Barking dogs: A Fr\'echet distance variant for detour detection

Imagine you are a dog behind a fence $Q$ and a hiker is passing by at constant speed along the hiking path $P$. In order to fulfil your duties as a watchdog, you desire to bark as long as possible at the human. However, your barks can only be heard in a fixed radius $\rho$ and, as a dog, you have bounded speed $s$. Can you optimize your route along the fence $Q$ in order to maximize the barking time with radius $\rho$, assuming you can run backwards and forward at speed at most $s$? We define the barking distance from a polyline $P$ on $n$ vertices to a polyline $Q$ on $m$ vertices as the time that the hiker stays in your barking radius if you run optimally along $Q$. This asymmetric similarity measure between two curves can be used to detect outliers in $Q$ compared to $P$ that other established measures like the Fr\'echet distance and Dynamic Time Warping fail to capture at times. We consider this measure in three different settings. In the discrete setting, the traversals of $P$ and $Q$ are both discrete. For this case we show that the barking distance from $P$ to $Q$ can be computed in $O(nm\log s)$ time. In the semi-discrete setting, the traversal of $Q$ is continuous while the one of $P$ is again discrete. Here, we show how to compute the barking distance in time $O(nm\log (nm))$. Finally, in the continuous setting in which both traversals are continuous, we show that the problem can be solved in polynomial time. For all the settings we show that, assuming SETH, no truly subquadratic algorithm can exist.