D Algorithm

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D Algorithm

• Vincent Lee-Shue Jr.
• Prasad Narendra Atkar

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D Algorithm
Goal

• Point to point path planning in unknown,
  partially known or changing environments in an
  efficient, optimal and complete manner.

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Abstraction

• Lets look at P2P problem on a Grid
• Problem is a graph search one
• Objective values (Time, Distance, Energy)
• Blind Search Wavefront (BFS)
• Highly inefficient

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Informed Search A

• Utilizes information obtained from the
  environment.
• Concept of a heuristic function (Estimation)
• conditions of heuristic
• Optimistic (must be less than or equal to the
  real cost)
• As close to the real cost as possible
• Algorithm
• Nodes expanded in order of (cost incurred in
  moving to current location from the start
  Estimated cost to goal from current location)

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Example (1/5)
Legend
h(x)
c(x)
Priority g(x) h(x)
Note
g(x) sum of all previous arc costs, c(x),
from start to x Example c(H) 2
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Example (2/5)
First expand the start node
If goal not found, expand the first node in the
priority queue (in this case, B)
Insert the newly expanded nodes into the priority
queue and continue until the goal is found, or
the priority queue is empty (in which case no
path exists)
Note for each expanded node, you also need a
pointer to its respective parent. For example,
nodes A, B and C point to Start
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Example (3/5)
No expansion
GOAL(5)
Weve found a path to the goal Start gt A gt E
gt Goal (from the pointers) Are we done?
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Example (4/5)
No expansion
GOAL(5)
There might be a shorter path, but
assuming non-negative arc costs, nodes with a
lower priority than the goal cannot yield a
better path. In this example, nodes with a
priority greater than or equal to 5 can be
pruned. Why dont we expand nodes with an
equivalent priority? (why not expand nodes D and
I?)
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Example (5/5)
No expansion
GOAL(5)
GOAL(4)
We can continue to throw away nodes with priority
levels lower than the lowest goal found. As we
can see from this example, there was a shorter
path through node K. To find the path,
simply follow the back pointers. Therefore the
path would be Start gt C gt K gt Goal
If the priority queue still wasnt empty, we
would continue expanding while throwing away
nodes with priority lower than 4. (remember,
lower numbers higher priority)
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Completeness, Optimality and Efficiency

• A yields an optimal path, if it exists. If goal
  is unreachable, it signals so. (i.e. A Complete
  and optimal)
• Efficient A is very efficient for static
  worlds (much better than Wavefront)
• What happens with Dynamic worlds, where the arc
  costs may change ?

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Dynamics Worlds

• A Replanner Plan by A using all the available
  information at the start.
• Start tracing the optimal path
• If there is a discrepancy between the initial map
  and the actual environment, update the new cost
  values for the corresponding arcs, run A again
  for planning between the current position and the
  goal. Trace new optimal path, if discrepancy
  between map and environment, update corresponding
  arc costs, run A again.

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Efficiency of A Replanner

• In expansive environments where the goal is far
  away, little changes may force the planner to use
  A over the whole environment, although the
  changes in the optimal path may be small, that
  is, the new optimal path may be very similar to
  the old assumed optimal path.
• Hence, A re-planner can be grossly inefficient
  computationally.
• What can we do to improve the efficiency of A
  Re-planner?

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Dynamic A (D ) Stentz, 1994

• Functionally equivalent to A replanner.
• Plan initial path using A to every state in the
  world (Dijkstras search ).
• Make local changes to the map and the resultant
  optimal path when a discrepancy between map and
  the environment is found.
• Essentially prunes the graph search.
• Optimal given the information at a time
• Omniscient optimal if exact representation of map
  is available.

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D Definitions

• X represents a state
• Qu D maintains a Priority Queue
• t(X) OPEN, if X is on Qu.
• NEW, if X has never been on Qu.
• CLOSED , if X is not currently on Qu but
  it was on Qu earlier.
• c(X,Y) cost of moving from X to Y c(Y,X).
• h(X) current cost of path from X to Goal.
• k(X) Estimated minimum cost from X to Goal.
• min unmodifed h(X), all h(X)
  value X takes when X is put on
• O.
• o(X) cost of optimal path from X to Goal.
• X path to goal, i.e.,sequence of states from
  X to Goal
• b(X) Y , Y is parent state of X, X-Y-Goal

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Procedures
MODIFY-COST(X,Y,cval)
MIN-STATE()
c(X,Y)cval if t(X) CLOSED then INSERT
(X,h(X)) Return GET-MIN ( )
Return X if k(X) is minimum for all states on Qu
INSERT(X,hnew)
GET-KMIN()
if t(X) NEW then k(X)hnew if t(X) OPEN then
k(X)min(k(X),hnew) if t(X) CLOSED then
k(X)min(k(X),hnew) and t(X) OPEN Sort Qu based
on increasing k values
Return the minimum value of k for all states on
Qu
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Procedures
PROCESS-STATE()
else for each neighbor Y of X if t(Y)
NEW or (b(Y) X and h(Y) ? h(X)c (X,Y) ) then
b(Y) X INSERT(Y, h(X)c(X,Y)) else
if b(Y) ? X and h(Y) gt h(X)c (X,Y)
then INSERT(X, h(X)) else
if b(Y) ? X and h(X) gt h(Y)c (X,Y) and
t(Y) CLOSED and h(Y) gt kold then
INSERT(Y, h(Y)) Return GET-KMIN ( )
X MIN-STATE( ) if X NULL then return 1 kold
GET-KMIN( ) DELETE(X) if koldlt h(X) then
for each neighbor Y of X if h(Y) lt kold and
h(X) gt h(Y) c(Y,X) then b(X) Y h(X)
h(Y)c(Y,X) if kold h(X) then for each
neighbor Y of X if t(Y) NEW or (b(Y) X and
h(Y) ? h(X)c (X,Y) ) or (b(Y) ? X and h(Y) gt
h(X)c (X,Y) ) then b(Y) X INSERT(Y,
h(X)c(X,Y))
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D Algorithm
h(G)0 do kminPROCESS-STATE() while(kmin
! -1 start state not removed from
Qu) if(kmin -1) goal unreachable
exit else do do trace
optimal path() while ( goal is not reached
map environment) if (
goal_is_reached) exit else
Y State of discrepancy reached trying
to move from some State X
MODIFY-COST(Y,X,newc(Y,X)) do
kminPROCESS-STATE()
while( k(Y) lt h(Y) kmin ! -1)
if(kmin-1) exit()
while(1)
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c(x1,x2)1
c(x1,x3)1.4
c(x1,x8)10000,if x8 is in obstacle,x1 is a
freecell
c(x1,x9)10000.4, if x9 is in obstacle, x1 is a
freecell
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Goal
Gate
Start
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Example Pictures
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Comparisons

• Comparison with A replanner
• Size of environment , number, size and
  placement of obstacles affect relative superior
  performance of D over A replanner. D is much
  better is expansive and more complex
  environments.
• Where do we go from here?

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Focussed D.

• Adds the focus heuristics to limit the search
  space. Stentz, 1995

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Applications

• Now we have an optimal, efficient and complete
  planner. So what?
• Planetary Exploration (Mars, etc)
• Search and Rescue (WTC)
• Any graph search problem where change is possible
• Financial Markets
• Artificial Intelligence