# Recursion problem - Graph theory - Algorithm needed

Discussion in 'C++' started by JimC, Jan 17, 2004.

1. ### JimCGuest

I pose a question here concerning what I think is a classic computer problem,
but I don't know of an algorithm for solving it. It resembles the Towers of
Hanoi prolem.

A few matrices follow. In order to make them more easily understood, it would
be helpful if the reader can disable proportional spacing, although this is not
essential.

Here is the problem. We have a matrix containing elements, represented here
as single upper case letters, and one element with value x. To make this
concrete, let this be the following 4 x 4 matrix in Figure 1:

G B C J
M E O L
x A N D
F H K I

Figure 1

We want to rearrange the elements so that they are ordered beginning with the x
element, followed by the upper-case letters arranged alphabetically, as
shown in Figure 2.

x A B C
D E F G
H I J K
L M N O

Figure 2

But we must use the following rule:

An element can be moved only if it is next to the x element,
in the same row or column, and it must be exchanged with the x element.

So, a possible re-configuration of Figure 1 would be as shown
in Figure 3:

G B C J
M E O L
A x N D
F H K I

Figure 3

where the element with value A in the third row has been exchanged with the
x element to its left.

Does anyone know of an efficient algorithm for producing the steps that
lead to the ordered arrangement in Figure 2?

JimC, Jan 17, 2004

2. ### Victor BazarovGuest

"JimC" <> wrote...
> I pose a question here concerning what I think is a classic computer

problem,
> but I don't know of an algorithm for solving it. [...]

I strongly suggest comp.programming for questions of this sort.

Victor

Victor Bazarov, Jan 17, 2004

3. ### David HarmonGuest

On Sat, 17 Jan 2004 03:55:15 GMT in comp.lang.c++, "JimC"
<> was alleged to have written:
>But we must use the following rule:
>
> An element can be moved only if it is next to the x element,
> in the same row or column, and it must be exchanged with the x element.

David Harmon, Jan 17, 2004
4. ### JimCGuest

"David Harmon" <> wrote in message
news:...

> On Sat, 17 Jan 2004 03:55:15 GMT in comp.lang.c++, "JimC"
> <> was alleged to have written:
> >But we must use the following rule:
> >
> > An element can be moved only if it is next to the x element,
> > in the same row or column, and it must be exchanged with the x element.

>
>

Thanks!

Ah.... so then the problem is informally known as "the 15-Puzzle" and is
solvable iff the sum of inversions in the initial layout is an even number. The
puzzle seems to have been around at least since the 1870s.

Of the many articles turned up by the preceding Google search, I thought
the following following two were quite good:

http://www.javaonthebrain.com/java/puzz15/
http://mathworld.wolfram.com/15Puzzle.html

JimC, Jan 17, 2004