search’ - northeastern university · 2 search as problem solving ! need to search a space of...

ì Search Chapter 3 (acknowledgement goes to Gillian Smith(NEU) and Hwee Tou Ng’s for slides used in this lecture)

2

Search as Problem Solving

ì  Need to search a space of possible solutions

ì  There are many sequences of actions, each with their own utility.

ì  We want to find, or search for, the best one.

Framework for Search Problems

ì  Problem formula1on ì  A set of discrete states (usually finite, usually huge) ì  Start state ì  Subset of states designated as goals (usually) ì  A set of ac1ons the agent can perform

ì  Objec1ve ì  Find a sequence of operators that lead to the goal

state

4

Example: Romania

ì  On holiday in Romania; currently in Arad.

ì  Flight leaves tomorrow from Bucharest

ì  Formulate problem: ì  Goal: at a city; e.g., at Bucharest ì  states: at various cities ì  actions: choose next city

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Example: Romania

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Problem Formulation

A problem is defined as follows:

initial state e.g., “at Arad”

actions set of possible actions in current state x.

transition model Result(x,a) = state that follows from applying action

a in state x. e.g., Result(X2 (Arad), A4 (Arad Zerind)) = X3 (Zerind)

goal test, e.g., x = “at Bucharest”

path cost (additive) ì  e.g., sum of distances, number of actions executed, etc. ì  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

Search Algorithm

ì  Start with a specific state

ì  State space forms a tree structure ì  Root = start state ì  Each node represents a state ì  Ac1ons are branches, children are all possible

next-‐states

ì  Search involves expanding a fron%er of poten1al next states

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Tree search example

9

Implementation: general tree search

fron%er

Fron%er)

(fron%er)

fron%er

Fron%er)

Kinds of Search

ì  Uninformed

ì  Informed

ì  Op1miza1on

ì  Adversarial

Uninformed Search Strategies

ì  Breadth-‐First Search

ì  Depth-‐First Search

ì  Limited Depth Search

ì  Itera1ve Deepening

Breadth-‐First Search

ì  Order of node expansion? ì  Shallowest unexpanded node

A

B

D E

C

F G

Fron%er (FIFO Queue)

A


ì  Order of node expansion? ì  Shallowest unexpanded node

Fron%er (FIFO Queue)

B C

A

B

D E

C

F G

Breadth-‐First Search ì  Order of node expansion?

ì  Shallowest unexpanded node Fron%er (FIFO Queue)

C D E

A

B

D E

C

F G

Breadth-‐First Search ì  Order of node expansion?

ì  Shallowest unexpanded node Fron%er (FIFO Queue)

D E F G

A

B

D E

C

F G

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Example BFS

Evaluating a Search Strategy ì  Search strategy is defined by picking the order of node

expansion

ì  Evalua1on criteria ì  Completeness ì  Time complexity ì  Space Complexity ì  Op1mality

ì  Time and space complexity measured in ì  b – maximum branching factor for the search tree ì  d – depth of least-‐cost solu1on ì  m – maximum depth of the state space


ì  Complete? ì  Yes! As long as b is finite

ì  Op1mal? ì  Yes

ì  Time cost? ì  O(bd)

ì  Space cost? ì  O(bd)


ì  Complete? ì  Yes! As long as b is finite

ì  Op1mal? ì  Yes! Assuming?

ì  Time cost? ì  O(bd)

ì  Space cost? ì  O(bd)

Suppose b=10, a node uses 1000 bytes, 1m nodes/sec can be generated Depth 10: ~3 hours, ~10 terabytes Depth 12: ~13 days, ~1 petabyte Depth 14: ~3.5 years, ~99 petabytes

Depth-‐First Search

ì  Order of node expansion? ì  Deepest unexpanded node

A

B

D E

C

F G

Fron%er (LIFO Queue)

A


A

B

D E

C

F G


C B


A

B

D E

C

F G


C E D


A

B

D E

C

F G


C E


A

B

E

C

F G


C


A

C

F G


G F


A

C

F G


G

Properties of depth-first search

ì  Complete? No: fails in infinite-depth spaces

Can modify to avoid repeated states along path

ì  Time? O(bm) with m=maximum depth

ì  terrible if m is much larger than d ì  but if solutions are dense, may be much faster than breadth-first

ì  Space? O(bm), i.e., linear space! (we only need to

remember a single path + expanded unexplored nodes)

ì  Optimal? No (It may find a non-optimal goal first)

Depth-‐Limited Search

ì  Depth first search, but don’t go past a pre-‐set limit (L) while searching

A

B

D E

C

F G

H I J K L M N O

L

Limited Depth Search

ì  Combines benefits of depth-‐ and breadth-‐first

ì  Number of nodes generated: ì  (d+1)b0 + (d)b1 + (d-‐1)b2 + … + (2)bd-‐1 + (1)bd = O(bd)

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Iterative deepening search L=0

31


32


33

Iterative Deepening Search L=3

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Iterative deepening search

ì  Complete? Yes

ì  Time? O(bd)

ì  Space? O(bd)

ì  Optimal? Yes, if step cost = 1 or increasing function of depth.

