Reconstruction of Weakly Simple Polygons from their Edges

If $G=(V,E)$ is undirected, the algorithm construct an Euler tour $P$ [12]; and the tour is weakly simple by construction. In the remainder of the proof, we consider a directed Eulerian geometric graph $G$. First, we show that if $G$ satisfies the conditions of Theorem 3.1, then Algorithm A returns a weakly simple Euler tour. If the circular order of edges around each vertex alternates between incoming and outgoing, then all edges on the boundary of a face of $G$ have the same orientation (ccw or cw), and adjacent faces have opposite orientations. Without loss of generality, the edges on the boundaries of white (resp., gray) faces are oriented ccw (resp., cw). Hnece, the condition in step 2 of the algorithm is satisfied.

We show that the Euler tour $P$ constructed by Algorithm A is weakly simple, that is, it can be perturbed into a simple polygon. Initially, each circuit $C=(p_{0},\ldots,p_{k-1})$ in $L$ is the boundary of a gray face, and hence it is a simple polygon. Let $C^{\prime}=(p_{0}^{\prime},\ldots,p_{k-1}^{\prime})$ be perturbation obtained by moving each point $p_{i}$ to the interior of the face along an angle bisector of $\angle p_{i-1}p_{i}p_{i+1}$. Initially $P$ is a weakly simple polygon (one of the circuits). It is enough to show that Step 4.2 maintains a weakly simple polygon, that is, when we merge $P$ and a circuit $C$, their Jordan curve perturbations $P^{\prime}$ and $C^{\prime}$ can also be combined. Edges $(u,v)$ and $(v,w)$ are adjacent to a common white face $f_{0}$; they correspond to an edge $(p_{u},p_{v})$ in $P^{\prime}$ and $(q_{v},q_{w})$ in $C^{\prime}$, where both $p_{v}$ and $q_{v}$ lie in the $\varepsilon$-neighborhood of $v$ in two different gray faces adjacent to $f_{0}$. We can modify $P^{\prime}$ and $C^{\prime}$ in the $\varepsilon$-neighborhood of $v$, by removing a short Jordan arc from each and reconnecting them across the white face $f_{0}$ into a single Jordan curve. By induction, we can obtain a Jordan curve within $\varepsilon$ Frèchet distance from the output polygon $P$. Hence, the algorithm returns a weakly simple Euler tour $P$.

Now, we show that if $G$ has a vertex $v$ with two consecutive incoming (resp., outgoing) edges $(a,v)$ and $(c,v)$, then $G$ does not admit a weakly simple Euler tour. Suppose, for contradiction, that there exists a weakly simple Euler tour $P$. Since both $(a,v)$ and $(c,v)$ are directed into $v$, the tour $P$ contains edge-disjoint paths $(a,v,b)$ and $(c,v,d)$. Since $P$ is weakly simple, the circular order of these four edges incident to $v$ must be as shown in Figure 2(b). The polygon must contain edge-disjoint paths $\pi_{1}=(v,b,\ldots,c,v)$ and $\pi_{2}=(d,\ldots,a)$. The perturbation of $\pi_{1}$ is $\pi_{1}^{\prime}=(v^{\prime},b^{\prime},\ldots,c^{\prime},v^{\prime\prime})$ where $v^{\prime}\neq v^{\prime\prime}$. Note that $a$ and $d$ are on opposite sides of the cycle $\pi^{\prime}\cup v^{\prime}v^{\prime\prime}$. The perturbation of $\pi_{2}$, path $\pi_{2}^{\prime}$, can intersect neither $\pi_{1}^{\prime}$ nor $v^{\prime}v^{\prime\prime}$, because $(a,v)$ and $(c,v)$ are adjacent to the same face. Hence $P$ is not weakly simple.

Finally, Algorithm A runs in $O(|E|)$ time. Step 1 and 2 can be done by traversing the dual graph of $G$. Step 4 executes $O(|E|)$ merges, each of which takes constant time.

Replace every edge $e$ of multiplicity $k$ by $k$ edge-disjoint paths of length two whose interior points are close to the midpoint of $e$. We obtain a simple Eulerian geometric graph with $|V|+2|E|$ vertices. Theorem 3.1 completes the proof.

The proof of the upper bound is constructive. The algorithm subdivides edges within each bar independently. Let $b$ be a bar containing $m$ vertices. We apply $O(m)$ edge subdivisions and partition the edges in $b$ into subsets: Subsets $\mathcal{M}^{+}$ and $\mathcal{M}^{-}$ will consists of subdivision paths between the intersection points of $b$ with other bars lying above and below $b$, respectively; all remaining edges will be partitioned into tours (each of which is a weakly simple polygon by Theorem 3.6). The algorithm is divided into three phases: Phase 1 creates $\mathcal{M}^{+}$ and $\mathcal{M}^{-}$; phase 2 forms circuits; and phase 3 establishes common vertices between the subdivision paths and circuits. Refer to Figure 6.

