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, \(d\) known
verifiedA robot that knows the distance of a target moving toward the origin goes for time \(d\) on one side and turns: ratio exactly 3, for every speed.
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: distance known
- The robots know \(d\) but not \(u\).
- 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
- a slow target (\(u = \varepsilon/3\)) placed on the side opposite to the first of the points \(\pm(d-\varepsilon)\) that the robot reaches (Theorem 14)
- a robot that never visits \(\pm d\) has an unbounded ratio for small \(u\)
Results, in order of publication
Introductory and current best (Algorithm 6, Theorem 14): go for time \(d\) on one side, then turn.
Open questions
None recorded for this variant.
Notes
Two robots do no better, whether they communicate face-to-face Jawhar & Kranakis 2026b or as a sender and a receiver Jawhar & Kranakis 2025b. 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
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.