How to Walk Your Dog in the Mountains with No Magic Leash

Abstract

We describe a $O(\log n )$-approximation algorithm for computing the homotopic Fréchet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic Fréchet distance, or the minimum height of a homotopy, is in NP.

Appendix: Sweeping a Convex Polytope, Star Unfolding, and Banana Peels

Consider a convex polytope $\mathcal {P}$ in three dimensions, a base point $b$ on its boundary, and the problem of finding the minimum length leash needed for a guard that walks on the polytope such that the leash sweeps over all the points on the surface of the polytope. Specifically, at any point in time, the guard maintains a connection to the base point $b$ via a path (i.e., the leash) connecting it to the base point, and the leash has to move continuously as the guard moves around.

For a point $p$ on the boundary of $\mathcal {P}$, let $d_{\mathcal {P}}\left( {p}\right)$ be the geodesic distance from $b$ to $p$ (i.e., the length shortest path $\zeta$ that lies on the boundary of $\mathcal {P}$ connecting $b$ to $p$). Let $M$ be the medial axis of this distance—formally, a point $p$ is on the medial axis if there are two distinct shortest paths $\zeta$ and $\psi$ from $b$ to $p$, such that $\Vert \zeta \Vert = \Vert \psi \Vert = d_{\mathcal {P}}\left( {p}\right) .$ It is known that $M$ is a tree in this case [1].

Now, let ${\varPi }$ be the union of all the shortest paths from $b$ to the vertices of $\mathcal {P}$ (we assume that no vertex is on the medial axis, which holds under general position assumption). The set ${\varPi }$ is also a tree. Surprisingly, if you cut $\partial {\mathcal {P}}$ along ${\varPi }$, then the resulting polygon can be flattened on the plane. Maybe even more surprisingly, this even holds if one cuts $\partial {\mathcal {P}}$ along $M$. This is known as star unfolding of a polytope, see Agarwal et al. [1] for details.

Consider cutting $\partial {\mathcal {P}}$ along both $M$ and ${\varPi }$. This breaks $\mathcal {P}$ into a collection of polygons $\mathcal {Q}$, where each polygon ${Q}\in \mathcal {Q}$, has no vertices of $\mathcal {P}$ in its interior, and has $b$ as a vertex. As such, one can unfold this ${Q}$ into the plane. Here, the two paths of ${\varPi }$ adjacent to $b$ that belong to the boundary of ${Q}$ maps in this unfolding to two straight edges. The rest of the boundary ${Q}$ is a closed connected portion of $M$. One can think about ${Q}$ as being a “leaf” in a decomposition of $\partial {\mathcal {P}}$ (i.e., think about the sides of a banana peel). Here, shortest paths from $b$ to any point on $p\in \partial {\mathcal {P}}$ that belongs to ${Q}$ results in a straight segment in (the planar embedded version of) the polygon ${Q}$. As such, the polygons of $\mathcal {Q}$ completely capture the structure of all the shortest paths on $\partial {\mathcal {P}}$ to $b$.

Back to the problem of sweeping $\partial \mathcal {P}$. For the points of $M_{Q}= M\cap \partial {Q}$, we can sweep the region of $\partial {Q}$ that corresponds to ${Q}$, by walking along the curve $M_{Q}$ (say counterclockwise), and the leash being the shortest path in $\partial \mathcal {P}$ (that lies inside ${Q}$). This completely sweeps over the region of ${Q}$. We then continue this sweeping in the next polygon of $\mathcal {Q}$ adjacent to ${Q}$ around $b$. We continue in this fashion till all the boundary of the polytope is swept over. Note that the leash is moving continuously, and during this motion, the maximum length of the leash is the distance to the furthest point on $\partial \mathcal {P}$ from $b$. We conclude that this is an optimal solution and using the known algorithms for computing shortest paths [1].

Let us recap the algorithm: We compute the medial axis $M$ of $b$ on $\partial \mathcal {P}$, under the shortest path distance on the boundary of the polytope. Next, parameterize a point $p(t)$ to move continuously around the tree $M$ (i.e., traversing along each edge twice, in both direction). At each point in time, the leash is connected via the shortest path to the base point $b$.

Lemma 17

Given a convex polytope $\mathcal {P}$ in three dimensions, and a base point $p\in \partial \mathcal {P}$, one can compute in polynomial time, a continuous motion of a point $p(t)$, $t \in [0,1]$, and an associated leash $\ell \left( {t}\right)$ connecting $p(t)$ with $b$, such that (i) the leash sweeps over all the points of $\partial \mathcal {P}$, (ii) the leash moves continuously, (iii) a point of $\partial \mathcal {P}$ get swept over only once during this motion, and (iv) the maximum length of the leash is $\max _{ p\in \partial \mathcal {P}} d_{\mathcal {P}}\left( {p}\right)$, which is optimal.

The above is in sharp contrast to our original problem of computing the homotopy height, as the two ends of the leash must move along two prespecified curves $\mathsf {L}$ and $\mathsf {R}$. Furthermore, because of that we no longer have the property that the leashes do not jump, as the leash head no longer moves along the medial axis, as the resulting paths might be too long (compared to the optimal morph). Nevertheless, the above captures our basic strategy of breaking the input disk into smaller slivers, induced by shortest paths, and solving the problem in each sliver separately, and gluing the solutions together.