Problem solving by search

Tomas Svoboda

Department of Cybernetics, Vision for Robots and Autonomous Systems

March 5, 2018

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Outline

I Search problem.

I State space graphs.

I Search trees.

I Strategies, which tree branches to choose?

I Strategy/Algorithm properties?

I Programming instrastructure

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Example: Romania

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

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Example: Romania

Goal:be in Bucharest

Problem formulation:states: position in a city (cities)actions: drive between cities

Solution:Sequence of cities (path)

Giurgiu

UrziceniHirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75

120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

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Example: The 8-puzzle

2

Start State Goal State

51 3

4 6

7 8

5

1

2

3

4

6

7

8

5

states?actions?solution?cost?

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Example: Vacuum cleaner

R

L

S S

S S

R

L

R

L

R

L

S

SS

S

L

L

LL R

R

R

R

states?actions?solution?cost?

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A Search Problem

I State space (including Start/Initial state): position, boardconfiguration,

I Actions: drive to, Up, Down, Left . . .

I Transition model: Given state and action return state (and cost)

I Goal test: Are we done?

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State Space Graphs

State space graph: a representation of asearch problem

I Nodes are abstracted worldconfigurations

I Arcs represent action results

I Goal test is a set of goal nodes

Each state occurs only once in a state (search)space.

S b

c

d

G

a

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Search Trees

Possible futures

Here we startS

a b c

I A “what if” tree of plans and their outcomes

I Start node is the root

I Children are successors

I Nodes show states, but correspond to plans that achieve those states

What does the last item mean, actually?

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State Space Graphs vs. Search Trees

S b

c

d

G

a

S

a

S

a b c

b

a

S

d

S G

c

G a b

How big is the search tree?

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Search tree for Romania(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

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Search elements

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

I Expand plans - possible ways (tree nodes).

I Manage/Maintan fringe (or frontier) of plans under consideration.

I Expand new nodes wisely(?).

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Tree search algorithm

(a) The initial state

(b) After expanding Arad

(c) After expanding Sibiu

Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea

Rimnicu Vilcea Lugoj

ZerindSibiu

Arad Fagaras Oradea

Timisoara

AradArad Oradea

Lugoj AradArad Oradea

Zerind

Arad

Sibiu Timisoara

Arad

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

function tree search(problem) return a solution or failureinitialize by using the initial state of the problemloop

if no candidates for expansion then return failureelse choose a leaf node for expansionend ifif the node contains a goal state then return the solutionend ifExpand the node and add the resulting nodes to the tree

end loopend function

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Example of a tree search

a

S

b

c

d

e

f

G

S

a

b

e f

G

c d

Which nodes to explore?What are the properties of a strategy/algorithm?

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Search (algorithm) properties

I Complete? Guaranteed to find a solution (if exists)?

I Optimal? Guaranteed to find the least cost path?

I Time complexity? How many steps - an operation with a node?

I Space complexity? How many nodes to remember?

How many nodes in a tree? What are tree parameters?

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Depth-First Search (DFS)

a

S

b

c

d

e

f

G

S

a

b

e

f

f

e

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f d

G

d

What are the DFS properties?

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DFS properties

I Complete?

I Optimal?

I Time complexity?

I Space complexity?

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Breadth-First Search (BFS)

a

S

b

c

d

e

f

G

S

a

b e

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f d

d

G

What are the BFS properties?

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BFS properties

I Complete?

I Optimal?

I Time complexity?

I Space complexity?

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DFS vs BFS

S

a

b

e

f

f

e

f

c

f d

G

d

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a

b e

c

f d

d

G

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DFS with limited depth, maxdepth=2

Do not follow nodes with depth > maxdepth

a

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e f

e

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f d

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Iterative deepening DFS (ID-DFS)

I Start with maxdepth = 1

I Perform DFS with limited depth. Report success or failure.

I If failure, forget everything, increase maxdepth and repeat DFS

Is it not a terrible waste to forget everything between steps?

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Cost sensitive search

a

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b

c

d

e

f

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3

1

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1.5 S,0

a,1

b,2 e,2

c,1

f,2 d,2

d,1

G,2

I In BFS, DFS, node ±depth was thenode-value.

I How was the depth actuallycomputed?

I How to evaluate nodes with pathcost?

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Uniform Cost Search (UCS)

a

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1.5 S,0

a,1

b,2

e,3 f,4

e,2.5

c,1

f,2 d,2

G,3

d,3

When to check the goal (and stop) the search? When visiting orexpanding the node?

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When to stop, when visiting or expanding?

a

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b

c

d

e

f

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1

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11.2

0.5

1

1

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1.5S,0

a,1

b,2

e,3 f,4

e,2.5

f,3

c,1

f,2.2

G,2.7

d,2

G,3

d,3

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Example: Graph with costs

Uniform#Cost#Search#

S

a

b

d p

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c

e

p

h

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q c G

a

q e

p

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q c G

a

Strategy: expand a cheapest node first:

Fringe is a priority queue (priority: cumulative cost) S

G

d

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p q

c

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a

f

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3 9 1

16 4 11

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7 13

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0

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8 2

15

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Cost contours

2

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UCS properties

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Programming a Tree Search

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Infrastructure for (tree) search algorithms

What should a tree node n now?

I n.state

I n.parent

I n.pathcost

Perhaps we may add something later, if needed . . .

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How to organize nodes?

The Python examples are just suggestions, . . .

I A dynamically linked structure (list()).

I Add a node (list.insert(node)).

I Take a node and remove from the structure (node=list.pop()).

I Check the Python modules heapq1 and queue2 for inspiration.

1https://docs.python.org/3.5/library/heapq.html2https://docs.python.org/3.5/library/queue.html

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What is the solution?

I We stop when Goal is reached.

I How do we contruct the path?

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