UNINFORMED SEARCH

Problem - solving agents

Example : Romania  On holiday in Romania ; currently in Arad.  Flight leaves tomorrow from Bucharest

What do we need to define ?

Problem Formulation  The process of defining actions, states and goal.  States :  Cities ( e. g. Arad, Sibiu, Bucharest, etc )  Actions :  GoTo ( adjacent city )  Goal :  Bucharest Why not “turn left 5 degrees” or “walk 100 meters forward…”?

Abstraction  The process of removing details from a representation.  Simplifies the problem  Makes problems tractable ( possible to solve )  Humans are great at this !  Imagine a hierarchy in which another agent takes care of the lower level details, such as navigating from the city center to the highway.

Back to Arad…  We are in Arad and need to find our way to Bucharest.

Step 1 – Check Goal Condition  Check, are we at the goal ?  ( obviously not in this case, but we need to check in case we were )

Step 2 – Expand Current Node  Enumerate all the possible actions you could take from the current state  Formally : apply each legal action to the current state, thereby generating a new set of states.  From Arad can go to :  Sibiu  Timisoara  Zerind

Step 3 – Select which action to perform  Perform one of the possible actions ( e. g. GoTo ( Sibiu ))  Then go back to Step 1 and repeat.

This is an example of Tree Search  Exploration of state space by generating successors of already - explored states ( a. k. a. expanding states )  Usually performed offline, as a simulation  Returns the sequence of actions that should be performed to reach the goal, or that the goal is unreachable.

Example : Romania

Tree Search

Tree search example

Tree search example : start with Sibiu

Tree search example : need to process the descendants of Sibiu Note that we can loop back to Arad. Have to make sure we don’t go in circles forever!

Tree search algorithms

Implementation : general tree search a.k.a. frontier

This is the part that distinguishes different search strategies

Search strategies  A search strategy is defined by picking the order of node expansion

Uninformed search strategies  Uninformed search strategies use only the information available in the problem definition  What does it mean to be uninformed ?  You only know the topology of which states are connected by which actions. No additional information.  Later we ’ ll talk about informed search, in which you can estimate which actions are likely to be better than others.

Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

BFS on a Graph

Search Strategy Evaluation : finding solutions  Strategies are evaluated along the following dimensions :  completeness : does it always find a solution if one exists ?  optimality : does it always find a least - cost solution ?

Search Strategy Evaluation : complexity ( cost )  Two types of complexity  time complexity : number of nodes visited  space complexity : maximum number of nodes in memory  Time and space complexity are measured in terms of  b : maximum branching factor of the search tree ( may  ∞ )  d : depth of the least - cost solution  m : maximum depth of the state space ( may be ∞ )

Properties of breadth - first search  Complete ?  Yes ( if b is finite )  Optimal ?  Yes ( if cost = 1 per step )  Time ?  1+ b + b 2 + b 3 + … + b d = O ( b d )  Space ?  O ( b d ) ( keeps every node in memory )

Problems of breadth first search  Space is the biggest problem ( more than time )  Example from book, BFS b =10 to depth of 10  3 hours ( not so bad )  10 terabytes of memory ( really bad )  Only reason speed is not a problem is you run out of memory first

Problems of breadth first search  BFS is not optimal if the cost of some actions is greater than others…

Uniform - cost search  For graphs with actions of different cost  Equivalent to breadth - first if step costs all equal  Expand least - cost unexpanded node  Implementation :  fringe = queue sorted by path cost g ( n ), from smallest to largest ( i. e. a priority queue )

Uniform - cost search

 Complete ?  Yes, if step cost ≥ ε  Time ?  O ( b ceiling ( C */ ε ) ) where C * is the cost of the optimal solution  Space ?  # of nodes with g ≤ cost of optimal solution, O ( b ceiling ( C */ ε ) )  Optimal ?  Yes – nodes expanded in increasing order of g ( n ) See book for detailed analysis.

Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front ( i. e. a stack )

Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

This is the part that distinguishes different search algorithms

Search Solution  Each node needs to keep track of its parent  Once the goal is found, traverse up the tree to the root to find the solution

Properties of depth - first search  Complete ?  No : fails in infinite - depth spaces  Yes : in finite spaces  Optimal ?  No  Time ?  O ( b m ): ( m is max depth of state space )  terrible if m is much larger than d  but if solutions are plentiful, may be much faster than breadth - first  Space ?  O ( bm ), i. e., linear space !

Depth - limited search  depth - first search with depth limit l ( i. e., don ’ t expand nodes past depth l )  … will fail if the goal is below the depth limit

Iterative deepening search

Iterative deepening search l =0

Iterative deepening search l =1

Iterative deepening search l =2

Iterative deepening search l =3

Properties of iterative deepening search  Complete ?  Yes  Time ?  ( d +1) b 0 + d b 1 + ( d -1) b 2 + … + b d = O ( b d )  Space ?  O ( bd )  Optimal ?  Yes, if step cost = 1

Bidirectional Search  Run two simultaneous searches  One forward from the start  One backward from the goal  Hope that the searches meet in the middle  b d /2 + b d /2 << b d

Summary of algorithms

Graph search The closed set keeps track of loops in the graph so that the search terminates.

Questions ? Waitlisted ? Talk to me after class.