Quick review Problem definition
Initial state, goal state, state space, actions, goal function, path cost function
Factors Branching factor, depth of shallowest solution,
maximum depth of any path in search state, complexity, etc.
Uninformed techniques BFS, DFS, Depth-limited, UCS, IDS
Informed search What we’ll look at:
Heuristics
Hill-climbing
Best-first search Greedy search A* search
Heuristics A heuristic is a rule or principle used to guide
a search It provides a way of giving additional knowledge of
the problem to the search algorithm Must provide a reasonably reliable estimate of
how far a state is from a goal, or the cost of reaching the goal via that state
A heuristic evaluation function is a way of calculating or estimating such distances/cost
Heuristics and algorithms A correct algorithm will find you the best
solution given good data and enough time It is precisely specified
A heuristic gives you a workable solution in a reasonable time It gives a guided or directed solution
Evaluation function There are an infinite number of possible
heuristics Criteria is that it returns an assessment of the
point in the search
If an evaluation function is accurate, it will lead directly to the goal
More realistically, this usually ends up as “seemingly-best-search”
Traditionally, the lowest value after evaluation is chosen as we usually want the lowest cost or nearest
Heuristic evaluation functions Estimate of expected utility value from a
current position E.g. value for pieces left in chess Way of judging the value of a position
Humans have to do this as we do not evaluate all possible alternatives These heuristics usually come from years of
human experience
Performance of a game playing program is very dependent on the quality of the function
Heuristics?
Heuristics?
Heuristic evaluation functions Must agree with the ordering a utility function
would give at terminal states (leaf nodes)
Computation must not take long
For non-terminal states, the evaluation function must strongly correlate with the actual chance of ‘winning’
The values can be learned using machine learning techniques
Heuristics for the 8-puzzle Number of tiles out of place (h1)
Manhattan distance (h2) Sum of the distance of each tile from its goal
position Tiles can only move up or down city blocks
0 1 2
3 4 56 7
The 8-puzzle Using a heuristic evaluation function:
h2(n) = sum of the distance each tile is from its goal position
0 1 2
3 4 56 7
0 1 2
3 4 56 7
Goal state
h1=1h2=1
h1=5h2=1+1+1+2+2=7
Current state Current state
0 2 5
3 1 76 4
Search algorithms Hill climbing
Best-first search Greedy best-first search A*
Iterative improvement Consider all states laid out on the surface of a
landscape
The height at any point corresponds to the result of the evaluation function
Iterative improvement Paths typically not retained - very little
memory needed
Move around the landscape trying to find the lowest valleys - optimal solutions (or the highest peaks if trying to maximise) Useful for hard, practical problems where the state
description itself holds all the information needed for a solution
Find reasonable solutions in a large or infinite state space
Hill-climbing (greedy local) Start with current-state = initial-state Until current-state = goal-state OR there is
no change in current-state do: a) Get the children of current-state and apply
evaluation function to each child b) If one of the children has a better score, then
set current-state to the child with the best score
Loop that moves in the direction of decreasing (increasing) value Terminates when a “dip” (or “peak”) is reached If more than one best direction, the algorithm can
choose at random
Hill-climbing (gradient ascent)
Hill-climbing drawbacks Local minima (maxima)
Local, rather than global minima (maxima)
Plateau Area of state space where the evaluation function
is essentially flat The search will conduct a random walk
Ridges Causes problems when states along the ridge are
not directly connected - the only choice at each point on the ridge requires uphill (downhill) movement
Best-first search Like hill climbing, but eventually tries all paths
as it uses list of nodes yet to be explored
Start with priority-queue = initial-state While priority-queue not empty do:
a) Remove best node from the priority-queue b) If it is the goal node, return success. Otherwise
find its successors c) Apply evaluation function to successors and
add to priority-queue
Best-first example
Best-first search Different best-first strategies have different
evaluation functions
Some use heuristics only, others also use cost functions: f(n) = g(n) + h(n)
For Greedy and A*, our heuristic is: Heuristic function h(n) = estimated cost of the
cheapest path from node n to a goal node For now, we will introduce the constraint that if n
is a goal node, then h(n) = 0
Greedy best-first search Greedy BFS tries to expand the node that is
‘closest’ to the goal assuming it will lead to a solution quickly f(n) = h(n) aka “greedy search”
Differs from hill-climbing – allows backtracking
Implementation Expand the “most desirable” node into the frontier
queue Sort the queue in decreasing order
Route planning: heuristic??
Route planning - GBFS
Greedy best-first search
Route planning
Greedy best-first search
Greedy best-first search
This happens to be the same search path that hill-climbing would produce, as there’s no backtracking involved (a solution is found by expanding the first choice node only, each time).
