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Towers of Hanoi

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Presentation on theme: "Towers of Hanoi"— Presentation transcript:

1 Towers of Hanoi

2 Towers of Hanoi Move all the discs from the leftmost peg to the rightmost one. –Only one disc may be moved at a time. –A disc can be placed either on empty peg or on top of a larger disc. startfinish Demo

3 Recursive Solution Move n-1 smallest discs right. Move largest disc left. cyclic wrap-around

4 The Recursive Process

5 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(3, 1) n = 3 d = 1

6 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(3, 1) n = 3 d = 1

7 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(2, 0) n = 2 d = 0

8 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(2, 0) n = 2 d = 0

9 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(1, 1) n = 1 d = 1

10 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(1, 1) n = 1 d = 1

11 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(1, 1) n = 1 dir = 1 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(0, 0) n = 0 d = 0

12 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(1, 1) n = 1 d = 1

13 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(1, 1) n = 1 d = 1

14 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(1, 1) n = 1 d = 1 Print “Move disk 1 right”

15 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(2, 0) n = 2 d = 0

16 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(2, 0) n = 2 d = 0

17 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - d); if (dir == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - dir); move(2, 0) n = 2 d = 0 Print “Move disk 2 left”

18 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(3, 1) n=3 dir = 1 if (n == 0) return; move(n - 1, 1 - dir); if (dir == 1) System.out.println("Move disk " + n + " to right"); else System.out.println("Move disk " + n + " to left"); move(n - 1, 1 - dir); move(2, 0) n = 2 dir = 0 if (n == 0) return; move(n - 1, 1 - d); if (d == 1) System.out.println("Move disk " + n + “ right"); else System.out.println("Move disk " + n + “ left"); move(n - 1, 1 - d); move(1, 1) n = 1 d = 1

19 3, 0 2, 1 1, 0 2, 1 1, 0 1 left2 right 1 left 3 left 2 right 1 left n, left Recursive Tree

20 Recursive Graphics

21 21

22 22 Htree H-tree of order n –Draw an H. –Recursively draw 4 H-trees of order n-1, one connected to each tip. and half the size order 1 order 2 order 3 tip size


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