Problem Solving by Search by Jin Hyung Kim Computer Science Department KAIST

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Problem Solving by Searchby Jin Hyung
KimComputer Science DepartmentKAIST
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Example of Representation

• Euler Path

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Graph Theory

• Graph consists of
• A set of nodes may be infinite
• A set of arcs(links)
• Directed graph, underlying graph, tree
• Notations
• node, start node(root), leaf (tip node), root,
  path, ancestor, descendant, child(children, son),
  parent(father), cycle, DAG, connected, locally
  finite graph, node expansion

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State Space Representation

• Basic Components
• set of states s
• set of operators o s -gt s
• control strategy c sn -gt o
• State space graph
• State -gt node
• operator -gt arc
• Four tuple representation
• N, A, S, GD, solution path

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Examples of SSR

• TIC_TAC_TOE
• n2-1 Puzzle
• Traveling salesperson problem (TSP)

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Search Strategies

• A strategy is defined by picking the order of
  node expansion
• Search Directions
• Forward searching (from start to goal)
• Backward searching (from goal to start)
• Bidirectional
• Irrevocable vs. revocable
• Irrevocable strategy Hill-Climbing
• Most popular in Human problem solving
• No shift of attention to suspended alternatives
• End up with local-maxima
• Commutative assumption
• Applying an inappropriate operators may delay,
  but never prevent the eventual discovery of
  solutions.
• Revocable strategy Tentative control
• An alternative chosen, others reserve

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Evaluation of Search Strategies

• Completeness
• Does it always find a solution if one exists ?
• Time Complexity
• Number of nodes generated/expanded
• Space complexity
• Maximum number of nodes in memory
• Optimality
• Does it always find a least-cost solution ?
• Algorithm is admissible if it terminate with
  optimal solution
• Time and Space complexity measured by
• b maximum branching factors of the search tree
• d depth of least-cost solution
• m - maximum depth of the state space

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Implementing Search Strategies

• Uninformed search
• Search does not depend on the nature of solution
• Systematic Search Method
• Breadth-First Search
• Depth-First Search (backtracking)
• Depth-limited Search
• Uniform Cost Search
• Iterative deepening Search
• Informed or Heuristic Search
• Best-first Search
• Greedy search (h only)
• A search (g h)
• Iterative A search

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X-First Search Algorithm
yes
Expand n. Put its successors to OPEN
yes
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Comparison of BFS and DFS

• Selection strategy from OPEN
• BFS by FIFO Queue
• DFS by LIFO - stack
• BFS always terminate if goal exist
• cf. DFS on locally finite infinite tree
• Guarantee shortest path to goal - BFS
• Space requirement
• BFS - Exponential
• DFS - Linear,
• keep children of a single node
• Which is better ? BFS or DFS ?

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Depth-limited Search

• depth-first search with depth limit
• Nodes at depth have no successors

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Uniform Cost Search

• A Generalized version of Breadth-First Search
• C(ni, nj) cost of going from ni to nj
• g(n) (tentative minimal) cost of a path from s
  to n.
• Guarantee to find the minimum cost path
• Dijkstra Algorithm

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Uniform Cost Search Algorithm
yes
yes
n goal ?
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Iterative Deepening Search

• Compromise of BFS and DFS
• Iterative Deepening Search depth first search
  with depth-limit increment
• Save on Storage, guarantee shortest path
• Additional node expansion is negligible
• Can you apply this idea to uniform cost search ?

proc Iterative_Deeping_Search(Root) begin
Success 0 for (depth_bound 1
depth_bound Success 1) depth_first_search
(Root, depth_bound) if goal found, Success
1 end
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Iterative Deeping (l0)
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Iterative Deeping (l1)
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Iterative Deeping (l2)
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Iterative Deeping (l3)
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Properties of IDS

• Complete ??
• Time Complexity
• db1 (d-1)b2 bd O(bd)
• Bottom level generated once, top-level generated
  dth
• Space Complexity O(bd)
• Optimal ?? Yes, if step cost 1
• Can be modified to explore uniform cost tree ?
• Numerical comparison of speed ( of node
  expanded)
• b10 and d5, solution at far right
• N(IDS) 50 400 3,000 20,000 100,000
  123,450
• N(BFS) 10 100 1000 10000 100000
  999,990 1,111,100

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Repeated States

• Failure to detect repeated states can turn a
  linear problem into an exponential one !
• Search on Graph

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Summary of Algorithms
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Informed Search
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8-Puzzel Heuristics

• Which is the best move among a, b, c ?

