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(a) An x-monotone polygon. (b) The pocket pocket( v, r) is the shaded sub-polygon. (c) Since E is not visible, capture condition is not satisfied.
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In the original version of the lion and man game, a lion tries to capture a man who is trying to escape in a circular arena. The players have equal speeds. They can observe each other at all times. We study a new variant of the game in which the lion has only line-of-sight visibility. That is it can observe the man's position only if the line segme...
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... that since P has a deterministic strategy, the evader can simulate the pursuer's moves, and hence it knows the location of P at all time steps. (5) The pursuer captures E if at any time, the distance between them is less than or equal to one (the step size) while P can see E, see Figure 1(c). ...
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... leftmost vertex and the rightmost vertex are denoted by O L and O R , respec- tively. The boundary of the polygon connects these vertices by two x-monotone chains denoted by Chain L and Chain U , see Figure 1(a). ...
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... direction is also important we refer to the ray pointing from u to v as uv. We define a local refer- ence frame whose origin coincides with P. Its axes X P and Y P are parallel to the axes of the reference frame, see Figure 1(a). We refer to the boundary of Q as ∂Q, and the number of vertices in Q as n. ...
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... the rest of the paper, we consider the shortest path tree rooted at o = O L . We denote π( O L , O R ) by (Figure 1(a)). For simplicity we denote d( O L , p) by R( p) for a point p ∈ Q. ...
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... that it is the evader's turn to move. See Figure 1(b). Imagine that the pursuer and the evader can see each other before the evader's move but the evader dis- appears behind a vertex v after moving to E . ...
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... 1. In the rest of the paper, we refer to the pocket that the evader is hidden inside of as pocket( v, r) (also the contaminated region) where r = = pv and p is the location of the pursuer at the time that the evader has disappeared behind the vertex v. See Figure 1(b). ...
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... general idea for the pursuer's strategy is the follow- ing. See Figure 10, and let p ref be the current reference vertex used for tracking progress. Let v be the vertex that defines the contaminated pocket right before this guard state. ...
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... let p 0 be the location of the pursuer at the begin- ning of the simple guard state. The pursuer moves back toward the vertex p ref along the line segment that connects p 0 to p ref (P 1 in Figure 10(a)). It continues moving toward p ref until it reaches p ref , or E crosses the ray shot from P in the direction of p ref to P (P 2 and E 2 in Figure 10(b)). ...
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... pursuer moves back toward the vertex p ref along the line segment that connects p 0 to p ref (P 1 in Figure 10(a)). It continues moving toward p ref until it reaches p ref , or E crosses the ray shot from P in the direction of p ref to P (P 2 and E 2 in Figure 10(b)). In both of these cases, P follows E by lion's move with respect to the center p ref . ...
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... pursuer continues with this lion's move until one of the following configurations hold: (1) either P reaches π ( O L , E) while it is inside h( p ref ) or (2) E moves to the region which is to the left of p ref , i.e. it crosses the vertical line which passes through p ref to the left and re-contaminates h( p ref ). See P 3 and E 3 in Figure 10(b). ...
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... illustrative example of the auxiliary vertex p aux is shown in Figure 11(a). The interested reader is referred to the Appendix D, for the definition of the auxiliary vertex p aux based on the structure of the polygon and the location of the pursuer. ...
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... if v / ∈ h( v), the pocket pocket( v , r ) defined by v will be a simple pocket, and thus, by performing the simple pocket strategy (Appendix A), the pursuer can force the evader to exit pocket( v , r ) and con- tinue the simple pocket strategy. See Figure 13(a) for an illustration. ...
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... complete description of simple guard strategy is given in Algorithm 4. Notice that when the evader disap- pears while P is moving back toward p ref , the pursuer's reac- tion depends on whether the hiding vertex v is from Chain L or Chain U (lines 12-16 in Algorithm 4). An example is shown in Figure 10(c) parts (c) and (d). When v ∈ Chain L , the pursuer moves toward v until it reaches v at which time it switches to the S state (P 4 and E 4 in Figure 10(c)). ...
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... example is shown in Figure 10(c) parts (c) and (d). When v ∈ Chain L , the pursuer moves toward v until it reaches v at which time it switches to the S state (P 4 and E 4 in Figure 10(c)). When v ∈ Chain U , the pursuer continues moving back toward p ref , and if in the meantime E crosses − → ray (line 3) the pursuer performs the same strategy from line 4. ...
