Deterministic Leader Election Among Disoriented Anonymous Sensors

Consider a set of $n>1$ sensors arbitrarily scattered on the plane such that no two sensors are located at the same position. A configuration ${\mathcal{C}}$ is a set of positions $p_{1},\ldots,p_{n}$ occupied by the sensors. The sensors are uniform and anonymous, i.e, they all execute the same program using no local parameter (such as an identity) allowing to differentiate any of them. However, we assume that each sensor is a computational unit having the ability to determine the positions of the $n$ sensors within an infinite decimal precision. We assume no kind of communication medium. Each sensor has its own local $x$-$y$ Cartesian coordinate system defined by two coordinate axes ($x$ and $y$), together with their orientations, identified as the positive and negative sides of the axes.

In this paper, we assume that the sensors have no Sense of Direction and we discuss the influence of Chirality in a sensor network.

In Figure 2, the sensors have sense of direction in the cases ($a$) and ($b$), whereas they have no sense of direction in the cases ($c$) and ($d$).

Given an $x$-$y$ Cartesian coordinate system, the handedness is the way in which the orientation of the $y$ axis (respectively, the $x$ axis) is inferred according to the orientation of the $x$ axis (resp., the $y$ axis).

In Figure 2, the sensors have chirality in the cases ($a$) and ($c$), whereas they have no chirality in the cases ($b$) and ($d$).

Let an alphabet $A$ be a finite set of letters. A non empty word $w$ over $A$ is a finite sequence of letters $a_{1},\ldots,a_{i},\ldots,a_{\ell}$, $\ell>0$. The concatenation of two words $u$ and $v$, denoted simply by $uv$, is equal to the word $a_{1},\ldots,a_{i},\ldots,a_{k},b_{1},\ldots,b_{j},\ldots,b_{\ell}$ such that $u=a_{1},\ldots,a_{i},\ldots,a_{k}$ and $v=b_{1},\ldots,b_{j},\ldots,b_{\ell}$. Let $\varepsilon$ be the empty word such that for every word $w$, $w\varepsilon=\varepsilon w=w$. The length of a word $w$, denoted by $|w|$, is equal to the number of letters of $w$ ($|\varepsilon|=0$).

Suppose the alphabet $A$ is totally ordered by the relation $\prec$. Then a word $u$ is said to be lexicographically smaller than or equal to a word $v$, denoted by $u\preceq v$, iff there exists either a word $w$ such that $v=uw$ or three words $r,s,t$ and two letters $a,b$ such that $u=ras$, $v=rbt$, and $a\prec b$. We write $u\prec v$ when $u\preceq v$ and $u\neq v$.

Let $k$ and $j$ be two positive integers. The $k^{th}$ power of a word $s$ is the word denoted by $s^{k}$ such that $s^{0}=\varepsilon$, and $s^{k}=s^{k-1}s$. A word $u$ is said to be primitive if and only if $u=v^{k}\Rightarrow k=1$. Otherwise ($u=v^{k}$ and $k>1$), $u$ is said to be strictly periodic.

The $j^{th}$ rotation of a word $w$, notation $Rot_{j}(w)$, is defined by:

Note that $Rot_{1}(w)=w$.

A word $w$ is said to be minimal if and only if $\forall j\in 1,\ldots,\ell$, $w\preceq Rot_{j}(w)$.

For instance, if $A=\{a,b\}$, then $a$, $b$, $ab$, $aab$, $abb$ are Lyndon words, whereas $aba$, and $abab$ are not — $aba$ is not minimal ($aab\preceq aba$) and $abab$ is not primitive ($abab=(ab)^{2}$).

The leader election problem considered in this paper is stated as follows: Given a configuration ${\mathcal{C}}$ of $n>1$ sensors, the $n$ sensors are able to deterministically agree on a same sensor $L$ called the leader.

