56 research outputs found
An improved lion strategy for the lion and man problem
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.Comment: Preprint submitted to IEEE Control Systems Letter
Straight Line Movement in Morphing and Pursuit Evasion
Piece-wise linear structures are widely used to define problems and to represent simplified
solutions in computational geometry. A piece-wise linear structure consists of straight-line
or linear pieces connected together in a continuous geometric environment like 2D or 3D
Euclidean spaces. In this thesis two different problems both with the approach of finding
piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion
and straight-line morphing.
Straight-line pursuit evasion is a geometric version of the famous cops and robbers game
that is defined in this thesis for the first time. The game is played in a simply connected
region in 2D. It is a full information game where the players take turns. The copâs goal
is to catch the robber. In a turn, each player may move any distance along a straight
line as long as the line segment connecting their current location to the new location is
not blocked by the regionâs boundary. We first prove that the cop can always win the
game when the players move on the visibility graph of a simple polygon. We prove this by
showing that the visibility graph of a simple polygon is âdismantlableâ (the known class of
cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other
settings of the game are also studied in this thesis: when the players are free to move on
the infinitely many points inside a simple polygon, and inside a splinegon. In both cases
we show that the cop can always win the game. For the case of polygons, the proposed cop
strategy gives an asymptotically tight linear bound on the number of steps the cop needs
to catch the robber. For the case of splinegons, the cop may need a quadratic number of
steps with the proposed strategy, while our best lower bound is linear.
Straight-line morphing is a type of morphing first defined in this thesis that provides a
nice and smooth transformation between straight-line graph drawings in 2D. In straight-
line morphing, each vertex of the graph moves forward along the line segment connecting
its initial position to its final position. The vertex trajectories in straight-line morphing
are very simple, but because the speed of each vertex may vary, straight-line morphs are
more general than the commonly used âlinear morphsâ where each vertex moves at uniform
speed. We explore the problem of whether an initial planar straight-line drawing of a graph
can be morphed to a final straight-line drawing of the graph using a straight-line morph
that preserves planarity at all times. We prove that this problem is NP-hard even for
the special case where the graph drawing consists of disjoint segments. We then look at
some restricted versions of the straight-line morphing: when only one vertex moves at a
time, when the vertices move one by one to their final positions uninterruptedly, and when
the edges morph one by one to their final configurations in the case of disjoint segments.
Some of the variations are shown to be still NP-complete while some others are solvable
in polynomial time. We conjecture that the class of planar straight-line morphs is as
powerful as the class of planar piece-wise linear straight-line morphs. We also explore
a simpler problem where for each edge the quadrilateral formed by its initial and final
positions together with the trajectories of its two vertices is convex. There is a necessary
condition for this case that we conjecture is also sufficient for paths and cycles
Chasing Puppies: Mobile Beacon Routing on Closed Curves
We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired
by his and others' work on beacon-based routing. Consider a human and a puppy
on a simple closed curve in the plane. The human can walk along the curve at
bounded speed and change direction as desired. The puppy runs with unbounded
speed along the curve as long as the Euclidean straight-line distance to the
human is decreasing, so that it is always at a point on the curve where the
distance is locally minimal. Assuming that the curve is smooth (with some mild
genericity constraints) or a simple polygon, we prove that the human can always
catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure
Geometric Pursuit Evasion
In this dissertation we investigate pursuit evasion problems set in geometric environments. These games model a variety of adversarial situations in which a team of agents, called pursuers, attempts to catch a rogue agent, called the evader. In particular, we consider the following problem: how many pursuers, each with the same maximum speed as the evader, are needed to guarantee a successful capture? Our primary focus is to provide combinatorial bounds on the number of pursuers that are necessary and sufficient to guarantee capture. The first problem we consider consists of an unpredictable evader that is free to move around a polygonal environment of arbitrary complexity. We assume that the pursuers have complete knowledge of the evader's location at all times, possibly obtained through a network of cameras placed in the environment. We show that regardless of the number of vertices and obstacles in the polygonal environment, three pursuers are always sufficient and sometimes necessary to capture the evader. We then consider several extensions of this problem to more complex environments. In particular, suppose the players move on the surface of a 3-dimensional polyhedral body; how many pursuers are required to capture the evader? We show that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints, where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights. Next we consider a more general problem with a less restrictive sensing model. The pursuers' sensors are visibility based, only providing the location of the evader if it is in direct line of sight. We begin my making only the minimalist assumption that pursuers and the evader have the same maximum speed. When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Î(n^1/2 ) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n^2/3 ) and an upper bound of O(n^5/6 ) pursuers, where n includes the vertices of the hole boundaries. However, we show that with realistic constraints on the polygonal environment these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the features of the environment, we give an algorithm with a worst case upper bound of O(log n) pursuers for simply-connected n-gons and O(âh + log n) for polygons with h holes. The final problem we consider takes a small step toward addressing the fact that location sensing is noisy and imprecise in practice. Suppose a tracking agent wants to follow a moving target in the two-dimensional plane. We investigate what is the tracker's best strategy to follow the target and at what rate does the distance between the tracker and target grow under worst-case localization noise. We adopt a simple but realistic model of relative error in sensing noise: the localization error is proportional to the true distance between the tracker and the target. Under this model we are able to give tight upper and lower bounds for the worst-case tracking performance, both with or without obstacles in the Euclidean plane
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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