The Cat and The Mouse -- The Case of Mobile Sensors and Targets David K. Y. Yau Lab for Advanced Network Systems Dept of Computer Science Purdue University (Joint work with J. C. Chin, Y. Dong, and W. K. Hon) David K. Y. Yau Lab for Advanced Network Systems Dept of Computer Science Purdue University (Joint work with J. C. Chin, Y. Dong, and W. K. Hon)

Why Mobile?  The mouse  Evasion of detection  Nature of “mission”  The cat  Improved coverage with fewer sensors  Robustness against contingencies  Planned or random movement (randomness useful)  The mouse  Evasion of detection  Nature of “mission”  The cat  Improved coverage with fewer sensors  Robustness against contingencies  Planned or random movement (randomness useful)

Mobility Model  Four-tuple  N: network area  M: accessibility constraints -- the “map”  T: trip selection  R: route selection  Random waypoint model is a special case  Null accessibility constraints  Uniform random trip selection  Cartesian straight line route selection  Four-tuple  N: network area  M: accessibility constraints -- the “map”  T: trip selection  R: route selection  Random waypoint model is a special case  Null accessibility constraints  Uniform random trip selection  Cartesian straight line route selection

Problem Formulation  Two player game  Payoff is time until detection (zero sum)  Cat plays detection strategy  Stochastic, characterized by per-cell presence probabilities  Mouse plays evasion strategy  Knows statistical process of cat’s movement, but not necessarily exact routes (exact positions at given times)  Two player game  Payoff is time until detection (zero sum)  Cat plays detection strategy  Stochastic, characterized by per-cell presence probabilities  Mouse plays evasion strategy  Knows statistical process of cat’s movement, but not necessarily exact routes (exact positions at given times)

Best Mouse Play  Cat’s presence matrix given  Network region divided into 2D cells  P i,j gives probability for mouse to find cat in cell (i, j)  Expected detection time “long” compared with trip from point A to point B  Dynamic programming solution to maximize detection time  Local greedy strategy does not always work  Cat’s presence matrix given  Network region divided into 2D cells  P i,j gives probability for mouse to find cat in cell (i, j)  Expected detection time “long” compared with trip from point A to point B  Dynamic programming solution to maximize detection time  Local greedy strategy does not always work

Optimal Escape Path Formulation  For each cell j, mouse decides whether to stay or to move to a neighbor cell (and which one)  If stay, expected max time until detection is E j [T stay ]  If move to neighbor cell k, expected max time until detection is E j [T move(k) ]  For cell j, expected max time until detection, E j [T], is largest of E j [T stay ] and E j [T move(k) ] for each neighbor cell k of j  E j [T stay ] determined by cat’s presence matrix and expected cat’s sojourn time in each cell  Optimal escape path is sequence of safest neighbors to move to, until mouse decides to stay  How to compute E j [T] for each cell j?  For each cell j, mouse decides whether to stay or to move to a neighbor cell (and which one)  If stay, expected max time until detection is E j [T stay ]  If move to neighbor cell k, expected max time until detection is E j [T move(k) ]  For cell j, expected max time until detection, E j [T], is largest of E j [T stay ] and E j [T move(k) ] for each neighbor cell k of j  E j [T stay ] determined by cat’s presence matrix and expected cat’s sojourn time in each cell  Optimal escape path is sequence of safest neighbors to move to, until mouse decides to stay  How to compute E j [T] for each cell j?

Computing E j [T]  Initialize E j [T] as E j [T stay ]  Insert all the cells into heap sorted by decreasing E j [T]  Delete root cell 0 from heap  For each neighbor cell k of 0, update E k [T] as E k [T] := max(E k [T], E k [T move(0) ])  Reorder heap in decreasing E j [T] order  Repeat until heap becomes empty  Initialize E j [T] as E j [T stay ]  Insert all the cells into heap sorted by decreasing E j [T]  Delete root cell 0 from heap  For each neighbor cell k of 0, update E k [T] as E k [T] := max(E k [T], E k [T move(0) ])  Reorder heap in decreasing E j [T] order  Repeat until heap becomes empty

Example Optimal Paths Path when mouse moves slowly Path when mouse moves quickly

Comparison with Local Greedy Strategy Local greedy strategy: mouse will stay Dynamic programming strategy: mouse moves to cell with small probability of cat’s presence (0.0075) Current mouse position

If Cat Plays Random Waypoint Strategy  Highest presence probability at the center of the network area  Lowest presence probabilities at the corners and perimeters  Good “safe havens” for mouse to hide  Sum of presence probabilities is one  n cats  sum of probabilities  n  Equality for disjoint cats’ surveillance areas  Highest presence probability at the center of the network area  Lowest presence probabilities at the corners and perimeters  Good “safe havens” for mouse to hide  Sum of presence probabilities is one  n cats  sum of probabilities  n  Equality for disjoint cats’ surveillance areas