Given a configuration ${\mathcal{C}}$, $SEC$ denotes the smallest enclosing circle of the positions of the sensors in ${\mathcal{C}}$. The center of $SEC$ is denoted by $\mathtt{O}$. The distance from $\mathtt{O}$ to any point on $SEC$, the radius of $SEC$, is denoted by $\sigma$. In any configuration ${\mathcal{C}}$, $SEC$ is unique and can be computed by any sensor in linear time [Meg83, Wel91]. It passes either through two of the positions that are on the same diameter (opposite positions), or through at least three of the positions in ${\mathcal{C}}$. Note that if $n=2$, then $SEC$ passes both sensors and no sensor can be located inside $SEC$, in particular at $\mathtt{O}$.

Given a smallest enclosing circle $SEC$, the radii are the line segments from the center $\mathtt{O}$ of $SEC$ to the boundary of $SEC$. Let $\mathcal{R}$ be the finite set of radii such that a radius $r$ belongs to $\mathcal{R}$ iff at least one sensor is located on $r$ but $\mathtt{O}$. Denote by $\sharp\mathcal{R}$ the number of radii in $\mathcal{R}$. In the sequel, we will abuse language by considering radii in $\mathcal{R}$ only. Given two distinct positions $p_{1}$ and $p_{2}$ located on the same radius $r$ ($\in\mathcal{R}$), $d(p_{1},p_{2})$ denotes the Euclidean distance between $p_{1}$ and $p_{2}$.

Since $\mathcal{R}$ and the set of positions are finite, the set of different distances between positions of sensors on the radii in $\mathcal{R}$ is finite. Every robot codes each of these distances $x$ using a fonction $Code(x)$ such that the output of $Code(x)$ is a letter of an arbitrary alphabet provided with an order relation, and the code respects the natural order on distances (i.e. for letters $a_{1}$, $a_{2}$ and distances $d_{1}$, $d_{2}$, $a_{1}\leq a_{2}$ if and only if $d_{1}\leq d_{2}$). A similar code is used for the angles in Definition 3.3.

Note that all the distances are computed by each sensor with respect to its own coordinate system, i.e., proportionally to its own measure unit. So, for any radius $r\in\mathcal{R}$, all the sensors compute the same word $\rho_{r}$. For instance, in Figure 3, for every sensor, $\rho_{r_{1}}=c$ and $\rho_{r_{2}}=\rho_{r_{3}}=ab$.

Denote by $\circlearrowright$ an arbitrary orientation of $SEC$, i.e., $\circlearrowright$ denotes either the clockwise or the counterclockwise direction¹¹1Note that the clockwise (resp. counterclockwise) direction may be the counterclockwise (resp. clockwise) direction for the robots depending on their handedness. However, without loss of generality, we can assume that both directions are those we commonly use.. Given an orientation $\circlearrowright$ of $SEC$, $\circlearrowleft$ denotes the opposite orientation. Let $r$ be a radius in $\mathcal{R}$, the successor of $r$ in the $\circlearrowright$ direction, denoted by $Succ(r,\circlearrowright)$, is the next radius in $\mathcal{R}$, according to $\circlearrowright$. The $i^{th}$ successor of $r$, denoted by $Succ_{i}(r,\circlearrowright)$, is the radius such that $Succ_{0}(r,\circlearrowright)=r$, and $Succ_{i}(r,\circlearrowright)=Succ(Succ_{i-1}(r,\circlearrowright),\circlearrowright)$. Given $r$ and its successor $r^{\prime}=Succ(r,\circlearrowright)$, $\sphericalangle(r\mathtt{O}r^{\prime})^{\circlearrowright}$ denotes the angle between $r$ and $r^{\prime}$ following the $\circlearrowright$ direction.

In the sequel, by abuse of language, we will refer to distances and angles without explicitely mentionning the code, to ease the reading.

In Figure 3, assuming that $\circlearrowright$ denotes the clockwise direction, then:
$\omega(r_{1})^{\circlearrowright}=(c,\beta)(ab,\alpha)(ab,\beta)$, $\omega(r_{2})^{\circlearrowright}=(ab,\alpha)(ab,\beta)(c,\beta)$, $\omega(r_{3})^{\circlearrowright}=(ab,\beta)(c,\beta)(ab,\alpha)$,
$\omega(r_{1})^{\circlearrowleft}=\omega(r_{1})^{\circlearrowright}$, $\omega(r_{2})^{\circlearrowleft}=\omega(r_{3})^{\circlearrowright}$, and $\omega(r_{3})^{\circlearrowleft}=\omega(r_{2})^{\circlearrowright}$.