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... v ∈ Chain L : In this case, if the pursuer keeps moving back toward p ref , it cannot keep track of the hiding ver- tex v . For example, in Figure 11(c), at the time that E disappears, P defines the hiding pocket with respect to v = v 1 . If P continues moving back toward p ref , at P 2 the hiding pocket with respect to v 1 doesn't include E (Figure 11(d)). ...
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... example, in Figure 11(c), at the time that E disappears, P defines the hiding pocket with respect to v = v 1 . If P continues moving back toward p ref , at P 2 the hiding pocket with respect to v 1 doesn't include E (Figure 11(d)). In other words, P cannot keep track of the hiding vertex v . ...
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... if the pursuer moves toward v the evader can cross the segment between p ref and P (Figure 11(b)), and thus it escapes to the previously cleared region. ...
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... let us proceed with the definition of the center c. Let I be the intersection between the horizontal line pass- ing through p ref and ∂P (see Figure 12(a)). Then c is the intersection between the bisector of PI and the line l v . ...
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... the previous state is zig-zag guard then Proof. Let β be the angle between PI and the horizontal line passing from c aux (see Figure 12(a)). Also let 2l = PI. ...
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... is because: the evader is inside the fourth quadrant of P (zig-zag guard state), v ∈ h[p ref ] accord- ing to invariant (I2), and the search path used for searching pocket( v, r) ensures that the evader cannot cross p ref to the left. See Figure 13(b). Proof. ...
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... of theorem 3.4. Suppose P is currently in a com- bined ( SG) state. In Lemma 5.1, Lemma 5.2, and Lemma 5.5, we showed that after finite time this combined Fig. 13. (a) When v ∈ Chain L and E disappears behind v ∈ Chain L so that x( v ) < x( v), the resulting pocket is a simple pocket. (b) When v ∈ Chain L and E appears in the fourth quad- rant, E cannot cross p ref to the left since it is confined with ...
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... turns out that if we slightly relax the monotonicity constraint and consider the class of weakly monotone polygons 9 , capture is no longer guaranteed. Fig- ure 14 shows a weakly monotone polygon in which the evader can escape forever. ...
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... time for the lion and man game is ( D log D) for the case of a circular environment and full visibility ( Alonso et al., 1992), and O( nD) for the case of line-of-sight and gen- eral polygons (Isler et al., 2005). One interesting question is whether the capture time for monotone polygons can be improved (perhaps using a randomized strategy). Fig. 14. A weakly monotone polygon with respect to s and t. The upper chain that connects s to t is a repetition of the chain from s to y 4 . The number of repetitions can be arbitrarily large. The chains from s to x 1 , from x 2 to x 3 , and from x 4 to x 5 are x-monotone chains. Also, the chains from x 1 to y 1 , from y 1 to x 2 , from x 3 ...
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... following the MPC pur- suit strategy on a simple pocket pocket( v, r), after at most O( n 2 D 3 ) time steps, E is forced to cross the entrance and exit the pocket in order to prevent being captured. At the crossing time, P and E both lie on r and P is in between v and E, see Figure 12(d). The only difference is that during search the pursuer moves along r. ...
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... during lion's move, P lies on the edge parent( E) E, the entrance − → Pv is in direction of the tree edge parent( v ) v . See Figure 15(a). Therefore, it is enough to show that, in the shortest path tree rooted at v, all edges have positive slope, and hence, if the evader disappears during L state, the new pocket pocket( v , − → Pv ) ...
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... will show that there exists a path from v to w which is shorter than π( v, w), which is a contradiction. See Figure 15(c). Since the original pocket had a positive slope, there exists a vertex u below vw such that w is visible to u . ...
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... p is in the 1st-type critical sub-polygon: Note that all points in this part are descendants of v i−1 , see Lemma B.1. For the sake of contradiction let us assume that parent( p) is in the third quadrant of p, see Figure 17. In the following we will show that there exist a shorter path than ...
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... p is in the 2nd-type critical sub-polygon: The same as the previous case. See Figure 17(c). Note that all points in this part are descendants of the summit vertex s, see Lemma B.1. ...
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... we only present the argument for the 2nd type. See Figure 18 and consider the step from A to D: ...
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... If E is inside the fourth quadrant of P: See Fig- ure 18(a). Observe that the pocket formed by the segment uD and ∂Q from u to D is a simple pocket. ...