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Road Map Problem

• To go Bucharest, which city do you choose to
  visit next from Arad? Zerind, Siblu, Timisoara?
• Your rationale ?

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Best-First Search

• Idea use of evaluation function for each node
• Estimate of desirability
• We use notation f( )
• Special cases depending on f()
• Greedy search
• Uniform cost search
• A search algorithm

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Best First Search Algorithm( for tree search)
yes
yes
n goal ?
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Generic form of Best-First Search

• Best-First Algorithm with f(n) g(n) h(n)
• where
• g(n) cost of n from start to node n
• h(n) heuristic estimate of the cost from n to
  a goal
• h(G) 0, h(n) gt 0
• Also called A algorithm
• Uniform Cost algorithm
• When h(n) 0 always
• Greedy Search Algorithm
• When g(n) 0 always
• Expands the node that appears to be close to goal
• Complete ??
• Local optimal state
• Greedy may not lead to optimality
• Algorithm A if h(n) lt h(n)

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Examples of Admissible heuristics

• h(n) lt h(n) for all n
• Air-distance never overestimate the actual road
  distance
• 8-tile problem
• Number of misplaced tiles
• Sum of Manhattan distance
• TSP
• Length of spanning tree

Current node
Goal node
1
3
3
6
2
1
8
4
4
8
2
5
5
6
7
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Number of misplaced tiles 4 Sum of Manhattan
distance 1 2 0 0 0 2 0 1
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Algorithm A
Start node

• f(n) g(n) h(n), when h(n) lt h(n) for all n
• Find minimum f(n) to expand next
• Role of h(n)
• Direct to goal
• Role of g(n)
• Guard from roaming due to not perfect heuristics

g(n)
Current Node n
h(n)
Goal
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A Search Example
Romania with cost in km
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Nodes not expanded (node pruning)
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Algorithm A is admissible

• Suppose some suboptimal goal G2 has been
  generated and is in the OPEN. Let n be an
  unexpended node on a shortest path to an oprimal
  goal G1
• f(G2) g(G2) since h(G2) 0
• gt g(G1) sinee G2 is suboptimal
• gt f(n) since h is admissible
• Since f(G2) gt f(n), A will bever select G2 for
  expansion

start
n
G2
G1
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A expands nodes in the order of increasing f
value

• Gradually adds f-contours of nodes (cf.
  breadth-first adds layers)
• Contour i has all nodes with f fi, where fi lt
  fi1
• f(n) g(n) h(n) lt C will be expanded
  eventually
• A terminate even in locally finite graph
  completeness

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Monotonicity (consistency)

• A heuristic function is monotone if
• for all states ni and nj suc(ni)
• h(ni) - h(nj) cost(ni,nj)
• and h(goal) 0
• Monotone heuristic is admissible

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Uniform Cost Searchs f-contours
Start
Goal
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As f-contours
Start
Goal
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Greedys f-contours
Start
Goal
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More Informedness (Dominate)

• For two admissible heuristic h1 and h2, h2 is
  more informed than h1 if
• h1(n) h2(n) for all n
• for 8-tile problem
• h1 of misplaced tile
• h2 sum of Manhattan distance
• Combining several admissible heuristics
• h(n) max h1(n), , hn(n)

h(n)
h1(n)
h2(n)
0
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Semi-admissible heuristics andRisky heuristics

