The Role of Visibility in Pursuit/Evasion Games
- We use the following notations for sets: denotes ; denotes ; denotes ; ; denotes the cardinality of A (i.e., the number of its elements).
- A graph consists of a node set V and an edge set E, where every has the form . In other words, we are concerned with finite, undirected, simple graphs; in addition we will always assume that G is connected and that G contains n nodes: . Furthermore, we will assume, without loss of generality, that the node set is . We let . We also define by (it is the set of “diagonal” node pairs).
- A directed graph (digraph) consists of a node set V and an edge set E, where every has the form . In other words, the edges of a digraph are ordered pairs.
- In graphs, the (open) neighborhood of some is ; in digraphs it is . In both cases, the closed neighborhood of is .
- Given a graph , its line graph is defined as follows: the node set is , i.e., it has one node for every edge of G; the edge set is defined by having the nodes connected by an edge if and only if (i.e., if the original edges of G are adjacent).
- We will write if and only if . Note that in this asymptotic notation n denotes the parameter with respect to which asymptotics are considered. So in later sections we will write , etc.
2.2. The CR Game Family
|Adversarial Visible Robber||av-CR|
|Adversarial Invisible Robber||ai-CR|
|Drunk Visible Robber||dv-CR|
|Drunk Invisible Robber||di-CR|
3. Cop Number and Capture Time
3.1. The Node av-CR Game
- Nodes of the form correspond to positions (in the original CR game) with the cops located at , the robber at and player being next to move.
- There is single node which corresponds to the starting position of the game: neither the cops nor the robber have been placed on G; it is C’s turn to move (recall that λ denotes the empty sequence).
- Finally, there exist n nodes of the form : the cops have just been placed in the graph (at positions ) but the robber has not been placed yet; it is R’s turn to move.
3.2. The Node dv-CR Game
3.3. The Node ai-CR Game
3.4. The Node di-CR Game
3.5. The Edge CR Games
4. The Cost of Visibility
4.1. Cost of Visibility in the Node CR Games
- With probability , the robber starts on the same ray as the cop but farther away from the center. Conditioning on this event, the expected capture time is .
- With probability , the robber starts on the same ray as the cop but closer to the center. Conditioning on this event, the expected capture time is .
- With probability , the robber starts on different ray than the cop. Conditioning on this event, the expected capture time is .
- With probability , the robber starts on the same ray as the cop. Conditioning on this event, the expected capture time is .
- With probability , the robber starts on the -th ray visited by the cop. Conditioning on this event, the expected capture time is . ( steps are required to move from the end of the first ray to the center, steps are `wasted’ to check rays, and steps are needed to catch the robber on the -th ray, on expectation.)
4.2. Cost of Visibility in the Edge CR Games
5. Algorithms for COV Computation
5.1. Algorithms for Visible Robbers
5.1.1. Algorithm for Adversarial Robber
- , the optimal game duration when the cop/robber configuration is and it is C’s turn to play;
- , the optimal game duration when the cop/robber configuration is and it is R’s turn to play.
|Algorithm 1: Cops Against Adversarial Robber (CAAR)|
|01 For All|
|05 For All|
|11 For All|
|15 If And|
|Output: C, R|
5.1.2. Algorithm for Drunk Robber
|Algorithm 2: Cops Against Drunk Robber (CADR)|
|Input: , ε|
|01 For All|
|04 For All|
|09 For All|
5.2. Algorithms for Invisible Robbers
5.2.1. Algorithms for Adversarial Robber
5.2.2. Algorithm for Drunk Robber
|Algorithm 3: Pruned Cop Search (PCS)|
|Input: , , , ε|
|02 , ,|
|07 For All|
|08 , ,|
|09 For All|
|13 , ,|
|Output: , .|
6. Experimental Estimation of the Cost of Visibility
6.1. Experiments with Node Games
6.2. Experiments with Edge Games
Conflicts of Interest
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Kehagias, A.; Mitsche, D.; Prałat, P. The Role of Visibility in Pursuit/Evasion Games. Robotics 2014, 3, 371-399. https://doi.org/10.3390/robotics3040371
Kehagias A, Mitsche D, Prałat P. The Role of Visibility in Pursuit/Evasion Games. Robotics. 2014; 3(4):371-399. https://doi.org/10.3390/robotics3040371Chicago/Turabian Style
Kehagias, Athanasios, Dieter Mitsche, and Paweł Prałat. 2014. "The Role of Visibility in Pursuit/Evasion Games" Robotics 3, no. 4: 371-399. https://doi.org/10.3390/robotics3040371