Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary
aa r X i v : . [ m a t h . M G ] M a y Two-Dimensional Pursuit-Evasion in a Compact Domain withPiecewise Analytic Boundary
Andrew Beveridge ∗ and Yiqing Cai † Abstract
In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players al-ternate turns, move with equal speed, and have full information about the state of the game. Weconsider the most restictive capture condition: a pursuer must become colocated with the evaderto win the game. We prove two general results about pursuit-evasion games in topological spaces.First, we show that one pursuer has a winning strategy in any
CAT (0) space under this restrictivecapture criterion. This complements a result of Alexander, Bishop and Ghrist, who provide a win-ning strategy for a game with positive capture radius. Second, we consider the game played ina compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Eulercharacteristic. We show that three pursuers always have a winning strategy by extending recentwork of Bhadauria, Klein, Isler and Suri from polygonal environments to our more general setting.
A pursuit-evasion game in a domain D is played by a team of pursuers p , p , . . . , p k and an evader e . The pursuers win if some p i becomes colocated with the evader after a finite number of turns,meaning that the distance d ( p i , e ) = 0 . When this occurs, we say that p i captures e . We consider thediscrete time version of the game, which proceeds in turns. Initially, the pursuers choose their positions p , p , . . . , p k , and then the evader chooses his initial position e . In turn t ≥ , each pursuer p i movesfrom her current position p t − i to a point p ti ∈ B ( p t − i ,
1) = { x ∈ D | d ( p t − i , x ) ≤ } . If the evaderhas been captured, then the game ends with the pursuers victorious. Otherwise, the evader moves from e t − to a point e t ∈ B ( e t − , . The evader wins if he remains uncaptured forever. We consider thefull-information (full-visibility) game in which each player knows the environment and the locationsof all the other players. Furthermore, the pursuers may coordinate their movements.Turn-based pursuit games in simply connected domains have been well-characterized: one pursueris sufficient to capture the evader. Winning pursuer strategies have been found for environments in R n [21, 16], and in simply connected polygons [14]. Taking a topological viewpoint and using the ∗ Department of Mathematics, Statistics and Computer Science, Macalester College, Saint Paul, MN 55105. [email protected] † The Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, [email protected] d ( p, e ) < ǫ for some constant ǫ > , Alexander, Bishop and Ghrist [3] provedthat a single pursuer can capture an evader in any compact CAT (0) by heading directly towards theevader at maximum speed. We provide an alternate strategy for a compact
CAT (0) space that achieves d ( p, e ) = 0 in a finite number of turns. Our winning pursuer strategy is the topological version of lion’s strategy , which has been used successfully in E n [21] and in simple polygons [14]. We defer thedescription of this strategy to Section 2. Theorem 1.1.
A pursuer p using lion’s strategy in a compact CAT (0) space D , captures the evader e by achieving d ( p, e ) = 0 after at most diam( D ) turns. Theorem 1.1 implies that a single pursuer can become colocated with an evader in a simply con-nected, compact domain
D ⊂ E . But it is easy to construct compact domains that are evader win:removing one large open set from the middle of a simply connected domain tips the game in theevader’s favor. Indeed, the evader can keep this large obstruction between himself and the pursuer,indefinitely. Such an open set is called an obstacle or hole in the environment. It is not hard to showthat adding a second pursuer to this two-dimensional domain gives the game back to the pursuers.Adding multiple obstacles creates a distinct topology, and it is natural to wonder how many pursuersare needed to capture an evader in such an environment. The analogous question has been resolved forpursuit-evasion games in certain two-dimensional environments. Aigner and Fromme [1] proved thatthree pursuers are sufficient for pursuit-evasion on a planar graph. Bhadauria, Klein, Isler and Suri [8]showed that the analogous result holds in a two-dimensional polygonal environment with polygonalholes. We generalize this three-pursuer result to a broader class of topological spaces.Our pursuit game takes place in a compact and path-connected domain D ⊂ E . The set D containsa finite set of disjoint open obstacles O = { O , O , . . . , O k } . The boundary of the domain is ∂ D = { ∂O , ∂O , ∂O , . . . , ∂O k } where we define ∂O to be the outer boundary of D , for convenience.We place two conditions on the boundary. First, ∂ D can be decomposed into a finite number ofanalytic curves γ ( t ) = ( x ( t ) , y ( t )) for ≤ t ≤ , where each of x ( t ) , y ( t ) can locally be expandedas convergent power series. Second, we require that ∂D is a 1-manifold: for any x ∈ ∂D , thereexists an ǫ > , such that B ( x, ǫ ) ∩ ∂D is homeomorphic to R . In other words, we forbid self-intersections. For brevity, we say that a domain D satisfying these properties is piecewise analytic .We list three consequences of these conditions. First, the number of singular points on the boundary isfinite. Second, the absolute value of the curvature at the nonsingular points of ∂D is bounded aboveby some constant κ max > . Third, there is a minimum separation d min > between boundarycomponents: d ( O i , O j ) > d min for all ≤ i < j ≤ k . During the game, the pursuers will guard asequence of geodesics; crucially, we will see in Section 4 that each of these geodesics is also piecewiseanalytic. This brings us to our main result. Theorem 1.2.
Three pursuers can capture an evader in a compact domain in E with piecewise ana-lytic boundary. The number of turns required to capture the evader for a domain with k obstacles is (2 k · diam( D ) + diam( D ) ) . At a high level, our winning three-pursuer strategy builds directly on those found in [1, 8], and weare indebted to those previous papers. However, our geometric and topological approach is entirelynew. In particular, our arguments are grounded in a careful investigation of the convexity, curvatureand homotopy classes of geodesic curves in our domain. And of course, Theorem 1.2 significantlyextends the class of known three-pursuer-win domains.
