Easy, Hard, and Impossible

Easy, Hard, and Impossible Elaine Rich

Easy

Tic Tac Toe

Hard

Chess

The Turk Unveiled in 1770.

Searching for the Best Move A B C D E F G H I J K L M (8) (-6) (0) (0) (2) (5) (-4) (10) (5)

How Much Computation Does it Take? • Middle game branching factor  35 • Lookahead required to play master level chess  8 • 358

How Much Computation Does it Take? • Middle game branching factor  35 • Lookahead required to play master level chess  8 • 358 2,000,000,000,000 • Seconds in a year 

How Much Computation Does it Take? • Middle game branching factor  35 • Lookahead required to play master level chess  8 • 358 2,000,000,000,000 • Seconds in a year  31,536,000 • Seconds since Big Bang  300,000,000,000,000,000

Growth Rates of Functions

The Turk Still fascinates people.

How Did It Work?

A Modern Reconstruction Built by John Gaughan. First displayed in 1989. Controlled by a computer. Uses the Turk’s original chess board.

Chess Today In 1997, Deep Blue beat Garry Kasparov.

Seems Hard But Really Easy

Nim At your turn, you must choose a pile, then remove as many sticks from the pile as you like. The player who takes the last stick(s) wins.

Nim Now let’s try:

Nim Now let’s try: Oops, now there are a lot of possibilities to try.

Nim http://www.gamedesign.jp/flash/nim/nim.html

Binary Numbers

Nim My turn: 10 (2) 10 (2) 11(3) 11 • For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise.

Nim My turn: 100 (4) 010 (2) 101 (5) 011 • For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise.

Nim Your turn: 100 (4) 001 (1) 101 (5) 000 • For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise.

Following Paths

Seven Bridges of Königsberg

Seven Bridges of Königsberg Seven Bridges of Königsberg: 1 3 4 2

Seven Bridges of Königsberg Seven Bridges of Königsberg: 1 3 4 2 As a graph:

Eulerian Paths and Circuits Cross every edge exactly once. All these people care:  Bridge inspectors Road cleaners Network analysts Leonhard Euler 1707 - 1783

Eulerian Paths and Circuits Cross every edge exactly once. Leonhard Euler 1707 - 1783

Eulerian Paths and Circuits Cross every edge exactly once. Leonhard Euler 1707 - 1783 There is a circuit if every node touches an even number of edges.

So, Can We Do It? 1 3 4 2 As a graph:

The Good King and the Evil King The good king wants to build exactly one new bridge so that: • There’s an Eulerian path from the pub to his castle. • But there isn’t one from the pub to the castle of his evil brother on the other bank of the river.

Here’s What He Starts With 1 3 4 2 As a graph:

Here’s What He Ends Up With 1 3 4 2 As a graph:

Unfortuntately, There Isn’t Always a Trick Suppose we need to visit every vertex exactly once.

Visting Nodes Rather Than Edges ● A Hamiltonian path: visit every node exactly once. ● A Hamiltonian circuit: visit every node exactly once and end up where you started. • All these people care: • Salesmen, • Farm inspectors, • Network analysts

The Traveling Salesman Problem 15 25 10 28 20 4 8 40 9 7 3 23 Given n cities: Choose a first city n Choose a second n-1 Choose a third n-2 … n!

The Traveling Salesman Problem Can we do better than n! ● First city doesn’t matter. ● Order doesn’t matter. So we get (n-1!)/2.

The Growth Rate of n!

Putting it into Perspective

Growth Rates of Functions

Getting Close Enough • Use a technique that is guaranteed to find an optimal solution and likely to do so quickly. • Use a technique that is guaranteed to run quickly and find a “good” solution. The Concorde TSP Solver found an optimal route that visits 24,978 cities in Sweden. The World Tour Problem

Is This The Best We Can Do? • It is generally believed that there’s no efficient algorithm that finds an exact solution to: • The Travelling Salesman problem • The question of whether or not a Hamiltonian circuit exists. Would you like to win $1M? The Millenium Prize

Impossible

Easy, Hard, and Impossible