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Spirals and Beyond: Competitive Plane Search with Multi-Speed Agents

Konstantinos Georgiou, Caleb Jones, Matthew Madej

TL;DR

This paper analyzes plane search for a hidden point target using multiple mobile agents with heterogeneous speeds starting from a common origin, quantified by the worst-case normalized exposure time. It extends the classical unit-speed single-agent result (logarithmic spiral with cost $\mathcal{U}_1$) to $n$ unit-speed agents, obtaining $\mathcal{U}_n$ and a closed-form parameter $\kappa_n$; more broadly, it provides an upper bound for arbitrary speeds via a speed-aware Off-Set Spiral framework that uses a subset of fastest agents to achieve cost $\mathcal{U}_\nu/\mathcal{G}_\nu$. The work also derives specialized upper bounds for cone-search and conic-complement search with a unit-speed agent, and introduces a Cone-Wedge Hybrid strategy that beats spiral-based methods in a moderate-speed regime, demonstrating that spiral-type trajectories may be suboptimal in the multi-speed setting. Overall, the results reveal a separation between optimal strategies in the unit-speed and heterogeneous-speed contexts and show that partitioning the search domain by geometry (cones, wedges) can yield tangible gains in certain speed ranges. These findings motivate further exploration of mixed-strategy designs and tightened lower bounds for multi-speed planar search.

Abstract

We consider the problem of minimizing the worst-case search time for a hidden point target in the plane using multiple mobile agents of differing speeds, all starting from a common origin. The search time is normalized by the target's distance to the origin, following the standard convention in competitive analysis. The goal is to minimize the maximum such normalized time over all target locations, the search cost. As a base case, we extend the known result for a single unit-speed agent, which achieves an optimal cost of about $\mathcal{U}_1 = 17.28935$ via a logarithmic spiral, to $n$ unit-speed agents. We give a symmetric spiral-based algorithm where each agent follows a logarithmic spiral offset by equal angular phases. This yields a search cost independent of which agent finds the target. We provide a closed-form upper bound $\mathcal{U}_n$ for this setting, which we use in our general result. Our main contribution is an upper bound on the worst-case normalized search time for $n$ agents with arbitrary speeds. We give a framework that selects a subset of agents and assigns spiral-type trajectories with speed-dependent angular offsets, again making the search cost independent of which agent reaches the target. A corollary shows that $n$ multi-speed agents (fastest speed 1) can beat $k$ unit-speed agents (cost below $\mathcal{U}_k$) if the geometric mean of their speeds exceeds $\mathcal{U}_n / \mathcal{U}_k$. This means slow agents may be excluded if they lower the mean too much, motivating non-spiral algorithms. We also give new upper bounds for point search in cones and conic complements using a single unit-speed agent. These are then used to design hybrid spiral-directional strategies, which outperform the spiral-based algorithms when some agents are slow. This suggests that spiral-type trajectories may not be optimal in the general multi-speed setting.

Spirals and Beyond: Competitive Plane Search with Multi-Speed Agents

TL;DR

This paper analyzes plane search for a hidden point target using multiple mobile agents with heterogeneous speeds starting from a common origin, quantified by the worst-case normalized exposure time. It extends the classical unit-speed single-agent result (logarithmic spiral with cost ) to unit-speed agents, obtaining and a closed-form parameter ; more broadly, it provides an upper bound for arbitrary speeds via a speed-aware Off-Set Spiral framework that uses a subset of fastest agents to achieve cost . The work also derives specialized upper bounds for cone-search and conic-complement search with a unit-speed agent, and introduces a Cone-Wedge Hybrid strategy that beats spiral-based methods in a moderate-speed regime, demonstrating that spiral-type trajectories may be suboptimal in the multi-speed setting. Overall, the results reveal a separation between optimal strategies in the unit-speed and heterogeneous-speed contexts and show that partitioning the search domain by geometry (cones, wedges) can yield tangible gains in certain speed ranges. These findings motivate further exploration of mixed-strategy designs and tightened lower bounds for multi-speed planar search.

Abstract

We consider the problem of minimizing the worst-case search time for a hidden point target in the plane using multiple mobile agents of differing speeds, all starting from a common origin. The search time is normalized by the target's distance to the origin, following the standard convention in competitive analysis. The goal is to minimize the maximum such normalized time over all target locations, the search cost. As a base case, we extend the known result for a single unit-speed agent, which achieves an optimal cost of about via a logarithmic spiral, to unit-speed agents. We give a symmetric spiral-based algorithm where each agent follows a logarithmic spiral offset by equal angular phases. This yields a search cost independent of which agent finds the target. We provide a closed-form upper bound for this setting, which we use in our general result. Our main contribution is an upper bound on the worst-case normalized search time for agents with arbitrary speeds. We give a framework that selects a subset of agents and assigns spiral-type trajectories with speed-dependent angular offsets, again making the search cost independent of which agent reaches the target. A corollary shows that multi-speed agents (fastest speed 1) can beat unit-speed agents (cost below ) if the geometric mean of their speeds exceeds . This means slow agents may be excluded if they lower the mean too much, motivating non-spiral algorithms. We also give new upper bounds for point search in cones and conic complements using a single unit-speed agent. These are then used to design hybrid spiral-directional strategies, which outperform the spiral-based algorithms when some agents are slow. This suggests that spiral-type trajectories may not be optimal in the general multi-speed setting.

