1. CS301
Peter Kemper
R 104A, phone 221-3462, email:kemper@cs.wm.edu
Today: Knight’s Tour Project
Reference: Project Assignment 1
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2. Knight’s Tour Problem: A Chess Problem
Source: http://en.wikipedia.org/wiki/Knight's_tour 2

3. Project Overview
GUI =
Graphical
User Interface

provides
elementar
ydescribes available methods for
graphic
for i/o
any

GUI
 class that implements Tour interface
shows 5 x 5

board
picks
random initial
position

has
a
Interface:

4. Project Assignment
Class KnightsTourP1 does not
work!
Add a solution algorithms that
given an initial position
on an n x n board (here
n=5)

compute an open Knight’s tour if one
exists
 recognize if a tour does not exist

Typical questions in math
Does a given problem have a
solution?
 If it does, is a solution unique?
 How do you find a solution?

Typical questions in CS

5. Knight’s Tour Problem --> Programming Problem
What is a good point
of view to implement
an algorithm that
solves this problem?
Map to a formalism
that is closer to
programming!
Suggestions?
Source: http://en.wikipedia.org/wiki/Knight's_tour 5

6. Knight’s Tour Problem
How do you solve this?
5 x 5 matrix

permutation of numbers
1,2,3,...,25

distance between
consecutive numbers i and
i+1 is 2 steps in one
direction plus 1 step in
orthogonal direction

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open tour means that
distance between 25 and 1
is arbitrary

7. Knight’s Tour Solution

Note
 Not all configurations on a 5 x5 board have a solution!
 Chess board is larger such that computation takes longer!
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8. Knight’s Tour Problem
How do you solve this?
How do you solve this
with a program?
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9. How to solve this on a computer?
5 x 5 = 25 fields with (1,2,...,25) in some order
Naive:
Try all permutations of (1,2,...,25) and see
which one fits all constraints
 Constraints:




rules of the game
some fields have given, fixed values
Why is this naive?
 Does not involve any real insight + better concepts exist
 Number of permutations is high
Other ideas?
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10. How to solve this on a computer
Backtracking algorithm
function backtrack(P, c)
if (invalid(P,c)) then return
// recognize failure, backtrack
if (complete(P,c) then output(P,c) ; stop // termination
while (hasMoreCandidates(P,c))
through all candidates
// go
{
s = nextCandidate(P,c) ;
// get next candidate backtrack(P,s)
;
// recursion, try next candidate undoChanges(P,s)
;
// failed, need to recover state of P
}
Problem P, candidate c
Note: undoChanges() only necessary if function manipulates shared
data, e.g. P
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11. Backtracking
Illustrate Knight’s Tour backtracking as tree!
Do you see ways to improve the performance of the
algorithm?
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12. Recursion
A method calls itself
Classic example: Factorial function

Computes: N! = N x (N-1) x ... x 1
Shows two essential
components
Base case: returns value
without recursion, i.e. terminates recursive
method calls

Reduction step: relates function at particular
input to same function at some other inputs.

Key for termination is that the sequence of
Source : Sedgewick & Wayne: http://introcs.cs.princeton.edu/java/23recursithe base
parameter values in reduction step converges to
1
case.
2


13. Pitfalls of Recursion
Missing base case
Recursion will not terminate
 Example: Faulty method for Harmonic numbers

No guarantee of convergence

Recursive call to a problem that is not smaller
Source : Sedgewick & Wayne: http://introcs.cs.princeton.edu/java/23recursi on/
1
3

14. Pitfalls of Recursion
Excessive space requirements
Recursive methods calls build up on execution stack
 Java: eventually results in StackOverflowError, e.g. at N=5000

Excessive recomputation
Recursive calls may imply that results are
computed over and over again
 Example: Fibonacci sequence
 Note: storing results leads to concept of Dynamic
Programming

Source : Sedgewick & Wayne: http://introcs.cs.princeton.edu/java/23recursion1/4

15. Problem solving techniques
Divide and conquer
split a problem into a set
of smaller ones
 solve smaller problems

combine solutions of smaller problems to
solution of main problem
 matches with recursion

Change point of view

Alain Kay: “A change in perspective is worth 80 IQ points.”

other quotes from Kay see http://en.wikiquote.org/wiki/Alan_Ka
y
 e.g. “Actually I made up the term "object-oriented", and I can
tell you I did not have C++ in mind.”
Try to solve more restricted special case(s)

get a solution, try to generalize from here
Try to solve a more general problem
Nice to read: George Polya, How to solve it,

16. Getting back to CS 301
Learning objectives for Project 1
Develop Java programming skills:
basic UI,
 recursion

Understand Unit testing with Junit

Project contains a substantial set of test cases as examples
Recognize Test-driven development

Develop code to pass a set of tests
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