Breadth first search Uniform cost search Robert Platt Northeastern - - PowerPoint PPT Presentation

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Breadth first search Uniform cost search Robert Platt Northeastern - - PowerPoint PPT Presentation

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Breadth first search Uniform cost search Robert Platt Northeastern University Some images and slides are used from: 1. CS188 UC Berkeley 2. RN, AIMA What is graph search? Goal state Start state What is a graph? Graph: Vertices: Edges:

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Breadth first search Uniform cost search

Robert Platt Northeastern University Some images and slides are used from:

  • 1. CS188 UC Berkeley
  • 2. RN, AIMA
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What is graph search?

Start state Goal state

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What is a graph?

Graph: Edges: Vertices: Directed graph

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What is a graph?

Graph: Edges: Vertices: Undirected graph

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Graph search

Given: a graph, G Problem: find a path from A to B – A: start state – B: goal state

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A search tree

Start at A

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A search tree

Successors of A

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A search tree

Successors of A parent children

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A search tree

Let's expand S next

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A search tree

Successors

  • f S
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A search tree

A was already visited!

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A search tree

A was already visited!

So, prune it!

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A search tree

In what order should we expand states? – here, we expanded S, but we could also have expanded Z or T – different search algorithms expand in different orders

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Breadth first search (BFS)

Slide: Adapted from Berkeley CS188 course notes (downloaded Summer 2015)

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Breadth first search (BFS)

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Breadth first search (BFS)

Start node

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Breadth first search (BFS)

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Breadth first search (BFS)

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Breadth first search (BFS)

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Breadth first search (BFS)

We're going to maintain a queue called the fringe – initialize the fringe as an empty queue Fringe

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Breadth first search (BFS)

– add A to the fringe fringe Fringe A

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Breadth first search (BFS)

  • - remove A from the fringe
  • - add successors of A to the fringe

fringe Fringe B C

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Breadth first search (BFS)

  • - remove B from the fringe
  • - add successors of B to the fringe

fringe Fringe C D E

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Breadth first search (BFS)

fringe Fringe D E F G

  • - remove C from the fringe
  • - add successors of C to the fringe
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Breadth first search (BFS)

fringe Fringe D E F G Which state gets removed next from the fringe?

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Breadth first search (BFS)

fringe Fringe D E F G Which state gets removed next from the fringe? What kind of a queue is this?

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Breadth first search (BFS)

fringe Fringe D E F G Which state gets removed next from the fringe? What kind of a queue is this?

FIFO Queue! (first in first out)

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Breadth first search (BFS)

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Breadth first search (BFS)

What is the purpose of the explored set?

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BFS Properties

Is BFS complete? – is it guaranteed to find a solution if one exists?

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BFS Properties

Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a sol'n? – b: branching factor – d: depth of shallowest solution – complexity = ???

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BFS Properties

Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a sol'n? – b: branching factor – d: depth of shallowest solution – complexity =

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BFS Properties

Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a sol'n? – b: branching factor – d: depth of shallowest solution – complexity = What is the space complexity of BFS? – how much memory is required? – complexity = ???

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BFS Properties

Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a sol'n? – b: branching factor – d: depth of shallowest solution – complexity = What is the space complexity of BFS? – how much memory is required? – complexity =

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BFS Properties

Is BFS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of BFS? – how many states are expanded before finding a sol'n? – b: branching factor – d: depth of shallowest solution – complexity = What is the space complexity of BFS? – how much memory is required? – complexity = Is BFS optimal? – is it guaranteed to find the best solution (shortest path)?

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Another BFS example...

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Uniform Cost Search (UCS)

Slide: Adapted from Berkeley CS188 course notes (downloaded Summer 2015)

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Uniform Cost Search (UCS)

Notice the distances between cities

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Uniform Cost Search (UCS)

Notice the distances between cities – does BFS take these distances into account?

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Uniform Cost Search (UCS)

Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage?

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Uniform Cost Search (UCS)

Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage? – compare S-F-B with S-R-P-B. Which costs less?

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Uniform Cost Search (UCS)

Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage? – compare S-F-B with S-R-P-B. Which costs less?

How do we fix this?

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Uniform Cost Search (UCS)

Notice the distances between cities – does BFS take these distances into account? – does BFS find the path w/ shortest milage? – compare S-F-B with S-R-P-B. Which costs less?

How do we fix this? UCS!

