Capturing an Evader in Polygonal Environments: A Complete Information Game
aa r X i v : . [ c s . G T ] O c t Capturing an Evader in Polygonal Environments:A Complete Information Game ∗ Kyle Klein and Subhash SuriDepartment of Computer ScienceUC Santa Barbara, CA 93106, USA { kyleklein, suri } @cs.ucsb.edu Abstract
Suppose an unpredictable evader is free to move around in a polygonal environmentof arbitrary complexity that is under full camera surveillance. How many pursuers, eachwith the same maximum speed as the evader, are necessary and sufficient to guaranteea successful capture of the evader? The pursuers always know the evader’s current po-sition through the camera network, but need to physically reach the evader to captureit. We allow the evader the knowledge of the current positions of all the pursuers aswell—this accords with the standard worst-case analysis model, but also models a prac-tical situation where the evader has “hacked” into the surveillance system. Our mainresult is to prove that three pursuers are always sufficient and sometimes necessary to capture the evader. The bound is independent of the number of vertices or holesin the polygonal environment. The result should be contrasted with the incomplete information pursuit-evasion where at least Ω( √ h + log n ) pursuers are required [4] justfor detecting the evader in an environment with n vertices and h holes. Pursuit-evasion games provide an elegant setting to study algorithmic and strategic questionsof exploration or monitoring by autonomous agents. Their mathematical history can betraced back to at least 1930s when Rado posed the now-classical Lion-and-Man problem: alion and a man in a closed arena have equal maximum speeds; what tactics should the lionemploy to be sure of his meal? The problem was settled by Besicovitch who showed that theman can escape regardless of the lions strategy. An important aspect of this pursuit-evasion ∗ The research on this paper was supported in part by the National Science Foundation Grant IIS-0904501.A preliminary version of this paper appeared at the 25th Conference on Artificial Intelligence. n -gons that may require Ω(log n ) pursuersin the worst-case to detect a single, arbitrarily fast moving evader, and O (log n ) pursuersalso always suffice for all n vertex simple polygons [4]. When the polygon has h holes, thenumber of necessary and sufficient pursuers turns out to be O ( √ h + log n ) [4]. However,these results hold only for detection of the evader, not for the capture. For capturing the evader, it is reasonable to assume that the pursuers and the evaderall have the same maximum speed. Under this assumption, it is shown by Isler et al. [8]that two pursuers can capture the evader in a simply-connected polygon using a randomized strategy whose expected search time is polynomial in n and the diameter of the polygon.When the polygon has holes, no non-trivial upper bound is known for capturing the evader.For instance, we do not even know if O ( h ) pursuers are able to capture the evader. Becausevisibility-based pursuit allows unbounded line-of-sight visibility regardless of the distance, itis unclear how to map a detection strategy to a capture strategy.In this paper, we attempt to disentangle these two orthogonal issues inherent in pursuitevasion: localization , which is purely an informational problem, and capture , which is aproblem of planning physical moves. In particular, we ask how complex is the captureproblem if the evader localization is available for free ? In other words, suppose the pursuershave complete information about the evader’s current position, how much does it help themto capture the evader? Besides being a theoretically interesting question, the problem is alsoa reasonable model for many practical settings. Given the rapidly dropping cost of electronicsurveillance and camera networks, it is now both technologically and economically feasible tohave such monitoring capabilities. These technologies enable cheap and ubiquitous detectionand localization, but in case of intrusion, a physical capture of the evader is still necessary. Our main result is that under such a complete information setting, three pursuers are alwayssufficient to capture an equally fast evader in a polygonal environment with holes, using a deterministic strategy. The bound is independent of the number of vertices n or the holes ofthe polygon, although the capture time depends on both n and the diameter of the polygon.Complementing this upper bound, we also show that there exists polygonal environmentsthat require at least three pursuers to capture