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... From A to u: Note that E has to be in the first quadrant of P because otherwise P must have seen him sooner. The configuration that π ( O L , E) is above P, shown in Fig- ure 18(b), is similar to the case (b) above. The configu- ration that π ( O L , E) is below P, shown in Figure 18(c), is as follows. ...
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... configuration that π ( O L , E) is above P, shown in Fig- ure 18(b), is similar to the case (b) above. The configu- ration that π ( O L , E) is below P, shown in Figure 18(c), is as follows. The pursuer moves toward π ( O L , E) along X P . ...
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... ensures that E is in the first quadrant of P until E crosses X P . At this time, P moves toward π ( O L , E) along − Y P , see Figure 18(d). Note that P is becoming closer and closer to π ( O L , E) while π ( O L , E) is con- fined in the triangular region ABu. ...
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... simple guard strategy, we define a local variable called the auxiliary vertex p aux which is used as a landmark to by guest on January 11, 2016 ijr.sagepub.com Downloaded from Fig. 18. The zig-zag moves when P invokes zig-zag guard inside the 2nd-type critical sub-polygons while v in the preceding S state was in the 1st type. See Lemma ...
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... Figure 19 • If p 0 is in the portion of the search path after the floor point: See Figure 19 parts (b) and (c). Let a be the inter- section point between the α line passing through p 0 and . ...
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... Figure 19 • If p 0 is in the portion of the search path after the floor point: See Figure 19 parts (b) and (c). Let a be the inter- section point between the α line passing through p 0 and . ...
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... if a is inside the next criti- cal sub-polygon, p aux is the second critical endpoint that defines the current critical sub-polygon. If w 1 = w 1 , i.e. a and p ceil are not in between the endpoints of the same edge on , then p aux = w 1 , see Figure 19(b). Oth- erwise, p aux is the bottommost vertex from the upper chain which is inside h[p ceil ] and to the left of the line that connects p ceil to p 0 , see Figure 19(c). ...
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... w 1 = w 1 , i.e. a and p ceil are not in between the endpoints of the same edge on , then p aux = w 1 , see Figure 19(b). Oth- erwise, p aux is the bottommost vertex from the upper chain which is inside h[p ceil ] and to the left of the line that connects p ceil to p 0 , see Figure 19(c). First observe that the angle p aux p 0 is equal to or smaller than α, see Lemma E.1. ...
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Citations
... Lion and Man game occurs in different areas of mathematics, e.g., it appears in [7,8] to study Brownian motions, in [13] to study the safe path planning, in [1,14] to examine the behavior of pursuit curves on the CAT(k) spaces, and in [16,21,22] to obtain new geometric relations. Besides, pursuit-evasion games can be used in robotic systems, but such tasks usually require a more detailed analysis of spatial peculiarities (e.g., see [12,23,24]). ...
This paper addresses the robustness of the capture radii with respect to perturbation in the phase space on the example of the so-called Lion and Man game. This is a two-person pursuit-evasion game with equal players’ top speeds. The existence of \(\alpha \)-capture by a time T in one compact geodesic space is proved to yield the existence of \(\bigl (\alpha + (20T +8)\sqrt{\delta }\bigr )\)-capture by the time T in any compact geodesic space that is \(\delta \)-close to the given one. In this way, a pursuer’s strategy in one space is transferred to another space that is close to the given one in the sense of the Gromov–Hausdorff distance. It means that the capture radii (in similar spaces) tend to the given one as the distance between spaces tends to zero. In particular, this result justifies the consideration of Lion and Man game on the finite metric graphs instead of complicated original spaces.
... Stealth and pursuit-evasion problems tend to be computationally hard. Even a basic problem with a single pursuer and single evader in a confined but otherwise empty space (Rado's famous "Lion and Man" scenario) has significant complexity, with ideal strategies for both parties depending on relative speed, time bounds, and arena shape assumptions (Noori and Isler 2014). The addition of simple obstacles adds further complication (Karnad and Isler 2009) to movement choices. ...
The need to stay hidden from opponents is a common feature of many games. Defining algorithmic strategies for hiding, however, is difficult, and thus not usually a non-player character activity outside of very simple or scripted behaviours. In this work we explore several algorithmic approaches for ensuring a character can remain hidden with respect to another, moving agent. We compare these strategies with an upper-bound solution based on a known opponent path, giving us a mechanism for evaluating both relative efficacy and for understanding the different factors that affect success. Experimental evaluation considers multiple levels, including ones adapted from commercial games, and also examines the impact of relative movement speed and different observer movements. Our analysis shows that simple cost-effective approaches to hiding are feasible, but success strongly depends on level geometry, with a large gap remaining between heuristic and optimal performance.
