Commit f4e69c19 authored by Michael Keller's avatar Michael Keller
Browse files

Added reference

parent 09c5fb2e
......@@ -49,7 +49,8 @@ size $X \cdot Y$ and number of crops $C$.
It is known that finding Hamiltonian Paths is NP
complete. Thus, reducing the decision version
complete \cite[Page 199, GT39]{alma990005774300205503}.
Thus, reducing the decision version
of hamiltonian paths (explained below) to the decision
version of pixel farming demonstrates that pixel
farming is NP complete.
......@@ -112,10 +113,13 @@ The following example illustrates this:
\end{minipage}
\end{figure}
\FloatBarrier
Now we pass the Pixel Farming problem formulated above to our algorithm
that solves pixel farming problems in polynomial time.
It should be noted that is problem can be built
in polynomial time in relation to the field size and number
of crops $C$. Now we pass the pixel farming problem formulated
above to our algorithm
that solves the decision version of the pixel farming problem.
If we get the answer that no solution
with value at least $s$ exists we return that no longest path of
with value at least $s$ exists, we return that no longest path of
length $\abs{V} - 1$ exists. If on the other hand a solution
of at least $s$ exists, we return that a hamiltonian path
must exist in $G$.
......@@ -124,9 +128,8 @@ The rationale for this is as follows. Let us assume
a solution of value at least $s$ exists. This would
imply that every pixel likes all of their neighbors.
As "crop $a$ likes crop $b$" is the same function as
"vertex $a$ is connected to vertex $b$" and we have
a value for our arrangement of $2 \cdot (\abs{V} - 1)$, we must have
found $n-1$ edges that connect our $\abs{V}$ vertices. Because we also
"vertex $a$ is connected to vertex $b$", we must have
found $\abs{V}-1$ edges that connect our $\abs{V}$ vertices. Because we also
know that every crop can only appear once in our field and all crops
have two neighbors (except the ends) we must have found a
path of length $\abs{V} - 1$.
......
......@@ -4,13 +4,10 @@
year={1996},
publisher={Hartley \& Marks}
}
@book{alma991170823569005501,
author = {Cormen, Thomas H.},
edition = {Third edition},
isbn = {978-0-262-03384-8},
keywords = {Computer programming},
language = {eng},
publisher = {The MIT Press},
title = {Introduction to algorithms},
year = {2009},
@book{alma990005774300205503,
title = {COMPUTERS AND INTRACTABILITY A Guide to the Theory of NP-Completeness},
author = {Michael R. Garey, David S. Johnson},
isbn = {0-7167-1044-7},
year = {1979},
publisher = {Bell Telephone Laboratories, Incorporated},
}
\begin{thebibliography}{}
\begin{thebibliography}{1}
\bibitem{alma990005774300205503}
David S.~Johnson Michael R.~Garey.
\newblock {\em COMPUTERS AND INTRACTABILITY A Guide to the Theory of
NP-Completeness}.
\newblock Bell Telephone Laboratories, Incorporated, 1979.
\end{thebibliography}
......@@ -2,47 +2,45 @@ This is BibTeX, Version 0.99d (TeX Live 2021)
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