Pursuit-evasion games have enjoyed a long research history. In the 1930s, Rado posed the Lion andMan game in which a lion hunts the man in a circular arena. The players move with equal speeds,and the lion wins if it achieves colocation. At first blush, it seems that lion should be able to captureman, regardless of the man’s evasive strategy. However, Besicovitch showed that when the game isplayed in continuous time, the man can follow a spiraling path so that lion can get arbitrarily close, butcannot achieve colocation [17]. However, when lion and man move in discrete time steps, our intuitionprevails: lion does have a winning strategy [21].The interdisciplinary literature on pursuit-evasion games spans a range of settings and variations.Pursuit games have been studied in many environments, including graphs, in polygonal environmentsand in topological spaces. Researchers have considered motion constraints such as speed differentialsbetween the players, constraints on acceleration, and energy budgets. As for sensing models, theplayers may have full information about the positions of other players, or they may have incompleteor imperfect information. Typically, the capture condition requires achieving colocation, a proximitythreshold, or sensory visibility (such as a non-obstructed view of the evader). For an overview ofpursuit-evasion on graphs, see the monograph [9]. The papers [11] and [16] provide a nice introductionto pursuit in the polygonal setting.The past decade has witnessed a renaissance of pursuit-evasion results in multiple disciplines.Prominent research efforts come from the robotics community, where pursuit-evasion in polygonal en-vironments is a productive setting for exploring autonomous agents. Pursuit-evasion has also thrivedin the graph theory community, where it is known as the game of Cops and Robbers. More recently,researchers have started exploring pursuit-evasion games in topological spaces. This is a natural evo-lution for the study of pursuit-evasion games. Indeed, determining the number of pursuers requiredto capture an evader in a given environment becomes a question about its topology since the variousloops and holes of the environment provide escape routes for the evader.The classic paper of Aigner and Fromme [1] initiated the study of multiple pursuers versus a singleevader on a graph. In this turn-based game, agents can move to adjacent vertices, and the cops win ifone of them becomes co-located with the robber. This paper introduced the cop number of a graph,which is the minimum number of pursuers (cops) needed to catch the evader (robber). Aigner and3romme proved that the cop number of any planar graph is at most 3. This bound is tight, as thedodecahedron graph requires three cops. At a high level, their winning pursuer strategy proceeds asfollows. Two cops guard distinct ( u, v ) -paths where u, v are vertices of the graph G . This restricts thepursuer movement to a subgraph of G . The third pursuer then guards another ( u, v ) -path, chosen sothat (1) the robber’s movement is further restricted, and (2) one of the other cops no longer needs toguard its path. This frees up that cop to continue the pursuit. This process repeats until the evader iscaught.More recently, an analogous result was proven by Bhadauria, Klein, Isler and Suri [8] for pursuit-evasion games in a two-dimensional polygonal environment with polygonal holes. In this turn-basedgame, an agent can move to any point within unit distance of its current location. Like Aigner andFromme, they use colocation as their capture criterion. Bhadauria et al. prove that three pursuers aresufficient for pursuit-evasion in this setting, and that this bound is tight. The pursuer strategy is inspiredby the Aigner and Fromme strategy for planar graphs: two pursuers guard paths that confine the evaderwhile the third pursuer takes control of another path that further restricts the evader’s movement. Ofcourse, the details of the pursuit and the technical proofs are quite different from the graph case. Theirproofs make heavy use of the polygonal nature of the environment, both to find the paths to guard andto guarantee that their pursuit finishes in finite time.Just as the proofs of Bhadauria et al. were inspired by Aigner and Fromme, our proof of Theorem1.2 is inspired by those for the polygonal environment. Bhadauria et al. actually give two differentwinning strategies for three pursuers. At a high level, these strategies progress in the same way, butthe tactics for choosing paths and how to guard them are different. Herein, we adapt their shortestpath strategy to our more general setting. The topological environment introduces a distinctive set ofchallenges to overcome. In particular, we do not have a finite set of polygonal vertices to use as abackbone for our guarded paths. Instead, we rely on homotopy classes to differentiate between pathsto guard. Looking beyond the high-level structure of our pursuer strategy, the arguments (and theirtechnical details) in this paper are wholly distinct from those found in [8], and our result applies to amuch broader class of environments.Finally, we note that our result follows in the footsteps of other recent explorations of pursuit-evasion games in general geometric and topological domains. Pursuit-evasion games in such spaceshave applications in robotics, where agents must navigate and coordinate in high dimensional config-uration spaces. Alexander, Bishop and Ghrist helped to pioneer this subject, studying pursuit-evasiongames with the capture condition d ( p, e ) < ǫ for some constant ǫ > (rather that colocation). In [3],Alexander, Bishop and Ghrist prove that a single pursuer can capture an evader in any compact CAT (0) space: the simple pursuit strategy of heading directly towards the evader is a winning strategy. In [5],these authors explore the simple pursuit strategy in unbounded
CAT ( K ) spaces with positive curvature K > , developing connections between evader-win environments and the total curvature of the pur-suer’s trajectory. Finally, in [4], they consider pursuit games in unbounded Euclidean domains using4ultiple pursuers. They provide conditions on the initial configuration of the players that guaranteecapture, generalizing (and amending) results of Sgall [21] and Kopparty and Ravishankar [16]. We introduce some notation and review some concepts and results from algebraic topology [13]. Wethen prove three lemmas about convex paths in two-dimensional compact regions with piecewise ana-lytic boundary.A topological space is a set X along with a collection of subsets of X , called open sets that satisfya sequence of axioms [7]. A map f : X → Y between two spaces is continuous when the inverseimage of every open set in Y is open in X . A path Π : [0 , → D is a continuous map from interval [0 , to D , with initial point Π(0) and terminal point
Π(1) . The length l (Π) of this path is its arclength in Euclidean space E . A path Π is a loop when Π(0) = Π(1) . A simple path has no self-intersections, meaning that Π is injective. By abusing notation, we write x ∈ Π when x = Π( t ) forsome t ∈ [0 , . For x, y ∈ Π , we use Π( x, y ) to denote the subpath connecting these points. The space X is path-connected if there exists a path between any pair of points x, y ∈ X .A homotopy of paths is a family of maps f t : [0 , → X, t ∈ [0 , , such that the associated map F : [0 , × [0 , → X given by F ( s, t ) = f t ( s ) is continuous, and the endpoints f t (0) = x and f t (1) = x are independent of t . The paths f and f are called homotopic . The relation of homotopyon paths with fixed endpoints is an equivalence relation and we use [ f ] to denote the homotopy class ofthe curve f under this relation. The set [ f ] of loops in X at the basepoint x forms a group under pathcomposition, called the fundamental group of X at the basepoint x . The space X is simply connected when it is path-connected and its fundamental group is trivial. For example, a subspace X of E issimply connected if and only if it has the same homotopy type as a 2-disc.We now turn to some geometric properties of paths in E . The distance d ( x, y ) between points x, y ∈ X is the length of a shortest ( x, y ) -path in X . When restricting ourselves to R ⊂ X , we use d R ( x, y ) to denote the distance between these points in the subdomain. We will frequently consider asubdomain R enclosed by two simple ( u, v ) -paths Π , Π . We denote such a set as R [Π , Π ] ⊂ X .For a C path γ : [0 , T ] → E , its curvature at γ ( t ) is defined as κ ( t ) = ± || γ ′ ( t ) × γ ′′ ( t ) |||| γ ′ ( t ) || , withthe sign positive if the tangent turns counterclockwise, and negative if the tangent turns clockwise.The smoothness of a piecewise analytic curve γ : [0 , → E ensures that its absolute curvature isbounded at its nonsingular points. If γ is piecewise C and continuous, with t < t < · · · < t n as thepreimages of the singular points, then its total curvature is κ total ( γ ) = n X i =1 Z t t i − κ ( t ) dt + n X i =1 θ i where θ i is the exterior angle at γ ( t i ) , and θ n = 0 when γ ( t ) = γ ( t n ) . This brings us to the celebratedGauss-Bonnet Theorem which relates the total curvature of a closed curve with the Euler characteristic5 u v xy y u v Π Π Λ (a) (b)Figure 1.1: (a) A piecewise convex ( u, v ) -path Π . (b) Shortcutting a non-convex ( u, v ) -path Π .of its enclosed region. In our setting, the Euler characteristic equals − k , where k is the number ofobstacles in the region R . Theorem 1.3. [Gauss-Bonnet Theorem, cf. [12]] Given a compact region R ⊂ E with boundary ∂R ,we have κ total ( ∂R ) = 2 πχ ( R ) , where χ ( R ) is the Euler characteristic of R . We use the Gauss-Bonnet Theorem to understand the effect of obstacles on shortest paths. Inparticular, we will consider pairs of paths Π , Π with shared endpoints u, v . These paths will bepiecewise analytic (so that they have a finite number of singular points). Our goal is to prove thatif the shortest ( u, v ) -path is not unique, then each shortest path must touch an obstacle in the givenregion. We begin with a definition of convexity, which we define for the broader family of piecewise C smooth curves; an example is shown in Figure 1.1(a). Definition 1.
