Rendezvous search with markers that can be dropped at chosen times

The Rendezvous search problem asks how two noncomunnicating players, who move at unit-speed after a random placement in the search region (in this paper, the line), can minimize their expected meeting time. Baston and Gal cite{bastongal2001} solved a version where each player leaves a marker at his starting point, for example parachutes left after jumping into the search region. Here we consider that one or both players have markers which they can drop at chosen times. When the players are placed facing random directions at a distance $D$ on the line the optimal expected meeting times $R$ are known to be as follows: With no markers, $R=13D/8$; with one each dropped at the start, $R=3D/2$. Here we show that when only one player has a marker, but it can be dropped at any time, we still have $R=3D/2$, obtained by dropping the marker at time $D/4$. Having both players drop markers at chosen times does not further reduce $R$. We adopt a new algorithmic approach which first fixes the dropping times and then reduces the resulting problem to a finite one. We also consider evanescent markers, which are detectable for a specified time $T$ after being dropped, modeling pheromone scent markers used by some species in mate search or stain dropped by sailors. From a less theoretical and more practical point of view, we can see our problem as that faced by two hikers who get separated when they are walking not too far from a coast. A reasonable rendezvous strategy is for each of them to head in the general direction of the coastline. On reaching the sea, they can calculate (based on their speed and time taken to reach to coast) the maximum distance along the coastline where the other one arrives. Then they can use our solution, taking that distance as our parameter $D$. Of course if it turns out that they reached the sea at points closer than $D$, their rendezvous time will only be shorter. So our solution can be interpreted as a worst case expected meeting time. The marker might be a note scribbled on the sand, or a pile of rocks. Perhaps before getting separated they would have decided which roles (I or II) each would take up, being careful and cautious hikers. So in particular the hiker taking the role of the marker placer would put down a marker of some sort when he had walked along the coast one quarter the maximum distance to his partner. If the other finds it, he would know for sure the direction of the hiker who dropped it. Another thing to take away from this article is our result that says “two markers are not better than one”. This observation has some significance in the mate search context, where males and females use pheromone deposits to help find each other. In most species, it is only the females who can make such deposits, and the males can detect it. Giving both species the depositing facility would be more costly in terms of energy and, given our negative result, might not significantly reduce rendezvous times. So in a speculative sense, our result has some explanatory value in this context.