the evader even with full information.2 .2 Related Work There is an enormous literature on pursuit evasion and related problems [1, 2, 3, 6, 9, 10, 11,13, 14, 15, 17]. The research tends to fall into two distinct categories: geometry-based andgraph-based. The former assumes a continuous model of space, typically a polygon, while thelatter assumes a discrete graph model where agents move along edges. The graphs providea very general setting but can suffer from two shortcomings: one, the generality leads toweak upper bounds and, two, they fail to model many restrictions imposed by the geometryof physical world. Thus, for instance, determining the search number of cop-number or ageneral graph remains a difficult open problem despite decades of research.In visibility-based pursuit, a seminal paper [4] shows that Θ(log n ) pursuers are bothnecessary and sufficient in worst-case for a simply-connected n -vertex polygon. Most of theexisting work in polygon searching, however, is on detection and not capture. The onlyrelevant result on capture is by Isler et al. [8] showing that in polygons without holes twopursuers can achieve both detection and capture. When the environment has holes, it isnot even known how many pursuers are sufficient to capture an evader, even though a tightbound of Θ( √ h +log n ) for detection is known. In one important aspect, polygon searching isfundamentally different from graph searching: re-contamination is unavoidable in polygons,in general, while graphs can always be searched optimally without re-contamination [4].Our work bears some resemblance to, and is inspired by, the result of Aigner andFromme [1] on planar graphs, showing that graph searching on planar graph requires 3cops. In that work, the graph is unweighted, does not deal with Euclidean distances, andrequire players to move to only neighboring nodes. Unlike the graph model, our search oc-curs in continuous Euclidean plane, and players can move to any position within distanceone. Thus, while our bounds are similar, the proof techniques and technical details are quitedifferent. We assume that an evader e is free to move in a two-dimensional closed polygon P , whichhas n vertices and h holes. A set of pursuers, denoted p , p , . . . , wish to capture the evader.All the players have the same maximum speed, which we assume is normalized to 1. Thebounds in our algorithm depend on the number of vertices n and the diameter of the polygon,diam( P ), which is the maximum distance between any two vertices of P under the shortestpath metric.For the sake of notational brevity, we also use e to denote the current position of theevader, and p i to denote the position of the i th pursuer. We model the pursuit-evasion as acontinuous space, discrete time game: the players can move anywhere inside the polygon P ,but they take turns in making their moves, with the evader moving first. In each move, aplayer can move to any position whose shortest path distance from its current position is atmost one; that is, within geodesic disk of radius one. On the pursuers’ move, all the pursuerscan move simultaneously and independently. We say that e is successfully captured when3ome pursuer p i becomes collocated with e .In order to focus on the complexity of the capture, we assume a complete informationsetup: each pursuer knows the location of the evader at all times. We also endow the evaderthe same information, so e also knows the locations of all the pursuers. In addition, bothsides know the environment P , but neither side knows anything about the future moves orstrategies of the other side. We begin with a high level description of the capture strategy,followed by its technical details and proof of correctness in the next section. We show that three pursuers, denoted p , p , p , can always capture an evader using a deter-ministic strategy, regardless of the evader’s strategy and the geometry of the environment.Our overall strategy is to progressively trap the evader in an ever-shrinking region of thepolygon P . The pursuit begins by first choosing a path Π that divides the polygon into sub-polygons (see Figure 1(a))—we will use the notation P e to denote the sub-polygon containingthe evader. We show that, after an initialization period, the pursuer p can successfully guardthe path Π , meaning that e cannot move across it without being captured.In a general step, the sub-polygon P e containing the evader is bounded by two paths Π and Π , satisfying a geometric property called minimality , each being guarded by a pursuer.We then choose a third path Π splitting the region P e into two non-empty