1. Workshop: A* Search

2. Goals Understand the A* algorithm Reason about its properties
Know that it applies to many problems
4/6/2019
Workshop: A* Search

3. Contents A* basics Heuristics Properties and proofs
Solving other problems
Variants of A*
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Workshop: A* Search

4. Getting started
A* Basics
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Workshop: A* Search

5. 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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Workshop: A* Search

6. 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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Workshop: A* Search

7. 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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Workshop: A* Search

8. A* versus Dijkstra A* steers towards the goal using heuristics
An estimate of the path cost to the goal
“Dijkstra is A* without heuristics”
Best-first search
Expand the most promising vertex
Choice of heuristic determines the behavior
Optimal vs. sub-optimal paths
Running time
Possibility for speed-ups
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Workshop: A* Search

9. Formalized Edge costs Heuristics OPEN set CLOSED set
Each edge (v,v’) has a cost c(v,v’) ≥ 0
Heuristics
For each vertex v, a function h(v) estimates the path cost to vg
We assume that h(v) never overestimates the actual cost
(we’ll explain why later)
OPEN set
A “wavefront” of vertices to expand, starting at vs
Sorted by f(v) = g(v) + h(v)
g(v) = cost of the optimal path to v found so far
CLOSED set
A set of already expanded vertices
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Workshop: A* Search

10. Assignment 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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Workshop: A* Search

11. 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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Workshop: A* Search

12. Solution
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Workshop: A* Search

13. εὑρίσκω (“I find / discover”)
Heuristics
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Workshop: A* Search

14. 3 important properties 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
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
A heuristic h is consistent/monotone if:
the decrease in h is never bigger than the increase in g
Formally: 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’)
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Workshop: A* Search

15. Assignment 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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Workshop: A* Search

16. Reasoning about A*
Properties and proofs
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Workshop: A* Search

17. Every consistent heuristic
Assignment 3
Prove the following:
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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Workshop: A* Search

18. Other properties 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
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 cannot (safely) use a closed set!
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Workshop: A* Search

19. Solving other problems
Graph search in other places
Solving other problems
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Workshop: A* Search

20. Navigation meshes 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
Multi-layered environments
Heuristics?
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Workshop: A* Search

21. Abstract graph search Remember: A* can find optimal path in graphs
Many search problems can be defined as graph problems
Can be non-geometric
Example: Sliding puzzle
Vertex = puzzle state
Edge = move (sliding 1 piece)
Heuristic = ???
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Workshop: A* Search

22. Variants of A*
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Workshop: A* Search

23. 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
Check out:
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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Workshop: A* Search

24. 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
Check out:
D* Lite; Koenig & Lihkachev (2002)
Lifelong Planning A*; Koenig et al. (2004)
Dynamically Pruned A*; van Toll & Geraerts (2015)
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Workshop: A* Search

25. 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
Optimality?
Use coarse graph plan as heuristic
Precomputed or on the fly?
Check out:
Hierarchical A*: Searching Abstraction Hierarchies Efficiently
Holte et al. (1996)
Near-Optimal Hierarchical Pathfinding (HPA*)
Botea et al. (2004)
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Workshop: A* Search

26. Conclusions
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Workshop: A* Search

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

28. Conclusions A* is widely used and analyzed 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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Workshop: A* Search