Sweep and Seek Strategies

• Sweep and Seek is a design pattern that couples a broad sweeping phase to constrain uncertainty with a subsequent targeted seeking phase to resolve residual challenges.
• It underpins applications from smart-evader search and searchlight scheduling in robotics to strategic moves in triangular peg solitaire, supported by rigorous performance bounds.
• Advanced implementations include pincer, spiral, and divide-and-search protocols that optimize critical speeds and coverage in diverse sensing and geometric environments.

Across the cited literature, Sweep and Seek is used for a family of search, coverage, and acquisition problems in which an agent or a coordinated team performs a structured sweep of a geometric domain, frontier, or feasible set and then resolves the remaining uncertainty by detection, interception, visitation, grasping, or exhaustive completion. In robotics and control, the term is most strongly associated with guaranteed search for smart evaders in planar regions using line sensors, fan-shaped sensors, or searchlights, with emphasis on confinement, critical-speed bounds, trajectory design, and total search time (Francos et al., 2019, Francos et al., 2020, Francos et al., 2023). In adjacent work, it also denotes divide-and-search with directional sensing (Borri et al., 2011), distributed searchlight scheduling in polygonal environments [0701077], optimal allocation of guards along a monotone sweep frontier (Feng et al., 2023), and dual-mode indoor cleaning in which a mobile robot must both sweep debris and seek graspable objects (Li et al., 7 Aug 2025). A broader, non-robotic usage appears in triangular peg solitaire, where a “sweep” is a long consecutive capture sequence and “seek” corresponds to minimizing the total number of moves (0711.0486).

1. Terminological scope and problem classes

The phrase does not denote a single canonical model. The literature instead clusters around several technical problem classes, each with a distinct sensing model, adversary model, and performance criterion.

Domain	Core model	Representative sources
Smart-evader search	Moving evaders in a circular region; guaranteed confinement and detection	(Francos et al., 2019, Francos et al., 2020, Francos et al., 2023)
Defensive sweeping	Prevent invaders from entering a protected region; possible region expansion	(Francos et al., 2022)
Directional sensing	Mobile searcher queries sensors that return half-plane information	(Borri et al., 2011)
Searchlight scheduling	Fixed guards rotate rays in polygonal environments	[0701077]
Sweep-line coverage allocation	Robots placed along a time-parameterized frontier with probabilistic sensing	(Feng et al., 2023)
Embodied cleaning	A service robot alternates between sweeping and grasping	(Li et al., 7 Aug 2025)

In the smart-evader literature, the environment is typically a planar disk of initial radius $R_0$ or $R$, evaders move arbitrarily with bounded speed $v_e$ or $V_T$, and sweepers must move sufficiently fast to ensure that no evader slips through an unswept gap (Francos et al., 2019, Francos et al., 2023). In directional-sensing and searchlight formulations, the core difficulty is not radial shrinkage of an evader disk but sequential information acquisition under sensing geometry and communication constraints (Borri et al., 2011), [0701077]. In embodied cleaning, “Sweep + Seek” is operationalized as the joint execution of sweeping loose debris and grasping discrete objects in time-constrained indoor tasks (Li et al., 7 Aug 2025).

2. Guaranteed pursuit–evasion in circular regions

A central line of work studies must-win search against smart mobile evaders initially located somewhere inside a known circular region. In the earliest line-formation model, the sweeper is a straight sensor of total length $2r$ moving around the boundary of the current evader disk; success means that evaders with maximum speed $v_e$ cannot escape the sweeping formation and are eventually detected (Francos et al., 2019). The swarm generalization replaces the single line formation by $n$ identical agents, each with a line sensor of length $2r$, usually arranged in cooperative pincer pairs (Francos et al., 2020). A later extension replaces line sensors with fan-shaped sensors that model a pin-hole camera’s field of view, with half-angle $\alpha$ and maximal radial reach $2r$, again assuming an even number $R$0 of identical sweepers (Francos et al., 2023).

The basic decomposition of the task is stable across these papers: confinement is the easier task, requiring that no evader ever leaves the current admissible region, while detection requires that every evader is eventually intersected by some sensor footprint (Francos et al., 2019, Francos et al., 2023). The resulting feasibility conditions are expressed as lower bounds on sweeper speed. For a single line formation, a protocol-independent lower bound follows from comparing the area swept by a line of length $R$1 moving at speed $R$2 with the worst-case area expansion rate of a circular evader region, yielding

$R$3

An exact no-escape critical speed is then obtained from a nonlinear condition involving the sensor-tip trajectory, with a conservative explicit bound

$R$4

The inward step after a circular sweep is

$R$5

and the search repeats until the residual radius is at most $R$6, after which a final small-circle pass completes detection (Francos et al., 2019).