Paper Structure

This paper contains 15 sections, 26 theorems, 108 equations, 7 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Definitions and Main Results
  4. Problem Definition, Notation & Terminology
  5. Main Contributions & Paper Organization
  6. Searching the Plane with $n$ Unit-Speed Agents
  7. Searching the Plane with $n$ Agents of Arbitrary Speeds
  8. The Speed-Dependent Off-Set Spiral Algorithm for $\mathcal{P}\text{-}\textsc{MSP}_n(c)$
  9. Properties of the Off-Set Trajectories
  10. Proof & Discussion of Theorem \ref{['thm: opt off-set algo and performance']}
  11. A Simplified Algorithm for $\mathcal{P}\text{-}\textsc{MSP}_n(c)$
  12. Cone Search with a Unit-Speed Agent
  13. Conic Complement Search with a Unit-Speed Agent
  14. Sub-Optimality of the Spiral-Type Off-Set Algorithm
  15. Discussion

Key Result

theorem thmcountertheorem

For every $n \in \mathbb{N}$, the search problem $\mathcal{P}\text{-}\textsc{MSP}_n(\textbf{1})$ admits a solution with search cost where $\kappa_n$ admits a closed-form expression, obtained as the root of an explicit cubic polynomial (see Lemma lem: closed form of k_i).

Figures (7)

  • Figure 1: The orange curve represents the upper bound from Theorem \ref{['thm: simplified general spiral upper bound']}, achieved by the Off-Set Algorithm \ref{['algo: Off-Set Trajectory']} for searching the plane with agent speeds $1,c$, as a function of $c \in (0,1)$. The transition value where the search cost is not differentiable in the speed is $c= (\mathcal{U}_2/\mathcal{U}_1)^2\approx 0.266687$, and note that past this threshold speed value, the induced search cost is strictly lower than $\mathcal{U}_1$. The blue curve represents the upper bound from Theorem \ref{['thm: hybrid upper bound']}, achieved by the Cone-Wedge Hybrid Algorithm \ref{['algo: cone wedge hybrid']} for searching the plane with agent speeds $1,c$. The right panel zooms in on the region where the two bounds converge, showing that the latter strategy is strictly better for searching the plane even for $c \in (0.266687,0.268872)$ in which both the speed-$1$ and speed-$c$ agents expose new targets following the Off-Set Algorithm \ref{['algo: Off-Set Trajectory']}.
  • Figure 2: The depiction of trajectories of Algorithm \ref{['algo: uniform spiral trajectory']}. In this example, we use $k=0.356$, $n=30$, and we show the trajectories of agents $0,1,2,29$, along with their positions when $t=t_s = 2\pi s$, for some $s\in \mathbb{Z}$. All agents lie on the circle of radius $e^{k t_s}$, while agent-$0$ lies on ray $\mathcal{L}_0$. The last agent who previously hit $\mathcal{L}_0$ was agent-$(n-1)$.
  • Figure 3: The depiction of trajectories of Algorithm \ref{['algo: Off-Set Trajectory']}. In this example, we use $n=3$ agents with speeds $c_0=1$, $c_1=0.9$, and $c_2=0.7$, and we plot the corresponding offset logarithmic spirals (as defined by the algorithm). We also show two snapshots of the agents, at two distinct parameter values, together with the triangles formed by connecting the three agent positions in each snapshot. In contrast to the unit speed case (Figure \ref{['fig: uniform spiral']}), where all agents lie on the same circle at every snapshot and hence form a regular $3$-gon, the multispeed agents lie at different distances from the origin; these distances remain proportional, and the two triangles are similar (their side lengths scale by the same factor between snapshots).
  • Figure 4: A depiction of the Bouncing Trajectory for $\mathcal{C}_\phi\text{-}\textsc{MSP}_1(1)$. Angle $\theta$ determines the sequence of points $\Gamma_i$, as per Lemma \ref{['lem: choice of sequence g']}, using sequence $q_i = \left(\tfrac{\sin\left(\theta\right)}{\sin\left(\theta-\phi\right)}\right)^i$.
  • Figure 5: A depiction of the Shortcut Spiral Trajectory to $\mathcal{W}_\phi\text{-}\textsc{MSP}_1(1)$. The shaded cone corresponds to $\mathcal{C}_\phi$ with extreme rays $\mathcal{L}_0,\mathcal{L}_\phi$, while the shortcut is represented by the dashed arrow within the cone.
  • ...and 2 more figures

Theorems & Definitions (48)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 38 more