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 away, speed known
standard settingdraftOne robot that knows the speed \(u < 1\) of a target moving away from the origin, but not its distance, zigzags with turning distances that grow by the factor \(2(1+u)/(1-u)\); its ratio \((3+u)^2/(1-u)^2\) is optimal (Alpern and Gal, as reported).
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: away
- The target moves away from the origin at constant speed \(u < 1\), and the robots know it.
- Knowledge of the target: speed known
- The robots know \(u\) but not \(d\).
- 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
- one robot
- as reported by Coleman et al. 2022 (Theorem 6) and Jawhar & Kranakis 2025b; the book was not read
Results, in order of publication
Introductory and current best, as reported by Coleman et al. 2022 (Theorem 6, which writes it \(1 + 8\frac{1+u}{(1-u)^2}\)) and Jawhar & Kranakis 2025b (who cite p. 134, Equation 8.25): the zigzag of Algorithm 3 of Coleman et al. 2022, which turns at distances \(1, a, a^2, \dots\) on alternate sides, with \(a = 2(1+u)/(1-u)\). Not read.
Open questions
Tight bounds when the robot does not know whether the target moves toward or away from the origin Coleman et al. 2022.
Notes
Draft because the source of the bound, the book of Alpern and Gal, was not read; Coleman et al. 2022 state the result without proof. At \(u = 0\) the zigzag is the doubling strategy of linear search.
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
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