Distribution of Movement Direction in 150 m by 150 m Network Area

Cat’s Presence Matrix for Random Waypoint Movement

Cat’s Presence Matrix in 500  500 m Network for Random Waypoint Movement

Distribution of Movement Direction (a) Calculated probabilities of sensor moving towards the center cell from different current cells (b) Measured probabilities of sensor moving towards the center cell from different current cells

Analytical Cell Coverage Statistics (a) Expected number of trips before covering a cell (average = , maximum = ) (b) Expected time before covering a cell (average = s, maximum = s)

Measured Cell Coverage Statistics (b) Expected number of trips before covering a cell (average = , maximum = ) (b) Expected time before covering a cell (average = s, maximum = s)

Optimal Cat Strategy  Maximize minimum presence probability among all the cells  Eliminate safe haven  Achieved by equal presence probabilities in each cell  Will lead to Nash Equilibrium  Zero sum game  Pareto optimality  Maximize minimum presence probability among all the cells  Eliminate safe haven  Achieved by equal presence probabilities in each cell  Will lead to Nash Equilibrium  Zero sum game  Pareto optimality

Minimum Sensing Range for Expected Random Waypoint Coverage  Stationary mouse; cat in random waypoint movement  Expected coverage desired by given deadline  What is minimum sensing distance required?  Stochastic analysis of shortest distance between cat and mouse within deadline  Stationary mouse; cat in random waypoint movement  Expected coverage desired by given deadline  What is minimum sensing distance required?  Stochastic analysis of shortest distance between cat and mouse within deadline

Lower Bound Cat-mouse Distance Network divided into m by n cells; each has fixed size s by s D(i, j): Euclidean distance between cell i and cell j N sets of cells sorted by set’s distance to mouse Each set of cells denoted as S j, 0 ≤ j ≤ N - 1 Each cell in S j is equidistant from the mouse; distance is D S j Distances sorted in increasing order; i.e., D S j < D S j+1 Network divided into m by n cells; each has fixed size s by s D(i, j): Euclidean distance between cell i and cell j N sets of cells sorted by set’s distance to mouse Each set of cells denoted as S j, 0 ≤ j ≤ N - 1 Each cell in S j is equidistant from the mouse; distance is D S j Distances sorted in increasing order; i.e., D S j < D S j+1

Example Equidistant Sets of Cells Mouse located at center of network area

Correlation between Cells Visited P i : probability that cat may visit cell i P S j : probability that cat may visit any cell in set S j P i : probability that cat may visit cell i P S j : probability that cat may visit any cell in set S j

Shortest Distance Probability Matrix from Cell i to Cell j 3-D probability matrix B Each element b i,j gives cat’s shortest distance distribution from mouse after trip from cell i to j is a size N vector: b i,j [k] is the probability that the shortest distance during the trip is D S k

where is the probability that D S 0 is shortest distance for trip l, and is probability that D S n is shortest distance for the trip, 1 ≤ n ≤ N - 1 Let denote, then is calculated as: Shortest Distance Probability Matrix after l Trips B l is the shortest distance probability matrix after l trips Computed by * operator B l is the shortest distance probability matrix after l trips Computed by * operator Each element of B l is calculated as:

Expected shortest distance  The expected shortest distance between cat and mouse after l trips:

Approximate Expected Shortest Distance  Approximate expected shortest distance from mouse after cat has visited k cells:  P D j (k) is probability that after visiting k cells, a cell in S j is visited, but no cell in S i, i< j, is visited  Approximate expected shortest distance from mouse after cat has visited k cells:  P D j (k) is probability that after visiting k cells, a cell in S j is visited, but no cell in S i, i< j, is visited

Lower Bound Cat-mouse Distance for Random Waypoint Model (a) Expected speed = 5 m/s (b) Expected speed = 10 m/s(c) Expected speed = 25 m/s

Lower Bound Cat-mouse Distance for Indiana Map-based Model

Conclusions  Considered cat and mouse game between mobile sensors and mobile target  For random waypoint model, other coverage properties can be obtained analytically  Expected cell sojourn time, expected time to cover general AOI, number of sensors to achieve coverage by given deadline, …  Considered cat and mouse game between mobile sensors and mobile target  For random waypoint model, other coverage properties can be obtained analytically  Expected cell sojourn time, expected time to cover general AOI, number of sensors to achieve coverage by given deadline, …

Conclusions (cont’d)  Many extensions possible  Explicit account for plume explosion / dispersion models  Model for sensor (un)reliability, interference, etc  Explicit quantification of sensing uncertainty and its reduction  Validation with empirical data  Many extensions possible  Explicit account for plume explosion / dispersion models  Model for sensor (un)reliability, interference, etc  Explicit quantification of sensing uncertainty and its reduction  Validation with empirical data