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

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Gurantee shortest path to goal - BFS. Space requirement. BFS - Exponential ... Save on Storage, guarantee shortest path. Additional node expansion is negligible ... – PowerPoint PPT presentation

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


1
Problem Solving by Search by Jin Hyung
Kim Computer Science Department KAIST
2
Example of Representation
  • Euler Path

3
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

4
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

5
Examples of SSR
  • TIC_TAC_TOE
  • n2-1 Puzzle
  • Traveling salesperson problem (TSP)

6
Search Strategies
  • A strategy is defined by picking the order of
    node expansion
  • Search Direction s
  • 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

7
Evaluation of 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 ?
  • 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 (may be

8
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

9
X-First Search Algorithm
yes
Expand n. Put successors at the end of
OPEN pointers back to n
yes
10
Comparison of BFS and DFS
  • BFS always terminate if goal exist
  • cf. DFS on locally finite infinite tree
  • Gurantee shortest path to goal - BFS
  • Space requirement
  • BFS - Exponential
  • DFS - Linear,
  • keep children of a single node
  • Which is better ? BFS or DFS ?

11
Uniform Cost Search
  • A Genaralized 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

12
Uniform Cost Search Algorithm
yes
yes
n goal ?
13
Iterative Deepening Search
  • Compromise of BFS and DFS
  • Save on Storage, guarantee shortest path
  • Additional node expansion is negligible

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
14
Iterative Deeping (l0)
15
Iterative Deeping (l1)
16
Iterative Deeping (l2)
17
Iterative Deeping (l3)
18
Properties of IDS
  • Complete ??
  • Time ??
  • (d1)b0 db1 (d-1)b2 … bd O(bd)
  • Space ?? O(bd)
  • Optimal ?? Yes, if step cost 1
  • Can be modified to explore uniform cost tree ?
  • Numerical comparison 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

19
Informed Search
20
Use of Heuristics to select the Best
  • Tic-tac-toe

21
Tic-tac-toe
  • Most-Win Heuristics

x
x
x
2 win
3 win
4 win
22
8-Puzzel Heuristics
1
2
3
3
2
1
8
4
4
8
5
6
5
6
7
7
  • of Misplaced tiles
  • Sum of Manhattan distance

3
2
3
8
2
2
3
8
1
1
4
4
4
1
8
6
7
7
5
6
7
5
5
6
a
c
b
23
Heuristics for Road Map Problem
24
Best First Search Algorithm( for tree search)
yes
yes
n goal ?
25
Algorithm A
  • 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
  • Algorithm is admissible if it terminate with
    optimal solution
  • What if f(n) f(n) where f(n) g(n) h(n)
  • where g(n) shortest cost to n
  • h(n) shortest actual cost from n to goal

26
Algorithm A (Branch and Bound method)
  • Algorithm A becomes A if h(n) h(n)
  • Algorithm A is admissible
  • can you prove it ?
  • If h(n) 0, A algorithm becomes uniform cost
    algorithm
  • Uniform cost algorithm is admissible
  • If n is on optimal path, f(n) C
  • f(n) gt C implies that n is not on optimal path
  • A terminate in finite graph

27
Examples of Admissible Heuristics
  • 8 puzzle heuristic
  • N queen problem, Tic-tac-toe
  • Air distance heuristic
  • Traveling Salesperson Problem
  • Minimum spanning tree heuristics
  • ….

28
Iterative Deeping A
  • Modification 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

29
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
30
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.

31
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

32
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

h(n)
h1(n)
h2(n)
0
33
Generation of Heuristics
  • Relaxed problem solution is an admissible
    heuristics
  • Manhattan distance heuristic
  • Solution of subproblems
  • combining several admissible heuristics
  • h(n) max h1(n), …, hn(n)
  • Use of Pattern databases
  • Max of heuristics of sub-problem pattern database
  • 1/ 1000 in 15 puzzle compared with Manhattan
  • Addition of heuristics of disjoint sub-problem
    pattern database
  • 1/ 10,000 in 24 puzzle compared with Manhattan
  • disjoint subdivision is not possible for Rubics
    cube

34
Semi-addmissible heuristics Dynamic Weighting,
Risky heuristics
  • If h(n) h(n) e , then C(n)
    (1e) C(n)
  • f(n) g(n) h(n) e1-d(n)/N h(n)
  • At shallow level depth first excursion
  • At deep level assumes admissibility
  • 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
35
Performance Measure
  • Penetrance
  • how search algorithm focus on goal rather than
    wander off in irrelevant directions
  • P L / T
  • Effective Branching Factor (B)
  • B B2 B3 ..... BL T
  • less dependent on L

36
Planning Monkey and Banana
  • Monkey is on floor at (x1, y1), banana is hanging
    at (x2, y2), and box is at (x3, y3). Monkey can
    grab banana if he push box under banana and climb
    on it. Develop a state-space search
    representation for this situation and show how
    monkey can grab banana.

37
Local Search and Optimization
  • Local 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

38
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

39
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

40
State space landscape
Real World
41
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
42
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

43
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
  • Implementation
  • Metropolis algorithm
  • Simulated Annealing algorithm
  • Genetic algorithm

44
Metropolis algorithm
  • Modified Monte Carlo method
  • Suppose our objective is to reach the state
    minimizing energy function
  • 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

45
From Statistical Mechanics
  • In thermal equilibrium, probability of state i
  • energy of state i
  • absolute temperature
  • Boltzman constant
  • In NN
  • define

46
Simulated AnnealingProbability distribution
47
Simulated Annealing algorithm
  • 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

contraction cause stress
48
Simulated Annealing
  • Simulates slow cooling of annealing process
  • Solves combinatorial optimization
  • variant of Metropolis algorithm
  • by S. Kirkpatric (83)
  • finding minimum-energy solution of a neural
    network finding low temperature state of
    physical system
  • To overcome local minimum problem
  • Instead always going downhill, try to go downhill
    most of the time

49
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

50
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

51
Simulated Annealing Schedule example
  • 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

52
Simulated Annealing parameters
  • Temperature T
  • Used to determine the probability
  • High T large changes
  • Low T small changes
  • 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

53
Simulated Annealing(10)
  • 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

54
Simulated Annealing Local Maxima
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