Let
Π : [0 , → E be a piecewise C smooth curve in E . Then Π is convex when thefollowing holds for any point x ∈ Π \{ u, v } :(a) If Π is C smooth at x , then the curvature at x is nonpositive;(b) If Π is not C smooth at x , then the tangent line at x turns clockwise by an angle ≤ θ ≤ π . Lemma 1.4.
Let Π and Π be two piecewise analytic ( u, v ) -paths with Π ∩ Π = { u, v } . If Π is ashortest path in R [Π , Π ] , and touches no obstacle inside R [Π , Π ] , then Π is convex in R [Π , Π ] .Proof. We prove the lemma by contradiction. Suppose that there exists an x ∈ Π \{ u, v } where theconvexity of Π in R [Π , Π ] is violated. Either (a) Π is C smooth at x , but the curvature at x ispositive, or (b) Π is not C smooth at x , and the tangent line turns counterclockwise by an angle < θ < π at x , creating a non-convex corner. Let d denote the minimum distance between Π andany obstacle O ∈ R [Π , Π ] with Π ∩ ∂O = ∅ . 6uppose that the curvature at x is positive, see Figure 1.1(b). There must be a C subpath Π x between y and y of Π around x with positive curvature. Using the lower bound d on the separationbetween Π and any obstacles inside R [Π , Π ] and Π , we may take y , y to be close enough so thatthe line segment Λ connecting y and y lies inside R [Π , Π ] and does not encounter any obstacles.Replacing Π x with Λ creates a path that is strictly shorter than Π , contradicting the minimality of Π .Next suppose there is a non-convex corner at x . By an analogous argument to the previous case,we can create a short-cut Λ around x to make a shorter path than Π , a contradiction. Lemma 1.5.
Let Π , Π be two ( u, v ) -paths with Π ∩ Π = { u, v } . Suppose that Π is a convex andpiecewise analytic ( u, v ) -path in R [Π , Π ] , and let Π be a convex and piecewise analytic ( v, u ) -pathin R [Π , Π ] . Then Π and Π are both straight lines connecting u, v .Proof. Let Q = Q [Π , Π ] , which is the closed region between Π and Π , pretending there beingno obstacles. If we consider ∂Q , the concatenation of Π ( u, v ) and Π ( v, u ) , it is in fact a loopbounding a simply connected region. By the Gauss-Bonnet Theorem 1.3 , the total curvature along ∂Q equals πχ ( Q ) ≥ π . We decompose the value πχ ( Q ) as the sum of total curvature of Π and Π respectively, and the exterior angles at u and v . Because of convexity, both Π and Π have totalcurvature no greater than . As for the two angles at u, v , neither can exceed π . Therefore the totalcurvature of the loop does not exceed π , and could only achieve π when κ total (Π ) = κ total (Π ) =0 . Therefore, Π and Π are both straight lines connecting u and v . Lemma 1.6.
