Graph Search Rob Platt Northeastern University Some images and - - PowerPoint PPT Presentation

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Graph Search Rob Platt Northeastern University Some images and - - PowerPoint PPT Presentation

Dec 31, 2023 357 likes •1.42k views

Graph Search Rob Platt Northeastern University Some images and slides are used from: AIMA CS188 UC Berkeley What is graph search? Start state Goal state Graph search: find a path from start to goal what are the states? what are

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Graph Search

Rob Platt Northeastern University Some images and slides are used from: AIMA CS188 UC Berkeley

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What is graph search?

Graph search: find a path from start to goal – what are the states? – what are the actions (transitions)? – how is this a graph? Start state Goal state

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What is graph search?

Graph search: find a path from start to goal – what are the states? – what are the actions (transitions)? – how is this a graph? Start state Goal state

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What is graph search?

Graph search: find a path from start to goal – what are the states? – what are the actions (transitions)? – how is this a graph? Start state Goal state

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What is graph search?

Graph search: find a path from start to goal – what are the states? – what are the actions (transitions)? – how is this a graph? 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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What is a graph?

Graph: Edges: Vertices: Also called states Also called transitions

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Defining a graph: example

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Defining a graph: example

How many states?

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Defining a graph: example

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Defining a graph: example

Pairs of states that are “connected” by one turn of the cube.

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Example: Romania

  • On holiday in Romania; currently

in Arad. Flight leaves tomorrow from Bucharest

  • Formulate goal: Be in Bucharest
  • Formulate problem:
  • states: various cities
  • actions: drive between cities
  • Find solution:
  • sequence of cities, e.g., Arad,

Sibiu, Fagaras, Bucharest

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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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Graph search

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

How?

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Problem formulation

A problem is defined by four items:

  • initial state e.g., “at Arad”
  • successor function S(x) = set of action–state pairs

e.g., S(Arad) = { Arad → Zerind, Zerind , . . .} ⟨ ⟩

  • goal test, can be explicit, e.g., x = “at Bucharest” implicit, e.g.,

NoDirt(x)

  • path cost (additive)

e.g., sum of distances, number of actions executed, etc. c(x, a, y) is the step cost, assumed to be ≥ 0

  • A solution is a sequence of actions leading from the initial state to a

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)

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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 solution? – 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 solution? – 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 solution? – 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 solution? – 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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Uniform Cost Search (UCS)

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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 solution? – b: branching factor – C*: cost of optimal solution – 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 cheapest node first: Fringe is a priority queue (priority: cumulative cost) 3 9 1 16 4 11 5 7 13 8 10 11 17 11 6 Cost contours

UCS vs BFS

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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 Search Tiers Strategy: expand a shallowest node fjrst Implementation: Fringe is a FIFO queue

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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!
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Depth First Search (DFS)

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DFS

fringe Fringe A

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DFS

fringe Fringe A B C

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DFS

fringe Fringe A B C F G

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DFS

Fringe A B C F G H I

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DFS

Fringe A B C F G H I Which state gets removed next from the fringe?

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DFS

Fringe A B C F G H I Which state gets removed next from the fringe? What kind of a queue is this?

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DFS

Fringe A B C F G H I Which state gets removed next from the fringe? What kind of a queue is this?

LIFO Queue! (last in first out)

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DFS vs BFS: which one is this?

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DFS vs BFS: which one is this?

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BFS/UCS: which is this?

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BFS/UCS: which is this?

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DFS Properties: Graph search version

Is DFS complete? – only if you track the explored set in memory What is the time complexity of DFS (graph version)? – how many states are expanded before finding a solution? – complexity = number of states in the graph What is the space complexity of DFS (graph version)? – how much memory is required? – complexity = number of states in the graph Is DFS optimal? – is it guaranteed to find the best solution (shortest path)? This is the “graph search” version of the algorithm

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DFS Properties: Graph search version

Is DFS complete? – only if you track the explored set in memory What is the time complexity of DFS (graph version)? – how many states are expanded before finding a solution? – complexity = number of states in the graph What is the space complexity of DFS (graph version)? – how much memory is required? – complexity = number of states in the graph Is DFS optimal? – is it guaranteed to find the best solution (shortest path)? This is the “graph search” version of the algorithm

So why would we ever use this algorithm?

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm What is the space complexity of DFS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity =

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm What is the space complexity of DFS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = This is why we might want to use DFS

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm What is the space complexity of DFS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity =

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm What is the space complexity of DFS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity = Is it complete?

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm What is the space complexity of DFS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity = Is it complete?

NO!

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DFS: Tree search version

Suppose you don't track the explored set. – why wouldn't you want to do that? This is the “tree search” version of the algorithm What is the space complexity of DFS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity = Is it complete?

NO! What do we do???

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IDS: Iterative deepening search

What is IDS? – do depth-limited DFS in stages, increasing the maximum depth at each stage

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IDS: Iterative deepening search

What is IDS? – do depth-limited DFS in stages, increasing the maximum depth at each stage What is depth limited search? – any guesses?

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IDS: Iterative deepening search

What is IDS? – do depth-limited DFS in stages, increasing the maximum depth at each stage What is depth limited search? – do DFS up to a certain pre-specified depth

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IDS: Iterative deepening search

… b

  • Idea: get DFS’s space advantage

with BFS’s time / shallow-solution advantages

  • Run a DFS with depth limit 1. If

no solution…

  • Run a DFS with depth limit 2. If

no solution…

  • Run a DFS with depth limit 3.

…..

  • Isn’t that wastefully redundant?
  • Generally most work happens in

the lowest level searched, so not so bad!

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IDS

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IDS

What is the space complexity of IDS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity = Is it complete?

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IDS

What is the space complexity of IDS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity = Is it complete? YES!!! Is it optimal?

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IDS

What is the space complexity of IDS (tree version)? – how much memory is required? – b: branching factor – m: maximum depth of any node – complexity = What is the time complexity of DFS (tree version)? – how many states are expanded before finding a solution? – complexity = Is it complete? YES!!! Is it optimal? YES!!!

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General thoughts about search

If your model is wrong, then your solution will be wrong.

  • In November 2010, Nicaraguan troops

unknowingly crossed the border to Costa Rica, removed that country's flag and replaced it with their own. The reason: Google Maps told the troops' commander the territory belonged to Nicaragua.