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Wentworth Institute of Technology COMP3770 Artificial Intelligence | Summer 2017 | Derbinsky Uninformed Search Lecture 4 What are common search strategies that operate given only a search problem ? How do they compare ? Uninformed Search

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Uninformed Search

Lecture 4

What are common search strategies that operate given only a search problem? How do they compare?

May 22, 2017 Uninformed Search 1

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Agenda

  • A quick refresher
  • DFS, BFS, ID-DFS, UCS
  • Unification!

May 22, 2017 Uninformed Search 2

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Search Problem Formalism

Defined via the following components:

  • The initial state the agent starts in
  • A successor/transition function

– S(x) = {action+cost->state}

  • A goal test, which determines whether a given state is

a goal state

  • A path cost that assigns a numeric cost to each path

A solution is a sequence of actions leading from initial state to a goal state (optimal = lowest path cost) Together the initial state and successor function implicitly define the state space, the set of all reachable states

May 22, 2017 Uninformed Search 3

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

State Space Graph

  • State space graph: A

mathematical representation of a search problem

– Nodes are (abstracted) world configurations – Arcs represent successors (action results) – The goal test is a set of goal node(s)

  • In a search graph, each state
  • ccurs only once!
  • We can rarely build this full

graph in memory (i.e. it’s too big), but it’s a useful idea

May 22, 2017 Uninformed Search 4

S

G d b p q c e h a f r

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Search Tree

  • A “what if” tree of plans and

their outcomes

  • The start state is the root node
  • Children correspond to

successors

  • Nodes show states, but

correspond to PLANS that achieve those states

  • For most problems, we can

never actually build the whole tree

May 22, 2017 Uninformed Search 5

“E”, 1.0 “N”, 1.0

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Searching for Solutions

Basic idea: incrementally build a search tree until a goal state is found

  • Root = initial state
  • Expand via transition function to

create new nodes

  • Nodes that haven’t been

expanded are leaf nodes and form the frontier (open list)

  • Different search strategies

choose next node to expand (as few as possible!)

  • Use a closed list to prevent

expanding the same state more than once

May 22, 2017 Uninformed Search 6

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

General Algorithm

May 22, 2017 Uninformed Search 7

Queue (FIFO) Stack (LIFO) Priority Queue

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Evaluating a Search Strategy

  • Completeness: does it

always find a solution if

  • ne exists?
  • Optimality: does it

always find a least-cost solution?

  • Time Complexity:

number of nodes generated/expanded

  • Space Complexity:

maximum number of nodes in memory

May 22, 2017 Uninformed Search 8

Solution Efficiency

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Depth-First Search (DFS)

May 22, 2017 Uninformed Search 9

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

DFS Example

May 22, 2017 Uninformed Search 10

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 q p h f d b a c e r

Strategy: expand a deepest node first Implementation: Fringe is a LIFO stack

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Let’s Evaluate!

May 22, 2017 Uninformed Search 11

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Search Tree

Number of nodes in the tree? 1 + b + b2 + …. bm = O(?)

bm

May 22, 2017 Uninformed Search 12

… b 1 node b nodes b2 nodes bm nodes m tiers

Properties

  • Branching factor
  • Maximum depth
  • Solutions at

various depths

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

DFS Evaluation

Time

  • Expands left

– Could be whole tree!

  • Assuming finite depth,

O(bm) Space

  • Only siblings on path,

O(bm) Complete

  • Only if finite

Optimal

  • No, ”left-most” w/o

regard to cost/depth

May 22, 2017 Uninformed Search 13

… b 1 node b nodes b2 nodes bm nodes m tiers

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Breadth-First Search (BFS)

May 22, 2017 Uninformed Search 14

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

BFS Example

May 22, 2017 Uninformed Search 15

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 first Implementation: Fringe is a FIFO queue

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

BFS Evaluation

Time

  • Processes all nodes

above shallowest solution, O(bs) Space

  • Has roughly the last

tier, so O(bs) Complete

  • Yes!