Suppose Π , Π are two shortest ( u, v ) -paths in region R = R [Π , Π ] , with Π ∩ Π = { u, v } . Then each of Π , Π touches at least one obstacle in R .Proof. Suppose that the conclusion is false. Without loss of generality, Π does not touch any obstaclesin R . By Lemma 1.4, the ( u, v ) -path Π is convex in R . Let Q be the simply connected region obtainedby removing the obstacles in R . We have l (Π ) = l (Π ) , so they are both shortest ( u, v ) -paths in thesimply connected environment Q . Therefore Π is also convex by Lemma 1.4, if parameterized as apath from v to u . By Lemma 1.5, Π and Π are both straight lines connecting u, v , which contradicts Π ∩ Π = { u, v } . This proves that when there is more than one shortest ( u, v ) -path, each of thesepaths must touch an obstacle.This concludes our topological and geometric preliminaries. CAT (0) space
In this section, we describe a winning strategy for a single pursuer in a compact
CAT (0) domain, andprove Theorem 1.1. Our strategy generalizes lion’s strategy for pursuit in E , introduced by Sgall [21].This strategy was adapted for pursuit in polygonal regions by Isler, Kannan and Khanna [14]. Their7 p p ′ e ′ e Figure 2.1: Lion’s strategy in E . On each move, the pursuer moves on the line segment connectingthe center c to the evader, and increases her distance from c .adaptation relies heavily on the vertices of the polygon P and gives a capture time of n · diam( P ) ,where n is the number of vertices of P . We give a topological version of lion’s strategy that succeedsin any compact CAT(0) domain D (including polygons) with capture time bounded by diam( D ) .Sgall’s version of lion’s strategy proceeds as follows. Fix a point c as the center of our pursuit, seeFigure 2.1. The pursuer starts at p = c and the evader starts at some point e . On her first move, thepursuer moves directly towards e along the line ce . Considering a general round, the pursuer will beon the line segment between c and e prior to the evader move. After the evader moves to e ′ ∈ B ( e, ,the pursuer looks at the circle centered at p with radius ǫ . If e is inside this circle, then the pursuercan capture the evader. Otherwise, this circle intersects the line segment ce ′ at two points. The pursuermoves to the point p ′ that is closer to e . Lemma 2.1 (Sgall [21]) . A pursuer using lion’s strategy in E re-establishes her location on the linesegment between c and the evader. Furthermore, if the evader moves from e to e ′ and the pursuermoves from p to p ′ then d ( c, p ′ ) ≥ d ( c, p ) + 1 . (cid:3) Before generalizing lion’s strategy, we introduce of the basics of a
CAT (0) geometry; see [10] fora thorough treatment. A complete metric space ( X, d ) is a geodesic space when there is a uniquepath Π( x, y ) whose length is the metric distance d ( x, y ) . This path Π( x, y ) is called a geodesic(or shortest path). A triangle △ xyz between three points x, y, z ∈ X is the triple of geodesics Π( x, y ) , Π( y, z ) , Π( z, x ) . To each △ xyz ∈ X , we associate a comparison triangle △ ˜ x ˜ y ˜ z ⊂ E whose side lengths in E are the same as the lengths of the corresponding geodesics in X . The com-plete geodesic metric space ( X, d ) is CAT (0) when no triangle in X has a geodesic chord that is longerthan the corresponding chord in the comparison triangle. In other words, pick any triangle △ xyz andany points u ∈ Π( x, y ) and v ∈ Π( y, z ) . Let ˜ u ∈ ˜ x ˜ y and ˜ v ∈ ˜ y ˜ z be the corresponding points, chosendistancewise on the edges ˜ x ˜ y and ˜ y ˜ z . If the space X is CAT (0) then d X ( u, v ) ≤ d E (˜ u, ˜ v ) . Colloqui-ally, this is called the “no fat triangles” property, since it also implies that the sum of the angles of thetriangle is not greater than π .Our CAT (0) lion’s strategy generalizes the extended lion’s strategy for polygons of Isler et al. [14]by taking a more topological approach. The pursuer starts at a fixed center point c and her goal isto stay on the shortest path Π( c, e ) at all times. In particular, assume that p t is on the shortest path8 ( c, e t ) and that the evader moves from e t to e t +1 . If d ( p t , e t +1 ) ≤ then the purser responds bycapturing the evader. Otherwise, the pursuer draws the unit circle C centered at p t and moves to thepoint in C ∩ Π( c, e t +1 ) that is closest to e t +1 . Lemma 2.2 (Lion’s Strategy) . A pursuer using lion’s strategy in a
CAT (0) space ( X, d ) re-establishesher location on the line segment between c and the evader. Furthermore, if the evader moves from e to e ′ and the pursuer moves from p to p ′ then d ( c, p ′ ) ≥ d ( c, p ) + 1 .Proof. Suppose that p ∈ Π( c, e ) and then the evader moves to e ′ . Consider the CAT (0) triangle △ cee ′ and its comparison triangle △ ˜ c ˜ e ˜ e ′ in E . Look at the corresponding E pursuit-evasion game with thepursuer at ˜ p ∈ ˜ c ˜ e . By Lemma 2.1, the pursuer can move to a point ˜ p ′ ∈ ˜ c ˜ e ′ such that d E (˜ c, ˜ p ′ ) ≥ d E (˜ c, ˜ p ) +1 . Since there are no fat triangles in X , we have d X ( p, p ′ ) ≤ d E (˜ p, ˜ p ′ ) where p ′ ∈ Π( c, e ′ ) is the point corresponding to ˜ p ′ . Therefore, in our original game, the pursuer can move to the point p ′ ∈ Π( c, e ′ ) , which satisfies d X ( c, p ′ ) ≥ d X ( c, p ) + 1 .Finally, we prove Theorem 1.1: lion’s strategy succeeds in a CAT (0) domain.
Proof of Theorem 1.1.
Consider a pursuit-evasion game in the compact
CAT (0) domain D . Pick any c ∈ ∂D as our center point. Using lion’s strategy, the pursuer increases her distance from c with everystep by Lemma 2.2, so she captures the evader after at most diam( D ) rounds. (cid:3) The key to our pursuit strategy is the ability of one pursuer to guard a shortest path, meaning that theevader cannot cross this path without being caught by a pursuer. When this shortest path splits thedomain into two subdomains, the evader will be trapped in a smaller region. We refer to this region asthe evader territory . In fact, we will be able to also guard a “second shortest path” when the shortestpath is already guarded. The definitions and lemmas in this section are adaptations of the minimal pathformulations introduced in [8] and further developed in [6]. Recall that we use d ( x, y ) to denote thelength of a shortest ( x, y ) -path in D . In addition, we will use ˚ X and X to denote the interior and theclosure of a set X , respectively. Definition 2.
Let X ⊂ D be a path-connected region. The simple path Π ⊂ X is minimal in X whenfor any y , y ∈ Π and any z ∈ X , we have d Π ( y , y ) ≤ d X ( y , z ) + d X ( z, y ) . Definition 3.
Let Z ⊂ X and let Π be a minimal ( u, v ) -path in Z where u, v ∈ ∂Z . Then the pathprojection with anchor u is the function π : Z → Π defined as follows. If d Z ( u, z ) < d Z ( u, v ) = d Π ( u, v ) , then π ( z ) is the unique point x ∈ Π with d Π ( u, x ) = d Z ( u, z ) . For the remaining z ∈ Z \ Π ,we set π ( z ) = v .
9e make a few observations. If X = D , then a shortest ( x, y ) -path is always a minimal pathin D . In this case, we can define the path projection π : D → Π . When X ( D , we might have d X ( x , x ) > d ( x , x ) . In this case, a shortest path in X will be minimal in X , but it will not beminimal in D . Next, we show that a path projection is non-expansive, meaning that distances cannotincrease. Lemma 3.1.