We recall some notation introduced in [6]. For every vertex $v\in V$, let $D_{v}$ be a disk centered at $v$ of radius $\frac{\varepsilon}{4}$. For a bar $b$ between $u_{0}$ and $u_{k}$, let $D_{b}$ be the $\varepsilon^{2}$ neighborhood of $b$ setminus $D_{u_{0}}\cup D_{u_{k}}$. Assume that $\varepsilon\in(0,\frac{1}{4})$ is so small that the disks $D_{v}$ are pairwise disjoint; a disk $D_{v}$ intersects $D_{b}$ only if $v\in b$, and the neighborhoods $D_{b}$ are pairwise disjoint.

For each bar $b$ with vertices $u_{0},\ldots,u_{k}$, we perturb the edges of $E^{*}$ contained in $b$ into noncrossing simple polygons and polygonal chains. Embed each subdivided path in $\mathcal{M}^{+}$ (resp., $\mathcal{M}^{-}$) $\widehat{u_{i}u_{j}}$ in the upper (resp., lower) boundary of the region $D_{b}$ such that $u_{i}$ is on the boundary of $D_{u_{i}}$ and $u_{j}$ is on the boundary of $D_{u_{j}}$. Subdivide $D_{b}$ with $\ell+1$ horizontal lines where $\ell$ is the number of components of the subgraph induced by $\mathcal{B}^{\prime}$. Embed all edges in $\mathcal{O}^{+}$ (resp., $\mathcal{O}^{-}$) in the first (resp., $\ell+1$-th) such line.

Recall that every vertex of $G$ has even degree. If $b$ contains a $b$-odd vertex $p$, there must exist a bar $b^{\prime}$ such that $p$ is $b^{\prime}$-odd. Because $\mathcal{M}^{+}$ and $\mathcal{M}^{-}$ matches such points, the subgraph induced by $\mathcal{B}$ contain only even degree vertices. By construction the edges in $\mathcal{O}^{+}\cup\mathcal{O}^{-}$ form nonoverlapping disjoint circuits. Hence, the subgraph induced by $\mathcal{B}^{\prime}$ contains only even degree vertices. Consequently, each of its components is Eulerian and forms a weakly simple polygons that we denote by $\gamma_{1}(b),\ldots,\gamma_{\ell}(b)$, sorted by the $x$-coordinates of their left endpoints. Perturb $\gamma_{1}(b),\ldots,\gamma_{\ell}(b)$ into simple polygons that lie in the interior of $D_{b}$, separated by one of the $\ell+1$ lines, in this linear order (ties are broken arbitrarily). For $i=0,\ldots,k$, consider all polygons $\gamma_{j}(b)$ whose leftmost vertex is $u_{i}$. Connect the left endpoints of each such $\gamma_{j}(b)$ to the copy of $u_{i}$ in $\mathcal{O^{+}}$ by two polygonal paths within $D_{u_{i}}$ (these paths connect different copies of vertex $u_{i}\in V^{*}$, see Figure 6(f)). Similarly, for each subdivision performed in phase 2 at $u_{i}$ of an edge in $\mathcal{O^{+}}$ (resp., $\mathcal{O^{-}}$) in a path $\widehat{r_{1}r_{2}}$, connect the copy of $u_{i}$ in this path to a copy in $\mathcal{M}^{+}$ (resp., $\mathcal{O^{-}}$) by two polygonal paths within the disk $D_{u_{i}}$. Connect the endpoints of the overlapping paths in $\mathcal{O^{+}}$ and $\mathcal{O^{-}}$ (forming a cycle of each), and if the right endpoint of $b$ is not $b$-odd and was added to $B^{+},B^{-}$, connect the copies of $u_{k}$ in $\mathcal{M}^{+}$ and $\mathcal{M}^{-}$. For each matching in $B^{+}$ and $B^{-}$ involving a point $u_{i}$ that is a $b^{\prime}$-odd endpoint of a bar $b^{\prime}$, connect the paths in $\mathcal{M}^{+}$ or $\mathcal{M}^{-}$ that correspond to a match in $b$ to the path in $\mathcal{M}^{+}$ of $b^{\prime}$ that contains $p$. Finally, for each point $u_{i}$ that is the endpoint of a bar $b^{\prime}$ and is $b^{\prime}$-even, connect the corresponding copies of $u_{i}$, making the graph induced by all edges containing a point on $b$ connected. This concludes the construction of $G^{\prime}$.

Theorem 3.1 completes the proof: An Euler tour $\widehat{P}$ of $G^{\prime}$ can be perturbed into a simple polygon $P$ such that ${\rm dist}_{F}(P,\widehat{P})<\frac{\varepsilon}{2}$. The tour $\widehat{P}$ maps to an Euler tour $P^{*}$ of $G^{*}$ by identifying the vertices that lie in the same disk $D_{v}$, $v\in V^{*}$; and ${\rm dist}_{F}(\widehat{P},P^{*})<\frac{\varepsilon}{2}$.

