CS 312: Algorithm Analysis

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CS 312 Algorithm Analysis

• Lecture 35 Heuristic Search Algorithms A

Credit These slides are primarily by Stuart
Russell of UC Berkeley.
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Announcements

• Homework 24 due today
• Project 6 Solving the TSP using Branch and
  Bound
• Early day Friday (4/13)
• Due date next Monday (4/16)
• No Improvement
• This project report is worth 10 of final grade
• Give this project your best effort!
• Clarification 10 rows in the results table
• Another Help Session
• Tuesday (4/10)
• 5-6pm
• Location meet at CS 312 desk

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HW 24 Problem 1
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Objectives

• Lecture 35 Heuristic Search Algorithms A
• To understand Best-first search as a class of
  algorithms
• To reconsider Greedy best-first search
• To understand Branch Bound in this setting
• To introduce A Search
• To understand Admissible Heuristics
• To consider the Relaxed Problem

• Lecture 34 Solving the TSP with BB, II
• Consider an alternate way to search the solution
  space.
• i.e., an alternate way to generate children in
  your search for a TSP tour.
• Put the pieces together

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Credit

• Russell Norvig 4.1-4.2

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Review Tree search

• Basic idea
• offline, simulated exploration of state space by
  generating successors of already-explored states
  (a.k.a. expanding states)
• 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
• Breadth-first search
• Uniform-cost search
• Depth-first search
• Depth-limited search
• Iterative deepening search

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Best-first search

• Idea use an evaluation function f(n) for each
  node
• estimate of "desirability"
• Expand most desirable unexpanded node
• Implementation
• Order the nodes in fringe in decreasing order of
  desirability
  
• Special cases
• Greedy best-first search
• Branch and Bound search
• A search

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Romania with step costs in km
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Greedy best-first search

• Evaluation function f(n) h(n) (heuristic)
• estimate of cost from n to goal
• e.g., hSLD(n) straight-line distance from n to
  Bucharest
• Greedy best-first search expands the node that
  appears to be closest to goal
• Youve seen this before! Recall the simple
  greedy algorithm from your Intelligent Scissors
  project

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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Search strategies

• A search strategy is defined by picking the order
  of node expansion
• Strategies are evaluated along the following
  dimensions
• completeness does it always find a solution if
  one exists?
• time efficiency number of nodes generated
• space efficiency maximum number of nodes in
  memory
• optimality does it always find a least-cost
  solution?
• Time and space efficiency are measured in terms
  of
• b maximum branching factor of the search tree
• d depth of the least-cost solution
• m maximum depth of the state space (may be 8)

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Properties of greedy best-first search

• Complete?
• No can get stuck in loops, e.g., Iasi ? Neamt ?
  Iasi ? Neamt ?
• Time?
• O(mb)
• Space?
• O(b)
• Optimal?
• No

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Branch and Bound Search

• Idea avoid expanding paths that are already
  expensive at best
• Bound function f(n)
• estimated total cost of path through state n to
  goal
• BSSF best solution so far
• You know this well!

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Properties of Branch and Bound

• Complete?
• Yes
• (unless there are infinitely many nodes with f
  f(G) )
• Time?
• O(bm), Exponential
• Space?
• O(bm), At worst, keeps all nodes in memory and
  does not prune. At best, ?
• Optimal?
• Yes

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A search

• Idea avoid expanding paths that are already
  expensive
• Evaluation function f(n) g(n) h(n)
• g(n) cost so far to reach n
• h(n) estimated cost from n to goal
• f(n) estimated total cost of path through n to
  goal

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A search example
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A search example
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A search example
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A search example
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A search example
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A search example
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Admissible heuristics

• A heuristic h(n) is admissible if for every node
  n,
• h(n) h(n), where h(n) is the true cost to
  reach the goal state from n.
  
• An admissible heuristic never overestimates the
  cost to reach the goal, i.e., it is optimistic
• Example hSLD(n) (never overestimates the actual
  road distance)
• Theorem If h(n) is admissible, A using
  TREE-SEARCH is optimal

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Optimality of A (proof)

• Suppose some suboptimal goal G2 has been
  generated and is in the fringe. Let n be an
  unexpanded node in the fringe such that n is on a
  shortest path to an optimal goal G.
• f(G2) g(G2) since h(G2) 0
• g(G2) gt g(G) since G2 is suboptimal
• f(G) g(G) since h(G) 0
• f(G2) gt f(G) from above

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Optimality of A (proof continued)

• f(G2) gt f(G) from above
• h(n) h(n) since h is admissible
• g(n) h(n) g(n) h(n)
• f(n) f(G)
• Hence f(G2) gt f(n), and A will never select G2
  for expansion
  

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Consistent heuristics

• A heuristic is consistent if for every node n,
  every successor n' of n generated by any action
  a,
• h(n) c(n,a,n') h(n')
• If h is consistent, we have
• f(n') g(n') h(n')
• g(n) c(n,a,n') h(n')
• g(n) h(n)
• f(n)
• i.e., f(n) is non-decreasing along any path.
• Theorem If h(n) is consistent, A using
  GRAPH-SEARCH is optimal

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Optimality of A

• A expands nodes in order of increasing f value
• Gradually adds "f-contours" of nodes
• Contour i has all nodes with ffi, where fi lt fi1

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Properties of A

• Complete?
• Yes (unless there are infinitely many nodes with
  f f(G) )
• Time?
• O(bm), Exponential
• Space?
• O(bm), Keeps all nodes in memory
• Optimal?
• Yes

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Example The 8-puzzle

• states?
• actions?
• goal test?
• path cost?

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Example The 8-puzzle

• states? locations of tiles
• actions? move blank left, right, up, down
• goal test? goal state (given)
• path cost? 1 per move
• Note optimal solution of n-Puzzle family is
  NP-hard
  

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Admissible heuristics?
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Admissible heuristics

• E.g., for the 8-puzzle
• h1(n) number of misplaced tiles
• h2(n) total Manhattan distance
• (i.e., no. of squares from desired location of
  each tile)
  
• h1(S) ?
• h2(S) ?

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Admissible heuristics

• E.g., for the 8-puzzle
• h1(n) number of misplaced tiles
• h2(n) total Manhattan distance
• (i.e., no. of squares from desired location of
  each tile)
  
• h1(S) ? 8
• h2(S) ? 31222332 18

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Dominance

• If h2(n) h1(n) for all n (both admissible)
• then h2 dominates h1
• h2 is better for search
• Typical search costs (average number of nodes
  expanded)
• d12 IDS 3,644,035 nodes A(h1) 227 nodes
  A(h2) 73 nodes
• d24 IDS too many nodes A(h1) 39,135 nodes
  A(h2) 1,641 nodes

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Relaxed problems

• A problem with fewer restrictions on the actions
  is called a relaxed problem
• The cost of an optimal solution to a relaxed
  problem is an admissible heuristic for the
  original problem
• If the rules of the 8-puzzle are relaxed so that
  a tile can move anywhere, then h1(n) gives the
  shortest solution
• If the rules are relaxed so that a tile can move
  to any adjacent square, then h2(n) gives the
  shortest solution

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In Summary

• A and Branch and Bound
• Reasoning about an admissible heuristic in A is
  like reasoning about tight bounds in BB
• A doesnt include a BSSF, by default
• Thus, BB is potentially more memory efficient

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Assignment

• None
• Work hard on Project 6!