Lecture 1: Recursion and Induction

Guess solution for M(n) from table of values and prove by induction
Unwind recurrence to obtain equivalent sum and reprove result using induction again
Geometric series (exc. use induction to prove the formula for geometric series)
Use induction to prove that the number of moves used by the program is optimal

An n-bit Gray code is a sequence of 2^n binary numbers where consecutive numbers differ by exactly one bit.
E.G. [000,001,011,010,110,111,101,100]
Shank Encoding

G_1 = [0,1]
G_n = [0G_{n-1},1G_{n-1}], where bG_k is the sequence obtained by prepending the bit

Let G_n(i) be a fuction from the interval [0,...,2^n-1] defined by G_n(i) = i ^ (i >>1)

Use induction to prove that the sequence G_n(i), i= 0,...,2^n-1 is the binary-reflected Gray code (exercise)

For the n-disk problem let d_0,...,d_{n-1} be n coordinates that can be either 0 or 1. Initialize d_i = 0, and set d_i to the complement (the complement of 0 is 1 and the complement of 1 is 0) of d_i whenever the ith disk is moved.
Retrace the solution to the 3-disk problem and keep track of (d_0,d_1,d_2). Observe that the sequence, when interpreted as a sequence of 3-bit binary numbers is equal to the binary-reflected Gray code. Why? Can you prove this? How can this be used to provide an iterative solution to the Tower of Hanoi problem.

Definition - an n-dimensional hypercube is a graph (a collection of vertices and edges) with 2^n nodes with the property that if the nodes are labeled by the binary representation of the number 0,...,2^n-1, there is an edge between every two nodes that differ by exactly one bit in their binary representation.
Recursive definition

A one-dimensional hypercube is a single node labeled by the number 0.
An n-dimensional hypercube H_n is obtained from two copies of an (n-1)-dimensional hypercube by connecting identical nodes in each of copy of H_{n-1} by an edge. The binary representation of the labels of the nodes in the first copy should be prepended with a 0-bit and the labels of the matching node in the second copy should be prepended with a 1-bit

How can you prove that the two definitions are equivalent?
Derive a recurrence relation for the number of edges in an n-dimensional hypercube.
A Hamiltonian path is a sequence of edges that visit each node exactly once. Can you find a Hamiltonian path for a 3-dimensional hypercube? Such a path provides a Gray code. Why?
Can you generalize your construction, using the recursive definition of a hypercube, to construct a Hamiltonian path for an n-dimensional hypercube? How does your construction relate to the Tower of Hanoi problem?

Created: Sept. 26, 2004 by jjohnson@cs.drexel.edu