subsets. If bothregions have holes, then we argue that the pursuer p can guard Π , thereby trapping e either between Π and Π (see Figure 1(b)), or between Π and Π , in which case the pursuititerates in a smaller region. If one of the regions is hole-free, then we show that the pursuer p can evict the evader from this region, forcing it into a smaller region where the searchresumes. In order for this strategy to work, the paths Π i need to be carefully chosen and must satisfycertain geometric conditions, which we briefly explain. First, although the pursuit occurs incontinuous space, our paths will be computed from a discrete space, namely, the visibilitygraph of the polygon. The visibility graph G ( P ) of a polygon P is defined as follows: thenodes are the vertices of the polygonal environment (including the holes), and two nodesare joined by an edge if the line segment joining them lies entirely in the (closed) interiorof the polygon. (In other words, the two vertices joined by an edge must have line of sightvisibility.) This undirected graph has n vertices and at most O ( n ) edges. We assign eachedge a weight equal to the Euclidean distance between its two endpoints. See Figure 1(a)for an example.One can easily see that, given two vertices u and v of P , the shortest path from u to v in G ( P ) is also the shortest Euclidean path constrained to lie inside P . (The shortest Euclideanpath has corners only at vertices of G ( P ).) However, we cannot make such a claim for the second , or in general the k th, shortest path—one can create an infinitesimal “bend” in theshortest path Π to create another path that is arbitrarily close to the first shortest path but4 Π x y zu v (a) uv Π Π Π e (b) Figure 1: (a) A polygonal environment with two holes (a rectangle and a triangle). xy isa visibility edge of G ( P ), while xz is not. Π and Π are the first and the second shortestpaths between anchors u and v . The figure (b) illustrates the main strategy of trapping theevader through three paths.does not belong to G ( P ). Therefore, we will only consider paths that belong to G ( P ) andare “combinatorially distinct” from Π —that is, they differ in at least one visibility edge.However, even then the k th shortest path between two nodes can exhibit counter-intuitivebehavior. For instance, while in graphs with non-negative weights the first shortest pathis always loop-free, the second , or more generally k th, shortest path can have loops—thismay happen if repeatedly looping around a small-weight cycle (to make the path distinctfrom others) is cheaper than taking a different but expensive edge [7]. Therefore, we willconsider only shortest loop-free paths. One of our technical lemmas proves that these pathsare also geometrically non-self-intersecting. (This is obvious for the shortest path Π but notfor subsequent paths.) In addition, we argue that these paths also satisfy a key geometricproperty, called minimality , which allows a pursuer to guard them against an evader. We begin with the discussion of how a single pursuer can guard a path in P , trapping theevader on one side. We then discuss the technically more challenging case of guarding thesecond and the third paths. In order to guarantee that a path in P can be guarded, it mustsatisfy certain geometric properties. We begin by introducing two key ideas: a minimal path and the projection of evader on a path. In the following, we use the notation d ( x, y ) todenote the shortest path distance between points x and y . When we require that distanceto be measured within a subset, such as restricted to a path Π, we write d Π ( x, y ). That is, d Π ( x, y ) is the length of path Π between its points x and y . Occasionally, we also use thenotation Π( x, y ) to denote subpath of Π between points x, y . We use the notation x ≺ y toemphasize that the point x precedes y on the path Π: that is, if Π is the path from node u to node v , then x ≺ y means that d Π ( u, x ) < d Π ( u, y ). The following property is important5or patrolling of paths. Definition 1. ( Minimal Path: ) Suppose Π is a path in P dividing it into two sub-polygons,and P e is the sub-polygon containing the evader e . We say that Π is minimal if, for all points x, z ∈ Π and y ∈ ( P e \ Π) , the following holds: d Π ( x, z ) ≤ d ( x, y ) + d ( y, z )Intuitively, a minimal path cannot be shortcut: that is, for any two points on the path,it is never shorter to take a detour through an interior point of P e . (This is a weak form oftriangle inequality, which excludes detours only through points contained in P e . ) The nextdefinition introduces the projection