# Rendezvous on the Line with Different Speeds and Markers That Can Be Dropped at Chosen Time
## 3 Problem Formulation  
We begin by presenting the formalization of the problem as given in [1]. Two players, I and II, are placed at distance \(D=1\) (although the results depend linearly on the initial distance and are stated for a general \(D\)). apart on the real line, and faced in random directions which they call \"forward\". Their common aim is to minimize the expected amount of time required to meet. They each know the distance 1 but not the direction the other player is facing. It is not a restriction to assume that player I's starting point is located at position 0 of the line and their speed is bounded by \(v\leq 1\). Their position is given by a function \(f(t)\in\mathcal{F}(\alpha)\), where  
\[\mathcal{F}(\alpha)\ =\ \{f:[0,T]\rightarrow\mathbb{R},\ f(0)=0,\big{|}f(t)-f \big{(}t^{\prime}\big{)}\big{|}\leq\alpha\big{|}t-t^{\prime}\big{|}\}, \tag{1}\]  
for some \(T\) sufficiently large so that the rendezvous will have taken place. What are unknown are the initial position of player II that may be \(\pm 1\) and the forward direction of player II that may point to the positive or negative side of the infinite line. Again, without restriction of generality, we assume that the speed of player II is bounded by \(1\). Hence, depending on the initial conditions of player II their position at time \(t\) is given by \(\pm 1\pm g(t)\) with \(g\in\mathcal{F}(1)=\mathcal{F}\). The rendezvous times are defined by:  
\[t^{1}=min\{t:f(t)=1+g(t)\}, \tag{2}\]  
when player II is originally located at \(+1\) and their forward direction points to the positive side of the line. \[t^{2}=min\{t:f(t)=1-g(t)\}, \tag{3}\]  
when player II is originally located at \(+1\) and their forward direction points to the negative side of the line. \[t^{3}=min\{t:f(t)=-1+g(t)\}, \tag{4}\]  
when player II is originally located at \(-1\) and their forward direction points to the positive side of the line. \[t^{4}=min\{t:f(t)=-1-g(t)\}, \tag{5}\]  
when player II is originally located at \(-1\) and their forward direction points to the negative side of the line. It is common in the literature to speak of four agents (of player II) located at positions \(\pm 1\) and with forward direction \(\pm 1\) and moving concurrently. Player I needs to rendezvous with the four agents to end the game [26]. Concretely,  
* Agent 1 is located at \(+1\) with forward direction \(+1\) and its rendezvous time is \(t^{1}\),
* Agent 2 is located at \(+1\) with forward direction \(-1\) and its rendezvous time is \(t^{2}\),
* Agent 3 is located at \(-1\) with forward direction \(+1\) and its rendezvous time is \(t^{3}\),
* Agent 4 is located at \(-1\) with forward direction \(-1\) and its rendezvous time is \(t^{4}\). We show in Figure 1 the configurations of the four agents and the optimal solution of the problem without marker. We observe on this figure that the speeds of both players are always maximal (equal to 1). Then, a complete description of the optimal strategy is provided by the order with which player I rendezvous with the agents, i.e., agent \(2,3,1\) and finally 4. **Definition 1**. : _We introduce the notation \(t_{1}\leq t_{2}\leq t_{3}\leq t_{4}\) to denote the rendezvous times in the order they occur, \((o_{i},b_{i})\) denotes the agent with origin \(o_{i}=\pm 1\) and forward direction \(b_{i}=\pm 1\). The order of the rendezvous times is given by the index \(i\), \(t_{1}\) is the rendezvous with agent \((o_{1},b_{1})\), \(t_{2}\) is the rendezvous with agent \((o_{2},b_{2})\), \(t_{3}\) is the rendezvous with agent \((o_{3},b_{3})\) and \(t_{4}\) is the rendezvous with agent \((o_{4},b_{4})\). When necessary, we use the convention \(t_{0}=0\)_  
For instance, for the solution in Figure 1 it holds that \(t_{1}=t^{2},\ t_{2}=t^{3},\ t_{3}=t^{1},\ t_{4}=t^{4}\), and \((o_{1},d_{1})=(1,-1),\ (o_{2},d_{2})=(-1,1),\ (o_{3},d_{3})=(1,1),\ (o_{4},d_{4})=(-1,-1)\),. We do not make any assumptions on how the functions \(f(t)\) and \(g(t)\) are computed. A different formalism than what we are proposing here may be possible.