Finding Paths in Large Spaces - A* and Hierarchical A*

Searching large spaces Finding Paths Efﬁciently with A* presented by Nicolas Bettenburg 1

2

3

What is the fastest way to get to TD? 4

B A 5

A student’s mental model of Kingston... Domino's Pizza 350 Grizzly Grill 500 TD 50 Pizza Pizza 60 140 A&P 700 100 Hospital Starbucks 900 400 750 1000 Goodwin Hall 100 400 Gordon Hall Biosciences 6

Describe the Problem as a State Space Gordon Hall Goodwin Domino's Hall Pizza Gordon Hall Biociences Hospital Starbucks Gordon Hall Grizzly Grill Goodwin Goodwin Goodwin Hospital TD Canada A&P Pizza PIzza TD Canada Hall Hall Hall Starbucks Grizzly Grill Starbucks Grizzly Grill 7

Uninformed Search Breadth-First Search will ﬁnd the path through the search tree to the shallowest solution. Gordon Hall Domino's Pizza Goodwin Hall Grizzly Grill Biociences Starbucks Hospital TD Pizza TD Hospital A&P Canada PIzza Canada 1,75 km 1,55 km TD Grizzly Grizzly Canada Grill Grill 8

Breadth-First Search will need to keep O(bd+1 ) nodes in memory! b is the branching factor d is the depth of the solution b=10 Depth Nodes Time Memory 2 1100 1/10 s 1 MB 4 111,100 11 s 106 MB 6 10^7 19 m 10 GB 8 10^9 31 h 1 TB 10 10^11 129 d 101 TB 12 10^13 35 y 10 PB 14 10^15 3,523 y 1 EB 9

Uninformed Search Depth-First Search will always expand the deepest unexplored node. Gordon Hall Domino's Pizza Goodwin Hall Grizzly Grill Biociences Starbucks Hospital TD Pizza TD Hospital A&P Canada PIzza Canada TD Grizzly Grizzly Canada Grill Grill TD TD Canada Canada 10

Uninformed Search Depth-First Breadth-First Search Search Space complexity O(b · m) O(bd+1 ) Time complexity O(bm ) O(bd+1 ) Complete? No Yes 11

Can you compute a solution? Yes. Can you compute a best solution? Yes. With enough time and memory Can you compute a best solution more efﬁciently? This rest of this presentation. 12

Idea: Add problem speciﬁc information using some heuristic. 13

For each node, measure the distance from that node to the goal “as the bird ﬂies”. Domino's Pizza 350 Grizzly Grill 500 TD 50 Pizza Pizza 60 140 A&P 700 100 Starbucks Hospital 900 400 750 1000 Goodwin Hall 100 400 Gordon Hall Biosciences 14

Greedy Best-First Search Domino's Pizza 350 Grizzly Grill 500 TD 50 Always to expand the node Pizza Pizza 140 60 A&P 700 that is closest to the goal. 100 Starbucks Hospital 900 400 750 1000 Goodwin Hall 100 400 Gordon Hall Biosciences 15

Greedy Best-First Search Gordon Hall 900 Goodwin Domino's Hall Pizza 780 1000 Gordon Hall Biociences Hospital Starbucks Gordon Hall Grizzly Grill 900 820 300 Goodwin Goodwin Goodwin Hospital TD Canada A&P Pizza PIzza TD Canada Hall Hall Hall 780 0 Starbucks Grizzly Grill Starbucks Grizzly Grill 16

Domino's Pizza 350 Grizzly Grill 500 TD 50 Pizza Pizza 60 140 A&P 700 100 Starbucks Hospital 900 400 750 1000 100 Goodwin Hall 1,55km! Not the best solution 400 Gordon Hall Biosciences but much faster with less memory consumption! 17

Problems • follows single path all the way to goal • will back up when hitting dead end • not better than Depth-First Search 18

A* Search Deﬁne an evaluation function as: f(n) = g(n) + h(n) g(n) = cost(start, n) h(n) = distance(n, end) Thus f(n) describes the estimated cost of the cheapest solution through n. Expand the node with the smallest f(n) 19

