InformedSearch

Summary
Heuristics and Optimal search strategies
heuristics
hill-climbing algorithms
Best-First search
A*: optimal search using heuristics
Properties of A*
admissibility,
monotonicity,
accuracy and dominance
efficiency of A*
Branch and Bound
Iterative deepening A*
Automatic generation of heuristics

Problem: finding a Minimum Cost Path
Previously we wanted an arbitrary path to a goal or best cost.
Now, we want the minimum cost path to a goal G
Cost of a path = sum of individual transitions along path
Examples of path-cost:
Navigation
path-cost = distance to node in miles
minimum => minimum time, least fuel
VLSI Design
path-cost = length of wires between chips
minimum => least clock/signal delay
8-Puzzle
path-cost = number of pieces moved
minimum => least time to solve the puzzle

Best-first search
Idea: use an evaluation function f(n) for each node
estimate of "desirability"
Expand most desirable unexpanded node
Implementation:
Order the nodes in fringe in decreasing order of desirability
Special cases:
greedy best-first search
A* search

Heuristic functions
8-puzzle
8-queen
Travelling salesperson

Heuristic functions
8-puzzle
W(n): number of misplaced tiles
Manhatten distance
Gaschnig’s
8-queen
Travelling salesperson

Heuristic functions
8-puzzle
W(n): number of misplaced tiles
Manhatten distance
Gaschnig’s
8-queen
Number of future feasible slots
Min number of feasible slots in a row
Travelling salesperson
Minimum spanning tree
Minimum assignment problem

Best first (Greedy) search: f(n) = number of misplaced tiles

Romania with step costs in km

Greedy best-first search
Evaluation function f(n) = h(n) (heuristic)
= estimate of cost from n to goal
e.g., hSLD(n) = straight-line distance from n to Bucharest
Greedy best-first search expands the node that appears to be closest to goal

Greedy best-first search example

Greedy best-first search example

Greedy best-first search example

Greedy best-first search example

Problems with Greedy Search
Not complete
Get stuck on local minimas and plateaus,
Irrevocable,
Infinite loops
Can we incorporate heuristics in systematic search?

A* search
Idea: avoid expanding paths that are already expensive
Evaluation function f(n) = g(n) + h(n)
g(n) = cost so far to reach n
h(n) = estimated cost from n to goal
f(n) = estimated total cost of path through n to goal

A* search example

A* search example

A* search example

A* search example

A* search example

A* search example

A*- a special Best-first search
Goal: find a minimum sum-cost path
Notation:
c(n,n’) - cost of arc (n,n’)
g(n) = cost of current path from start to node n in the search tree.
h(n) = estimate of the cheapest cost of a path from n to a goal.
Special evaluation function: f = g+h
f(n) estimates the cheapest cost solution path that goes through n.
h*(n) is the true cheapest cost from n to a goal.
g*(n) is the true shortest path from the start s, to n.
If the heuristic function, h always underestimate the true cost (h(n) is smaller than h*(n)), then A* is guaranteed to find an optimal solution.

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, i.e., it is optimistic
Example: hSLD(n) (never overestimates the actual road distance)
Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal

A* on 8-puzzle with h(n) = w(n)

Algorithm A* (with any h on search Graph)
Input: a search graph problem with cost on the arcs
Output: the minimal cost path from start node to a goal node.
1. Put the start node s on OPEN.
2. If OPEN is empty, exit with failure
3. Remove from OPEN and place on CLOSED a node n having minimum f.
4. If n is a goal node exit successfully with a solution path obtained by tracing back the pointers from n to s.
5. Otherwise, expand n generating its children and directing pointers from each child node to n.
For every child node n’ do
evaluate h(n’) and compute f(n’) = g(n’) +h(n’)= g(n)+c(n,n’)+h(n)
If n’ is already on OPEN or CLOSED compare its new f with the old f and attach the lowest f to n’.
put n’ with its f value in the right order in OPEN
6. Go to step 2.