Let π : Z → Π be a path projection onto a minimal path in Z . Then d Π ( π ( z ) , π ( z )) ≤ d Z ( z , z ) for all z , z ∈ Z .Proof. The proof is a straight-forward argument using the triangle inequality. We consider the case z , z ∈ Z with d ( u, z ) ≤ d ( u, z ) ≤ d ( u, v ) . We have d Z ( z , z ) ≥ d Z ( z , u ) − d Z ( z , u ) = d Π ( π ( z ) , u ) − d Π ( π ( z ) , u ) = d Π ( π ( z ) , π ( z )) . The other non-trivial cases are argued similarly.A single pursuer can turn a minimal path Π into an impassable boundary: once the pursuer hasattained the position p = π ( e ) , the evader cannot cross Π without being captured in response. Theproof of the following lemma is similar to the analogous result in [6], but we include this brief argumentfor completeness. Lemma 3.2 (Guarding Lemma) . Let π : X → Π be a path projection onto the minimal ( u, v ) -path Π ⊂ X . Consider a pursuit-evasion game between pursuer p and evader e in the environment X .(a) After O (diam( X )) turns, the pursuer can attain p t = π ( e t − ) .(b) Thereafter, the pursuer can re-establish p s +1 = π ( e s ) for all s ≥ t .(c) If the evader moves so that a shortest path from e s − to e s intersects Π , then the pursuer cancapture the evader at time s + 1 .Proof. To achieve (a), the pursuer moves as follows. First, p travels to u , reaching this point in O (diam( X )) turns. Next, p traverses along Π until first achieving d ( u, p i ) ≤ d ( u, π ( e i − )) 10o (b) holds. As for (c), suppose that a shortest path from e s − to e s includes the point y ∈ Π . Then d ( p t , e t ) ≤ d Π ( π ( e t − ) , y ) + d ( y, e t ) ≤ d ( e t − , y ) + d ( y, e t ) = d ( e t − , e t ) = 1 . Therefore the pursuer can capture the evader on her next move.The Guarding Lemma is the cornerstone of our pursuer strategy. When the pursuer moves asspecified in the lemma, we say that she guards the path Π . In Section 4, our pursuers will repeatedlyguard paths chosen to reduce the number of obstacles in the evader territory. Once the evader is trappedin a region that is obstacle-free, we have reached the endgame of the pursuit. Lemma 3.3. Suppose that the evader is located in a simply connected region R whose boundaryconsists of subcurves of the original boundary ∂ D and two guarded paths Π and Π . If the evaderremains in R , then the third pursuer can capture him in finite time. If the evader tries to leave theregion through Π or Π , then he will be captured by the guarding pursuer.Proof. By Lemma 3.2, if the evader tries to leave this region, he will be caught by either p or p . Ifthe evader remains in this component, then Theorem 1.1 guarantees that pursuer p captures the evaderin a finite number of moves.The remainder of this section is devoted to identifying guardable paths that touch obstacles. Guard-ing such a path will neutralize the threat posed by the obstacle. First, we consider the case when p guards the unique shortest ( u, v ) -path Π that touches an obstacle O in the evader region. The objectiveof p is to guard another ( u, v ) -path Π of a different homotopy type. This path can be guarded evenwhen Π is longer than Π , provided that any path shorter than Π also intersects Π . Lemma 3.4. Suppose that the evader territory R = [Π , ∆] is bounded by the unique ( u, v ) -shortestpath Π and another boundary curve ∆ . Furthermore, suppose that Π touches an obstacle O andthat Π is guarded by p . Then we can find a ( u, v ) -path Π ⊂ R with the following properties: (a) O ⊂ R [Π , Π ] , so that the homotopy type of Π is different than that of Π ; and (b) Π is guardableby p , provided that Π remains guarded by p . A naive attempt to find such a path is to pick some x ∈ Π ∩ ¯ O and find a shortest path that doesnot include the point x . However, R \{ x } is not a closed set, which would complicate our argument.Furthermore, it could be that the next shortest path includes x without using this point as a shortcutaround the obstacle O , as shown in Figure 3.1(b). We handle both problems by removing a small andwell-chosen open region A near x , rather than removing the point x . The delicate choice of A relieson the existence of both κ max and d min . We note that this unusual circumstance is overlooked in [8], where it can occur during their minimal path strategy. Thiscase can be easily handled in a manner analogous to our approach, but based on the visibility graph of their environment. vx O ∆Π (a) uv xO Π Π (b) ∆ uv xAy z (c) xAy z (d)Figure 3.1: Finding the second shortest path. (a) When Π ∩ ∂O contains a curve, we can remove asmall open ball. (b) The shortest path Π touches obstacle O at x . The second shortest path Π goesaround O , but includes the point x . (c) Finding Π requires removing a small, open, triangular set A between Π and O , and then finding the shortest ( u, v ) -path in the closed set R \ A . (d) Any path thatcrosses the line segment yz can be short-cut. Proof. First, suppose that Π ∩ ∂O includes a continuous subcurve C ⊂ Π . Pick x ∈ ˚ C and ǫ > sothat B ( x, ǫ ) ∩ R ⊂ O . Let R ′ = R \ ˚ B ( x, ǫ ) , which effectively absorbs the obstacle into the boundary,see Figure 3.1 (a). The region R ′ is closed, so there is a well-defined shortest ( u, v ) -path Π ⊂ R ′ .The path Π is guardable in R ′ , and therefore it is guardable by p in R , provided that p guards Π .Indeed, any shorter path in R must go through the point x , so Lemma 3.2 guarantees that an evaderusing such a path will be caught by p . Finally, we note that Π , Π have distinct homotopy typesbecause O ⊂ R [Π , Π ] .Next, we consider the case where Π ∩ ¯ O contains no continuous curves: we just focus on the firstpoint x ∈ Π ∩ ¯ O encountered as we move from u to v . Locally around x , the path Π and the boundary ∂O separate R into two external regions (outside of Π and inside ∂O ) and two internal regions, seeFigure 3.1 (b). The shortest path Π does not self-intersect, so locally near x , this path consists of twoline segments meeting at x , create an interior angle smaller than π . Therefore, at least one of the twointerior angles made by Π and the obstacle tangent line(s) at x is strictly less than π . This local regionis where we will remove our triangular open set.Without loss of generality, suppose that the subpath Π ( x, v ) helps to bound this local region. Takepoints y ∈ Π ( x, v ) and z ∈ ∂O (traveling counterclockwise from x ) such that < d Π ( x, y ) = d ∂O ( x, z ) < d min / , and the angle ∠ yxz < π . Let A ⊂ B ( x, d min ) be the closed region withendpoints ( x, y, z ) , where the third curve is the unique shortest ( y, z ) -path Γ , see Figure 3.1 (c). Thebound on the absolute curvature κ max allows us to choose our y, z so that the region A is essentiallytriangular. Since d min is the minimum distance between obstacles, A is obstacle-free, so Γ is a straightline segment. 