For a swarm of $R$7 agents with line sensors, the universal lower bound scales as

$R$8

while the simplest circular cooperative sweep gives the stronger sufficient condition

$R$9

The spiral cooperative pattern is more efficient: its critical speed is obtained from a nonlinear equation and approaches the lower bound for large $v_e$0 or large $v_e$1 (Francos et al., 2020). For fan-shaped sensors, a protocol-independent lower bound of

$v_e$2

is derived geometrically, and the exact critical speed for the pincer protocol is obtained numerically after accounting for both spiral sweeping and inward regrouping (Francos et al., 2023).

A recurring consequence is that speed above the confinement threshold buys not only feasibility but also shorter completion times. The papers explicitly derive $v_e$3, $v_e$4, $v_e$5, and associated inward-motion terms as functions of $v_e$6, $v_e$7, $v_e$8 or $v_e$9, $V_T$0, and $V_T$1 (Francos et al., 2019, Francos et al., 2020, Francos et al., 2023).

3. Pincer, circular, and spiral trajectories

The most developed trajectory family is the pincer movement, in which sweepers are paired back-to-back and assigned disjoint angular sectors. In the fan-sensor formulation, each pair handles a sector of size $V_T$2, starts on the boundary of the initial evader region, and spirals toward the partner while maintaining the sensor’s outer tip tangent to the shrinking evader frontier (Francos et al., 2023). The angular offset $V_T$3 is chosen by

$V_T$4

which preserves the circular shape of the possible-evader region and minimizes sweep time. Each cycle consists of a spiral pass through angle $V_T$5, followed by an inward advance

$V_T$6

and the radius update $V_T$7 (Francos et al., 2023).

Comparison papers make the role of the pincer explicit. Same-direction circular and spiral sweeps require an additional safety angle to catch evaders spreading from dangerous boundary points, whereas pincer-based circular sweeps eliminate that extra angular overhead and thereby reduce both critical speed and search time (Francos et al., 2021). In that comparison, the ordering of required critical speeds is

$V_T$8

and the same ordering appears for total search times. This is why the spiral pincer process is characterized as “nearly optimal” with respect to the lower bound (Francos et al., 2021).

The defensive analogue in which invaders attempt to enter an initially empty protected disk uses the same geometric logic but reverses the operational objective. Circular-pincer defense has critical speed

$V_T$9

while the spiral defensive process solves a nonlinear equation and yields a critical speed only slightly above the universal lower bound

$2r$0

If defender speed exceeds the threshold, successive sweep-plus-push-out iterations expand the protected region, with maximal radii

$2r$1

for the circular case and

$2r$2

for the spiral case (Francos et al., 2022). The numerical plots in the pincer-search and fan-sensor papers also show diminishing returns as $2r$3 increases (Francos et al., 2023), and the swarm line-sensor study reports that gains taper rapidly beyond approximately $2r$4–$2r$5 agents (Francos et al., 2020).

4. Directional sensing, divide-and-search, and searchlights

A distinct Sweep-and-Seek usage appears in Hide-and-Seek with Directional Sensing, where the target is static, the environment is a bounded planar region $2r$6, the searcher starts at the center, and sensors return a random direction corresponding to a half-plane that contains the hidden treasure (Borri et al., 2011). The proposed heuristic, called Divide-and-Search, repeatedly moves to the sensor nearest the centroid of the current feasible polygon, intersects that polygon with the reported half-plane, and, once no sensors remain in the feasible set, executes an exhaustive Euclidean-TSP through the surviving candidate points. With $2r$7 sensors and

$2r$8

iterations, the expected distance obeys

$2r$9

and more explicitly

$v_e$0

The same paper also formulates the problem as a large zero-sum dynamic game and applies a sampled saddle-point procedure that gives $v_e$1-security with high confidence; when only very few samples are used, the heuristic is comparable, but as the number of samples grows, the randomized procedure provides a higher probabilistic security level (Borri et al., 2011).

Searchlight scheduling addresses yet another variant. In nonconvex polygonal environments, agents are fixed guards that rotate rays rather than mobile sweepers carrying finite-width sensors. The DOWSS algorithm is a distributed one-way sweep in which agents communicate “CLEAR-REQUEST” and “CLEAR-DONE” messages along a visibility-tree structure. Its worst-case time is

$v_e$2

where $v_e$3 is the number of agents, $v_e$4 the number of polygon-plus-hole vertices, and $v_e$5 the angular speed [0701077]. The PTSS algorithm partitions the environment into a tree of star-shaped cells and allows parallel sweeping; its time bound is

$v_e$6

with $v_e$7 the tree height. This directly contrasts sequential and parallel scheduling under local sensing and limited communication [0701077].