Our lower bound construction is shown in Figure 7(a). It consists of a graph $G=(V,E)$ containing a long edge $e_{R}$ (shown in red) and a path of $(|E|+5)/7$ non-overlapping collinear edges that connects the endpoints of $e_{R}$. Each vertex in the interior of $e_{R}$ is also incident to two small cycles above and below $e_{R}$ respectively. Although the graph is Eulerian, it does not admit a weakly simple Euler tour. Each vertex $p$ in the interior of the red edge $e_{R}$ is incident to two small triangles. Suppose that $e_{R}$ is not subdivided at $p$. Then $p$ has degree 6. In any perturbation of a weakly simple Euler tour, vertex $p$ is split into 3 copies, each of degree 2, and each lying above or below $e_{R}$. Suppose only one copy of $p$ lies below $e_{R}$. Then it is incident to two edges of a small triangle below $e_{R}$, which is then disconnected from the rest of the graph, a contradiction. Consequently, $e_{R}$ must be subdivided at all interior vertices. Figure 7(b) shows that $O(|E|)$ subdivisions of $e_{R}$ suffice in this case.

We construct $E^{+}$ as follows. Partition $E$ into bars (equivalence classes of the transitive closure of the overlap relation on $E$). Denote by $b$ the union of edges in a bar. W.l.o.g., we may assume that $b$ is horizontal. Denote by $u_{0},\ldots,u_{k}\in V$ the vertices of $V$ along $b$ sorted by $x$-coordinates (where $b=u_{0}u_{k}$). For $i=1,\ldots,k$, add an edge $u_{i-1}u_{i}$ to $E^{+}$ if the edges of $E$ in the bar cover the line segment $u_{i-1}u_{i}$ an odd number of times. The old and new edges in the bar $b$ jointly form a graph of even degree that we denote by $G(b)$. By Theorem 3.6, every component of $G(b)$ admits a weakly simple Euler tour. Finally, add two more copies of edge $u_{i-1}u_{i}$ to $E^{+}$ for all $i=1,\ldots k$. After repeating the above steps for every bar, we have $\|E^{+}\|\leq 3\|E\|$ and $G^{+}=(V,E\cup E^{+})$ is Eulerian.

We omit the proof of correctness (which is provided in Appendix A.), that shows that $G^{+}$ admits a weakly simple Euler tour, since it is similar to the proof of Theorem 5.1.

(1) Refer to Figs. 7(a)–(c). Let $n\in\mathbb{N}$ and $\delta\in(0,\frac{1}{3})$. Place vertices $v_{i}=(i,0)$, for $i=0,\ldots,n$, on the $x$-axis. A red edge of length $n$ connects $v_{0}$ and $v_{n}$. A black edge of length $1/n$ connects $v_{i-1}$ and $v_{i}$ for $1\leq i\leq n$. A small cycle of length $\delta<\frac{1}{3}$ is placed on each $v_{i}$, $1\leq i\in n-1$, on each side of the $x$-axis. The total length of the construction is $\|E\|=2n+2(n-1)\delta$.

Let $G^{+}=(V,E\cup E^{+})$ be a multigraph in which $P$ is a weakly simple Euler tour; and let $P^{\prime}$ be an $\varepsilon$-perturbation into a simple polygon, for some $0<\varepsilon<\delta$. We define a pair of vertical lines $\ell_{i}^{-}:x=i+\delta$ and $\ell_{i}^{+}:x=(i+1)-\delta$, for $0\leq i\leq n-1$. The portion of $P^{\prime}$ between any two of these lines consists of disjoint paths whose endpoints are on the lines. By Morse theory, $P^{\prime}$ contains an even number of paths between any two of these lines; and the length of such a path is at least the distance between the parallel lines. The input edges already contain two line segments between any two of these lines: a red and a black segment.

We claim that $G^{+}$ contains at least 4 paths between $\ell_{i}^{-}$ and $\ell_{i}^{+}$ for all but at most one index $0\leq i\leq n-1$. Indeed, suppose that there are two such paths between $\ell_{i}^{-}$ and $\ell_{i}^{+}$ and between $\ell_{j}^{-}$ and $\ell_{j}^{+}$ ($0\leq i<j\leq n-1$). We may assume w.l.o.g. that the black edge is above the red edge between $\ell_{i}^{-}$ and $\ell_{i}^{+}$. Then the black edge must be above the red edge between $\ell_{j}^{-}$ and $\ell_{j}^{+}$, as well. Consequently, $P^{\prime}$ cannot reach the small cycles at $v_{i+1},\ldots,v_{j}$. This confirms the claim. It follows that $\|E\cup E^{+}\|\geq(4n-2)(1-2\delta)$. This lower bound tends to $2\|E\|$ as $n\rightarrow\infty$ and $\delta n\rightarrow 0$.

Due to space restrictions, we omit the proofs for cases 2 and 3.