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A '''zone diagram''' is a certain geometric object which a variation on the notion of [[Voronoi diagram]].
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It was introduced by [[Tetsuo Asano]], [[Jiří Matoušek (mathematician)|Jiri Matousek]], and [[Takeshi Tokuyama]] in 2007]
.<ref name=AMT2007zd/> <br />
Formally, it is a fixed point of
a certain function. Its existence or uniqueness are not clear in advance  and have been established only in specific cases.
Its computation is not obvious too.
 
==A particular but informative case==
Consider a group of <math>n</math> different points <math>\{p_1,\ldots,p_n\}</math> in the [[Euclidean plane]].
Each point is called a site. When we speak about the [[Voronoi diagram]] induced by these
sites, we associate to the site <math>\displaystyle{p_k}</math> the set <math>\displaystyle{R_k}</math> of all points
in the plane whose distance to the given site <math>\displaystyle{p_k}</math> is not greater to their
distance to any other site <math>p_j,\,j\neq k</math>. The collection <math>(R_k)_{k=1}^n</math> of
these regions is the Voronoi diagram associated with these sites, and it induces a decomposition of the [[Plane (geometry)|plane]] into
regions: the Voronoi regions (Voronoi cells).
 
In a zone diagram  the region associated with the site <math>p_k</math> is defined a little bit differently: instead of
associating it the set of all points whose distance to <math>p_k</math> is not greater than their distance to the other
sites, we associate to <math>p_k</math> the set <math>R_k</math> of all points in the plane whose distance to <math>p_k</math>
is not greater than their distance to any other region. Formally,
 
:<math>R_k=\{x\,|\,\,d(x,p_k)\leq d(x,R_j),\,\text{for all}\, j\neq k\}</math>.
Here <math>\displaystyle{d(a,b)}</math> denotes the [[euclidean distance]] between the points <math>a</math> and <math>b</math> and
<math>d(x,A)=\inf\{d(x,a) \,|\, a\in A\}</math> is the distance between the point <math>x</math> and the set <math>A</math>.
In addition, <math>x=(x_1,x_2)\in \mathbb{R}^2</math>  since we consider the plane. The tuple <math>(R_k)_{k=1}^n</math> is the zone diagram associated with the sites.
 
The problem with this definition is that it seems circular:  in order to know <math>R_k</math> we should
know <math>\displaystyle{R_j}</math> for each index <math>j,\,j\neq k</math> but each such <math>\displaystyle{R_j}</math> is
defined in terms of <math>\displaystyle{R_k}</math>. On a second thought, we see that actually
the [[tuple]] <math>(R_k)_{k=1}^n</math> is a solution of the following system of equations:
 
:<math>
\begin{cases}
R_1=\{x\in \mathbb{R}^2\,|\,\,d(x,p_1)\leq d(x,R_j),\,\text{for all}\, j\neq 1\}\\
\vdots\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\vdots\\
R_n=\{x\in \mathbb{R}^2\,|\,\,d(x,p_n)\leq d(x,R_j),\,\text{for all}\, j\neq n\}
\end{cases}
</math>
 
Rigorously, a zone diagram is any solution of this system, if such a solution exists.
This definition can be extended without essentially any change to higher dimensions, to sites which are not
necessarily points, to infinitely many sites, etc.
 
==Interpretation==
In some settings, such as the one described above, a zone diagram can be interpreted as a certain equilibrium between
mutually hostile kingdoms,.<ref name=AMT2007zd/><ref name=RR2009/> In a discrete setting it can be interpreted
as a stable configuration in a certain combinatorial game.<ref name=RR2009/>
 
==Formal definition==
Let <math>\displaystyle{(X,d)} </math> be a [[metric space]] and let <math> \displaystyle{K} </math> be a set of at least 2 elements (indices), possibly
infinite. Given a tuple <math> (P_k)_{k\in K} </math> of nonempty subsets of <math> \displaystyle{X} </math>, called the sites, a zone diagram  with respect to this [[tuple]] is a tuple <math> R=(R_k)_{k\in K} </math> of subsets of <math> \displaystyle{X} </math> such that
for all <math> k\in K</math> the following equation is satisfied:
 
:<math>
R_k=\{x\in X\,|\,\,d(x,P_k)\leq d(x,R_j),\,\text{for all}\, j\neq k\}.
</math>
 
==Zone diagram as a fixed point==
The system of equations which defines the zone diagram can be represented as a fixed point of a function
defined on a product space. Indeed, for each index <math>k\in K</math> let <math> \displaystyle{X_k}</math>  be the
set of all nonempty subsets of <math>\displaystyle{X}</math>.
Let
 
:<math>Y=\prod_{k\in K} X_k </math>
 
and let  <math>\text{Dom}:Y\to Y </math> be the function defined by <math>\displaystyle{\text{Dom}(R)=S}</math>, where <math>S=(S_k)_{k\in K}</math> and
 
:<math>
S_k=\{x\in X\,|\,\,d(x,P_k)\leq d(x,R_j),\,\text{for all}\, j\neq k\}.
</math>
 
Then  <math>\displaystyle{R}</math> is a zone diagram if and only if it is a [[fixed point (mathematics)|fixed point]] of '''Dom''',
that is, <math>R=\displaystyle{\text{Dom}(R)}</math>. Viewing zone diagrams as fixed points is useful since in some settings known tools or approaches from [[fixed point theory]] can be used for investigating them and
deriving relevant properties (existence,  etc.).
 
