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, speed known
verifiedA robot that knows the speed but not the distance of a target moving toward the origin should wait at the origin if \(u \ge 1/3\) and otherwise zigzag with turning distances that grow by the factor \(2(1-u)/(1+u)\); both are 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: 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
- any strategy: the adversary takes the limit \(\beta\) of the ratio of time to distance from the origin along the robot's trajectory and places the target on the side the robot is not on, just beyond what the robot can have reached (Theorem 9, with techniques of an unpublished manuscript of Killick)
- a contracting strategy, which never goes beyond some distance on one side, cannot beat \(1 + 1/u\) (Theorem 7)
Results, in order of publication
Introductory and current best (Algorithm 4, Theorem 8): wait at the origin if \(u \ge 1/3\); otherwise zigzag, turning at distances \(1, a, a^2, \dots\) on alternate sides with \(a = 2(1-u)/(1+u)\).
Open questions
None recorded for this variant.
Notes
A target moving toward the origin comes to the robot, so waiting is a strategy; it is optimal down to speed \(1/3\), and the zigzag below. 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.