• If h(n) h(n)(1 e) , C(n)
  (1e) C(n)
• Small cost sacrifice save a lot in search
  computation
• Semi-admissible heuristics saves a lot in
  difficult problems
• In case when costs of solution paths are quite
  similar
• e-admissible
• Use of non-admissible heuristics with risk
• Utilize heuristic functions which are admissible
  in the most of cases
• Statistically obtained heuristics

PEARL, J., AND KIM, J. H. Studies in
semi-admissible heuristics. IEEE Trans. PAMI-4, 4
(1982), 392-399
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Dynamic Use of Heuristics

• f(n) g(n) h(n) e1-d(n)/N h(n)
• d(n) depth of node n
• N (expected) depth of goal
• At shallow level depth first excursion
• At deep level assumes admissibility
• Modify Heuristics during search
• Utilize information obtained in the of search
  process to modify heuristics to use in the later
  part of search

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Inventing Admissible Heuristics Relaxed
Problems

• An admissible heuristics is exact solution of
  relaxed problem
• 8-puzzle
• Tile can jump number of misplaced tiles
• Tile can move adjacent cell even if that is
  occupied - Manhattan distance heuristic
• Automatic heuristic generator ABSOLVER
  (Prieditis, 1993)
• Traveling salesperson Problem
• Cost of Minimum spanning tree lt Cost of TSP tour
• Minimum spanning tree can be computed O(n2)

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Inventing Admissible Heuristics SubProblems

• Solution of subproblems
• Take Max of heuristics of sub-problems in the
  pattern database
• 1/ 1000 of nodes are generated in 15 puzzle
  compared with Manhattan heuristic
• Disjoint sub-problems
• 1/ 10,000 in 24 puzzle compared with Manhattan

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Iterative Deeping A

• Iterative Deeping version of A
• use threshold as depth bound
• To find solution under the threshold of f(.)
• increase threshold as minimum of f(.) of
• previous cycle
• Still admissible
• same order of node expansion
• Storage Efficient practical
• but suffers for the real-valued f(.)
• large number of iterations

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Iterative Deepening A Search Algorithm ( for
tree search)
set threshold as h(s)
yes
threshold min( f(.) , threshold )
yes
n goal ?
Expand n. calculate f(.) of successor if f(suc) lt
threshold then Put successors to OPEN if
pointers back to n
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Memory-bounded heuristic Search

• Recursive best-first search
• A variation of Depth-first search
• Keep track of f-value of the best alternative
  path
• Unwind if f-value of all children exceed its best
  alternative
• When unwind, store f-value of best child as its
  f-value
• When needed, the parent regenerate its children
  again.
• Memory-bounded A
• When OPEN is full, delete worst node from OPEN
  storing f-value to its parent.
• The deleted node is regenerated when all other
  candidates look worse than the node.

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Performance Measure

• Penetrance
• how search algorithm focus on goal rather than
  wander off in irrelevant directions
• P L / T
• L depth of goal
• T total number of node expanded
• Effective Branching Factor (B)
• B B2 B3 ..... BL T
• less dependent on L

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Local Search and Optimization
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Local Search is irrevocable

• Local search irrevocable search
• less memory required
• Reasonable solutions in large (continuous) space
  problems
• Can be formulated as Searching for extreme value
  of Objective function
• find i ARGMAX Obj(pi)
• where pi is parameter

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Search for Optimal Parameter

• Deterministic Methods
• Step-by-step procedure
• Hill-Climbing search, gradient search
• ex error back propagation algorithm
• Finding Optimal Weight matrix in Neural Network
  training
• Stochastic Methods
• Iteratively Improve parameters
• Pseudo-random change and retain it if it improves
• Metroplis algorithm
• Simulated Annealing algorithm
• Genetic Algorithm

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Hill Climbing Search

• 1. Set n to be the initial node
• 2. If obj(n) gt max obj(childi(n)) then exit
• 3. Set n to be the highest-value child of n
• 4. Return to step 2
• No previous state information
• No backtracking
• No jumping
• Gradient Search
• Hill climbing with continuous, differentiable
  functions
• step width ?
• Slow in near optimal

