Download presentation

1
Notes Dijstra’s Algorithm Corrected syllabus

2
**Tree Search Implementation Strategies**

Require data structure to model search tree Tree Node: State (e.g. Sibiu) Parent (e.g. Arad) Action (e.g. GoTo(Sibiu)) Path cost or depth (e.g. 140) Children (e.g. Faragas, Oradea) (optional, helpful in debugging)

3
**Queue Methods: Empty(queue) Pop(queue) Insert(queue, element)**

Returns true if there are no more elements Pop(queue) Remove and return the first element Insert(queue, element) Inserts element into the queue InsertFIFO(queue, element) – inserts at the end InsertLIFO(queue, element) – inserts at the front InsertPriority(queue, element, value) – inserts sorted by value

4
informed Search

5
Search start goal Uninformed search Informed search

6
Informed Search What if we had an evaluation function h(n) that gave us an estimate of the cost associated with getting from n to the goal h(n) is called a heuristic

7
**Romania with step costs in km**

h(n)

8
**Greedy best-first search**

Evaluation function f(n) = h(n) (heuristic) e.g., f(n) = hSLD(n) = straight-line distance from n to Bucharest Greedy best-first search expands the node that is estimated to be closest to goal

9
**Romania with step costs in km**

h(n) f(n)

10
Best-First Algorithm

11
**Performance of greedy best-first search**

Complete? Optimal?

12
**Failure case for best-first search**

13
**Performance of greedy best-first search**

Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt Optimal? No

14
**Complexity of greedy best first search**

Time? O(bm), but a good heuristic can give dramatic improvement Space? O(bm) -- keeps all nodes in memory

15
What can we do better?

16
**A* search Ideas: Avoid expanding paths that are already expensive**

Consider Cost to get here (known) – g(n) Cost to get to goal (estimate from the heuristic) – h(n)

17
**A * Evaluation functions**

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 start goal n g(n) h(n) f(n)

18
n g(n) h(n) f(n)

20
**A* 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)

21
**What happens if heuristic is not admissible?**

Will still find solution (complete) But might not find best solution (not optimal)

22
**Properties of A* (w/ admissible heuristic)**

Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) ) Optimal? Yes Time? Exponential, approximately O(bd) in the worst case Space? O(bm) Keeps all nodes in memory

23
**The heuristic h(x) guides the performance of A***

Let d(x) be the actual distance between S and G h(x) = 0 : A* is equivalent to Uniform-Cost Search h(x) <= d (x) : guarantee to compute the shortest path; the lower the value h(x), the more node A* expands h(x) = d (x) : follow the best path; never expand anything else; difficult to compute h(x) in this way! h(x) > d(x) : not guarantee to compute a best path; but very fast h(x) >> g(x) : h(n) dominates -> A* becomes the best first search

24
**Admissible heuristics**

25
**Admissible heuristics**

E.g., for the 8-puzzle:

26
**Admissible heuristics**

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

27
**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) = ? = 18 Which is better?

28
**Dominance If h2(n) ≥ h1(n) for all n (both admissible)**

then h2 dominates h1 h2 is better for search What does better mean? All searches we’ve discussed are exponential in time

29
**Comparison of algorithms (number of nodes expanded)**

Iterative deepening A*(teleporting tiles) A* (manhattan distance) 2 10 6 112 13 12 680 20 18 364035 227 73 14 539 113 1.8 * 108 3056 363 24 8.6 * 1010 39135 1641

30
Visually

31
**Where do heuristics come from?**

From people Knowledge of the problem From computers By considering a simpler version of the problem Called a relaxation

32
Relaxed problems 8-puzzle 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 Consider the example of straight line distance (Romania navigation) Is that a relaxation?

33
**Iterative-Deepening A* (IDA*)**

Further reduce memory requirements of A* Regular Iterative-Deepening: regulated by depth IDA*: regulated by f(n)=g(n)+h(n)

34
Questions?

Similar presentations

OK

Slides by: Eric Ringger, adapted from slides by Stuart Russell of UC Berkeley. CS 312: Algorithm Design & Analysis Lecture #36: Best-first State- space.

Slides by: Eric Ringger, adapted from slides by Stuart Russell of UC Berkeley. CS 312: Algorithm Design & Analysis Lecture #36: Best-first State- space.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Science ppt on carbon and its compounds Ppt on low level language definition Ppt on spiritual leadership book Ppt on natural resources for class 11 Ppt on natural resources and conservation in south Ppt on earth day for kindergarten Ppt on seven wonders of india Ppt on loan against property Means of communication for kids ppt on batteries Download ppt on ecotourism in india