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UNINFORMED SEARCH

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Problem - solving agents

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

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What do we need to define ?

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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…”?

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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.

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

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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 )

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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

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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.

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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.

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Example : Romania

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Tree Search

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Tree search example

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Tree search example : start with Sibiu

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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!

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Tree search algorithms

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

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This is the part that distinguishes different search strategies

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

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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.

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

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

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

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

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BFS on a Graph

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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 ?

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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 ∞ )

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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 )

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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

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

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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 )

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Uniform - cost search

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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.

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

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

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

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

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

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

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This is the part that distinguishes different search algorithms

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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

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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 !

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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

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Iterative deepening search

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Iterative deepening search l =0

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Iterative deepening search l =1

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Iterative deepening search l =2

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Iterative deepening search l =3

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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

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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

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Summary of algorithms

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

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Questions ? Waitlisted ? Talk to me after class.

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