Greedy best-first search Complete
No, GBFS can get stuck in loops (e.g. bouncing back and forth between cities)
Time complexity O(bm) but a good heuristic can have dramatic
improvement
Space complexity O(bm) – keeps all the nodes in memory
Optimal No! (A – S – F – B = 450, shorter journey is
possible)
Practical 2 Implement greedy best-first search for
pathfinding
Look at code for AStarPathFinder.java
A* search A* (A star) is the most widely known form of
Best-First search It evaluates nodes by combining g(n) and h(n)
f(n) = g(n) + h(n)
where g(n) = cost so far to reach n h(n) = estimated cost to goal from n f(n) = estimated total cost of path through n
start
h(n)
g(n)
n
goal
A* search When h(n) = h*(n) (h*(n) is actual cost to
goal) Only nodes in the correct path are expanded Optimal solution is found
When h(n) < h*(n) Additional nodes are expanded Optimal solution is found
When h(n) > h*(n) Optimal solution can be overlooked
Route planning - A*
A* search
A* search
A* search
A* search
A* search
A* search Complete and optimal if h(n) does not
overestimate the true cost of a solution through n
Time complexity Exponential in [relative error of h x length of solution] The better the heuristic, the better the time
Best case h is perfect, O(d) Worst case h = 0, O(bd) same as BFS, UCS
Space complexity Keeps all nodes in memory and save in case of repetition This is O(bd) or worse A* usually runs out of space before it runs out of time
A* exercise
Node Coordinates SL Distance to KA (5,9) 8.0B (3,8) 7.3C (8,8) 7.6D (5,7) 6.0E (7,6) 5.4F (4,5) 4.1G (6,5) 4.1H (3,3) 2.8I (5,3) 2.0J (7,2) 2.2K (5,1) 0.0
Solution to A* exercise
GBFS exercise
Node Coordinates DistanceA (5,9) 8.0B (3,8) 7.3C (8,8) 7.6D (5,7) 6.0E (7,6) 5.4F (4,5) 4.1G (6,5) 4.1H (3,3) 2.8I (5,3) 2.0J (7,2) 2.2K (5,1) 0.0
Solution
To think about...f(n) = g(n) + h(n)
What algorithm does A* emulate if we set h(n) = -g(n) - depth(n) h(n) = 0
Can you make A* behave like Breadth-First Search?
A* search - Mario http://aigamedev.com/open/interviews/mario-
ai/ Control of Super Mario by an A* search Source code available Various videos and explanations Written in Java
Admissible heuristics A heuristic h(n) is admissible if for every node n,
h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n An admissible heuristic never overestimates the cost to
reach the goal Example: hSLD(n) (never overestimates the actual road
distance)
Theorem: If h(n) is admissible, A* is optimal (for tree-search)
Optimality of A* (proof) Suppose some suboptimal goal G2 has been generated and
is in the frontier. Let n be an unexpanded node in the frontier such that n is on a shortest path to an optimal goal G
f(G2) = g(G2) since h(G2) = 0 (true for any goal state)
g(G2) > g(G) since G2 is suboptimal f(G) = g(G) since h(G) = 0 f(G2) > f(G) from above
Optimality of A* (proof)
f(G2) > f(G) h(n) ≤ h*(n) since h is admissible g(n) + h(n) ≤ g(n) + h*(n) f(n) ≤ f(G) (f(G) = g(G) = g(n) + h*(n))
Hence f(G2) > f(n), and A* will never select G2 for expansion
Admissible heuristic example: for the 8-puzzle
h1(n) = number of misplaced tilesh2(n) = total Manhattan distance i.e. no of squares from desired location of each tile
h1(S) = ??h2(S) = ??
Heuristic functions
Admissible heuristic example: for the 8-puzzle
h1(n) = number of misplaced tilesh2(n) = total Manhattan distance i.e. no of squares from desired location of each tile
h1(S) = 6h2(S) = 4+0+3+3+1+0+2+1 = 14
Heuristic functions
Dominance/Informedness if h2(n) h1(n) for all n (both admissible)
then h2 dominates h1 and is better for search
Typical search costs: (8 puzzle, d = solution length) d = 12 IDS = 3,644,035 nodes
A*(h1) = 227 nodes A*(h2) = 73 nodes
d = 24 IDS 54,000,000,000 nodes A*(h1) = 39,135 nodes
A*(h2) = 1,641 nodes
Heuristic functions
Admissible heuristicexample: for the 8-puzzle
h1(n) = number of misplaced tilesh2(n) = total Manhattan distance i.e. no of squares from desired location of each tile
h1(S) = 6h2(S) = 4+0+3+3+1+0+2+1 = 14
But how do we come up with a heuristic?
Heuristic functions
Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem E.g. If the rules of the 8-puzzle are relaxed so that a
tile can move anywhere, then h1(n) gives the shortest solution
If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution
Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem
Relaxed problems
Choosing a strategy What sort of search problem is this?
How big is the search space?
What is known about the search space?
What methods are available for this kind of search?
How well do each of the methods work for each kind of problem?
Which method? Exhaustive search for small finite spaces when
it is essential that the optimal solution is found
A* for medium-sized spaces if heuristic knowledge is available
Random search for large evenly distributed homogeneous spaces
Hill climbing for discrete spaces where a sub-optimal solution is acceptable
Summary What is search for?
How do we define/represent a problem?
How do we find a solution to a problem?
Are we doing this in the best way possible?
What if search space is too large? Can use other approaches, e.g. GAs, ACO, PSO...
Finally Try the A* exercise on the course website
(solutions will be made available later)
Next seminar on Monday at 9:30am
See the algorithms in action: http://web.archive.org/web/20110719085935/http://www.s
tefan-baur.de/cs.web.mashup.pathfinding.html