3.0 State Space Representation of Problems

3.1 Graphs

3.2 Formulating Search Problems3.3 The 8-Puzzle as an example3.4 State Space Representation using graphs3.5 Performing a State Space Search3.6 Basic Depth First Search (DFS)3.7 Basic Breadth First Search (DFS)3.8 Best First Search (DFS)

3.1 Graphs• Definitions:• a graph consists of:

– A set of nodes N1, N2, N3,…Nn.

– A set of arcs that connect pairs of nodes.

• A directed graph has an indicated direction for traversing each arc.

• A labeled graph has its nodes labeled.

• A labeled directed graph is shown in figure 4.2

a

bc

d

e

1

23

4

5

Figure 4.2: Labeled directed graph

Nodes {a,b,c,d,e}

Arcs:{(a,b),(b,e),(c,a),(c,b), (d,c),(e,d)}

Figure 4.1:5 nodes, and 6 arcs graph.

• A path through a graph connects a sequence of nodes through successive arcs. It is represented by an ordered list of the nodes representing the path. – For example in figure 4.3, [a, b, e,

d] is a path through nodes a ,b ,e , d.

• A rooted graph has a unique node (called the root ) such that there is a path from the root to all nodes within the graph. i.e. all paths originate from the root ( figure 4.4).

a

bc

d

e

Figure 4.3: dotted curve indicates the path [a,b,e,d]

Figure 4.4: a rooted graph

a

bcd

The root

• A tree is a graph in which each two nodes have at most one path between them.– Figure 4.5 is an example of a

rooted tree.

• If a directed arc connects Ni to Nk then – Ni is the parent of Nk and– Nk is the child of Ni..– In figure 4.5: d is the parent

of e and f.– e and f are called siblings.

a

bcd

The root

e f g h i j

Figure 4.5: a rooted tree

• In a graph:1. An ordered sequence of nodes [ N1, N2, N3 ..,

Nn], where each Ni, Ni+1 in the sequence represent an arc (Ni,Ni+1), is called a path of length n-1.

2. If a path contains any node more than once it said to contain a cycle or loop.

3. Two nodes in a graph are said to be connected if there is a path that includes them both.

4. On a path on a rooted graph, a node is said to be the ancestor of all nodes positioned after it ( to its right) as well as descendent of all nodes before it ( to its left) -For example, in figure 4.5, d is the ancestor of e,

while it is the descendent of a in the path [a, d, e].

3.2 Formulating Search Problems • All search problems can be cast into the

following general form: – Starting State

E.g. • starting city for a route

– Goal State (or a test for goal state)E.g.

• destination city – The permissible operators

E.g. • go to city X

• A state is a data structure which captures all relevant information about the problem.E.g. – a node on a partial path

• 3.3 The 8-Puzzle as an example The eight puzzle consists of a 3 x 3 grid with 8

consecutively numbered tiles arranged on it. Any tile adjacent to the space can be moved on it. A number of different goal states are used.

5 4 .

6 1 8

7 3 2

1 2 3

8 . 4

7 6 5

Start State Goal State

A state for this problem needs to keep track of the position of all tiles on the game board, with 0 representing the blank position (space) on the board

The initial state could be represented as: ( (5,4,0), (6,1,8), (7,3,2) )

The final state could be represented as: ( (1,2,3) (8,0,4), (7,6,5) )

The operators can be thought of in terms of the direction that the blank space effectively moves. i.e.. up, down, left, right.

3.4 State Space Representation Using Graphs

• In the state space representation of a problem:– nodes of a graph correspond to partial

problem solution states.– arcs correspond to steps (application of

operators) in a problem solving process.– The root of the graph corresponds to the

initial state of the problem.– The goal node which may not exist, is a leaf

node which corresponds to a goal state.

• State Space Search is the process of finding a solution path from the start state to a goal state.

• The task of a search algorithm is to find a solution path through such a problem space.

• The generation of new states ( expansion of nodes) along the path is done by applying the operators (such as legal moves in a game).

• A goal may describe – a statea winning board in a simple game.– or some property of the solution path itself

(length of the path) shortest path for example.

3.5 Performing a State Space Search

State space search involves finding a path from the initial state of a search problem to a goal state.

To do this, 1-build a search graph, starting from the initial state (or the

goal state) 2- expand a state by applying the search operators to that

state, generating ALL of its successor states. These successors are in the next level down of the search

graph 3-The order in which we choose states for expansion is

determined by the search strategy Different strategies result in (sometimes massively) different

behaviour KEY CONCEPT: We want to find the solution while realizing

in memory as few as possible of the nodes in the search space.

3.6 Basic Depth First Search (DFS)

/* OPEN and CLOSED are lists */OPEN = Start node, CLOSED = emptyWhile OPEN is not empty do Remove leftmost state from OPEN, call it X If X is a goal return success Put X on CLOSED Generate all successors of X Eliminate any successors that are already on OPEN or CLOSED put remaining successors on LEFT end of OPENEnd whileNote: For depth first put successors on LEFT (i.e. acts like a STACK) For breadth first put successors on Right (i.e. acts like a QUEUE)

Consider the following segment of a search for a solution to the 8-Puzzle problem

a.The initial state

b. After expanding that state

c. After expanding "last" successor generated

In depth first search, the "last" successor generated will be expanded next .

Example: road map• Consider the following road map

S

AB

C

D

G3

4

4

5

23

3

4

Applying the DFS

1-Open=[S], closed=[]

2-Open=[AC], closed=[S]

3-Open=[BC], closed=[AS]

4-Open=[DC], closed=[BAS]

5-Open=[GC], closed=[DBAS]

6-Open=[C], closed=[GDBAS]

Report success

Path is SABDG

A

S

C

B C A D

D D BG

G G G

D

3.7 Basic Breadth First Search (BRFS)

/* OPEN and CLOSED are lists */OPEN = Start node, CLOSED = emptyWhile OPEN is not empty do Remove leftmost state from OPEN, call it X If X is a goal return success Put X on CLOSED Generate all successors of X Eliminate any successors that are already on OPEN or CLOSED put remaining successors on RIGHT end of OPENEnd whileNote: For depth first put successors on LEFT (i.e. acts like a STACK) For breadth first put successors on RIGHT (i.e. acts like a QUEUE)

Example: road map• Consider the following road map

S

AB

C

D

G3

4

4

5

23

3

4

Applying the BRFS

1-Open=[S], closed=[]

2-Open=[AC], closed=[S]

3-Open=[CB], closed=[AS]

4-Open=[BD], closed=[CAS]

5-Open=[D], closed=[CAS]

6-Open=[G], closed=[DCAS]

7-Open=[], closed=[GDCAS]

Report success

Path is SCDG

C

A

S

C

C A D

D BG

G G G

D

/* OPEN and CLOSED are lists */OPEN = Start node, CLOSED = emptyWhile OPEN is not empty do Remove leftmost state from OPEN, call it X If X is a goal return success Put X on CLOSED Generate all successors of X Eliminate any successors that are already on OPEN or CLOSED put remaining successors on OPEN sorted

according to their heuristic distance to the goal

( ascending from left to right) End while

3.8 Best First Search (BFS)

Example: road map• Consider the following road map

S

AB

C

D

G3

4

4

5

23

3

4

Applying the BFS

1-Open=[S], closed=[]

2-Open=[C7A8], closed=[S]

3-Open=[D3A8], closed=[CS]

4-Open=[G0B3A8], closed=[DCS]

5-Open=[B3A8], closed=[GDCS]

Report success

Path is SCDG

C

A

S

C

C A D

D BG

G G G

D