of the evader on to a path, which is an important conceptin our algorithm. Definition 2. ( Projection: ) Suppose Π is a path in P dividing it into two sub-polygons,and P e is the sub-polygon containing the evader e . Then, the projection of e on Π , denoted e π , is a point on Π such that, for all x ∈ Π , e is no closer to x than is e π . Thus, if a pursuer is able to position itself at the projection of e at all times, thenit guarantees that the evader cannot cross the path without being captured. With thesedefinitions in place, we now discuss how to guard the first path Π . We choose two non-neighbor vertices u and v on the outer boundary of P , and call them an-chors . We let Π be the shortest path from u to v in G ( P ); this is also the shortest Euclideanpath between u and v constrained to lie inside the environment. Our first observation is thatthis path Π is always minimal. Lemma 1.
The path Π between u and v is minimal.Proof. For the sake of contradiction, suppose there are two points x, z ∈ Π that violate theminimality. Let the point y / ∈ Π be the witness of this violation. That is, d ( x, y ) + d ( y, z ) Suppose Π is a minimal path between the anchor nodes u and v . Then, for everyposition of the evader e in P e , the projection e π exists. In fact, point on Π at distance d ( e, u ) from u along the path is a projection of e .Proof. On the path Π, starting from u , we pick the furthest point z such that for all x ≺ z we have d Π ( x, z ) ≤ d ( x, e ). We note that necessarily for some x = z , this must be equalitybecause otherwise there exists a point farther along Π satisfying the inequality. We claim6hat z is a projection of e . Suppose not. Then there exists a point x ≻ z such that d Π ( z, x ) > d ( e, x ). This violates the minimality of Π because d Π ( u, x ) = d Π ( u, z ) + d Π ( z, x ) = d ( u, e ) + d Π ( z, x ) > d ( u, e ) + d ( e, x )Next, we show that the point at d ( e, u ) along Π is a projection of e . In case we have d ( e, u ) > d Π ( u, v ), we choose v as the projection point as a convention. Otherwise, considerthe point z that lies on Π at distance d ( e, u ) from u . In order to show that z is a valid andunique projection, it suffices to show that for any other point x on Π, the point z is closerto it than e is. Indeed, if x is such that z ≺ x , then d Π ( z, x ) > d ( e, x ) would contradict theminimality of Π: d Π ( u, x ) = d Π ( u, z ) + d Π ( z, x ) = d ( u, e ) + d Π ( z, x ) > d ( u, e ) + d ( e, x )Similarly, if x is such that x ≺ z , then d Π ( x, z ) > d ( x, e ) contradicts the hypothesis that y satisfies d ( e, u ) = d Π ( u, z ): d ( e, u ) ≤ d ( e, x ) + d Π ( x, u ) < d Π ( z, x ) + d Π ( x, u ) = d Π ( z, u )Therefore, the chosen point z is closer to all points on Π than e is, and hence it is aprojection of e . This completes the proof.The next lemma shows how a pursuer can guard a minimal path. Lemma 3. Suppose Π is a minimal path between the anchors u, v in P , and a pursuer p is located at the current projection of e . Suppose on its turn the evader moves from e to e ′ .Then, the pursuer p can either capture the evader or relocate to the new projection e ′ π in onemove.Proof. First, suppose that the new position e ′ is on different side of the path Π than e ; thatis, the evader crosses the path. Because e moves a distance of at most one, and supposeit crosses the path at a point z , we have d ( e, z ) + d ( z, e ′ ) ≤ 1. On the other hand, since p is located at the projection of e before the move, d Π ( p, z ) ≤ d ( e, z ). Therefore, the newposition of the evader e ′ is within distance one of p , and the pursuer can capture the evaderon its move.Therefore, assume that the evader does not cross the path, and moves to a position e ′ such that e π = e ′ π . Consider any two points x ≺ e π ≺ y (one on either side of the projection),and let d ( x, e ′ ) = d Π ( x, e π ) + c and d ( y, e ′ ) = d Π ( y, e π ) + c ′ . Then we claim that c + c ′ ≥ c + c ′ ≥ d Π ( x, y ) ≤ d ( x, e ′ ) + d ( e ′ , y ) = d Π ( x, e π ) + d Π ( y, e π ) + c + c ′ = d Π ( x, y ) + c + c ′ Thus e can move closer to x or y but not both. Suppose it moves closer to x , meaning c < 0. Consider some x ≺ e π with the smallest value of c . Then suppose for any y ≻ e π ,7 ′ < | c | . This would imply c + c ′ < 0, and so if e moves a maximum of c closer to any x , itmoves at least c farther from any y . We claim that e ′ π is at the point c closer to x . To showthis we argue that at the position e ′ π , the pursuer is closer to all points on Π. But e ′ cannotbe closer to any x ≺ e ′ π because e ′ is no more than c closer than e to