In Figure 4, configuration words are strictly periodic (we can distinct two configuration words, $w=((d,\alpha),(ab,\alpha))^{2}$ and $w^{\prime}=((ab,\alpha)ad,\alpha))^{2}$) and the figure is periodic since the set of positions of the sensors is preserved by the rotation of center $\mathtt{O}$ and of angle $\pi$.

Notice the following important fact, which is fondamental for the statement of the following results:

Proof. Let $\mathcal{P}=\{p_{1},\ldots,p_{n}\}$ be a configuration of sensors. If $w(r)$ is strictly periodic for some radius $r$, that is $w(r)=u^{k}$ for some integer $k>2$ and a word $u$, then there exists $k$ rotations $Rot_{j}$, $1\leq j\leq k$ of center $\mathtt{O}$ and of angle $\frac{2\pi j}{k}$ such that $Rot_{j}(\mathcal{P})=\mathcal{P}$. $\Box$

In this section, we consider sensors with chirality. In particular, they are able to agree on a common orientation $\circlearrowright$ of $SEC$. Let $CW$ be the set of configuration words computed by the sensors over $\mathcal{R}$ according to $\circlearrowright$. Since for every sensor, there is no ambiguity on $\circlearrowright$, in the sequel of this section, we omit $\circlearrowright$ in the notations when it is clear in the context. For instance in Figure 3,

The lexicographic order $\preceq$ on the set of radius words over $\mathcal{R}$ is naturally built over the natural order $<$ on the set of codes of distances (we recall that is is compatible with the natural order on real numbers).

In this section, we show that the problems of the existence of a unique Lyndon word and of the existence of a deterministic Leader Election protocol are equivalent.

Proof. Directly follows from Corollary 3.10: If there is a sensor $s$ located on $\mathtt{O}$, then the $n$ sensors are able to agree on $L=s$. Otherwise, there exists a single radius $r\in\mathcal{R}$ such that $\omega(r)$ is a Lyndon word. In that case, all the sensors are able to agree on the sensor on $r$ that is the nearest one from $\mathtt{O}$. $\Box$

Proof. Assume by contradiction that no radius $r\in\mathcal{R}$ exists such that $\omega(r)$ is a Lyndon word and that there exists an algorithm $A$ allowing the $n$ sensors to deterministically agree on a same sensor $L$. From Remark 3.6, there is no sensor located on $\mathtt{O}$, otherwise for all $r\in\mathcal{R}$, $w(r)=(0,0)$ would be a Lyndon word.

Let $min_{\omega}$ be a word in $CW$ such that $\forall r\in\mathcal{R}$, $min_{\omega}\preceq\omega(r)$. That is, $min_{\omega}$ is minimal. Assume first that $min_{\omega}$ is primitive. Then, $min_{\omega}$ is a Lyndon word that contradicts the assumption. So, $min_{\omega}$ is a strictly periodic word (there exist $u$ and $k>1$ such that $min_{\omega}=u^{k}$) and, from Lemma 2.3, we deduce that for all $r\in\mathcal{R}$, $w(r)$ is also strictly periodic. Since by Remark 3.5 our construction is period-free, this implies that the configuration is periodic. Thus, for every $r\in\mathcal{R}$, there exists at least one radius $r^{\prime}\in\mathcal{R}$ such that $r\neq r^{\prime}$ and $\omega(r)=\omega(r^{\prime})$. So if an algorithm elects a leader on $r$ in a deterministical way, this algorithm distinguishes another leader on $r^{\prime}$. In that case, $A$ cannot allow the $n$ sensors to deterministically agree on a same sensor $L$. $\Box$

Figure 4 presents an example of a configuration where the sensors have the same measure unit and their $y$ axis meets the radius on which they are located. In this case, no radius $r\in\mathcal{R}$ exists such that $\omega(r)$ is a Lyndon word and, with respect to Lemma 3.12, sensors are not able to deterministically agree on a same sensor $L$.