12e remove the relatively open set A ′ = A \ Γ from our domain. We then find the shortest ( u, v ) -path Π in the closed set R = R \ A ′ . We claim that p can guard Π in R , provided that p guards Π .As in the previous case, the shorter paths that go through x are not available to the evader. Therefore,we must show that any path in R that visits A ′ is longer than Π . Such a path Λ must enter and leave A ′ through Γ , say at points a, b , see Figure 3.1 (d). However, the subpath Λ( a, b ) can be replaced with theunique shortest path Γ( a, b ) without changing the homotopy type, a contradiction. Once again, Π , Π have distinct homotopy types because O ⊂ R [Π , Π ] .We refer to the paths Π , Π from Lemma 3.4 as a guardable pair . Provided that the shortest ( u, v ) -path Π is guarded, the “second shortest ( u, v ) -path” Π can also be guarded. The followingcorollary is a variation of the lemma. Corollary 3.5. Let Π , Π be ( u, v ) -paths that are guarded by p , p , respectively. Suppose that for i = 1 , , the path Π i touches an obstacle O i , where O = O . Then we can find a path Π with thefollowing properties: (a) the homotopy type of Π is different than the homotopy types of Π and Π ,and in particular, O i ∈ R [Π i , Π ] for i = 1 , ; and (b) Π is guardable by p , provided that Π , Π remain guarded by p , p .Proof. The proof is similar to the proof of Lemma 3.4. This time, we must remove an open set A near x ∈ Π ∩ O and an open set B ′ near y ∈ Π ∩ O . We then find Π in R \ ( A ′ ∪ B ′ ) . This concludes our search for guardable paths that touch obstacles. The next section lays out thethree-pursuer strategy for capturing the evader in a two-dimensional domain. In this section, we prove Theorem 1.2: three pursuers can capture an evader in a two-dimensional com-pact domain with piecewise analytic boundary. We adapt the the shortest path strategy of Bhaudaria etal. [8] to our more general topological setting. In particular, our guardable path lemmas from Section3 supplant their use of polygon vertices to find successive paths. Their algorithm guarantees successby reducing the number of polygon vertices in the evader territory. Instead, we keep track of the threatlevel of obstacles to argue that the evader becomes trapped in a simply connected region.Our pursuit proceeds in rounds. At the start of a round, at most two pursuers guard paths. The thirdpursuer moves to guard another path with the goal of eliminating obstacles from the evader territory.This third path will either be a shortest path, or it will create a guardable pair with the currently guardedpath(s). Once this third path is guarded, the evader is trapped in a smaller region, which releases oneof the other pursuers to continue the process. This continues until the evader is trapped in a simplyconnected region, where the free pursuer can capture the evader by Lemma 3.3.13e start by showing that the boundary of the evader territory is always piecewise analytic, afterrecalling two definitions. First, the endpoints of a line segment touching the boundary ∂ D are called switch points . Second, a point x is an accumulation point (or limit point) of a set S when any open setcontaining x contains an infinite number of elements in S . We make use of the following result aboutthe interaction of a geodesic with the boundary of the domain. Theorem 4.1 (Albrecht and Berg [2]) . If M is a 2-dimensional analytic manifold with boundary em-bedded in E , and γ is a geodesic in M , then the switch points on γ have no accumulation points. We restrict ourselves to analytic boundary, instead of smooth ( C , or even C ∞ ) boundary, to avoidsome potentially pathological behavior of geodesics. For example, Albrecht and Berg [2] constructa geodesic in C ∞ environment, that achieves a Cantor set of positive measure as the accumulationof switch points. This unusual geometry hampers our ability to confine the evader in a well-definedconnected component. Theorem 4.1 ensures that our new evader territory will be bounded by piecewiseanalytic curves. Lemma 4.2. Let D be a compact domain with piecewise analytic boundary. If Π is a shortest path in D , then Π is piecewise analytic. Furthermore, if D\ Π is disconnected, then it contains finitely manyconnected components, and the boundary of each connected component is piecewise analytic.Proof. Let B ⊂ ∂ (Π \ ∂ D ) be the set of switch points . We claim that B is finite. Otherwise, there mustbe an accumulation point of B since l (Π) is finite, contradicting Theorem 4.1. Now we can use thefinite set B as endpoints to partition Π so that each subcurve is either in the boundary ∂ D , or in theinterior ˚ D . Since any shortest path in the interior ˚ D must be a line segment, the path Π is piecewiseanalytic. For each connected component of D\ Π , its boundary is a subset of Π ∪ ∂ D , hence is piecewiseanalytic.In order to prove Theorem 1.2, we will show that our pursuit succeeds in finite time. To aid in thiseffort, we assign a threat level to each of the k obstacles in the original domain. These threat levels willreliably decrease during pursuit. An obstacle is in one of three states: dangerous, safe, or removed.A removed obstacle lies outside the evader territory. A safe obstacle lies in the evader territory andtouches a currently guarded path. This obstacle is not a threat because the evader cannot circle aroundthe object without being captured. The remaining obstacles are dangerous . Finally, we say that theevader territory is dangerous if it contains at least one dangerous obstacle.At the start of pursuit, all obstacles are dangerous. So long as there are still dangerous obstacles, around consists of taking control of a guardable path. This effort succeeds in a finite number of movesby Lemma 3.2. We will show that after at most two rounds, either a dangerous obstacle transitions tosafe/removed, or a safe obstacle transitions to removed. This is our notion of progress: after at most k rounds, the evader territory is not dangerous. From here forward, we focus on the transition of thethreat levels of obstacles. 