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Uniform Cost Search (UCS)

Same as BFS except: expand node w/ smallest path cost Length of path

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Uniform Cost Search (UCS)

Same as BFS except: expand node w/ smallest path cost Length of path Cost of going from state A to B: Minimum cost of path going from start state to B:

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Uniform Cost Search (UCS)

Same as BFS except: expand node w/ smallest path cost Length of path Cost of going from state A to B: Minimum cost of path going from start state to B: BFS: expands states in order of hops from start UCS: expands states in order of

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Uniform Cost Search (UCS)

Same as BFS except: expand node w/ smallest path cost Length of path Cost of going from state A to B: Minimum cost of path going from start state to B: BFS: expands states in order of hops from start UCS: expands states in order of

How?

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Uniform Cost Search (UCS)

Simple answer: change the FIFO to a priority queue – the priority of each element in the queue is its path cost.

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Uniform Cost Search (UCS)

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UCS

Fringe A Path Cost Explored set:

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UCS

140 118 75

Explored set: A Fringe A S T Z Path Cost 140 118 75

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UCS

140 118 75 146

Explored set: A, Z Fringe A S T Z T Path Cost 140 118 75 146

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UCS

140 118 75 146 229

Explored set: A, Z, T Fringe A S T Z T L Path Cost 140 118 75 146 229

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UCS

140 118 75 239 220 146 229

Explored set: A, Z, T, S Fringe A S T Z T L F R Path Cost 140 118 75 146 229 239 220

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UCS

140 118 75 239 220 146 229

Explored set: A, Z, T, S Fringe A S T Z T L F R Path Cost 140 118 75 146 229 239 220

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UCS

140 118 75 239 220 336 317 146 229

Explored set: A, Z, T, S, R Fringe A S T Z T L F R C P Path Cost 140 118 75 146 229 239 220 336 317

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UCS

140 118 75 239 220 336 317 146 229 299

Explored set: A, Z, T, S, R, L Fringe A S T Z T L F R C P M Path Cost 140 118 75 146 229 239 220 336 317 299

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UCS

140 118 75 239 220 336 317 146 229 299

Explored set: A, Z, T, S, R, L Fringe A S T Z T L F R C P M Path Cost 140 118 75 146 229 239 220 336 317 299

When does this end?

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UCS

140 118 75 239 220 336 317 146 229 299

Explored set: A, Z, T, S, R, L Fringe A S T Z T L F R C P M Path Cost 140 118 75 146 229 239 220 336 317 299

When does this end? – when the goal state is removed from the queue

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UCS

140 118 75 239 220 336 317 146 229 299

Explored set: A, Z, T, S, R, L Fringe A S T Z T L F R C P M Path Cost 140 118 75 146 229 239 220 336 317 299

When does this end? – when the goal state is removed from the queue – NOT when the goal state is expanded

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UCS

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UCS Properties

Is UCS complete? – is it guaranteed to find a solution if one exists? What is the time complexity of UCS? – how many states are expanded before finding a sol'n? – b: branching factor – C*: cost of optimal sol'n – e: min one-step cost – complexity = What is the space complexity of BFS? – how much memory is required? – complexity = Is BFS optimal? – is it guaranteed to find the best solution (shortest path)?

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S

a b d p a c e p h f r q q c

G

a q e p h f r q q c

G

a Strategy: expand a cheapest node first: Fringe is a priority queue (priority: cumulative cost) S G

d b p q c e h a f r

3 9 1 16 4 11 5 7 13 8 10 11 17 11 6 3 9 1 1 2 8 8 2 15 1 2 Cost contours 2

UCS vs BFS

Slide: Adapted from Berkeley CS188 course notes (downloaded Summer 2015)

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UCS vs BFS

S

a b d p a c e p h f r q q c

G

a q e p h f r q q c

G

a

S

G d b p q c e h a f r Search Tiers Strategy: expand a shallowest node fjrst Implementation: Fringe is a FIFO queue Slide: Adapted from Berkeley CS188 course notes (downloaded Summer 2015)

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UCS vs BFS

Start Goal … c ≤ 3 c ≤ 2 c ≤ 1

  • Remember: UCS explores

increasing cost contours

  • The good: UCS is complete and
  • ptimal!
  • The bad:
  • Explores options in every

“direction”

  • No information about goal

location

  • We’ll fjx that soon!

Slide: Adapted from Berkeley CS188 course notes (downloaded Summer 2015)