Lecture Notes in Computer Science 4837

More documents

Recommendations

Info

Counting Targets with Mobile Sensors 41 having their vertices on the boundary of P with the property that every triangle has at least one side on the boundary of P .Wecallsuchatriangleabaseline triangle, and the edge of the triangle that lies on the boundary of P a baseline edge. A partition of a polygon into baseline triangles is called a baseline triangulation. We might want to require that the baseline triangles of a baseline triangulation are triangles in the classical sense, i.e., specified only by vertices of P –these are called baseline vertex triangles. A triangulation into baseline vertex triangles is called a Hamiltonian triangulation, as its dual is a path. 1 Unfortunately, a Hamiltonian triangulation does not always exist, see Fig. 7. Either of the two robot enhancements which we have introduced allows robots to use additional points of the boundary of P to compute a baseline triangulation. w v Fig. 6. At v, arobotchoosesvw as the direction of the robot’s walk. After reaching w, it can continue in the same direction until it hits the boundary at point w ′ . w ′ Fig. 7. A polygon and its unique triangulation with triangles specified solely of vertices of the polygon. The triangulation is depicted by dashed lines. In the case of a baseline triangulation a robot can count the targets with the following algorithm. For every baseline triangle the robot moves to the vertex of the triangle (recall that this might not be a vertex of the polygon) opposite to the baseline edge, and counts the targets that are visible between the two vertices of the edge. Clearly, in this way every target is counted exactly once. Hence, a general algorithm that allows a robot to solve the counting problem is as follows: it is composed of a procedure to produce a baseline triangulation and of a navigation scheme to visit every baseline triangle exactly once. Narasimhan [10] presents an algorithm that recognizes whether a polygon has a Hamiltonian triangulation and computes one. The algorithm can be adapted for a robot that can discern convex vertices from reflex vertices [9] (which is not directly possible in our model [1]). Hence, such a robot can resolve whether a polygon admits a Hamiltonian triangulation and exactly count the number of targets in that case. However, when the polygon is non-Hamiltonian, this approach does not give anything useful, whereas our scheme, given in the following section, works for general polygons. 3.3 Walking Along Edge- and Diagonal-Extensions In this section we consider robots that can walk along edge- and diagonalextensions. We show that such robots can partition any simple polygon into baseline triangles, and thus can count the targets. 1 A dual of a triangulation is a graph, where each triangle corresponds to one vertex and there is an edge between two vertices iff the two corresponding triangles share an edge in the triangulation.
42 B. Gfeller et al. v5 P2 w2 p2 = v3 v4 v2 v w1 v1 = p1 Fig. 8. The extensions of a vertex v define baseline triangles and pockets of v P1 4 3 Fig. 9. The resulting baseline triangulation of the algorithm and the visited pockets (grey). The labeled dots show the order of the recursion calles. Consider a robot at a vertex v of the polygon P .Letv1,v2,...,vi ... denote the visible vertices from v, cyclically ordered in the counterclockwise direction. Observe that the lines vvi partition the visible part of P into baseline triangles (all with a common point v), each with at least one baseline edge. See Fig. 8 for an illustration, where the triangles vw1v2, vv2w2, vv3v4 and vv4v5 partition the visible part of P . Thus we can partition the visible part of P . Observe that the invisible part of P is a set of disjoint simple sub-polygons. In the example from Fig. 8 the sub-polygons P1 and P2 form the invisible part of P .Wecallsuch a sub-polygon a pocket of P . Observe that a pocket is created by a line which is an edge-extension or a diagonal-extension. Applying a recursive partitioning approach on the pockets, we create a partition of P into triangles with at least one edge on the boundary of P (see Fig. 9). Let T denote this triangulation. The main idea of the algorithm is to count all targets from the robot’s position v and then proceed recursively in the corresponding pockets of the polygon, thus navigating through T and counting the targets in the triangles of T .Webegin with a high-level description. For a vertex v let P1,...,Pℓ denote pockets of P defined by all extensions originating at v. Letpi, i =1,...,ℓ, denote the visible vertex whose extension defines Pi. Letwi be the point of P for which vwi is the extension of vpi. Counting in Simple Polygons 1. Count all the targets that are visible from the robot’s position at vertex v. 2. Put a pebble at v and remember the position of v in the respective piv of every vertex pi and of every point wi. 3. Recursively count the targets in Pi, i =1,...,ℓ,bymarkingthe point wi with a pebble and going to pi. When a robot walks to vertex pi to start a recursive call for pocket Pi, it first checks the position of the pebble that marks the point wi. Nexttherobot determines which vertices (and targets) visible from pi belong to pocket Pi. Let k be the number of vertices (including wi) and targets visible from pi. Leth be the index of wi in the piv of vertex pi. IfpocketPi lies to the right of piwi, then 1 2
Page 1 and 2: Lecture Notes in Computer Science 4
Page 3 and 4: Volume Editors Mirosław Kutyłowsk
Page 5 and 6: Organization Conference and Program
Page 7 and 8: Table of Contents Algorithmic Chall
Page 9 and 10: Algorithmic Challenges for Sensor N
Page 11 and 12: Algorithmic Challenges for Sensor N
Page 13: Algorithmic Challenges for Sensor N
Page 16 and 17: Topology and Routing in Sensor Netw
Page 26 and 27: Acknowledgements Codes for Sensors:
Page 28 and 29: Efficient Sensor Network Design 19
Page 38 and 39: Theorem 2. The collection V obtaine
Page 40 and 41: References Efficient Sensor Network
Page 42 and 43: Counting Targets with Mobile Sensor
Page 48 and 49: yi−1 xi−1 yi xi P3 xi+1 P2 P1 F
Page 56 and 57: Asynchronous Training in Wireless S
Page 64 and 65: number of transitions 14 13 12 11 1
Page 68 and 69: Optimal Placement of Ad-Hoc Devices
Page 74 and 75: |u, ·| = r(u) u PR st uv F st uv (
Page 81 and 82: Analysis of the Bounded-Hops Conver
Page 93 and 94: Correlation, Coding, and Cooperatio
Page 95 and 96: 2 System Model Correlation, Coding,
Page 97 and 98: 3.1 Static Scheduling Correlation,
Page 99 and 100: Correlation, Coding, and Cooperatio
Page 101 and 102:
Correlation, Coding, and Cooperatio
Page 103 and 104:
Page 105 and 106:
5 Conclusions Correlation, Coding,
Page 107 and 108:
Page 109 and 110:
Local Approximation Algorithms for
Page 111 and 112:
Page 113 and 114:
2.2 Shifting Strategy Local Approxi
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123:
Page 126 and 127:
Assigning Sensors to Missions with
Page 128 and 129:
Page 130 and 131:
Algorithm 1. Δ-Approximation Greed
Page 132 and 133:
Page 134 and 135:
Algorithm 2. Shifting PTAS (error
Page 136 and 137:
Page 138 and 139:
Maximal Breach in Wireless Sensor N
Page 140 and 141:
Page 142 and 143:
a’ a i Maximal Breach in Wireless
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
6 Conclusions and Future Work Maxim
Page 150 and 151:
Counting-Sort and Routing in a Sing
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Intrusion Detection of Sinkhole Att
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
% neighbors with successfull detect
Page 172 and 173:
References Intrusion Detection of S
show all

Lecture Notes in Computer Science 4837

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?