state space representations and search strategies - 2 spring 2007, juris vīksna

State space representations and search strategies - 2

Spring 2007, Juris Vīksna

Search strategies - A*<S,P,I,G,W> - state space

h*(x) - a minimum path weight fromx to the goal state

h(x) - heuristic estimate of h*(x)

g*(x) - a minimum path weight fromI to x

g(x) - estimate of g*(x) (i.e. minimal weightfound as far

f*(x) = g*(x) + h*(x)f(x) = g(x) + h(x)

Search strategies - A*

[Adapted from J.Pearl]

Search strategies - A*A*Search(state space =<S,P,I,G,W>,h)

Open {<h(I),0,I>}Closed while Open do

<fx,gx,x> ExtractMin(Open) [minimum for fx] if Goal(x, ) then return xInsert(<fx,gx,x>,Closed)for y Child(x ,) do

gy = gx + W(x,y)fy = gy + h(y) if there is no <f,g,y>Closed with ffy then

Insert(< fy,gy,y>,Open) [replace existing<f,g,y>, if present]

return fail

Complete search

Definition

An algorithm is said to be complete if it terminates with a solution when one exists.

Admissible search

Definition

An algorithm is admissible if it is guaranteed to return an optimal solution (with minimal possible path weight from the start state to a goal state) whenever a solution exists.

Dominant search

Definition

An algorithm A is said to dominate algorithm B, if every node expanded by A is also expanded by B. Similarly, A strictly dominates B if A dominates B and B does notdominate A. We will also use the phrase “more efficient than” interchangeably with dominates.

Optimal search

Definition

An algorithm is said to be optimal over a class of algorithms if it dominates all members of that class.

Locally finite state spacesDefinition

A state space <S,P,I,G,W> is locally finite, if • for every xS, there is only a finite number of ySsuch that (x,y)P

• there exists > 0 such that for all (x,y)P we haveW(x,y) .

Completeness of A*

Theorem

A* algorithm is complete on locally finite state spaces.

Admissibility of A*

Definition

A heuristic function h is said to be admissible if

0 h(n) h*(n) for all nS.

Admissibility of A*

Theorem

A* which uses admissible heuristic function is admissibleon locally finite state spaces.

Admissibility of A*

Lemma

If A* uses admissible heuristic function h, then at any time before A* terminates there exists a node n’ in Open,such that f(n’) f*(I).

Admissibility of A*

Theorem

A* which uses admissible heuristic function is admissibleon locally finite state spaces.

Informedness of heuristic functions

Definition

A heuristic function h2 is said to be more informed than h1, if both h1 and h2 are admissible and

h2(n) > h1(n) for every non-goal node nS.

Similarly, an A* algorithm using h2 is said to be more informed than that using h1.

Dominance of A*

Theorem

If A2* is more informed than A1*, then A2* dominates A1*.

Dominance of A*

Lemma

Any node expanded by A* cannot have an f valueexceeding f*(I), i.e.

f(n) f*(I) for all nodes expanded.

Dominance of A*

Lemma

Every node n on Open for which f(n) < f*(I) will eventually be expanded by A*.

C-bounded paths

Definition

We say that path P is C-bounded if every node along thispath satisfies gP(n)+h(n) C. Similarly, if a strict inequalityholds for every n along P, we say that P is strictly C-bounded. When it becomes necessary to identify which heuristic was used, we will use the notation C(h)-bounded.

C-bounded paths

Theorem

A sufficient condition for A* to expand a node n is that there exists some strictly f*(I)-bounded path P from I to n.

C-bounded paths

Theorem

A necessary condition for A* to expand a node n is that there exists a f*(I)-bounded path P from I to n.

Dominance of A*

Theorem

If A2* is more informed than A1*, then A2* dominates A1*.

Consistent heuristic functions

Definition

A heuristic function h is said to be consistent, if

h(n) k(n,n’) + h(n’) for all nodes n and n’.

(where k(n,n’) denotes the weight of cheapest path from n to n’).

Monotone heuristic functions

Definition

A heuristic function h is said to be monotone, if

h(n) W(n,n’) + h(n’) for all (n,n’)P.

Monotonicity and consistency

Theorem

Monotonicity and consistency are equivalent properties.

Monotonicity and admissibility

Theorem

Every monotone heuristic is also admissible.

A* with monotone heuristic

Theorem

An A* algorithm with monotone heuristic finds optimal paths to all expanded nodes, i.e.

g(n) = g*(n) for all <fn,gn,n> Closed.

Some terminology

A* - we just discussed that

A - basically the same as A*, but we check whether wehave reached a goal state already at the time whennodes are generated

Z* - generalization of A*, instead of f(x) = g(x) + h(x)uses more general function f(x’) = F(E(x),f(x),h(X’))

Z - related to Z* similarly as A to A*

Implementation issuesA*Search(state space =<S,P,I,G,W>,h)

Open {<h(I),0,I>}Closed while Open do

<fx,gx,x> ExtractMin(Open) [minimum for fx] if Goal(x, ) then return xInsert(<fx,gx,x>,Closed)for y Child(x ,) do

gy = gx + W(x,y)fy = gy + h(y) if there is no <f,g,y>Closed with ffy then

Insert(< fy,gy,y>,Open) [replace existing<f,g,y>, if present]

return fail

Implementation issues - Heaps

• They are binary trees with all levels completed, except the lowest one which may have uncompleted section on the right side