14 v ∆ ∆ Π (a) u v ∆ ∆ ∆ Π O (b)Figure 4.1: The shortest ( u, v ) -path Π guarded in round one. (a) The path Π partitions the outerboundary to subcurves ∆ , ∆ of equal length. (b) The evader may be trapped in a pocket between thepath Π and the boundary subcurve ∆ of obstacle O .In general, our evader territory will be bounded by part of the domain boundary ∂ D and by at mosttwo guarded paths Π , Π . At the end of a round, the evader territory will be updated, bounded inpart by updated paths Π ′ , Π ′ . If these guarded paths intersect or share subpaths, then the evader isactually trapped in a smaller region by Lemma 3.2. When this is the case, we advance the endpoint(s)of our paths so that these are the only point(s) shared by our paths. This obviates the need to discussdegenerate cases.The first round is an initialization round, so all obstacles might still be dangerous when this roundcompletes. However, we will be able to neutralize at least one obstacle in the subsequent round. Tokick off the first round, we pick points u, v ∈ ∂ D , chosen so that they divide the outer boundary intotwo curves ∆ , ∆ of equal length, see Figure 4.1(a). Let Π be a shortest ( u, v ) -path; if there aremultiple shortest paths (in which case each touches an obstacle), then we pick one arbitrarily. UsingLemma 3.2, p moves to guard Π . The round ends when p has attained guarding position, trappingthe evader in a subdomain that is bounded by Π and one of ∆ , ∆ . The evader could be trapped ina smaller pocket region between Π and a subcurve ∆ of an obstacle O ⊂ D , see Figure 4.1(b). Inthe latter case, the obstacle O is marked as removed and we treat ∆ as the new outer boundary. Afterupdating the evader territory R , any obstacle O R is marked as removed. Any obstacle O ⊂ R thattouches Π or Π is marked as safe.For the remainder of the game, the boundary of the evader territory is one of the following types. • Type region: A region containing no dangerous obstacles. • Type region: A dangerous three-sided region bounded by a ( u, v ) -shortest path Π , a ( u, w ) -shortest path Π and a ( v, w ) -path ∆ ⊂ ∂D . No obstacle touches both Π , Π . • Type ′ region: A dangerous two-sided region bounded by a ( u, v ) -shortest path Π and a ( u, v ) -path ∆ ⊂ ∂ D . We treat this as a special case of the previous type, where Π consists of the singlepoint w = u . This point is on Π , so it is guarded by p .15 Type 2 region: A dangerous two-sided region bounded by ( u, v ) -paths Π , Π , each of whichtouches an obstacle in the evader territory. The path Π is a shortest ( u, v ) -path in this region.The path Π might also be a shortest ( u, v ) -path, or it could be a “second shortest path,” meaningthat it is a shortest ( u, v ) -path among the set of ( u, v ) -paths that are not homotopic to Π . Noobstacle touches both Π , Π . • Type 3 region: a dangerous 4-sided region bounded by a ( u, v ) -shortest path Π , a ( w, x ) -shortest path Π , a ( v, w ) -path ∆ from the boundary and a ( u, x ) -path ∆ from the boundary.These vertices are arranged so that they are ordered clockwise as u, v, w, x . No obstacle touchesboth Π , Π .For example, after the initialization round, the evader territory is a type ′ region, bounded by a guardedpath and part of the boundary ∂ D . Finally, we emphasize that Lemma 4.2 ensures that the boundary ofthe evader region is always piecewise analytic, since it consists of sub-curves of the piecewise analyticboundary along with one or more shortest paths.We now describe the different types of rounds. In regions of type 1, ′ and 2, we will alwaystransition at least one obstacle. At the end of such a round, the evader could now be trapped in a regionof any type. Type 3 rounds are slightly different. Our primary goal is to trap the evader in a type1 region, where we will surely make progress in the subsequent next round. However, it is possibleto transition an object via a type 3 move (just as in the initialization round). In this case, we makeimmediate progress, and the evader could then be trapped in a region of any type.First we consider type 1 regions. This also handles type ′ regions as a special case. We use thefollowing lemma to identify a point x ∈ ∆ and a shortest ( u, x ) -path to guard during this round. Lemma 4.3. Let shortest paths Π ( u, v ) , Π ( u, w ) and boundary path ∆( v, w ) bound a type en-vironment R . If R contains obstacles, then there exists a point x ∈ ∆ such that there are multipleshortest ( u, x ) -paths in R , each of which touches at least one obstacle.Proof. Parameterize the boundary path as ∆ : [0 , → R . We prove the lemma by contradiction.Suppose that for every t ∈ [0 , , the shortest ( u, ∆( t )) -path Π t is unique. Denote its length by l (Π t ) = d ( u, ∆( t )) . Define the function n ( t ) to be the number of obstacles in the region R t boundedby Π , Π t and ∆ . The function n ( t ) is well-defined by the uniqueness of each Π t . Furthermore, n (0) = 0 , and n (1) > , so there must be a jump discontinuity somewhere in [0 , . Let s = inf { t ∈ [0 , 1] : n ( t ) > } .Case 1: n ( s ) > where < s ≤ . (Recall that n (0) = 0 , so s > .) Let Γ be a shortest ( u, ∆( s )) -path that is of the same homotopy class as Π . The choice of s and the uniqueness of Π s guarantee that l (Γ) > l (Π s ) . Also no obstacles are contained in the region bounded by Π , Γ and ∆(0 , s ) . By the continuity of l (Γ t ) = d ( u, ∆( t )) with respect to t , we have l (Γ) ≤ l (Π t ) + l (∆( t, s )) ,16 v wx Π Π Π (a) uv wx Π Π Π (b) uv wx Π Π Π (c) u wv = x Π Π Π (d) uv wx Π Π Π (e)Figure 4.2: Representative examples of a type 1 move, where we transition to (a) a type 1 region, (b)a type 1 or type ′ region, (c) a type 1 or a type 3 region, (d) a type 1 or type 2 region, (e) a type 1 ortype 3 region.so l (Γ) ≤ lim t → s − (cid:0) l (Π t ) + l (∆( t, s )) (cid:1) = l (Π s ) + 0 = l (Π s ) . This contradicts l (Γ) > l (Π s ) , so Π s is not the unique shortest ( u, ∆( s )) -path.Case 2: n ( s ) = 0 , where ≤ s < . Let { s i } be an infinite sequence s i → s + , such that n ( s i ) > for all i . There are finite number of obstacles, so by taking a subsequence if necessary, we can assumethat the shortest paths { Π s i } are of the same homotopy class. Let Γ be the shortest ( u, ∆( s )) -path ofthis homotopy class. We have l (Γ) ≤ l (Π s i ) + l (∆( s, s i )) for all i , and therefore l (Γ) ≤ lim i →∞ (cid:0) l (Π s i ) + l (∆( s, s i )) (cid:1) = l (Π s ) , where the limit holds by the continuity of distances in the region. However, this contradicts the unique-ness of Π s which would require l (Π s ) < l (Γ) .Finally, we can conclude that are multiple shortest ( u, x ) -paths. By Lemma 1.6, each of these pathstouches at least one obstacle.Having found the next path to guard, we now prove that we transition an object during a type 1move. Lemma 4.4. Suppose that the evader is trapped in a type 1 (or type ′ ) region. Then the third pursuercan guard a path that transitions an obstacle state.Proof. By Lemma 4.3, there is some point x ∈ ∆ with multiple shortest ( u, x ) -paths, each of whichtouches an obstacle. Let Π be one of these shortest ( u, x ) -paths. If x = v then we take a path Π = Π . Similarly if x = w we choose Π = Π . When x / ∈ { v, w } , we can choose Π arbitrarilyfrom the collection of ( u, x ) -shortest paths. Pursuer p moves to guard Π , which traps the evader ineither R [Π , Π ] or R [Π , Π ] . Any obstacles in the other