Recursive Algorithms A recursive algorithm calls itself to do part of its work The “call to itself” must be on a smaller problem than the one originally attempted

Recursive Algorithms (cont’d) Recursive solutions consist of: Base case(s) The problem is explicitly solved for certain values of the input (generally small values) Recursive step Divide the problem into one or more simpler or smaller parts Solve each part recursively Combine the solutions of the parts into a solution to the problem IMPORTANT! If the "part" is as big as the whole, then your program will not terminate IMPORTANT! If this is missing, or not implemented correctly, your program will not terminate.

Recursive Algorithm Examples Add 1..n Factorial

Recursive Algorithms (cont’d) Easy to start from a recursive definition of a function to its implementation as a recursive algorithm EXAMPLE: f(n) = n! (the factorial of n) and the corresponding recursive algorithm

Time Complexity T(1) = 2 T(n) = T(n-1) + 4 Recurrence relation T(N) = 4n-2

The Towers of Hanoi Only one disc may be moved at a time A disc may be placed on top of another only if the disc being moved has a smaller diameter than the one upon which it is placed

3-disc Towers of Hanoi Solution Solving the 3-disc puzzle requires 23-1 = 7 moves Solving the n-disc puzzle requires 2n -1 moves

The 64-disc Towers of Hanoi The 64-disc puzzle requires 264 = 18,446,744,073,709,551,616 moves Suppose the monks can make 1 move per second. This translates into 60 moves per minute 3600 moves per hour 86400 moves per day 31536000 moves per year At this rate it will take about 584,942,417,355 years to solve the problem

Towers of Hanoi Recursively Suppose the pegs are numbered 1, 2, and 3 and the initial configuration is n discs on one of those pegs, call it start start goal tmp

Towers of Hanoi Recursively Write a recursive algorithm for solving the n-disc Towers of Hanoi puzzle Suppose we want to move the n discs from the start peg to the goal peg, denote the other peg by tmp

Algorithm To move n discs from A to C, B as tmp If n>0 then EndIf Move n-1 from A to B, C as tmp Move the remaining disc from A to C Move n-1 from B to C, A as tmp EndIf

Recursive Procedures Pros Cons Often intuitive, more elegant Result in shorter programs Sometimes, a recursive solution may result in a faster algorithm Usually easier to prove correctness Cons More overhead due to function calls More memory used at runtime Sometimes, not as fast as an iterative version of the algorithm

Recursive vs. Iteration Any problem that can be solved recursively can be solved iteratively Choose recursion when you have a recursive data structure the recursive solution is easier to understand/debug Do not choose recursion when performance is an issue Examples where recursion is better Towers of Hanoi, certain sorting algorithms