The following theorem follows from Lemmas 3.11 and 3.12:

The acute reader will have noticed that when $n=1$ electing a leader is straightforward. On the other hand, when $n=2$, it is impossible to elect a unique leader in a deterministic way: Our theorem confirms this fact as the two words obtained for the two radii are the same in such a configuration (actually each of both words consists in the concatenation of two identical letters).

Devoid of chirality, the sensors are no longer able to agree on a common orientation $\circlearrowright$ of $SEC$. In other words, with respect to their handedness, some of the $n$ sensors may assign $\circlearrowright$ to the actual clockwise direction, whereas some other may assign $\circlearrowright$ to the counterclockwise direction. As a consequence, given a radius $r\in\mathcal{R}$ and two sensors $s$ and $s^{\prime}$ on $r$, the configuration word $\omega(r)^{\circlearrowright}$ computed by $s$ may differ from the one computed by $s^{\prime}$, depending on whether $s$ and $s^{\prime}$ share the same handedness or not. For instance, in Figure 3, the configuration word $\omega(r_{2})^{\circlearrowright}$ (as defined by Definition 3.3) is equal to either $(ab,\alpha)(ab,\beta)(c,\beta)$ or $(ab,\beta)(c,\beta)(ab,\alpha)$, depending on the fact $\circlearrowright$ denotes the clockwise direction or the counterclockwise, respectively.

Furthermore, given a radius $r_{i}\in\mathcal{R}$, if there exists a radius $r_{j}\in\mathcal{R}$ such that $\omega(r_{i})^{\circlearrowleft}=\omega(r_{j})^{\circlearrowright}$, then there exists an axial symmetry which corresponds to the bissectrix of $\sphericalangle(r_{i}\mathtt{O}r_{j})$. Otherwise (i.e., no radius $r_{j}$ exists such that $\omega(r_{i})^{\circlearrowleft}=\omega(r_{j})^{\circlearrowright}$), no such axial symmetry exists. This observation leads us to provide the following definition in order to classify the radii.

We use the type of symmetry to define the strong configuration word of any radius in $\mathcal{R}$ as follows:

Note that the (weak) configuration words used in Section 3 (Definition 3.3) could have been defined as strong configuration words by using Constant $0$ as type of symmetry, i.e., by mapping each letter $(\rho,\alpha)$ to $(0,\rho,\alpha)$. We discuss this issue in more details in the next section.

Notice also that the construction of strong configuration words has obviously the same important property as the construction of weak configuration words:

Devoid of chirality, the sensors cannot agree on a common orientation of $SEC$. So, for each radius $r\in\mathcal{R}$, each sensor computes two strong configuration words, one for each direction, $W(r)^{\circlearrowright}$ and $W(r)^{\circlearrowleft}$. Let $CW$ be the set of strong configuration words computed in both directions by the sensors for each radius in $\mathcal{R}$. In Figure 3, all the sensors compute the following set:

Similarly to Section 3, we build the lexicographic order $\preceq$ on the set of radius words over $\mathcal{R}$ as follows:

The proof of the following lemma is similar to the one of Lemma 3.9:

Let $\mathcal{R}_{L}$ be the subset of radii $r\in\mathcal{R}$ such that $W(r)$ is a Lyndon word in the clockwise or in the counterclockwise orientation. Denote by $\sharp\mathcal{R}_{L}$ the number of radii in $\mathcal{R}_{L}$.

The following result directly follows from Lemma 4.7 and Corollary 4.8:

Note that Lemma 4.9 applies necessarily when $\sharp\mathcal{R}_{L}>2$. Indeed, this means that there exists at least two distinct radii $r_{1}$ and $r_{2}$ such that $w(r_{1})$ and $w(r_{2})$ are Lyndon words for a common orientation $\circ\in\{\circlearrowright,\circlearrowleft\}$ and by Lemma 3.9, this implies that $CW^{\circ}=\{(0,0)\}$, so that $CW=\{(0,0)\}$. So, in the remainder of this section, we study the case $\sharp\mathcal{R}_{L}\leq 2$.