5. Sweep-line coverage allocation and embodied cleaning

In sweep-line coverage, the sweep is assumed given in advance as a continuous monotone sweep schedule

$v_e$8

and the problem is to allocate many mobile robot guards along the current frontier so that every point is covered with at least a desired probabilistic guarantee when the frontier reaches it (Feng et al., 2023). Sensing along the frontier is probabilistic, with

$v_e$9

and if two robots are the nearest guards to a sample point, the coverage probability is

$n$0

For a frontier segment of length $n$1 and minimum guarantee $n$2, the minimum number of robots is

$n$3

The workspace is decomposed via a generalization of boustrophedon decomposition into a DAG of cells, node demands are set to $n$4, and the minimum-guard problem reduces to circulation with demands or min-flow on an auxiliary graph. The proposed algorithm runs in low polynomial time and completes in under two seconds for polygonal environments with over $n$5 vertices (Feng et al., 2023).

Embodied cleaning introduces a more literal Sweep + Seek coupling. CleanUpBench models a wheeled service robot in NVIDIA Isaac Sim equipped with a front-mounted sweeping roller or brush and a 6-DOF manipulator with a parallel gripper. The robot must integrate the modes $n$6 in realistic indoor cleaning tasks, using RGB-D, semantic masks, 360° LiDAR with 1,440 points, proprioception, and optionally a high-level affordance map (Li et al., 7 Aug 2025). The benchmark contains 20 manually designed scenes in five categories—Sparse Exploration, High-Density Sweeping, Narrow Corridors, Dynamic Interference, and Multi-Zone Coordination—plus procedurally generated layouts with obstacle densities sampled in $n$7 of floor area (Li et al., 7 Aug 2025).

Evaluation is decomposed into Task Completion Rate $n$8, Spatial Efficiency $n$9, Motion Smoothness $2r$0, and Control Performance $2r$1. The spatial term combines coverage ratio and sweep redundancy by

$2r$2

the smoothness term is

$2r$3

and control performance is the reciprocal of average action-computation time (Li et al., 7 Aug 2025). Baselines include vertical, horizontal, Manhattan, and Chebyshev sweeping heuristics, plus map-based target ordering and A* navigation for grasping. Experimentally, sweep-only heuristics achieve up to 30% sweep success but 0% grasp success; grasp planners such as REMANI reach approximately $2r$4 grasp completion but nearly zero coverage efficiency; and dual-mode multi-robot RL with PRIMAL2 attains the highest combined completion $2r$5, balanced sweep/grasp performance, and moderate spatial efficiency $2r$6. On unseen layouts, learning-based dual-mode agents degrade by less than 10%, whereas heuristics lose more than 25% coverage, and aggressive RL with CA-DRL exceeds 80 collisions (Li et al., 7 Aug 2025).

6. Related usages, misconceptions, and recurrent themes

A non-robotic but explicit Sweep-and-Seek usage appears in triangular peg solitaire. There, a sweep is a move in which one peg removes many pegs consecutively, and the associated “seek” problem is to minimize the total number of moves (0711.0486). On Triangle$2r$7 and Triangle$2r$8, the geometrically longest final sweeps can actually occur as the last move of a solution; on Triangle$2r$9, there exists a family of solutions whose final move has length

$\alpha$0

while the board has $\alpha$1 holes (0711.0486). This usage is combinatorial rather than geometric-control-theoretic, but it retains the same structural pairing of a broad clearing action followed by a residual optimization criterion.

Several recurring themes emerge across the research areas. First, sweeping is not synonymous with complete task success: in smart-evader search, confinement is weaker than detection (Francos et al., 2019, Francos et al., 2023); in embodied cleaning, sweep-only agents leave graspable objects untouched and therefore score poorly overall (Li et al., 7 Aug 2025). Second, geometry fixes the performance floor: lower bounds on speed or guard count arise from area expansion, sensor projection, frontier length, or probabilistic gap constraints (Francos et al., 2020, Feng et al., 2023). Third, parallelism improves completion time but not always proportionally: pincer and PTSS-style decompositions reduce critical speed or search time, yet both pincer-sweep studies and guard-allocation studies show diminishing returns once the team becomes large (Francos et al., 2023, Francos et al., 2020). Fourth, simple heuristics remain competitive in limited-information regimes: Divide-and-Search has logarithmic expected travel and is comparable to sampled game-theoretic search when only a very small number of samples is used (Borri et al., 2011).

This suggests that Sweep and Seek is best understood not as a single algorithm but as a design pattern: an outer sweep constrains uncertainty or clears accessible mass, and an inner seek phase resolves what remains under the specific sensing, adversarial, and geometric assumptions of the domain.