any such x . Similarly e ′ must still be farther from any y ≻ p e because e moved c or more away from those points.This leaves the points between e ′ π and e π , but if e ′ is closer to one of these points (say, z ′ ),then p can capture e because d ( e, z ′ ) < d ( e ′ π , z ′ ). Therefore, the pursuer can move to thenew projection because | c | < 1. This completes the proof.Finally, we show that a pursuer p requires O (diam( P ) ) moves to either reach the currentprojection of e or capture it. u vP ′ Figure 2: Example of sub-polygons P ′ created with diameter larger than diam( P ). Lemma 4. Suppose Π is a minimal path between anchors u, v in P , and a pursuer p islocated at u . Then in O (diam( P ) ) moves, p can move to e ’s projection.Proof. By Lemma 3, the projection of e can only shift by distance at most one along thepath Π. Thus, p ’s strategy is simply to move along the path from one end to the other untilit coincides with the current projection of e , or captures it. However, as Π is not the shortest u, v path, d Π ( u, v ) may be larger than diam( P ) (an example of progressively longer paths isdepicted in Figure 2), thus we show that d Π ( u, v ) ≤ diam( P ) .Notice that Π splits P into two or more sub-polygons, consider one such sub-polygon P ’. Trivially area ( P ′ ) < area ( P ), thus if for all P ′ diam( P ′ ) ≤ area ( P ′ ), then d Π ( u, v ) ≤ area ( P ) and necessarily d Π ( u, v ) ≤ diam( P ) . Suppose that diam( P ′ ) > area ( P ′ ), we caneasily reverse this inequality by a simple rescaling of the units . In particular, suppose werescale the unit of measurement from 1 to 1 + α . This increases the area of a triangle by afactor of (1 + α ) , while a segment only increases in length by a factor of 1 + α . A suitablylarge choice of α , therefore, always ensures that the polygon’s area exceeds its diameter,because the former grows by a factor of (1 + α ) while the latter grows only by a factor of1 + α . Thus, we assume that a suitable rescaling has been applied to the polygon P , ensuringthat all sub-polygons P ′ encountered during the algorithm satisfy diam ( P ′ ) ≤ area ( P ′ ).8ince p moves a distance of 1 in each turn, and the path Π is at most diam( P ) long, theentire initialization phase takes O (diam( P ) ) time. Meanwhile, if the projection ever “crossesover” the current position of p , the pursuer immediately can move to the new projection pointbecause at that moment it must be within distance one of the target location. This completesthe proof. We now come to the main part of our pursuit strategy. The key idea is to progressively trapthe evader in a region bounded by two minimal paths, which are guarded by two pursuers,and to use the third pursuer to further divide the current trap region. When the third pursuersubdivides the current region containing e , two possibilities emerge: either both regions ofthe subdivision have holes, in which case we show that the third path is necessarily minimaland thus guardable by the third pursuer, limiting the evader to a smaller region than before;or one of the regions is hole-free, in which case the third pursuer uses the capture strategy fora simply-connected polygon to evict the pursuer from this region (or capture it). In order toformalize our strategy, we first show a key geometric property of the second and third shortestpaths between the anchors in the visibility graph, namely, that they are non-self-intersecting,and therefore lead to well-defined closed regions. l l l l Π L Π ( v , v ) v v Π R v v u v Π B Figure 3: Non-self-crossing of shortest paths Π , Π , Π . Lemma 5. Let Π be the shortest path between two anchor points u and v on P ’s boundary,and focus on the sub-polygon P e that lies on one side of Π . Let Π and Π , respectively, bethe second and the third simple (loop-free) shortest paths in the visibility graph G ( P e ) between u and v . Then, Π and Π are non-self-crossing.Proof. Without loss of generality, suppose the path Π violates the lemma, and that twoof its edges ( v , v ) and ( v , v ) intersect. See Figure 3. We first note that the intersectionpoint cannot be a vertex of the visibility graph because otherwise the path has a cycle, andwe assumed that Π is loop-free. As shown in the figure, we break the segment ( v , v ) into9 and l , and ( v , v ) into l and l . By the triangle inequality of the Euclidean metric, itis easy to see that d ( v , v ) < l + l , and d ( v , v ) < l + l . Similarly, d ( v , v ) < l + l .Let Π L , Π R , Π B , respectively, denote the shortest paths (in the graph G ( P e )) realizing thesedistances. Now consider the following three paths between v and v , each contained in G ( P e )) and non-self-intersecting : Π L · Π ( v , v ) · Π R , Π B , and the shorter of Π L · ( v , v ) and( v , v ) · Π R . They are all shorter than Π and at least one of them must be distinct fromboth Π and Π , thus contradicting the choice of Π . This completes the proof. In a general step of the algorithm, assume that the evader lies in a region P e of the polygonbounded by two minimal paths Π and Π between two anchor vertices u and v . (Strictlyspeaking, the region P e is initially bounded by Π , which is minimal, and portion of P ’sboundary, which is not technically a minimal path. However, the evader cannot cross thepolygon boundary, and so we treat this as a special case of the minimal path to avoidduplicating our proof argument.) We also assume that Π and Π only share vertices u and v ; if they share a common prefix or suffix subpath, we can delete those and advance theanchor nodes to the last common prefix vertex and the first common suffix vertex. Thisensures that the region P e is non-degenerate . Furthermore, we assume that the region P e contains at least one hole—otherwise, the evader is trapped in a simply-connected region,where a single (the third) pursuer can capture it.The key idea of our proof is to show that, in the visibility graph G ( P e ), if we compute a shortest path from u to v that is distinct from both Π and Π , then it divides P e into only two regions, and that the evader is trapped in one of those regions. We will call this new paththe third shortest path Π . Specifically, Π is the simple (loop-free) shortest path from u to v in G ( P e ) distinct from Π and Π . (One can compute such a path using any of the algorithmsfor computing k loop-free shortest paths in a weighted undirected graph [7, 12, 18].) Lemma 6. The shortest path Π between the anchor nodes u and v divides the current evaderregion P e into two connected regions.Proof. If the path is disjoint from Π and Π except at endpoints, then P e is clearly subdi-vided into two regions. If Π shares vertices only with Π or only with Π , but in multipledisjoint subpaths creating multiple regions , then minimality of those paths means that wecan contract all but one region and shorten Π , contradicting the choice of this third path.Therefore, let us suppose that Π shares vertices with both the paths, and so “hops” be-tween Π and Π , sharing common subpaths with them, and creates three or more regions.In that case, Π must leave and rejoin Π and Π at least once, as shown by points x, y, z in Figure 4(a). We observe that d Π ( y, v ) is no longer than d ( y, z ) + d Π ( z, v ), otherwise wecan contradict the choice of Π . Thus the third region can be removed by altering Π to usethe subpath Π ( y, v ). (A symmetric case arises when the roles of Π and Π are swapped.)Thus, we conclude that Π can create only two subregions.Because P e contains at least one hole, one of the regions created by the third shortestpath Π also must contain a hole. The following lemma argues that Π is minimal with10 v Π Π x y z Π (a) P − e P + e e Π Π (b) Figure 4: The left figure illustrates the proof of Lemma 6; the right figure illustrates the twosubregions created by a path, Π in this case.respect to that region. (The other region may be hole-free, and we will argue that the evadercan be evicted from such a hole-free region or captured.) u u ′ x z v ′ v Π Π Π ′ Π R (a) u u ′ x z v ′ v Π Π Π ′ y Π R (b) Figure 5: The left figure illustrates the proof of Lemma 6; the right figure illustrates the twosubregions created by a path, Π in this case. Lemma 7. Suppose Π divides the region P e into two subregions P + e and P − e , and assumethat P + e contains at least one hole. Then, Π is a minimal path within the region P + e .Proof. Assume, for the sake of contradiction, that the minimality of Π is violated for twopoints x, z ∈ Π . Let u ′ be the vertex immediately preceding the point x , possibly x = u ′ ,and v ′ is the vertex immediately following z , possibly z = v ′ , on Π . Consider the shortestpath in G ( P e ) from u ′ to v ′ . If this path is not a subpath of either Π or Π , then we canimmediately improve the length of Π by using this subpath, thereby contradicting the choiceof Π . Therefore, assume without loss of generality that the shortest path from u ′ to v ′ is asubpath of Π . Further, let Π denote the shortest path from point x to point z in P + e , andconsider the region R bounded by Π ( u ′ , v ′ ), Π and the segments ( z, v ′ ) and ( x, u ′ ). If thereare any holes in R then there is a distinct path Π ′ shorter than Π obtained by tightening11 around