S
G
A
B
D
E
C
F
2
1
2
5
4
2
4
3
5

Example of A* Algorithm in action
S
A
D
B
D
C
E
E
B
F
G
2 +10.4 = 12..4
5 + 8.9 = 13.9
3 + 6.7 = 9.7
7 + 4 = 11
8 + 6.9 = 14.9
4 + 8.9 = 12.9
6 + 6.9 = 12.9
11 + 6.7 = 17.7
10 + 3.0 = 13
13 + 0 = 13
Dead End

Behavior of A* - Completeness
Theorem (completeness for optimal solution) (HNL, 1968):
If the heuristic function is admissible than A* finds an optimal solution.
Proof:
1. A* will expand only nodes whose f-values are less (or equal) to the optimal cost path C* (f(n) less-or-equal c*).
2. The evaluation function of a goal node along an optimal path equals C*.
Lemma:
Anytime before A* terminates there exists and OPEN node n’ on an optimal path with f(n’) <= C*.

Consistent heuristics
A heuristic is consistent if for every node n, every successor n' of n generated by any action a,
h(n) ≤ c(n,a,n') + h(n')
If h is consistent, we have
f(n') = g(n') + h(n')
= g(n) + c(n,a,n') + h(n')
≥ g(n) + h(n)
= f(n)
i.e., f(n) is non-decreasing along any path.
Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

Optimality of A* with consistent h
A* expands nodes in order of increasing f value
Gradually adds "f-contours" of nodes
Contour i has all nodes with f=fi, where fi < fi+1

Summary: Consistent (Monotone) Heuristics
If in the search graph the heuristic function satisfies triangle inequality for every n and its child node n’: h^(ni) less or equal h^(nj) + c(ni,nj)
when h is monotone, the f values of nodes expanded by A* are never decreasing.
When A* selected n for expansion it already found the shortest path to it.
When h is monotone every node is expanded once (if check for duplicates).
Normally the heuristics we encounter are monotone
the number of misplaced ties
Manhattan distance
air-line distance

Admissible heuristics
E.g., for the 8-puzzle:
h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location of each tile)
h1(S) = ?
h2(S) = ?

Admissible heuristics
E.g., for the 8-puzzle:
h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location of each tile)
h1(S) = ? 8
h2(S) = ? 3+1+2+2+2+3+3+2 = 18

Dominance
If h2(n) ≥ h1(n) for all n (both admissible)
then h2 dominates h1
h2 is better for search
Typical search costs (average number of nodes expanded):
d=12 IDS = 3,644,035 nodes
A*(h1) = 227 nodes A*(h2) = 73 nodes
d=24 IDS = too many nodes
A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes

Complexity of A*
A* is optimally efficient (Dechter and Pearl 1985):
It can be shown that all algorithms that do not expand a node which A* did expand (inside the contours) may miss an optimal solution
A* worst-case time complexity:
is exponential unless the heuristic function is very accurate
If h is exact (h = h*)
search focus only on optimal paths
Main problem: space complexity is exponential
Effective branching factor:
logarithm of base (d+1) of average number of nodes expanded.

Effectiveness of A* Search Algorithm
d IDS A*(h1) A*(h2)
2 10 6 6
4 112 13 12
8 6384 39 25
12 364404 227 73
14 3473941 539 113
20 ------------ 7276 676
Average number of nodes expanded
Average over 100 randomly generated 8-puzzle problems
h1 = number of tiles in the wrong position
h2 = sum of Manhattan distances

Properties of A*
Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )
Time? Exponential
Space? Keeps all nodes in memory
Optimal? Yes
A* expands all nodes having f(n) < C*
A* expands some nodes having f(n) = C*
A* expands no nodes having f(n) > C*

Relationships among search algorithms

Pseudocode for Branch and Bound Search
(An informed depth-first search)
Initialize: Let Q = {S}
While Q is not empty
pull Q1, the first element in Q
if Q1 is a goal compute the cost of the solution and update
L <-- minimum between new cost and old cost
else
child_nodes = expand(Q1),
,
For each child node n do:
evaluate f(n). If f(n) is greater than L discard n.
end-for
Put remaining child_nodes on top of queue in the order of their evaluation function, f.
end
Continue

Properties of Branch-and-Bound
Not guaranteed to terminate unless has depth-bound
Optimal:
finds an optimal solution
Time complexity: exponential
Space complexity: linear

Iterative Deepening A* (IDA*)
(combining Branch-and-Bound and A*)
Initialize: f <-- the evaluation function of the start node
until goal node is found
Loop:
Do Branch-and-bound with upper-bound L equal current evaluation function
Increment evaluation function to next contour level
end
continue
Properties:
Guarantee to find an optimal solution
time: exponential, like A*
space: linear, like B&B.