... The work by Katsev et al. [42] introduces a simple wall-following robot to map and solve pursuit-evasion strategies in an unknown polygonal environment. The authors [43] revisit the lion and man problem by introducing line-of-sight visibility. In [44], the number of pursuers which can guarantee to capture an equal speed evader in polygonal environment with obstacles, is investigated. ...
... holds for all x 0 Pj ∈ X 0 k \X 0 k . Combining (43) with (46) shows that there exists a point, i.e., p 1 , in T that p can reach before all pursuers in X 0 k with a speed ratio α. Thus, p ∈W n k E , implying thatWn k E ⊆W n k E . ...
This work considers a multiplayer reach-avoid game between two adversarial teams in a general convex domain which consists of a target region and a play region. The evasion team, initially lying in the play region, aims to send as many its team members into the target region as possible, while the pursuit team with its team members initially distributed in both play region and target region, strives to prevent that by capturing the evaders. We aim at investigating a task assignment about the pursuer-evader matching, which can maximize the number of the evaders who can be captured before reaching the target region safely when both teams play optimally. To address this, two winning regions for a group of pursuers to intercept an evader are determined by constructing an analytical barrier which divides these two parts. Then, a task assignment to guarantee the most evaders intercepted is provided by solving a simplified 0-1 integer programming instead of a non-deterministic polynomial problem, easing the computation burden dramatically. It is worth noting that except the task assignment, the whole analysis is analytical. Finally, simulation results are also presented.
... Many other discretetime settings have been considered in the literature, including multi-pursuer search in simply connected polygons [11], in generic polygons [12,13] and in topological spaces [14]. Several works have also addressed the presence of sensing limitations, see e.g., [15,16]. ...
... Nevertheless, as recalled at the end of Section 3.1, if the lion and man are sufficiently close at the beginning of the game, the man can play in such a way that the P-FCLS bound J t (C 0 ) decreases approximately by 1 at every move. In such cases, the leftmost inequality in (16) guarantees that the P-MCLS bound dominates the P-FCLS one, i.e., J t (C * t ) ≤ J t (C 0 ), ∀t. ...
This paper studies a discrete-time pursuit–evasiongame within a convex polygonal environment. Building on solutions of the classic lion-and-man problem, two strategies are proposed for the pursuer, which guarantee exact capture in finite time and provide upper bounds on the time-to-capture at each move of the game. A numerical procedure for updating the so-called center of the game, which is instrumental for computing the lion’s move, is devised. Numerical simulations show that optimizing the center position, with respect to a suitable cost function taking into account the structure of the environment, allows one to remarkably reduce the number of moves required to capture the evader.
... This opens up the question about which is the most efficient pursuit strategy. Even more importantly, these lion's moves are employed as lowlevel procedures for addressing pursuit problems in more complex settings, like polygonal bounded environments (Ames et al., 2015;Isler, Kannan, & Khanna, 2005), topological spaces (Beveridge & Cai, 2017), or searches with visibility limitations (Noori & Isler, 2014). ...
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... In particular, a slight variation of the solution proposed in [10] is adopted iteratively in [4], where it is instrumental to devise a strategy for two pursuers to capture an evader in simply connected polygonal environments. A similar variation is employed in [12], when dealing with pursuer evasion games in monotone polygons with line-of-sight visibility. ...
... The result follows directly by (9) and (12). ...
In this paper, a novel lion strategy for David Gale's lion and man problem is proposed. The devised approach enhances a popular strategy proposed by Sgall, which relies on the computation of a suitable "center". The key idea of the new strategy is to update the center at each move, instead of computing it once and for all at the beginning of the game. Convergence of the proposed lion strategy is proven and an upper bound on the game length is derived, which dominates the existing bounds.
... Closely related pursuit-evasion games assume that one agent/team actively tries to avoid capture by another agent/team [13,22], leading to an adversarial relationship between the searchers(s) and the target(s). Capture the flag [17] assumes that one team is attempting to steal a target that is guarded by the other team. ...
We study a search game in which two multiagent teams compete to find a stationary target at an unknown location. Each team plays a mixed strategy over the set of search sweep-patterns allowed from its respective random starting locations. Assuming that communication enables cooperation we find closed-form expressions for the probability of winning the game as a function of team sizes, and vs. the existence or absence of communication within each team. Assuming the target is distributed uniformly at random, an optimal mixed strategy equalizes the expected first-visit time to all points within the search space. The benefits of communication enabled cooperation increase with team size. Simulations and experiments agree well with analytical results.
... There are many interests on systems of agents with conflicting aims (for example, pursuit-evasion problems or games [1], [4], [7], [8], [12]- [14], [18]- [26], the path defense reach-avoid game of one attacker and one defender [22], the confinement-escape problem [2] and the cooperative-confinement-escape problem [9]), as well as cooperative control of agents [3], [5]. Among pursuitevasion problems, there are large collections of searching problems, which can be divided into two categories: one category is the graph-searching problems (or called pursuit-evasion on graphs) in which all players are located on the vertices and/or edges of a graph [10], [11]; the other category is the environment-searching problems, in which all players are located in a geometric environment [15]- [17]. ...
... The princess and monster game, proposed in [1], is an environment-searching problem, in which the monster wins if the distance between two agents is within a predefined radius. The lion and man game is also an environment-searching problem [18]. The reader may also refer to [2] for more references. ...
In this paper, we investigate some mathematical properties of the confinement-escape problem of a defender and an evader with respect to a circular region, which was proposed in the author's previous work. Initially, the evader is located inside the circle, the defender patrols on the circle and tries to seal it to prevent the evader' escape; while the evader attempts to escape with avoidance of the defender. Here, we adopt the same control laws of the agents and consider particularly the successful-escape conditions which ensure a monotone-increasing distance (MID) between the defender and the evader as the system evolves, for abbreviation, we call it the escape with the MID to the defender, or simply the MID escape. Then, we: 1) provide some sufficient conditions for the MID escape under different situations; 2) provide the corresponding upper-limit estimations of the escape time; and 3) discuss the characteristics of the analytical results.
... The proofs for each polygon family can be found in the three sections that follow, with our arguments building successively. Monotone pursuit is the simplest case, so we prove Theorem 1.1(a) in Section 2. Noori and Isler [16] proved that monotone polygons are pursuer-win for a line-of-sight pursuer. Their algorithm guarantees a capture time of O(n(Q) 7 · diam(Q) 13 ) where diam(Q) = max x,y∈Q d(x, y). ...
... We start with a qualitative description of the pursuit, tackling the details of this algorithm in the subsections that follow. Our pursuer algorithm in a monotone polygon has much in common with the algorithm presented by Noori and Isler [16]. ...
... Determining the full class of pursuer-win environments remains a challenging open question. It is known that there are weakly monotone polygons which are evader-win [16]. One milestone would be resolving whether the family of sweepable polygons is pursuer-win. ...
We study a turn-based game in a simply connected polygonal environment $Q$
between a pursuer $P$ and an adversarial evader $E$. Both players can move in a
straight line to any point within unit distance during their turn. The pursuer
$P$ wins by capturing the evader, meaning that their distance satisfies $d(P,
E) \leq 1$, while the evader wins by eluding capture forever. Both players have
a map of the environment, but they have different sensing capabilities. The
evader $E$ always knows the location of $P$. Meanwhile, $P$ only has
line-of-sight visibility: $P$ observes the evader's position only when the line
segment connecting them lies entirely within the polygon. Therefore $P$ must
search for $E$ when the evader is hidden from view.
We provide a winning strategy for $P$ in the family of strictly sweepable
polygons, meaning that a straight line $L$ can be moved continuously over $Q$
so that (1) $L \cap Q$ is always convex and (2) every point on the boundary
$\partial Q$ is swept exactly once. This strict sweeping requires that $L$
moves along $Q$ via a sequence of translations and rotations. We develop our
main result by first considering pursuit in the subfamilies of monotone
polygons (where $L$ moves by translation) and scallop polygons (where $L$ moves
by a single rotation). Our algorithm uses rook strategy during its active
pursuit phase, rather than the well-known lion strategy. The rook strategy is
crucial for obtaining a capture time that is linear in the area of $Q$. For
monotone and scallop polygons, our algorithm has a capture time of $O(n(Q) +
\mbox{area}(Q))$, where $n(Q)$ is the number of polygon vertices. The capture
time bound for strictly sweepable polygons is $O( n(Q) \cdot \mbox{area}(Q) )$.