those holes as shown in Figure 5(a). Thus the hole in P + e must be outside R ,however pick the closest vertex on a hole in P + e to Π, call it y . Then a path Π ′ shorter thanΠ can be obtained using y as shown in Figure 5(b). Thus in all cases, if P + e contains ahole, Π can be shortened, which contradicts its optimality. Thus Π ’s minimality cannotbe violated, and the proof is complete.When one of the regions created by Π is hole-free, then Π has a very simple structure,consisting of only two distinct edges as seen in Figure 4(b), allowing it to be cleared usingthe search strategy of a simply-connected polygon. Lemma 8. Suppose Π divides the region P e into two subregions, one of which is hole-free.Then, Π consists of precisely two edges.Proof. Arguing as in the preceding lemma, it can be shown that if P + e is hole-free, then theshortest path Π has a common prefix and a common suffix with Π , and only differs in asubpath of the form Π ( u ′ , v ′ ). Suppose for the sake of contradiction that Π ( u ′ , v ′ ) consistsof more than two edges. Consider the first three points of this subpath, u ′ , x and z . Noticethat because they are not co-linear and by assumption there are no holes contained betweenΠ ( u ′ , v ′ ) and Π that the shortest u ′ , z path would shortcut x , contradicting the choice ofΠ . Thus z = v ′ and Π consists of only two edges. This completes the proof.Now, if both regions created by Π have holes, then the minimality of Π allows a thirdpursuer to guard this path, and the pursuit continues in this smaller region. If one of theregions created is hole-free, then we no longer can assume minimality with respect to thatregion, so a different strategy is required. The following lemmas show how to either capturethe evader in such a region, or to force the evader out of (evict) this region, while guardingΠ so the evader cannot reenter this region.Capturing an evader in a hole-free region can be accomplished by advancing along theshortest path towards the current position of the evader. In particular, we can fix a origin O inthe region (say, some vertex in P ), and then letting the pursuer move along the shortest pathbetween O and the current evader position. It can be shown that the pursuer makes sufficientprogress towards the evader, as articulated in the following lemma (which paraphrases atechnical result of [10]). Lemma 9. After each move, (1) the pursuer p remains on the shortest path between O and e , and (2) its new position p ′ satisfies d ( p ′ , O ) ≥ d ( p, O ) + n . Using this result, we can derive the following lemma about capturing the evader in ahole-free region. Lemma 10. Suppose the evader lies in hole-free region of k vertices that is bounded by Π and another minimal path. Then, in O ( k · diam( P ) ) moves, a single pursuer p can eithercapture the evader or force it out of the region and place itself on e ’s projection on the path Π . y ze pu v Π Figure 6: An illustration of the pursuer’s eviction strategy. Dashed lines denote moves where e moved first. Proof. Assume, without loss of generality, that our hole-free region is bounded by a minimalpath Π and the path Π , which by Lemma 8 must consist of two edges, say, ( x, y ) and ( y, z ).The pursuer p ’s strategy is to move to y , and execute a simple-polygon search with y as theorigin with the following modification: if p ’s move takes it outside the region, then it movesalong Π toward e π until e reenters, at which point its resumes the pursuit.As the shortest path between any two vertices consists of at most two edges, this regioncan have diameter no larger than 2 · diam( P ). Thus if e never leaves the region, then by theknown result of Lemma 9, a successful capture occurs in O ( k · diam( P ) ) moves. Therefore,assume that e leaves the region at some point. Since Π is minimal, the evader cannot leavethe region through that path, and so assume without loss of generality that the evader crossesthe segment ( x, y ) of Π . Because p always stays on the shortest path between e and y , inan unmodified pursuit p ’s move would cross ( x, y ) as well. In the modified pursuit, p stopsat the point where it crosses ( x, y ) and advances toward the projection of e . See Figure 6for illustration.We note that the projection of e is within distance one of where it crossed ( x, y ). As aresult, because p crossed ( x, y ) at a point closer to y than e , if e π lies on the subpath Π ( p, z ),then p can reach it in one move. Otherwise, p need simply advance forward along Π toward x . If e never re-enters the hole free region, then by Lemma 4 p will reach the projectionwithin O (diam( P ) ) moves.In case e re-enters the hole-free region, we note that it must do so by crossing thesegment ( x, p ), and that for each turn e was outside the hole-free region p moved distanceone along the shortest path from y to e . Thus on its next turn p can resume its pursuit, whilehaving sufficiently increased its distance from y to guarantee a successful capture occurs in O ( k · diam( P ) ) moves should e remain within the hole-free region. Thus e may continuallymove back and forth between the hole-free region, but within O ( k · diam( P ) ) moves e willeither be captured, or the pursuer will successfully guard Π by reaching the projection. Thiscompletes the proof.We can now summarize our main result. Theorem 1. Three pursuers are always sufficient to capture an evader in O ( n · diam( P ) ) moves in a polygon with n vertices and any number of holes. roof. Whenever a new path is introduced, the size (number of vertices) of the region P e containing e shrinks by at least one. Thus, the number of different paths guarded during thecourse of the pursuit before e is trapped in a hole-free region is at most n . Guarding eachpath requires O (diam( P ) ) moves for a minimal path, and O ( k · diam( P ) ) moves when in ahole-free region with k vertices. Since the evader cannot reenter hole-free regions once theyhave been guarded, the total cost of guarding all the hole-free regions during the course ofthe algorithm sums to O ( n · diam( P ) ).Finally e will be confined between two minimal paths in a hole-free region consisting ofthree vertices, otherwise additional u, v paths can be found to further reduce the region.This sub-polygon clearly has diameter no larger than diam( P ), and thus the evader can becaptured in O (diam( P ) ) moves with the known result of Lemma 9, for a total of O ( n · diam( P ) ) moves over the entire pursuit. We show that any deterministic strategy requires at least 3 pursuers in the worst-case, andthus the upper bound of the previous section is tight. Theorem 2. There exists an infinite family of polygons with holes that require at least threepursuers to capture an evader even with complete information about the evader’s location.Proof. The proof is based on a reduction from searching in planar graphs . In particular,consider a planar graph G , with minimum degree 3, and without any cycles of length threeor four (see Figure 7(c)). Using Fary’s Theorem, we can embed such a graph so that eachedge maps to a straight line segment. By suitable scaling, assume that the longest edge inthe embedding has length 1. (See Figure 7(a) for an example.)We now transform this straight-line embedding into a polygon with holes, by convertingeach edge into a “corridor.” Each corridor is constructed to ensure that the shortest paththrough it has length 1. In particular, the edges of length 1 map to straight corridors, whileshorter edges correspond to corridors with multiple turns, as shown in Figure 7 (b). It iseasy to see that such a construction can ensure that all the corridors are non-overlapping.With this transformation, the outer face of the graph becomes the boundary of the polygon P , while each face of the plane graph becomes a hole.It is known that in any graph with minimum degree k and no cycles of length three orfour, the evader has a winning strategy against k − √ / √ / √ / √ / (a) ǫ √ / ǫ (b) (c) Figure 7: Embedding of a planar graph (a), corridor construction (b), and a planar graphwith min-degree 3 and no three or four cycles (c).either vertex it could still threaten both vertices and several more if the vertex had degreegreater than one. Thus the evader’s strategy on the graph to avoid capture must still beviable as pursuer’s gain no additional ability to threaten vertices by being within corridors.In Figure 7(c) we see a example planar graph with no 3 or 4 cycles and all vertices ofdegree three, thus an evader can always avoid capture in the transformed polygon from twopursuers. In this paper, we proved that three pursuers are always sufficient to capture an evader ina polygonal environment of arbitrary complexity, under the assumption that pursuers haveaccess to evader’s location at all times. We also proved a matching lower bound, showing thatthree pursuers are also necessary in the worst-case. Traditionally, the papers on continuousspace, visibility-based pursuit problem have focussed on simply detecting the evader, and noton capturing it. One of our contributions is to isolate the intrinsic complexity of the capturefrom the associated complexity of detection or localization. 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