Search for a moving target
A target starts at unknown distance \(d\) from the origin of a line (or of \(m\) rays) and moves at constant speed \(u\); how fast can robots capture it, depending on what they know about \(d\) and \(u\)?
Problem statement and measuresexpand
Robots of unit speed start at the origin of the line (or of \(m\) rays). The target starts at distance \(d \ge 1\) from the origin, on a side (ray) unknown to the robots, and moves along it at constant speed \(u\), either away from the origin (\(u < 1\), since a faster target can never be caught) or toward it (\(u \ge 0\)); the robots know which. The target cannot change its speed or direction. Depending on the setting the robots know \(d\), \(u\), both or neither. The target is captured when every robot stands on it. Time is the competitive ratio: the supremum, over the target's initial positions and speeds, of the capture time divided by the time the robots would need if they knew the target's initial position.
Measures: Competitive ratio, worst-case capture time divided by the capture time of robots that know the target's initial position.
One robot, line, target moving toward the origin, \(d\) and \(u\) known
verifiedA robot that knows \(d\) and \(u\) goes for time \(d/(1+u)\) on one side and turns if needed, with ratio \((3+u)/(1+u)\) for \(u \le 1\); for \(u \ge 1\) waiting at the origin, with ratio \(1 + 1/u\), is optimal.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Domain: line
- The real line; the target moves along one side of the origin.
- Number of robots: one robot
- A single robot.
- Direction of the target: toward
- The target moves toward the origin at constant speed \(u \ge 0\), and the robots know it.
- Knowledge of the target: both known
- The robots know \(d\) and \(u\); only the side (ray) of the target is unknown.
- Communication: face-to-face
- Face-to-face: two robots exchange information only when they are at the same point at the same time.
- Faults: no faults
- Every robot detects and reports the target.
- Speeds: equal speeds
- All robots move at most at unit speed.
- Randomness: deterministic
- Trajectories are fixed functions of the information received.
What is known
- the adversary puts the target on the side opposite to the first point \(\pm a\) that the robot occupies exactly when the target would reach it (Theorem 3, for \(u < 1\))
- for \(u \ge 1\) the same bound is smallest when the robot never leaves the origin (Theorem 4)
Results, in order of publication
Introductory and current best: Algorithm 2 (go for time \(d/(1+u)\) on one side, then turn; Theorem 2) for \(u < 1\), and the waiting algorithm, which stays at the origin until the target arrives (Theorem 4), for \(u \ge 1\).
Open questions
None recorded for this variant.
Notes
The waiting algorithm uses neither \(d\) nor \(u\), so it applies in every setting in which the target moves toward the origin (Remark 5 of Coleman et al. 2022). Checked against the full text of Coleman et al. 2022 (OPODIS 2022).
What carries over from neighbouring settings
Read off the order of assumptions, not checked case by case: results may carry extra assumptions (shown on their pages), and asynchronous time is measured in epochs.
Algorithms from weaker settings work here
Lower bounds from stronger settings hold here
No catalogued stronger setting.
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