Input: 1 OPEN = {n0} 2 CLOSED = {} G, n0, t 3 while (OPEN is not empty) 4 n = pop(OPEN) 5 6 if (n = t) terminate Output: 7 else Best path from n0 to t 8 M = Expand(n) 9 for each m in M do 10 if (exists m’ in OPEN with m’ = m) 11 if (f(m’) smaller or equal f(n) + f(m)) 12 continue 13 else if (exists m’ in CLOSED with m’ = m) 14 if (f(m’) smaller or equal f(n) + f(m)) 15 continue 16 else 17 OPEN = OPEN \\ {m} 18 CLOSED = CLOSED \\ {m} 19 parent (m) = n and OPEN = OPEN + {m} 20 end for 21 CLOSED = CLOSED + {n} 22 end if 23 end while 20

Gordon Hall 900 = 0 + 900 Domino's Pizza Goodwin Hall 1680 = 900 + 780 880 = 100 + 780 Biociences Starbucks Hospital Domino's Pizza 1320 = 500 + 820 1100 = 500+600 1150 = 850 + 300 350 Grizzly Grill 500 TD 50 Pizza Pizza 60 Pizza A&P PIzza 140 A&P 700 1170 = 640+530 1110 = 600+510 100 Starbucks Hospital Grizzly 900 1110 = 660+450 Grill 400 750 TD 1160 = 1160 + 0 Canada 1000 Goodwin Hall 100 400 Gordon Hall Biosciences 21

Properties of A* A* is optimal if h(n) is consistent h(n)is consistent, if it never over-estimates the true cost and it fullﬁls the triangle inequality. f(n) = g(n) + h(n) with g(n) is the true cost from start to n. Hence f(n) never over-estimates the true cost of a solution through n. 22

Properties of A* Suppose an sub-optimal goal node n’ appears on the fringe. Let the cost of the optimal solution be C*. Since n’ is sub-optimal and h(n’) = 0 f(n’) = g(n’) + h(n’) = g(n’) + 0 = g(n’) > C* (a) For a fringe node n on the optimal path f(n) = g(n) + h(n) ≤C* (b) Combining (a) and (b) shows f(n) ≤ C* < f(g’) So n’ will not be expanded and A* will return an optimal solution. 23

Properties of A* A* is complete The f-costs along any path are non-decreasing. Since A* always expands the the fringe node with lowest f-cost, we will eventually ﬁnd a path where the f-cost equals the cost of an optimal solution C*, if such a path exists. 24

Properties of A* A* is optimally efﬁcient For any heuristic function h(n), no other algorithm can expand fewer nodes with f(n) < C* without running the risk of missing the optimal solution. 25

A* can be used for many other problems: 8-Puzzle Rubik’s Cube 26

Start State Goal State average solution cost: 22 steps average branching factor: 3 complete state space: 322 ≈ 3.1 x 1010 distinct state space: 181,440 27

Good heuristics to use with A* and the 8-Puzzle: h1 : number of misplaced tiles h2: sum of manhattan distances from current positions to goal positions Depth Cost h1 Cost h2 4 13 12 22 18,094 1,219 28

There are much harder Problems 15-Puzzle: 1013 distinct states Rubik’s Cube: 8! x 37 x 12! x 210 ≈ 4.3 x 1019 states branching factor: 13 8! ways to arrange corner cubes seven indenpendent, eighth dependent on previous choices 12! ways to arrange edges eleven can be ﬂipped independently, twelfth depending on previous choices 29

Image from http://todaysinsidescoop.blogspot.com/2008/04/snowball-effect.html 30

Divide into Subproblems! * 2 4 1 2 * * 3 4 * * 3 1 * * * Start Goal Cost of C* for this subproblem forms lower bound on the cost of complete problem 31

Calculate costs for every possible subproblem instance Store these in a database Use the recorded values as h(n) Can solve 15-puzzles in a few milliseconds! 32

Pattern databases allowed to ﬁnd the ﬁrst optimal solutions to Rubik’s cube in 1997! (18 moves) 33

Computing a pattern database is expensive! Usually BFS backwards from goal to start. 34

Compute needed pattern database entries on demand! 5 2 4 * 2 4 transformation 8 7 * * 9 3 1 * 3 1 real state s abstract state Φ(s) Search on abstraction levels 35

Hierarchical A* * * 4 state Øi(s) * * h(Øi(s)) * * * . . . . . . . . . * 2 4 state Ø(s) * * h(Ø(s)) * 3 1 5 2 4 state s 8 7 h(s) 9 3 1 36

Conclusions 37

Finding Paths in Large Spaces - A* and Hierarchical A*

0 comments

Notes on slide 1

Favorites, Groups & Events