Optimal

  • Only if all costs

equal (more later)

May 22, 2017 Uninformed Search 16

… b 1 node b nodes b2 nodes bm nodes s tiers bs nodes

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

DFS vs. BFS

May 22, 2017 Uninformed Search 17

Empty-DFS Empty-BFS Maze-DFS Maze-BFS

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Grounding the Branching Factor

Depth Nodes Time Memory 2 110 0.11 msecs 107 KB 4 11,110 11 msecs 10.6 MB 6 106 1.1 secs 1 GB 8 108 2 mins 103 GB 10 1010 3 hours 10 TB 12 1012 13 days 1 PB 14 1014 3.5 years 99 PB 16 1016 350 years 10 EB

May 22, 2017 Uninformed Search 18

Assumptions

  • b = 10
  • 1 million nodes/second
  • 1000 bytes/node

Memory often becomes the limiting factor

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Iterative Deepening DFS (ID-DFS)

  • Basic idea: DFS

memory with BFS time/shallow solution

– DFS up to 1 – DFS up to 2 – ….

  • Generally most work

happens in the lowest level searched, so not too wasteful

May 22, 2017 Uninformed Search 19

… b

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Cost-Sensitive Search

  • BFS finds the shortest path in terms of number of actions, but it

does not find the least-cost path.

  • We will now cover a similar algorithm which does!

May 22, 2017 Uninformed Search 20

START

GOAL

d b p q c e h a f r 2 9 2 8 1 8 2 3 2 4 4 15 1 3 2 2

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Uniform-Cost Search (UCS)

May 22, 2017 Uninformed Search 21

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

UCS Example

May 22, 2017 Uninformed Search 22

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

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

UCS Evaluation

Time

  • O(bC*/𝜁)

Space

  • O(bC*/𝜁)

Complete

  • Yes!

Optimal

  • Yes!

May 22, 2017 Uninformed Search 23

… b C*/e “tiers” (effective depth) c £ 3 c £ 2 c £ 1 Assume optimal solution costs C* and arcs cost at least 𝜁

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

UCS vs. DFS vs. BFS

  • UCS is good and
  • ptimal
  • However, it still

moves in every direction – it’s not informed about goal direction…

May 22, 2017 Uninformed Search 24

Empty-UCS Maze-UCS MazeCost-DFS MazeCost-BFS MazeCost-UCS … c £ 3 c £ 2 c £ 1 Start Goal

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Unification

  • All these search

algorithms are the same except for fringe strategies

  • Conceptually, all fringes

are priority queues (i.e. collections of nodes with attached priorities)

  • Practically, for DFS and

BFS, you can avoid the log(n) overhead from an actual priority queue, by using stacks and queues

May 22, 2017 Uninformed Search 25

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Uninformed Search

Search given only the problem definition

May 22, 2017 Uninformed Search 26

DFS BFS UCS Fringe LIFO (stack) FIFO (queue) PQ (path cost) Complete X X Optimal X Time 𝒫(𝑐%) 𝒫(𝑐') 𝒫(𝑐(∗/+) Space 𝒫(𝑐𝑛) 𝒫(𝑐') 𝒫(𝑐(∗/+) Assumptions: potentially infinite depth, arbitrary positive action costs

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

A Reminder

  • Search operates over

models of the world

  • The agent doesn’t

actually try all the plans

  • ut in the real world!
  • Planning is all “in

simulation”

  • Your search is only as

good as your models…

May 22, 2017 Uninformed Search 27

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Search Gone Wrong

May 22, 2017 Uninformed Search 28

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Wentworth Institute of Technology COMP3770 – Artificial Intelligence | Summer 2017 | Derbinsky

Summary

  • We evaluated several uninformed

strategies to solve a search problem

– DFS, BFS, ID-DFS, UCS

  • DFS, BFS, and UCS can all be

implemented via a generic graph-search algorithm over a search tree by simply changing how the fringe is organized

May 22, 2017 Uninformed Search 29