Iterative Deepening Search

ì  Combines benefits of depth-‐ and breadth-‐first

ì  Number of nodes generated: ì  (d+1)b0 + (d)b1 + (d-‐1)b2 + … + (2)bd-‐1 + (1)bd = O(bd)

ì  Spa1al complexity: O(bd)

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Summary of algorithms

preferred uninformed search strategy

Third edition has O(bd)

ì Classwork

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Example: 8-Queens

ì  states? -any arrangement of n<=8 queens -or arrangements of n<=8 queens in leftmost n columns, 1 per column, such that no queen attacks any other. ì  initial state? no queens on the board ì  actions? -add queen to any empty square -or add queen to leftmost empty square such

that it is not attacked by other queens. ì  goal test? 8 queens on the board, none attacked. ì  path cost? 1 per move

Environment types ì  Fully observable (vs. partially observable): An agent's

sensors give it access to the complete state of the environment at each point in time.

ì  Deterministic (vs. stochastic): The next state of the environment is completely determined by the current state and the action executed by the agent. (If the environment is deterministic except for the actions of other agents, then the environment is strategic)

ì  Episodic (vs. sequential): An agent’s action is divided into atomic episodes. Decisions do not depend on previous decisions/actions.

Environment types

ì  Static (vs. dynamic): The environment is unchanged while an agent is deliberating. (The environment is semidynamic if the environment itself does not change with the passage of time but the agent's performance score does)

ì  Discrete (vs. continuous): A limited number of distinct, clearly defined percepts and actions.

ì  Single agent (vs. multi-agent): An agent operating by itself in an environment. Does the other agent interfere with my performance measure?

BFS, DFS, IDFS, LDFS

ì  What type of environment/problem space would these algorithms be op1mal for?

Quizzes and Classwork

ì  “What type of agent?” ì  An intelligent agent

ì  “Can you formulate a poker agent in terms of a search problem?” ì  Sorry I cannot

ì  Show the state space ì  O(bd)

ì informed Search

Informed Search

ì  Use problem-‐specific knowledge beyond the problem defini1on to find solu1ons more efficiently

ì  Heuris%c: which node should be expanded next?

Heuristic Search

ì  Use an evalua1on func1on f(n) for each node ì  Es1mate of “desirability” ì  Includes a heuris1c func1on, h(n)

ì  Expand the most desirable unexpanded node

Greedy best-‐first search

ì  Evalua1on func1on f(n) = h(n) (heuris1c) = es1mate of cost from n to goal

ì  e.g., hSLD(n) = straight-‐line distance from n to Bucharest

ì  Greedy best-‐first search expands the node that appears to be closest to goal

Example: Heuristic for Path-‐Finding

Greedy best-‐first search example

Properties of greedy best-‐first search

ì  Complete? No – can get stuck in loops, e.g., Iasi à Neamt à Iasi à Neamt à

ì  Time? O(bm), but a good heuris1c can give drama1c improvement

ì  Space? O(bm) -‐-‐ keeps all nodes in memory

ì  Op1mal? No

A* Search

ì  Avoid expanding paths that are already expensive

ì  Evalua1on func1on: f(n) = g(n) + h(n) ì  g(n) = cost of path so far ì  h(n) = es1mated cost from n to goal ì  f(n) = es1mated total cost of path through n to goal

A* Search Example: Arad to Bucharest

A* Search

ì  Complete? ì  Yes

ì  Op1mal? ì  Yes!

ì  Time? ì  Exponen1al

ì  Space? ì  Keeps all nodes in memory

Admissible Heuristics

ì  A heuris1c is admissible if, for every node n, h(n) ≤ h*(n) ì  h*(n): true cost to reach the goal from state n

ì  Example: path-‐finding heuris1c never overes1mates actual road distance

ì  Generate from a relaxed problem ì  Fewer restric1ons than original problem

ì Classwork

A* and Games Given A* algorithm, define h(n) and f(n), environment, trace the algorithm

Resource: hop://unitygems.com/astar-‐1-‐journeys-‐start-‐single-‐step/, hop://en.wikipedia.org/wiki/A*_search_algorithm

Video – Summary and outlook

ì  hop://videolectures.net/aaai2010_thayer_bis/

search’ - northeastern university · 2 search as problem solving ! need to search a space of...

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