Informed Search Methods Copyright, 1996 © Dale Carnegie & Associates, Inc. Chapter 4 Spring 2005

CSE 471/598, CBS 598 by H. Liu2 What we’ll learn Informed search algorithms are more efficient in most cases What are informed search methods How to use problem-specific knowledge How to optimize a solution

CSE 471/598, CBS 598 by H. Liu3 Best-First Search Evaluation function gives a measure about which node to expand Minimizing estimated cost to reach a goal Greedy search at node n heuristic function h(n)  an example is straight-line distance (Fig 4.1) The simple Romania map Finding the route using greedy search – example (Fig 4.2)

CSE 471/598, CBS 598 by H. Liu4 Best-first search (2) h(n) is independent of the path cost g(n) Minimizing the total path cost f(n) = g(n) + h(n) estimated cost of the cheapest solution thru n Admissible heuristic function h never overestimates the cost optimistic

CSE 471/598, CBS 598 by H. Liu5 A* search How it works (Fig 4.3) Characteristics of A* Monotonicity (Consistency) - h is nondescreasing  How to check – using triangle inequality Tree-search to ensure monotonicity Contours (Fig 4.4) - from circle to oval (ellipse) Proof of the optimality of A* The completeness of A* (Fig 4.4 Contours) Complexity of A* (time and space) For most problems, the number of nodes within the goal contour search space is till exponential in the length of the solution

CSE 471/598, CBS 598 by H. Liu6 Different Search Strategies Uniform-cost search minimize the path cost so far Greedy search minimize the estimated path cost A* minimize the total path cost Time and space issues of A*  Designing good heuristic functions  A* usually runs out of space long before it runs out of time

CSE 471/598, CBS 598 by H. Liu7 Heuristic Functions An example (the 8-puzzle, Fig 4.7) How simple can a heuristic be?  The distance to its correct position  Using Manhattan distance What is a good heuristic? Effective branching factor - close to 1 (Why?) Value of h  not too large - must be admissible (Why?)  not too small - ineffective (oval to circle) (expanding all nodes with f (n) < f*) Goodness measures - no. of nodes expanded and branching factor (Fig 4.8)

CSE 471/598, CBS 598 by H. Liu8 Domination translates directly into efficiency Larger h means smaller branching factor If h2 >= h1, is h2 always better than h1?  Proof? (h1 <= h2 <= C* - g) Inventing heuristic functions Working on relaxed problems  remove some constraints

CSE 471/598, CBS 598 by H. Liu9 8-puzzle revisited Definition: A tile can move from A to B if A is horizontally or vertically adjacent to B and B is blank Relaxation by removing one or both the conditions A tile can move from A to B if A ~ B A tile can move from A to B if B is blank A tile can move from A to B Deriving a heuristic from the solution cost of a sub-problem Fig 4.9

CSE 471/598, CBS 598 by H. Liu10 If we have admissible h 1 … h m and none dominates, we can have for node n h = max(h 1, …, h m ) Feature selection and combination use only relevant features  “number of misplaced tiles” as a feature The cost of heuristic function calculation <= the cost of expanding a node otherwise, we need to rethink. Learning heuristics from experience Each optimal solution to 8-puzzle provides a learning example

CSE 471/598, CBS 598 by H. Liu11 Improving A* - memory-bounded heuristic search Iterative-deepening A* (IDA*) Using f-cost(g+h) rather than the depth Cutoff value is the smallest f-cost of any node that exceeded the cutoff on the previous iteration; keep these nodes only Space complexity O(bd) Recursive best-first search (RBFS) Best-first search using only linear space (Fig 4.5) It replaces the f-value of each node along the path with the best f- value of its children (Fig 4.6) Space complexity O(bd) Simplified memory bounded A* (SMA*) IDA* and RBFS use too little memory – excessive node regeneration Expanding the best leaf until memory is full Dropping the worst leaf node (highest f-value) by backing up to its parent

CSE 471/598, CBS 598 by H. Liu12 Local Search Algorithms and Optimization Problems Global and local optima Fig 4.10, from current state to global maximum Hill-climbing (maximization) Well know drawbacks (Fig 4.13)  Local maxima, Plateaus, Ridges Random-restart Simulated annealing Gradient descent (minimization) Escaping the local minima by controlled bouncing Local beam search Keeping track of k states instead of just one Genetic algorithms Selection, cross-over, and mutation

CSE 471/598, CBS 598 by H. Liu13 Online Search Offline search – computing a complete solution before acting Online search – interleaving computation and action Solving an exploration problem where the states and actions are unknown to the agent Good for domains where there is a penalty for computing too long, or for stochastic domains An example – robot is placed in a new building: explore it to build a map that it can use for getting A to B

CSE 471/598, CBS 598 by H. Liu14 Online search problems An agent knows: e.g., Fig 4.18 Actions(s) in state s Step-cost function c(s,a,s’)  c() cannot be used until the agent knows s’ is the outcome  In order to know c(), a must be actually tried Goal-Test(s) Others: with memory of states visited, and admissible heuristic from current state to the goal state Objective: Reaching a goal state while minimizing cost

CSE 471/598, CBS 598 by H. Liu15 Measuring its performance Competitive ratio: the true path cost over the path cost if it knew the search space in advance The best achievable competitive ratio can be 1 If some actions are irreversible, it may reach a dead-end ( Fig 4.19 (a)) An adversary argument – Fig 4.19 (b) No bounded competitive ratio can be guaranteed if there are paths of unbounded cost

CSE 471/598, CBS 598 by H. Liu16 Online search agents It can expand only a node that it physically occupies, so it should expand nodes in a local order Online Depth-First Search (Fig 4.20) Backtracking requires that actions are reversible Hill-climbing search keeps one current state in memory It can get stuck in a local minimum Random restart does not work here Random walk selects at random one of the available actions from the current state  It can be very slow, Fig 4.21 Augmenting hill climbing with memory rather than randomness is more effective  Learning real-time agent, Fig 4.22 H(s) is updated as the agent gains experience Encourages to explore new paths

CSE 471/598, CBS 598 by H. Liu17 Summary Heuristics are the key to reducing research costs f(n) = g(n)+h(n) Understand their variants A* is complete, optimal, and optimally efficient among all optimal search algorithms, but... Iterative improvement algorithms are memory efficient, but... Local search There is a cost associated withit Online search is different from offline search Mainly for exploration problems