region are marked as removed.17ithout loss of generality, let O ⊂ R [Π , Π ] be an obstacle touched by Π , see Figure 4.2(a).Suppose that prior to p guarding Π , the object O was dangerous. If e ∈ R [Π , Π ] then O transitionsto removed. If e ∈ R [Π , Π ] then O transitions to safe. However, we may be in a more advantageousposition, shown in Figure 4.2(b): the evader could be trapped in a pocket between obstacle O and path Π . In this case, the new evader territory is type ′ and the obstacle O is marked as removed, since itis now part of the outer boundary of the evader territory.Next, suppose that O was already safe, touched by Π . If e ∈ R [Π , Π ] after p guards Π , then O transitions from safe to removed. If the evader is trapped in a pocket region between O and Π ,we proceed as in the previous case. Otherwise, we have e ∈ R [Π , Π ] and the obstacle O separates R [Π , Π ] into disjoint regions, as shown in Figure 4.2(c). The evader is trapped in one of these twosubregions because both Π , Π are guarded. Let ∆ ′ be the subcurve of ∂O that bounds the effectiveevader territory. We update the evader territory appropriately, bounded by ∆ ′ and subpaths of Π , Π ,and perhaps part of ∆ . The result is a region of type 1 or 3. The obstacle O is marked as removed: itis now part of the boundary. This reduces the number of safe obstacles.When Π touches multiple obstacles, each of them transitions to a lower threat level. Figures 4.2(d)and (e) show that we can also end up in a type 2 or 3 region, depending on the configuration of theseobstacles and the location of the evader at the end of the round.Next, we consider a type 2 region. Such a region is bounded by ( u, v ) -paths Π , Π that form aguardable pair, where Π , Π touch safe obstacles O , O , respectively. Without loss of generality, Π is a shortest ( u, v ) -path in the region, and Π is either another shortest path, or a “second shortest path”as found in Lemma 3.4. (A type 1 move can lead to the first case. A type 2 move can lead to the secondcase, as we are about to see.) Lemma 4.5. Suppose that the evader is trapped in a type 2 region. Then the third pursuer can guarda path that transitions an obstacle state.Proof. Use Corollary 3.5 to find a guardable ( u, v ) -path Π in R [Π , Π ] whose homotopy type isdistinct from that of both Π , Π . Pursuer p establishes a guarding position on Π . The evader isnow trapped in either R [Π , Π ] or R [Π , Π ] , so one of O , O transitions from safe to removed.Furthermore, Π must touch at least one obstacle in each of R [Π , Π ] or R [Π , Π ] . Otherwise, Π would be shorter than one of Π , Π , which contradicts the minimality of that path in R [Π , Π ] .Depending on the configuration of the obstacles, we may be able to restrict the evader territory further.After doing so, the evader territory may be of any possible type, as shown in Figure 4.3 (a).This brings our discussion to a type 3 region, with p guarding shortest ( u, v ) -path Π and p guarding shortest ( w, x ) -path Π . Our primary goal is to trap the evader in a type 1 region, but wemight end up transitioning an obstacle instead. In the latter case, the new evader territory can be of anytype, as explained below. 18 ′ 31 1 u v Π Π Π (a) u Π vx Π w Π ′ (b)Figure 4.3: Examples of moves where the new guarded path Π divides the region into five subregions,each identified by its type. (a) A type 2 move. (b) A type 3 move. Lemma 4.6. Suppose that the evader is trapped in a type 3 region. Then the third pursuer can guard apath so that either (a) the evader is trapped in a type 1 region, or (b) an obstacle transitions to a lowerthreat level.Proof. Let Π be a ( u, w ) -shortest path. Pursuer p moves to guard this path using Lemma 3.2. Thistraps the evader in a smaller region: without loss of generality, this region is bounded by Π , ∆ , Π . If Π does not touch any obstacles in this region, then the evader is now in a type 1 region. If Π touchesan obstacle O , then this obstacle transitions to either safe or removed. The evader could be trapped ina region of any type, as shown in Figure 4.3 (b).We can now prove our main theorem: three pursuers can capture the evader in a two-dimensionalcompact domain with piecewise analytic boundary. Proof of Theorem 1.2. The first round traps the evader in a type ′ region, or transitions an obstaclestate. If we are in a region of type 1, ′ or 2 then we transition an obstacle state in the current round byLemma 4.4 and Lemma 4.5. When we are in a type 3 region, Lemma 4.6 ensures that we either trapthe evader in a type 1 region, or we transition an obstacle. With each path that we guard, the boundaryof the updated evader territory is still piecewise analytic by Lemma 4.2. At the end of the round, weupdate the evader territory and our value for minimum obstacle separation since our new guarded pathmight be closer to an obstacle than the current value d min . (Note that the maximum boundary curvature κ max never increases since all additions to the boundary are line segments.)After at most k rounds, we have transitioned all k obstacles to either safe or removed. Once allobstacles have been transitioned, the evader is trapped in a simply connected type 0 region. Lemma 3.3shows that the evader will then be caught. Each round completes in finite time, so the three pursuerswin the game. The capture time upper bound of O (2 k · diam( D ) + diam( D ) ) follows easily. Thetime required to guard any shortest path is diam( D ) by Lemma 3.2 and lion’s strategy completes intime diam( D ) by Theorem 1.1. (cid:3) Conclusion In this paper, we described a winning pursuer strategy for a single pursuer in a CAT (0) space whenthe capture criterion is d ( p, e ) = 0 . We then restricted our attention to compact domains in E withpiecewise analytic boundary. We showed that three pursuers are sufficient to catch an evader in suchenvironments.There are plenty of avenues for reseach in topological pursuit-evasion. Pursuit-evasion results onpolyhedral surfaces are an active area of current research [15, 18, 19]. For example, Klein and Suri[15] have proven that g + 4 pursuers have a winning strategy on a polyhedral surface of genus g .Meanwhile, Schr¨oder [20] has proven the that at most ⌊ g/ ⌋ + 2 pursuers are needed for a graph ofgenus g (meaning that such a graph can be drawn on a surface of genus g without edge crossings).It would be natural to consider this question for topological surfaces. Likewise, there are a wealth ofmotion and sensory constraints to consider. Most of these variations of pursuit-evasion have a naturalanalog in a topological setting. This work was completed while the first author was a long-term visitor at the Institute for Mathematicsand its Applications. 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