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State space landscape
Real World
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Hill-climbing Drawbacks

• Local maxima
• At Ridge
• Stray in Plateau
• Slow in Plateau
• Determination of proper Step size
• Cure
• Random restart
• Good for Only few local maxima

Global Maximum
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Local Beam Search

• Keep track of best k states instead of 1 in
  hill-climbing
• Full utilization of given memory
• Variation Stochastic beam search
• Select k successors randomly

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Iterative Improvement Algorithm

• Basic Idea
• Start with initial setting
• Generate a random solution
• Iteratively improve the quality
• Good For hard, practical problems
• Because it keeps current state only and No
  look-ahead beyond neighbors
• Advanced Implementation
• Metropolis algorithm
• Simulated Annealing algorithm
• Genetic algorithm

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Metropolis algorithm

• A Monte Carlo method (statistical simulation with
  random number)
• objective is to reach state minimizing energy
  function
• Instead always going downhill, try to go downhill
  most of the time
• Escape local minima by allowing some bad moves
• 1. Randomly generate a new state, Y, from state X
• 2. If ?E(energy difference between Y and X) lt 0
• then move to Y (set Y to X) and goto 1
• 3. Else
• 3.1 select a random number, ?
• 3.2 if ? lt exp(- ?E / T)
• then move to Y (set Y to X) and goto 1
• 3.3 else goto 1

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From Statistical Mechanics

• In thermal equilibrium, probability of state i
• energy of state i
• absolute temperature
• Boltzman constant
• In NN
• define

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Probability of State Transition by DE
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Simulated Annealing

• Simulates slow cooling of annealing process
• Applied for combinatorial optimization problem by
  S. Kirkpatric (83)
• To overcome local minima problem
• Escape local minima by allowing some bad moves
  but gradually decrease their size and frequency
• Widely used in VLSI layout and airline
  scheduling, etc.

• What is annealing?
• Process of slowly cooling down a compound or a
  substance
• Slow cooling let the substance flow around ?
  thermodynamic equilibrium
• Molecules get optimum conformation

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Simulated Annealing algorithm

• function Simulated-Annealing(problem, schedule)
  returns a solution state
• inputs problem, a problem
• local variables current, a node
• next, a node
• T, a temperature controlling the probability
  of downward steps
• current ? Make-Node(Initial-Stateproblem)
• for t?1 to infinity do
• T ? schedulet
• if T0 then return current
• next ? a randomly selected successor of current
• DE ? Valuenext Valuecurrent
• if DEgt0 then current?next
• else current?next only with probability eDE/T

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Simulated Annealing parameters

• Temperature T
• Used to determine the probability
• High T large changes
• Low T small changes
• Cooling Schedule
• Determines rate at which the temperature T is
  lowered
• Lowers T slowly enough, the algorithm will find a
  global optimum
• In the beginning, aggressive for searching
  alternatives, become conservative when time goes
  by

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Simulated Annealing Cooling Schedule
T0
T(t)
Tf
t

• if Ti is reduced too fast, poor quality
• if Tt gt T(0) / log(1t) - Geman
• System will converge to minimun configuration
• Tt k/1t - Szu
• Tt a T(t-1) where a is in between 0.8 and 0.99

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Tips for Simulated Annealing

• To avoid of entrainment in local minima
• Annealing schedule by trial and error
• Choice of initial temperature
• How many iterations are performed at each
  temperature
• How much the temperature is decremented at each
  step as cooling proceeds
• Difficulties
• Determination of parameters
• If cooling is too slow ?Too much time to get
  solution
• If cooling is too rapid ? Solution may not be the
  global optimum

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Iterative algorithm comparison

• Simple Iterative Algorithm
• 1. find a solution s
• 2. make s, a variation of s
• 3. if s is better than s, keep s as s
• 4. goto 2
• Metropolis Algorithm
• 3 if (s is better than s) or ( within Prob),
  then keep s as s
• With fixed T
• Simulated Annealing
• T is reduced to 0 by schedule as time passes

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Simulated Annealing Local Maxima