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Online Drone Coverage of Targets on a Line

Stefan Dobrev, Konstantinos Georgiou, Evangelos Kranakis, Danny Krizanc, Lata Narayanan, Jaroslav Opatrny, Denis Pankratov, Sunil Shende

Abstract

We study a problem of online targets coverage by a drone or a sensor that is equipped with a camera or an antenna of fixed half-angle of view $α$. The targets to be monitored appear at arbitrary positions on a line barrier in an online manner. When a new target appears, the drone has to move to a location that covers the newly arrived target, as well as already existing targets. The objective is to design a coverage algorithm that optimizes the total length of the drone's trajectory. Our results are reported in terms of an algorithm's competitive ratio, i.e., the worst-case ratio (over all inputs) of its cost to that of an optimal offline algorithm. In terms of upper bounds, we present three online algorithms and prove bounds on their competitive ratios for every $α\in [0, π/2]$. The best of them, called \FA is significantly better than the other two for $π/6 < α< π/3$. In particular, for $α=π/4$, its worst case, \FA has competitive ratio $1.25$, while the other two have competitive ratio $\sqrt{2}$. Finally, we prove a lower bound on the competitive ratio of online algorithms for a drone with half-angle $α\in [0, π/4]$; this bound is a function of $α$ that achieves its maximum value at $α= π/4$ equal to $(1+\sqrt{2})/2 \approx 1.207$.

Online Drone Coverage of Targets on a Line

Abstract

We study a problem of online targets coverage by a drone or a sensor that is equipped with a camera or an antenna of fixed half-angle of view . The targets to be monitored appear at arbitrary positions on a line barrier in an online manner. When a new target appears, the drone has to move to a location that covers the newly arrived target, as well as already existing targets. The objective is to design a coverage algorithm that optimizes the total length of the drone's trajectory. Our results are reported in terms of an algorithm's competitive ratio, i.e., the worst-case ratio (over all inputs) of its cost to that of an optimal offline algorithm. In terms of upper bounds, we present three online algorithms and prove bounds on their competitive ratios for every . The best of them, called \FA is significantly better than the other two for . In particular, for , its worst case, \FA has competitive ratio , while the other two have competitive ratio . Finally, we prove a lower bound on the competitive ratio of online algorithms for a drone with half-angle ; this bound is a function of that achieves its maximum value at equal to .

Paper Structure

This paper contains 12 sections, 21 theorems, 78 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Preliminaries
  5. Notation
  6. Two simple online algorithms
  7. $\beta$-Hedge Algorithm
  8. $\beta$-Hedge Algorithm Competitive Analysis
  9. On the Adversarial Choice $r$
  10. The Optimality of Angle $\beta_0$
  11. Lower bound on the competitive ratio
  12. Conclusions

Key Result

lemma 1

For any $0 \leq i \leq n$, consider the path $\mathcal{P}_i = T_0, T_1, \ldots ,T_i$ that proceeds sequentially through the apexes of the feasibility cones of the first $i$ subsequences of $\boldsymbol{X}$. Then the total length $\|\mathcal{P}_i\|$ of this path equals $|(x_i,0)T_i|$, the distance b

Figures (14)

  • Figure 1: Two dimensional representation of our problem with $X_0 = (0,0)$ and arrivals $X_1,X_2,X_3,X_4$ revealed one at a time in this order.
  • Figure 2: This plot depicts competitive ratios of algorithms studied in this paper as functions of $\alpha \le \pi/2$, as well as the lower bound (which holds for $\alpha \le \pi/4)$.
  • Figure 3: The cost of the offline algorithm is segment from $X_0$ to $T$. Points $X_i$ and $X_j$ are the extreme points in $\boldsymbol{X}$.
  • Figure 4: Algorithm Trajectories for $\alpha = \pi/4$: OPT (in black), Straight-Up algorithm (in red), Greedy algorithm (in solid blue, which is equivalent in distance to the dotted blue trajectory), and the $\beta$-Hedge algorithm (in green).
  • Figure 5: The segments $X_0 U$ and $U T'$ correspond to the path traveled by the Greedy algorithm when $\alpha > \pi/4$.
  • ...and 9 more figures

Theorems & Definitions (39)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • theorem 1
  • proof
  • theorem 2
  • proof
  • lemma 4
  • ...and 29 more