfirst(e), size(), makeset(e), union(s), insertatrank(e, r)

41 Recursion Advanced Problems Page 41 Problem 14: Differentiation Differentiate a mathematical function f (x) according to the following recursive rules: 0 if f (x) = C, C R 1 if f (x) = x f (x) = g (x) ± h (x) if f (x) = g(x) ± h(x) g (x)h(x) + g(x)h (x) if f (x) = g(x)h(x) 0 if f (x) = y, x y g (x)h(x) g(x)h (x) h(x)h(x) N x N 1 for f (x) = g(x) h(x) if f (x) = x N, N Z

42 Recursion Advanced Problems Page 42 Problem 14: Differentiation (cont.) Input f (the function) and v (the variable) Ouput f Examples differentiate( 2x 3 + 3y, x ) 6x 2 differentiate( 2x 2 + 3y, y ) 3 differentiate( 2x 2 + 3y, z ) 0 Auxiliary Functions isconst(), isvar(), issum(), isdiff(), isprod(), isfrac(), ispower(), makesum(f, g), makediff(f, g), makeprod(f, g), makefrac(f, g), makepower(v, n), firstarg(), secondarg(), base(), exp()