Recursion

Hanoi Tower Legend: Inside a Vietnamese temple there are three rods (labeled as r 1, r 2, and r 3 ), surrounded by n golden disks of different sizes. The monks of Hanoi wants to move these disks from r 1 to r 3 such that: at each step, take the upper disk of one rod and place it to the top of another rod no bigger disk can be placed on top of a smaller disk

Input and output format Input: A single integer n in [1, 20] Output: The steps to move the disks from r 1 to r 3 ; in the format of r i -> r j 3 r1 -> r3 r1 -> r2 r3 -> r1 r1 -> r3 r2 -> r1 r2 -> r3 r1 -> r3

Recursion Can we solve the problem by solving some smaller problems? To move n disks from r 1 to r 3 1.move n-1 disks from r 1 to r 2 2.move 1 disks from r 1 to r 3 3.move n-1 disks from r 2 to r 3 r1r1 r2r2 r3r3

Implementation Let hanoi( int n, int i, int j, int z ) be a function that will move n disks from rod i to rod j using rod k as the middle rod. void hanoi( int n, int i, int j, int k ){ if( n==1 ){ cout r" << j << endl; return; } hanoi( n-1, i, k, j ); hanoi( 1, i, j, k ); hanoi( n-1, k, j, i ); } int main(){ int n; cin >> n; hanoi( n ); return 0; }

How many moves are there? Let T(n) be the number of steps to move n disks T(1) = 1 T(n)= 2 T(n-1) + 1 = 4 T(n-2) =... = 2 n-1 T(1)+ 2 n = 2 n - 1 The running time is also Θ(2 n ) = exponential time Very slow But it is optimal already (prove by M.I. on n)

Other recursion problems Look for easy problems first POJ 1920

POJ 2386 Lake counting

POJ 2227

Topological sort

Summary We have seen some examples on recursion. ◦ Hanoi tower ◦ Reverse of hanoi tower Flood fill is a recursive method to identify connected region. ◦ Running time = size of the region We can order a partial ordered set by topological sort ◦ Running time = number of edges Exercises: UVA 900, 254, 699, 10940, 527, 572, 657, 11110, 10305, 11686, 200