==Existence and uniqueness==
Following the pioneering work of Asano et al.<ref name=AMT2007zd/> (existence and uniqueness of the zone diagram in the euclidean plane with respect to
finitely many point sites), several existence or existence and uniqueness results have been published.
As of 2012, the most general results which have been published are:
 
* '''2 sites (existence):''' there exists at least one zone diagram for any pair of arbitrary sites in any [[metric space]]. In fact, this existence result holds in any [[m-space]] <ref name=RR2009/> (a generalization of [[metric space]] in which, for instance, the distance function may be negative and may not satisfy the triangle inequality).
* '''More than 2 sites (existence):''' there exists a zone diagram of finitely many [[compact]]{{dn|date=November 2012}} and disjoint sites contained in the interior of a compact and [[convex subset]] of a [[uniformly convex space]]. This result actually holds in a more general setting.<ref name=KRR20123/>
 
* '''More than 2 sites (existence and uniqueness):''' there exists a unique zone diagram with respect to any collection (possibly infinite) of closed and positively separated sites in any finite dimensional [[euclidean space]]. Positively separated means that there exists a positive lower bound on the distance between any two sites. A similar result was formulated for the case in which the normed space is  finite dimensional and is both uniformly convex and uniformly smooth.<ref name=KMT20123/>
 
* '''non-uniqueness''': simple examples are known even for the case of two point sites,.<ref name=RR2009/><ref name=KMT20123/>
 
==Computation==
More information is needed.
 
==Related objects and possible applications==
In addition to [[Voronoi diagrams]], zone diagrams are closely related to other geometric objects such as [[double zone diagrams]],<ref name=RR2009/> [[trisectors]],<ref name=AMT2007dt/> [[k-sectors]],<ref name=IKMRT2010/>
[[mollified zone diagrams]]<ref name=dBKK2011/> and as a result may be used
for solving problems related to robot motion and VLSI design,.<ref name=AMT2007dt/><ref name=IKMRT2010/>
 
==References==
<references>
 
<ref name=AMT2007zd>Asano, Tetsuo; Matoušek, Jiří; Tokuyama, Takeshi (2007), "Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge". ''SIAM Journal on Computing'' '''37''' (4): 1182––1198, {{doi|10.1137/06067095X}}, a preliminary version in Proc. SODA 2007, pp. 756-765
</ref>
 
<ref name=RR2009>
Reem, Daniel; Reich, Simeon (2009). "Zone and double zone diagrams in abstract spaces". ''Colloquium Mathematicum'' '''115''': 129––145, {{doi|10.4064/cm115-1-11}}, arXiv:0708.2668 (2007)
</ref>
 
<ref name=KRR20123>
Kopecká, Eva; Reem, Daniel; Reich, Simeon (2012), "Zone diagrams in compact subsets of uniformly convex normed spaces", ''Israel Journal of Mathematics'' '''188''', 1--23, {{doi|10.1007/s11856-011-0094-5}}, preliminary versions in Proc. CCCG 2010, pp. 17-20, arXiv:1002.3583 (2010)
</ref>
 
<ref name=KMT20123>
Kawamura, Akitoshi; Matoušek, Jiří; Tokuyama, Takeshi (2012). "Zone diagrams in Euclidean spaces and in other normed spaces". ''Mathematische Annalen'' '''354''', 1201--1221, {{doi|10.1007/s00208-011-0761-1}},  preliminary versions in Proc. SoCG 2010,
pp. 216-221, arXiv:0912.3016 (2009)
</ref>
 
<ref name=AMT2007dt>
Asano, Tetsuo; Matousek, Jiří; Tokuyama, Takeshi (2007). "The distance trisector curve". ''Advances in Mathematics'' '''212''', 338--360, {{doi|10.1016/j.aim.2006.10.006}}, a preliminary version in Proc. STOC 2006, pp. 336--343
</ref>
 
<ref name=IKMRT2010>
Imai, Keiko; Kawamura, Akitoshi; Matoušek, Jiří; Reem, Daniel.; Tokuyama, Takeshi (2010), "Distance k-sectors exist". ''Computational Geometry'' '''43''' (9): 713--720, {{doi|10.1016/j.comgeo.2010.05.001}}, preliminary versions in Proc. SoCG 2010,
pp. 210--215, arXiv:0912.4164 (2009)
</ref>
 
<ref name=dBKK2011>
Biasi, Sergio C.; Kalantari, Bahman; Kalantari, Iraj (2011). "Mollified Zone Diagrams and Their Computation".
''Transactions on Computational Science XIV''. Lecture Notes in Computer Science. '''6970''', pp. 31--59, {{doi|10.1007/978-3-642-25249-5_2}}. ISBN 978-3-642-25248-8, a preliminary version in Proc. ISVD 2010, pp. 171--180
</ref>
 
</references>
 
<!--- After listing your sources please cite them using inline citations and place them after the information they cite. Please see http://en.wikipedia.org/wiki/Wikipedia:REFB for instructions on how to add citations. --->
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[[Category:Diagrams]]

Latest revision as of 14:54, 23 December 2014

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