random disambiguation paths

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Random Disambiguation PathsRandom Disambiguation Paths

Al AksakalliAl Aksakalli

In Collaboration withIn Collaboration with Carey Priebe Carey Priebe && Donniell Fishkind Donniell Fishkind

Department of Applied Mathematics and StatisticsDepartment of Applied Mathematics and Statistics

Johns Hopkins UniversityJohns Hopkins University

Adaptive Sensing MURI Workshop

June 28, 2006

Duke University

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1) Problem Description

2) Markov Decision Process Formulation

3) Simulated Risk Disambiguation Protocol

4) Computational Experiments

5) Ongoing Research

6) Summary and Conclusions

Outline:

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Spatial arrangement of detections:true detections ,false detections

Problem Description:

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Spatial arrangement of detections:true detections ,false detections

Assume for all that isthe probability that


We only see

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Given start and destination

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startt

destination

s


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About each detectionthere is a hazard region ,an open disk of fixed radius

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We seek a continuous curvefrom to inof shortest achievable arclength


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…and we assume the ability todisambiguate detections from the boundary of their hazard regions.



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true



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…or false



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the rest of the transversal…



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Definition: A disambiguation protocol is a function

# disambiguationsallowed

cost per disambiguation

which detection disambiguated next…

…and where the disambiguationperformed

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Example 1: Protocol gives rise to the RDP

Length=707.97, prob=.89670 Length=1116.19, prob=.10330

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(superimposed composite)

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Random Disambiguation Paths (RDP) Problem:

Given , find protocol

of minimum .

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Related work:• Canadian Traveller Problem (CTP): Graph theoretic RDP

• Given a finite graph – edges with specific probabilities of being traversable, and a starting and a destination vertex – each edge’s status is revealed only when one of the end points is visited: objective is to minimize expected traversal length

• Shown to be #P-hard

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Markov Decision Process (MDP) formulation:Let be the information vector keeping track of the decision maker’s current knowledge; be the set of all possible disambiguation points

RDP Problem can be cast as a K-stage finite horizon MDP with

States:

Actions: where v is a disambiguation point and i is a hazard region index

Rewards: the negative of the shortest path distance between the state vertex and the action vertex minus c, if not going to d - d is an absorbative state for which there is a one-time and very large reward for entering

Transitions: governed by ‘s

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Simulated Risk Protocol:For purpose of deciding next disambiguation point, we pretend that ambiguous disks are riskily traversable…

traversal ?

? ??

?

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is the surprise length of , which is the negative logarithm of the probability that is traversable in actuality.

Risk Simulation Protocol:For purpose of deciding next disambiguation point, we pretend that ambiguous disks are riskily traversable…

traversal ?

? ??

?

is the usual Euclidean length of .

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Given undesirability function(henceforth, monotonically non-decreasing in its arguments)

and, say,

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and, say,

Definition: The simulated risk protocol is defined as dictating that the next disambiguation be at the first ambiguous point of .

traversal ?

? ??

?

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and, say,

Definition: The simulated risk protocol is defined as dictating that the next disambiguation be at the first ambiguous point of .

traversal ?

? ??

?

How to proceed once this disambiguation is performed: update and , decrement , and set the new s to be y.

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How to navigate in this continuous setting: The Tangent Arc Graph (TAG) is the superimposition/subdivision of all visibility graphs generated by all subsets of disks.

For any undesirability function,

is an path in TAG !

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Linear undesirability functions:

• Because of the efficiency in their realization, we will consider simulated risk protocols generated by linear undesirability functions for a chosen parameter .

• As a further shorthand, denote such a protocol by .

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How (during the simulation of risk phase) can be affected by :

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Length=707.97, prob=.89670 Length=1116.19, prob=.10330

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(superimposed composite)

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Lattice Discretization:Discretization via a subgraph of the integer lattice with unit edge lengths:

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Example: Adapting the simulated risk protocol to lattice discretization:

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A 40 by 20 integer lattice is used

Each hazard region is a disk with radius 5.5

Disk centers sampled from a uniform distribution of integers in

‘s sampled from uniform distribution on (0,1)

Cost of disambiguation is taken as 1.5

For each N, K combination, 50 different instances were sampled

Optimal solutions found by solving the MDP model via value iteration

Computational experiments:

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Illustration with N=7, K=1:

Expected length:

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Runtime to findoverall optimal(SR-RDP runtimenegligible)

Comparison of optimal versus simulated risk:

• Simulated risk found the optimal solution 74% of the time

• Overall mean percentage error of simulated risk solutions was less than 1%

• For N=7, K=3; VI took more than an hour

• for N=10, K=1; VI did not run due to insufficient memory

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Ongoing Research: Pruning State Space via AO*

Implemented an enhanced version of AO-star algorithm

Preliminary results suggest up to 99% of the state space can be pruned

N=15, K=2 can be solved under 15 mins!

Not practical for K>2: N=15, K=3 takes 10.5 hours!!!

Simulated Risk protocol still seems to perform well

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Example: Enhanced AO* with N=15, K=2

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Ongoing Research: Multiple sensors & Neutralization

Deployment of multiple sensors with different accuracy rates & ranges at different costs

Also consider a limited neutralization capability

Develop and solve corresponding Partially Observable Markov Decision Process (POMDP) models

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Summary and Conclusions

RDP is an important, yet hard mine-countermeasures problem

Obtaining optimal solutions presently not feasible for realistic values of N and K

Simulated risk protocol is a sub-optimal yet efficient algorithm that performed well in computational experiments

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Q & AQ & A

random disambiguation paths

Documents

hazard regions

true detections

open disk of fixed radius

continuous curvefrom

starttdestinatio description

destination vertex

canadian traveller problem

rdp length