Inventing Heuristics automatically
Examples of Heuristic Functions for A*
the 8-puzzle problem
the number of tiles in the wrong position
is this admissible?
the sum of distances of the tiles from their goal positions, where distance is counted as the sum of vertical and horizontal tile displacements (“Manhattan distance”)
is this admissible?
How can we invent admissible heuristics in general?
look at “relaxed” problem where constraints are removed
e.g.., we can move in straight lines between cities
e.g.., we can move tiles independently of each other

Inventing Heuristics Automatically (continued)
How did we
find h1 and h2 for the 8-puzzle?
verify admissibility?
prove that air-distance is admissible? MST admissible?
Hypothetical answer:
Heuristic are generated from relaxed problems
Hypothesis: relaxed problems are easier to solve
In relaxed models the search space has more operators, or more directed arcs
Example: 8 puzzle:
A tile can be moved from A to B if A is adjacent to B and B is clear
We can generate relaxed problems by removing one or more of the conditions
A tile can be moved from A to B if A is adjacent to B
...if B is blank
A tile can be moved from A to B.

Generating heuristics (continued)
Example: TSP
Finr a tour. A tour is:
1. A graph
2. Connected
3. Each node has degree 2.
Eliminating 2 yields MST.

Relaxed problems
A problem with fewer restrictions on the actions is called a relaxed problem
The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem
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

Automating Heuristic generation
Use Strips representation:
Operators:
Pre-conditions, add-list, delete list
8-puzzle example:
On(x,y), clear(y) adj(y,z) ,tiles x1,…,x8
States: conjunction of predicates:
On(x1,c1),on(x2,c2)….on(x8,c8),clear(c9)
Move(x,c1,c2) (move tile x from location c1 to location c2)
Pre-cond: on(x1.c1), clear(c2), adj(c1,c2)
Add-list: on(x1,c2), clear(c1)
Delete-list: on(x1,c1), clear(c2)
Relaxation:
1. Remove from prec-dond: clear(c2), adj(c2,c3) ? #misplaced tiles
2. Remove clear(c2) ? manhatten distance
3. Remove adj(c2,c3) ? h3, a new procedure that transfer to the empty location a tile appearing there in the goal

Heuristic generation
The space of relaxations can be enriched by predicate refinements
Adj(y,z) iff neigbour(y,z) and same-line(y,z)
The main question: how to recognize a relaxed problem which is easy.
A proposal:
A problem is easy if it can be solved optimally by agreedy algorithm
Heuristics that are generated from relaxed models are monotone.
Proof: h is true shortest path I relaxed model
H(n) <=c’(n,n’)+h(n’)
C’(n,n’) <=c(n,n’)
? h(n) <= c(n,n’)+h(n’)
Problem: not every relaxed problem is easy, often, a simpler problem which is more constrained will provide a good upper-bound.

Improving Heuristics
If we have several heuristics which are non dominating we can select the max value.
Reinforcement learning.

Local search algorithms
In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution
State space = set of "complete" configurations
Find configuration satisfying constraints, e.g., n-queens
In such cases, we can use local search algorithms
keep a single "current" state, try to improve it
Constant space. Good for offline and online search

Hill-climbing search
"Like climbing Everest in thick fog with amnesia"

Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima

Hill-climbing search: 8-queens problem
h = number of pairs of queens that are attacking each other, either directly or indirectly
h = 17 for the above state

Hill-climbing search: 8-queens problem
A local minimum with h = 1

Simulated annealing search
Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency