5 research outputs found
Spy-Game on graphs
International audienceWe define and study the following two-player game on a graph G. Let k â N *. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s â N * is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d â N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s = 1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists > 0 such that âŠ(n 1+) guards are required to win in any n Ă n grid
To satisfy impatient Web surfers is hard
International audiencePrefetching is a basic mechanism for faster data access and efficient computing. An important issue in prefetching is the tradeoff between the amount of network's resources wasted by the prefetching and the gain of time. For instance, in the Web, browsers may download documents in advance while a Web surfer is surfing. Since the Web surfer follows the hyperlinks in an unpredictable way, the choice of the Web pages to be prefetched must be computed online. The question is then to determine the minimum amount of resources used by prefetching that ensures that all documents accessed by theWeb surfer have previously been loaded in the cache. We model this problem as a two-player game similar to Cops and Robber Games in graphs. Let k 1 be any integer. The first player, a fugitive, starts on a marked vertex of a (di)graph G. The second player, an observer, marks at most k vertices, then the fugitive moves along one edge/arc of G to a new vertex, then the observer marks at most k vertices, etc. The fugitive wins if it enters an unmarked vertex, and the observer wins otherwise. The surveillance number of a (di)graph is the minimum k such that the observer marking at most k vertices at each step can win against any strategy of the fugitive. We also consider the connected variant of this game, i.e., when a vertex can be marked only if it is adjacent to an already marked vertex. We study the computational complexity of the game. All our results hold for both variants, connected or unrestricted. We show that deciding whether the surveillance number of a chordal graph is at most 2 is NP-hard. We also prove that deciding if the surveillance number of a DAG is at most 4 is PSPACEcomplete. Moreover, we show that the problem of computing the surveillance number is NP-hard in split graphs. On the other hand, we provide polynomial time algorithms computing surveillance numbers of trees and interval graphs. Moreover, in the case of trees, we establish a combinatorial characterization of the surveillance number
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