1. Heuristic Search
Henry Kautz

2. Problem: Large Graphs
It is expensive to find optimal paths in large graphs, using BFS, IDS, or Uniform Cost Search (Dijkstra’s Algorithm)
How can we search large graphs efficiently by using “commonsense” about which direction looks most promising?

3. Plan a route from 9th & 50th to 3rd & 51st
Example
53nd St
52nd St G
51st St S
50th St
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
Plan a route from 9th & 50th to 3rd & 51st

4. Plan a route from 9th & 50th to 3rd & 51st
Example
53nd St
52nd St G
51st St S
50th St
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
Plan a route from 9th & 50th to 3rd & 51st

5. Best-First Search
The Manhattan distance ( x+  y) is an estimate of the distance to the goal
It is a search heuristic
Best-First Search
Order nodes in priority to minimize estimated distance to the goal
Compare: Uniform-Cost / Dijkstra’s
Order nodes in priority to minimize distance from the start

6. Best-First Search Fringe: Priority queue (heap)
Key: h(n) – heuristic estimate of distance from n to goal

7. Obstacles
Best-FS eventually will expand vertex to get back on the right track S G
52nd St
51st St
50th St
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave

8. Non-Optimality of Best-First
Path found by Best-first = 10
53nd St 2 3
52nd St 1 2 S G 5 4 3 2
51st St 1
50th St 5 4 3 2 1 7 6
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
Shortest Path = 8

9. Improving Best-First
Best-first is often tremendously faster than UCS, but might stop with a non-optimal solution
How can it be modified to be (almost) as fast, but guaranteed to find optimal solutions?
A* - Hart, Nilsson, Raphael 1968
One of the first significant algorithms developed in AI
Widely used in many applications

11. h values S G 2 3 1 2 5 4 3 2 1 5 4 3 2 1 7 6 53nd St 52nd St 51st St
50th St 5 4 3 2 1 7 6
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave

12. g values S G 8 7 9 6 1 2 3 4 5 3 4 5 6 7 1 2 53nd St 52nd St 51st St
50th St 3 4 5 6 7 1 2
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave

13. f = g + h
53nd St 10 10
52nd St 10 8 S G 6 6 6 6
51st St 6
50th St 8 8 8 8 8 8 8
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave

14. Maze Runner Demo

21. A* Graph Search
Optimal if h is consistent as well as admissible

22. A* Graph Search for Any Admissible Heuristic
if State[node] is not in closed OR
g[node] < g[LookUp(State[node],closed)]
then

26. Properties of Heuristics
Let h1 and h2 be admissible heuristics
if for all s, h1(s)  h2(s), then
h1 dominates h2
h1 is better than h2
h3(s) = max(h1(s), h2(s)) is admissible
h3 dominates h1 and h2

27. Exercise: Rubik’s Cube
State: position and orientation of each cubie
Corner cubie: 8 positions, 3 orientations
Edge cubie: 8 positions, 2 orientations
Center cubie: 1 position – fixed
Move: Turn one face
Center cubits never move!
Devise an admissible heuristic

28. Next
Heuristic functions for STRIPS planning
Games