Proof. The leader is the nearest sensor to $\mathtt{O}$, on $r_{\ell}$. $\Box$

Note that, in this case, $r_{\ell}$ is necessarily $0$-symmetric. An example of such a configuration is given by Figure 3: $W^{\circlearrowright}(r_{1})=W^{\circlearrowleft}(r_{1})=(0,c,\beta)(1,ab,\alpha)(1,ab,\beta)$ is a Lyndon word. The leader is the sensor on $r_{1}$.

Proof. Without loss of generality we can consider that $W(r_{1})^{\circlearrowright}$ and $W(r_{2})^{\circlearrowleft}$ are Lyndon words. We have three cases to consider:

1. 1.

   Radii $r_{1}$ and $r_{2}$ cannot be of different types. Assume by contradiction they can. Without loss of generality we can consider that $r_{1}$ is of type $0$ and $r_{2}$ is of type $1$. So, from Definition 4.1, we deduce there is another radius of type $1$ which is also a Lyndon word. So, we would have three radii corresponding to a Lyndon word, which contradicts Corollary 4.8.

2. 2.

   For the same reason, if $r_{1}$ and $r_{2}$ are both $1$-symmetric, then necessarily $W(r_{1})^{\circlearrowright}=W(r_{2})^{\circlearrowleft}$. In this case, if an algorithm elects a leader on $r_{1}$ in a deterministical way, this algorithm distinguishes another leader on $r_{2}$—refer to Figure 5 for an example of such a configuration. So an algorithm cannot allow the $n$ sensors to deterministically agree on a same sensor $L$.

3. 3.

   If $r_{1}$ and $r_{2}$ are both $0$-symmetric, then $W(r_{1})^{\circlearrowright}\neq W(r_{2})^{\circlearrowleft}$. Without loss of generality, we can consider that $W(r_{1})^{\circlearrowright}<W(r_{2})^{\circlearrowleft}$. Then the $n$ sensors are able to deterministically agree on the sensor that is the nearest to $\mathtt{O}$ on $r_{1}$.

$\Box$

Proof. Assume by contradiction that there exists no radius $r\in\mathcal{R}$ on the center of $SEC$ or such that $W^{\circlearrowleft}(r)$ or $W^{\circlearrowright}(r)$ is a Lyndon word and that there exists an algorithm $A$ allowing the $n$ sensors to deterministically agree on a same sensor $L$. Let $min^{\circ}_{W}$ be a word in $CW$ such that $\forall r\in\mathcal{R}$, $min^{\circ}_{W}\preceq W(r)$. Suppose w.l.o.g. that $\circ=\circlearrowleft$. That is, $min^{\circlearrowleft}_{W}$ is minimal. Assume first that $min^{\circlearrowleft}_{W}$ is primitive. Then it is a Lyndon word that contradicts the assumption. So, $min^{\circlearrowleft}_{W}$ is a strictly periodic word and from Lemma 2.3, we deduce that for all $r\in\mathcal{R}$, $w^{\circlearrowleft}(r)$ is also strictly periodic. Since by Remark 4.3 our construction is period-free, this implies that the configuration is periodic. Thus, for every $r\in\mathcal{R}$, there exists at least one radius $r^{\prime}\in\mathcal{R}$ such that $r\neq r^{\prime}$ and $W^{\circlearrowleft}(r)=W^{\circlearrowleft}(r^{\prime})$.

So if an algorithm elects a leader on $r$ in a deterministical way, this algorithm distinguishes another leader on $r^{\prime}$. In that case, $A$ cannot allow the $n$ sensors to deterministically agree on a same sensor $L$. $\Box$

The four previous lemmas lead to:

As noticed in this section, the (weak) configuration words built in Section 3 (Definition 3.3) can be easily defined in terms of strong configuration words (Definition 4.2) by mapping each pair $(\rho,\alpha)$ to $(0,\rho,\alpha)$. Thus, Theorem 3.13 also holds over the strong configuration words built on $CW$ assuming chirality.

So, we can formulate Theorems 3.13 and 4.15 more generally: