Fundamentals of Programming. Recursion - Part 2. October 30th, 2014
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1 Fundamentals of Programming Recursion - Part 2 October 30th, 2014
2 Important points from last lecture To understand recursion, you have to first understand recursion. A recursive function: a function that calls itself. Main usage: to solve a problem, recursively solve the problem on smaller instances.
3 Important points from last lecture Recursive Program Design Ask yourself: If I had the solutions to the smaller instances for free, how could I solve the original problem? Write the recursive relation: e.g. fib(n) = fib(n-1) + fib(n-2) Don t forget the base case: A small version of the problem that does not require recursive calls. Double check: All your recursive calls make progress towards the base case(s) and they don t miss it.
4 def factorial(n): Important points from last lecture How/Why does it work? if (n == 1): return 1 else: return n * factorial(n - 1) Does factorial(1) work (base case)? Does factorial(2) work? returns 2*factorial(1) Does factorial(3) work? returns 3*factorial(2)
5 Important points from last lecture fac(1) > fac(2) > fac(3) > fac(4) >..... fac(1) fac(2) fac(3) fac(4).
6 Important points from last lecture move (N, source, dest): if(n > 0): Let temp be the index of other pole. move(n-1, source, temp) print Move ring from pole + source + to pole + dest move(n-1, temp, destination)
7 Important points from last lecture move (N, source, dest): if(n > 0): Let temp be the index of other pole. move(n-1, source, temp) print Move ring from pole + source + to pole + dest move(n-1, temp, destination) Does N = 0 work (base case)? Does N = 1 work? Does N = 2 work? Does N = 3 work?
8 Important points from last lecture Recursion vs Iteration Elegance Performance Debugability Recursion Iteration
9 Important points from last lecture Getting comfortable with recursion 1. See lot s of examples 2. Practice yourself
10 Some easy practice questions Write a function to reverse a given string. Given integers x and y, write a recursive function to compute x**y. Write a recursive interleave function: f([6, 4, 2], [5, 3, 1]) > [6, 5, 4, 3, 2, 1] Write nthprime recursively. Write recursive functions for SelectionSort, BubbleSort.
11 def f5(x): if (x == 0): return 0 else: return 2*x-1 + f5(x-1) What do these do? def f1(x): if (x == 0): return 0 else: return 1 + f1(x-1) def f2(x): if (x == 0): return 40 else: return 1 + f2(x-1) def f6(x): if (x < 2): return 1 else: return f6(x-1) + f6(x-2) def f3(x): if (x == 0): return 0 else: return 2 + f3(x-1) def f4(x): if (x == 0): return 40 else: return 2 + f4(x-1)
12 Printing Call Stack def fact(n, depth=0): print " "*depth, "fact(", n, ")" if (n <= 1): result = 1 else: result = n*fact(n-1, depth+1) print " "*depth, "-->", result return result fact(4)
13 Printing Call Stack def fib(n, depth=0): print " "*depth, "fib(", n, " )" if (n < 2): result = 1 else: result = fib(n-1, depth+1) + fib(n-2, depth+1) print " "*depth, "-->", result return result fib(4)
14 Today Slightly more complicated examples
15 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]]
16 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]]
17 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] All subsets = All subsets that do not contain 1 +
18 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] All subsets = All subsets that do not contain 1 +
19 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] All subsets = All subsets that do not contain 1 + All subsets that contain 1
20 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] All subsets = All subsets that do not contain 1 + All subsets that contain 1 [1] + subset that doesn t contain a 1
21 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] def powerset(a): if (len(a) == 0): return [[]] else: allsubsets = [ ] for subset in powerset(a[1:]): allsubsets += [subset] allsubsets += [[a[0]] + subset] return allsubsets
22 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] def powerset(a): if (len(a) == 0): return [[]] else: allsubsets = [ ] for subset in powerset(a[1:]): allsubsets += [subset] allsubsets += [[a[0]] + subset] return allsubsets
23 Power set Given a list, return a list of all the subsets of the list. [1,2,3] -> [[], [1], [2], [3], [1,2], [2,3], [1,3], [1,2,3]] def powerset(a): if (len(a) == 0): return [[]] else: allsubsets = [ ] for subset in powerset(a[1:]): allsubsets += [subset] allsubsets += [[a[0]] + subset] return allsubsets
24 Permutations Given a list, return all permutations of the list. [1,2,3] -> [[1,2,3], [2,1,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1]]
25 Permutations Given a list, return all permutations of the list. [1,2,3] -> [[1,2,3], [2,1,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1]]
26 Permutations Given a list, return all permutations of the list. [1,2,3] -> [[1,2,3], [2,1,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1]] [1,3,2], [3,1,2], [3,2,1]
27 Permutations Given a list, return all permutations of the list. [1,2,3] -> [[1,2,3], [2,1,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1]] def permutations(a): if (len(a) == 0): return [[]] else: allperms = [ ] for subpermutation in permutations(a[1:]): for i in xrange(len(subpermutation)+1): allperms += [subpermutation[:i] + [a[0]] + subpermutation[i:]] return allperms
28 Permutations Given a list, return all permutations of the list. [1,2,3] -> [[1,2,3], [2,1,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1]] def permutations(a): if (len(a) == 0): return [[]] else: allperms = [ ] for subpermutation in permutations(a[1:]): for i in xrange(len(subpermutation)+1): allperms += [subpermutation[:i] + [a[0]] + subpermutation[i:]] return allperms
29 Permutations Given a list, return all permutations of the list. [1,2,3] -> [[1,2,3], [2,1,3], [2,3,1], [1,3,2], [3,1,2], [3,2,1]] def permutations(a): if (len(a) == 0): return [[]] else: allperms = [ ] for subpermutation in permutations(a[1:]): for i in xrange(len(subpermutation)+1): allperms += [subpermutation[:i] + [a[0]] + subpermutation[i:]] return allperms
30 Print files in a directory
31 Print files in a directory
32 Print files in a directory import os def printfiles(path): if (os.path.isdir(path) == False): # base case: not a folder, but a file, so print its path print path else: # recursive case: it's a folder for filename in os.listdir(path): printfiles(path + "/" + filename)
33 Flood fill click def floodfill(x, y, color): if ((not inimage(x,y)) or (getcolor(img, x, y) == color)): return img.put(color, to=(x, y)) floodfill(x-1, y, color) floodfill(x+1, y, color) Pixel version floodfill(x, y-1, color) floodfill(x, y+1, color)
34 Flood fill click U D L R Grid version floodfill(row-1, col, color, depth+1) floodfill(row+1, col, color, depth+1) floodfill(row, col-1, color, depth+1) floodfill(row, col+1, color, depth+1)
35 Solving a maze puzzle start finish
36 Solving a maze puzzle start finish
37 Solving a maze puzzle start finish def issolvable(maze, (r1,c1), (r2,c2)): > True or False Idea: if issolvable(maze, (r1,c1), (r2,c2)) == True, then for some neighbor of (r1,c1), say (nr1, nc1), issolvable(maze, (nr1,nc1), (r2,c2)) == True
38 Solving a maze puzzle def issolvable(maze, (r1, c1), (r2, c2)): if ((r1 == r2) and (c1 == c2)): return True U D R L for dir in [(-1,0), (1,0), (0,1), (0,-1)]: newcell = (r1, c1) + dir if (islegal(newcell) and issolvable(maze, newcell, (r2, c2))): return True return False No need to consider a cell more than once.
39 Solving a maze puzzle def issolvable(maze, (r1, c1), (r2, c2)): if ((r1, c1) in visited): return False visited = set() visited.add((r1,c1)) if ((r1 == r2) and (c1 == c2)): return True for dir in [(-1,0), (1,0), (0,1), (0,-1)]: newcell = (r1, c1) + dir if (islegal(newcell) and issolvable(maze, newcell, (r2, c2))): return True return False
40 Solving a maze puzzle def issolvable(maze, (r1, c1), (r2, c2)): if ((r1, c1) in visited): return False visited = set() visited.add((r1,c1)) if ((r1 == r2) and (c1 == c2)): return True for dir in [(-1,0), (1,0), (0,1), (0,-1)]: newcell = (r1, c1) + dir if (islegal(newcell) and issolvable(maze, newcell, (r2, c2))): return True visited.remove((r1, c1)) return False If we actually want the solution path.
41 nqueens Problem Place n queens on a n by n board so that no queen is attacking another queen. def solve(n): > [6, 4, 2, 3, 5, 7, 1, 3] list of rows
42 nqueens Problem Place n queens on a n by n board so that no queen is attacking another queen. n rows and n-1 columns one queen has to be on first column
43 nqueens Problem First attempt: - try rows 0 to 7 for first queen - for each try, recursively solve the red part Problem: Can t solve red part without taking into account first queen First queen puts constraints on the solution to the red part Need to be able to solve nqueens with added constraints. Need to generalize our function: def solve(n, m, constraints):
44 nqueens Problem def solve(n, m, constraints): n = number or rows m = number or columns constraints (in what form?) list of rows For the red part, we have the constraint [6]
45 nqueens Problem def solve(n, m, constraints): n = number or rows m = number or columns constraints (in what form?) list of rows For the red part, we have the constraint [6,4,2] The constraint tells us which cells are unusable for the red part. To solve original nqueens problem, call: solve(n, n, [])
46 nqueens Problem def solve(n, m, constraints): [?,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
47 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
48 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] [5,7,1,3] n = 8 m = 5 constraints = [6,4,2]
49 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] [5,7,1,3] > [0,5,7,1,3] n = 8 m = 5 constraints = [6,4,2]
50 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] no solution n = 8 m = 5 constraints = [6,4,2]
51 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
52 nqueens Problem def solve(n, m, constraints): NOT LEGAL [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
53 nqueens Problem def solve(n, m, constraints): NOT LEGAL [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
54 nqueens Problem def solve(n, m, constraints): NOT LEGAL [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
55 nqueens Problem def solve(n, m, constraints): NOT LEGAL [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
56 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
57 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] no solution n = 8 m = 5 constraints = [6,4,2]
58 nqueens Problem def solve(n, m, constraints): NOT LEGAL [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
59 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] n = 8 m = 5 constraints = [6,4,2]
60 nqueens Problem def solve(n, m, constraints): [0,?,?,?,?] no solution n = 8 m = 5 constraints = [6,4,2]
61 nqueens Problem n = 8 m = 5 [0,?,?,?,?] constraints = [6,4,2] def solve(n, m, constraints): if(m == 0): return [] for row in xrange(n): if (islegal(row, constraints)): newconstraints = constraints + [row] result = solve(n, m-1, newconstraints) if (result!= False): return [row] + result return False
62 nqueens Problem n = 8 m = 5 constraints = [6,4,2] def islegal(row, constraints): for ccol in xrange(len(constraints)): crow = constraints[ccol] shift = len(constraints) - ccol if ((row == crow) or (row == crow + shift) or (row == crow - shift)): return False return True
63 nqueens Problem n = 8 m = 5 constraints = [6,4,2] def islegal(row, constraints): for ccol in xrange(len(constraints)): crow = constraints[ccol] shift = len(constraints) - ccol if ((row == crow) or (row == crow + shift) or (row == crow - shift)): return False return True
64 nqueens Problem n = 8 m = 5 constraints = [6,4,2] def islegal(row, constraints): for ccol in xrange(len(constraints)): crow = constraints[ccol] shift = len(constraints) - ccol if ((row == crow) or (row == crow + shift) or (row == crow - shift)): return False return True
65 nqueens Problem n = 8 m = 5 constraints = [6,4,2] def islegal(row, constraints): for ccol in xrange(len(constraints)): crow = constraints[ccol] shift = len(constraints) - ccol if ((row == crow) or (row == crow + shift) or (row == crow - shift)): return False return True
66 Fractals Self similar image An image made up of smaller versions of itself. A change rule: length length/3
67 Fractals n = 1 n = 2 n = 3 n = 4 def kn(length, n): if (n == 1): turtle.forward(length) else: kn(length/3.0, n-1) turtle.left(60) kn(length/3.0, n-1) turtle.right(120) kn(length/3.0, n-1) turtle.left(60) kn(length/3.0, n-1)
68 Sierpinski Triangle level = 0 level = 1 level = 2 def drawst(x, y, size, level): if (level == 0): canvas.create_polygon(x, y, x+size, y, x+size/2, y-size*(3**0.5)/2, fill="black") else: drawst(x, y, size/2, level-1) drawst(x+size/2, y, size/2, level-1) drawst(x+size/4, y-size*(3**0.5)/4, size/2, level-1)
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