1. Crowd Simulation (INFOMCRWS) - A* Search
Wouter van Toll
January 15, 2018

2. Goals Understand the A* algorithm Reason about its properties
Know that it applies to many problems
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INFOMCRWS: A* Search

3. Contents A* basics Heuristics Properties and proofs
Solving other problems
Variants of A*
+ Pen-and-paper exercises!
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INFOMCRWS: A* Search

4. November 28, 2018
INFOMCRWS: A* Search

5. A* basics
Getting started
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INFOMCRWS: A* Search

6. A* is born General-purpose graph search Very popular, also in games
P. Hart, N. Nilsson, B. Raphael. A Formal Basis for the Heuristic Determination of Minimum Cost Paths (1968)
General-purpose graph search
Compute optimal paths
Sub-optimal paths if you want to
Easy to implement
Very popular, also in games
Many versions assume grids and distance-based costs
But it’s way broader than that
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INFOMCRWS: A* Search

7. Graph search 1959: Dijkstra’s algorithm Given a graph G = (V,E)
Edges e є E have non-negative costs
Start vertex vs є V
Compute minimum-cost paths from vs to all other vertices
O(|E| + |V| log |V|) time (with the right data structures)
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INFOMCRWS: A* Search

8. Graph search 1968: A* algorithm Given a graph G = (V,E)
Edges e є E have non-negative costs
Start vertex vs є V, goal vertex vg є V
Compute a minimum-cost path from vs to vg
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INFOMCRWS: A* Search

9. A* formalized Edge costs Each edge (v,v’) has a cost c(v,v’) ≥ 0
Heuristics
For each vertex v, h(v) estimates the path cost to vg
We assume that h(v) never overestimates the actual cost
OPEN set
A “wavefront” for expansion, starting at vs
Sorted by f(v) = g(v) + h(v)
g(v) = cost of best path to v found so far
CLOSED set
A set of already expanded vertices
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INFOMCRWS: A* Search

10. Exercise 1
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11. Exercise 1 Grid example Run A* by hand and see what happens
Cells are either free or blocked
Each cell is connected to 4 neighbours (if they are free)
Costs are uniform (e.g. 1 per cell connection)
Run A* by hand and see what happens
Pseudocode on next slide
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INFOMCRWS: A* Search

12. Pseudocode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
g(vs) = 0, parent(vs) = NULL OPEN = {vs}, CLOSED = Ø while OPEN ≠ Ø v = argmin v’ є OPEN f(v’) if v = vg return the path to vs via parent pointers Remove v from OPEN Add v to CLOSED for each outgoing edge (v,v’) є E if v’ є CLOSED continue if v’ є OPEN or g(v) + c(v,v’) < g(v’) g(v’) = g(v) + c(v,v’) f(v’) = g(v’) + h(v’) parent(v’) = v Add / update v’ in OPEN return 
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13. Solution
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14. Greek: εὑρίσκω (“I find / discover”)
Heuristics
Greek: εὑρίσκω (“I find / discover”)
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15. Properties of heuristics (1/2)
1. A heuristic h is perfect if:
it always equals the actual path cost to the goal
Formally: h(v) = g*(v,vg) for all v
2. A heuristic h is admissible if:
it never overestimates the actual path cost to the goal
Formally: h(v) ≤ g*(v,vg) for all v
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INFOMCRWS: A* Search

16. Properties of heuristics (1/2)
3. A heuristic h is consistent/monotone if:
the decrease in h is never bigger than the increase in g
Formally:
h(vg) = 0
For any vertex v and any successor v’ of v during the search: h(v) ≤ c(v,v’) + h(v’)
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INFOMCRWS: A* Search

17. Exercise 2
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18. Exercise 2 Look at various problems and heuristics
Check if the heuristics are perfect/admissible/consistent
Note: A heuristic can have multiple properties!
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19. Solution Admissible? Consistent? Perfect?
a) 4-neighbor grid / Manhattan
yes no
b) 4-neighbor grid / Euclidean
c) 8-neighbor grid / Manhattan
d) 8-neighbor grid / Euclidean
e) Graph example 1
f) Graph example 2
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20. Properties and Proofs Reasoning about A* November 28, 2018
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21. Exercise 3
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22. Every consistent heuristic
Exercise 3
Prove the following statement:
Hints
Admissible: h(v) ≤ g*(v,vg) for all v
Consistent: 1. h(vg) = 0 2. For any vertex v and any successor v’ of v during the search: h(v) ≤ c(v,v’) + h(v’)
You could use induction
Every consistent heuristic
is admissible!
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23. Solution
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24. Other properties (1/2) If h is admissible, A* returns an optimal path
An inadmissible heuristic may still be useful (faster?)
Anytime A*: Start non-optimal and improve while you have time
If h is perfect, A* does not expand any vertices that are not on an optimal path
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INFOMCRWS: A* Search

25. Other properties (2/2)
If h is consistent, the optimal path to a vertex v is known as soon as v is expanded
This allows us to use a closed set
If h is inconsistent, we may find even better paths to v later, so we cannot (safely) use a closed set!
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INFOMCRWS: A* Search

26. Solving other problems
Graph search in other places
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27. Navigation mesh Polygonal cells  Dual graph A* on the graph
h = 2D distance to goal vertex
Result: sequence of cells
Shorten the path within cells
Optimal path?
Graph: yes
Original geometry: not always
Solution: visibility graph (more expensive but optimal)
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28. Navigation mesh Multi-layered environments How to improve the search?
3D distance as heuristic?
Geodesic distance as heuristic?
Higher-level graph that says how the layers are connected?
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29. Abstract graph search Remember: A* can find optimal path in graphs
Many search problems can be defined as graph problems
“Optimal” doesn’t always have to be “shortest”
Can even be non-geometric
Example: Sliding puzzle
Vertex =
Edge =
Heuristic =
puzzle state
move (sliding 1 piece)
???
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30. Variants of A*
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31. Anytime A* Start with a fast and imperfect search
e.g. an inadmissible heuristic
e.g. limited search space
path cost ≤ (1+ε) * optimal cost
Improve the path while time is available
Resources:
ARA*: Anytime A* with Provable Bounds on Sub-Optimality (Likhachev et al., 2003)
ANA*: Anytime Nonparametric A* (van den Berg et al., 2011)
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32. Dynamic A* Related to / combined with anytime A*
Re-plan the path when the environment changes
Remember the A* search space from the initial plan
Update the vertices where g changes
+ good for unknown environments
- memory usage, structural changes
Resources:
D* Lite (Koenig & Lihkachev, 2002)
Lifelong Planning A* (Koenig et al., 2004)
Dynamically Pruned A* (van Toll & Geraerts, 2015)
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33. Hierarchical A* Exploit the environment’s hierarchy Various options
Multiple “levels” of graphs
e.g. highways versus city streets; multi-layered buildings
Various options
First plan on coarse graph, then on fine graph
Use coarse graph plan as heuristic
Resources:
Hierarchical A*: Searching Abstraction Hierarchies Efficiently (Holte et al., 1996)
Near-Optimal Hierarchical Pathfinding (Botea et al., 2004)
(Optimality?)
(Precomputed or on the fly?)
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34. Closing Comments */
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35. Recap: Goals Understand the A* algorithm Reason about its properties
Know that it applies to many problems
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36. Conclusions A* is widely used and analyzed Not just for grids
Many interesting (provable) properties
New variants are still being made today
Not just for grids
Modelling the problem is most of the work
Of course, there’s much more to say
Lots of resources online
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37. The final slide
Questions?
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INFOMCRWS: A* Search