• They satisfy so called Heap Property - for each subtree of heap the key for the root of subtree must be at least as large as the keys for its (left and right) children


2

45

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3

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13 1

T(n) = (h) = (log n)

Insert


Delete

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13 14

T(n) = (h) = (log n)


ExtractMin

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7 13 14

T(n) = (h) = (log n)

1

Implementation issues - BST

T is a binary search tree, if

• it is a binary tree (with a key associated with each node)

• for each node x in T the keys at all nodes of left subtree of x are not larger than key at node x and keys at all nodes of right subtree of x are not smaller than key at node x


3

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Insert

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142

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108

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11


Delete

3

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142

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108

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Delete

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142

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118

10


Delete

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352

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3018

3119

Implementation issues - AVL trees

T is an AVL tree, if

• it is a binary search tree

• for each node x in T we have

Height(LC(x)) – Height(RC(x)) {– 1, 0, 1}

Implementation issues - skip lists

4 5 9 11 13 1 63

“Perfect” Skip List

How to chose a heuristic? Original problem P Relaxed problem P'

A set of constraints removing one or more constraintsP is complex P' becomes simpler

Use cost of a best solution path from n in P' as h(n) for P Admissibility:

h* hcost of best solution in P >= cost of best solution in P'

Solution space of P

Solution space of P'

How to chose a heuristic - 8-puzzle Example: 8-puzzle

– Constraints: to move from cell A to cell B cond1: there is a tile on A cond2: cell B is empty cond3: A and B are adjacent (horizontally or vertically)

– Removing cond2: h2 (sum of Manhattan distances of all misplaced tiles)

– Removing cond2 and cond3: h1 (# of misplaced tiles)

– Removing cond3: h3, a new heuristic function

How to chose a heuristic - 8-puzzle h3:

repeat if the current empty cell A is to be occupied by tile x

in the goal, move x to A. Otherwise, move into A any arbitrary misplaced tile. until the goal is reached

h2>= h3 >= h1

h1(start) = 7h2(start) = 18h3(start) = 7

How to chose a heuristic - TSP• Example: TSP. A legal tour is a (Hamiltonian) circuit

How to chose a heuristic - TSP• Example: TSP. A legal tour is a (Hamiltonian) circuit

– It is a connected second degree graph (each node has exactly two adjacent edges)Removing the connectivity constraint leads to h1: find the cheapest second degree graph from the

given graph (with o(n^3) complexity)

The given complete

graph

A legal tour

Other second degree graphs

How to chose a heuristic - TSP– It is a spanning tree (when an edge is removed) with the constraint

that each node has at most 2 adjacent edges)Removing the constraint leads to h2:

find the cheapest minimum spanning tree from the given graph

(with O(n^2/log n)

The given graph A legal tour Other MST

How complicated heuristic to chose?

[Adapted from R.Shinghal]

Relaxing optimality requirements

• is f = g + h the best choice, if we want to minimize search efforts, not solution cost? • even if solution cost is important, admissible f can lead to non terminating A*. Can speed be gained by decreasing solution quality? • it may be hard to find good admissible heuristic. What happens, if we do not require admissibility?


Weighted evaluation function

fw(n) = (1–w)g(n) + w h(n)

w = 0 - uniform costw = 1/2 - A*w = 1 - BestFirst


Bounded decrease in solution quality?


Dynamic Weighting

f(n) = g(n) + h(n) + (1 – d(n)/N) h(n)

d(n) - depth of a nodeN - anticipated depth of a goal node


Dynamic Weighting

f(n) = g(n) + h(n) + (1 – d(n)/N) h(n)

Theorem

If h is admissible, then algorithm is -admissible, i.e.it finds a path with a cost at most (1+ )C*.


A* algorithm

Uses 2 lists - Open and Focal

Focal is a sublist of Open containing nodes that do not deviate from the lowest f node by a factor greater than 1+ .

A* selects the node from Focal with lowest hF value

hF(n) - a second heuristic function estimating the computational effort to complete the search starting from n


A* algorithm

Theorem

If h is admissible, then A* algorithm is -admissible, i.e.it finds a path with a cost at most (1+ )C*.

NotehF does not need to be admissible


Theorem

If h(n) - h*(n) , then A* algorithm is -admissible, i.e.it finds a path with a cost at most (1+ )C*.


Example

Consider SP with arc costs uniformly drawn from [0,1].

N - number of arcs between n and a goal

h*(n) tends to be close to N/2 for large N

The only admissible heuristic is h(n) = 0


R* algorithm selects node from Open with the lowest C(n) value

Three common risk measures:

R1 - the worst case risk

R2 - the probability of suboptimal termination

R3 - the expected risk


R* algorithm selects node from Open with the lowest C(n) value

Theorem

For risk measures R1, R2, R3 algorithm R* is -risk admissible, i.e. terminates with a solution cost C, such that R(C) for all nodes left in Open.

Some performance examples


Some performance

examples


Performance of search strategy

T - number of nodes in search graphD - number fo nodes in solution path

We define penetrance P as follows:

P = D/T

We have 0 < P < 1


T - number of nodes in search graphD - number fo nodes in solution path

We define branching factor B as follows:

T = B + B2 + ... + BD

T = B(BD – 1)/(B – 1)

We have B 1


[Adapted from R.Shinghal]

state space representations and search strategies - 2 spring 2007, juris vīksna

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