chapter 4 informed search algorithms 1. outline best-first search greedy best-first search a *...
TRANSCRIPT
![Page 1: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/1.jpg)
1
CHAPTER 4
Informed search algorithms
![Page 2: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/2.jpg)
2
Outline
Best-first searchGreedy best-first searchA* searchHeuristicsLocal search algorithmsHill-climbing searchSimulated annealing searchLocal beam searchGenetic algorithms
![Page 3: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/3.jpg)
3
Review: Tree search
A search strategy is defined by picking the order of node expansion
![Page 4: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/4.jpg)
4
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 A* search
![Page 5: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/5.jpg)
5
Romania with step costs in km
![Page 6: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/6.jpg)
6
Greedy best-first search
Evaluation function f(n) = h(n) (heuristic) = estimate of cost from n to goale.g., hSLD(n) = straight-line distance from n to Bucharest
Greedy best-first search expands the node that appears to be closest to goal
![Page 7: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/7.jpg)
7
Greedy best-first search example
![Page 8: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/8.jpg)
8
Greedy best-first search example
![Page 9: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/9.jpg)
9
Greedy best-first search example
![Page 10: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/10.jpg)
10
Greedy best-first search example
![Page 11: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/11.jpg)
11
Properties of greedy best-first search
Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt
Time? O(bm), but a good heuristic can give dramatic improvement
Space? O(bm) -- keeps all nodes in memoryOptimal? No
![Page 12: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/12.jpg)
12
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 nh(n) = estimated cost from n to goalf(n) = estimated total cost of path through n
to goal
![Page 13: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/13.jpg)
13
A* search example
![Page 14: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/14.jpg)
14
A* search example
![Page 15: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/15.jpg)
15
A* search example
![Page 16: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/16.jpg)
16
A* search example
![Page 17: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/17.jpg)
17
A* search example
![Page 18: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/18.jpg)
18
A* search example
![Page 19: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/19.jpg)
19
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
![Page 20: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/20.jpg)
20
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) > g(G) since G2 is suboptimal f(G) = g(G) since h(G) = 0 f(G2) > f(G) from above
![Page 21: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/21.jpg)
21
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) > 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) > f(n), and A* will never select G2 for expansion
![Page 22: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/22.jpg)
22
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 havef(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
![Page 23: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/23.jpg)
23
Optimality of A*
A* expands nodes in order of increasing f value Gradually adds "f-contours" of nodes Contour i has all nodes with f=fi, where fi < fi+1
![Page 24: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/24.jpg)
24
Properties of A*
Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )
Time? ExponentialSpace? Keeps all nodes in memoryOptimal? Yes
![Page 25: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/25.jpg)
Heuristic Functions
![Page 26: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/26.jpg)
Heuristics and A* Search
The heuristic function h(n) tells A* an estimate of the minimum cost from any vertex n to the goal.
The heuristic can be used to control A*’s behavior. If h(n) is 0, then only g(n) plays a role, and A* turns into
Dijkstra’s algorithm If h(n) is always lower than (or equal to) the cost of moving
from n to the goal, then A* is guaranteed to find a shortest path.
If h(n) is exactly equal to the cost of moving from n to the goal, then A* will only follow the best path and never expand anything else, making it very fast.
![Page 27: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/27.jpg)
Heuristics and A* Search
The heuristic can be used to control A*’s behavior. If h(n) is sometimes greater than the cost of moving
from n to the goal, then A* is not guaranteed to find a shortest path, but it can run faster.
At the other extreme, if h(n) is very high relative to g(n), then only h(n) plays a role, and A* turns into Greedy Best-First-Search.
![Page 28: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/28.jpg)
Heuristics and A* Search
Suppose your game has two types of terrain, Flat and Mountain
Movement costs are 1 for flat land and 3 for mountains
A* is going to search three times as far along flat land as it does along mountainous land
You can speed up A*’s search by using 1.5 as the heuristic distance between two map spaces.
Alternatively, you can speed up up A*’s search by decreasing the amount it searches for paths around mountains―just tell A* that the movement cost on mountains is 2 instead of 3.
![Page 29: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/29.jpg)
Heuristics
Heuristic (/hjʉˈrɪstɨk/; Greek: "Εὑρίσκω", "find" or "discover") refers to experience-based techniques for problem solving, learning, and discovery that find a solution, which is not guaranteed to be optimal, but good enough for a given set of goals. (Wikipedia)
Trial and Error
![Page 30: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/30.jpg)
Polya – How to solve it
If you are having difficulty understanding a problem, try drawing a picture.
If you can't find a solution, try assuming that you have a solution and seeing what you can derive from that ("working backward").
If the problem is abstract, try examining a concrete example.
Try solving a more general problem first (the "inventor's paradox": the more ambitious plan may have more chances of success).
![Page 31: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/31.jpg)
31
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) = ?
![Page 33: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/33.jpg)
33
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) = ? 8h2(S) = ? 3+1+2+2+2+3+3+2 = 18
![Page 34: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/34.jpg)
34
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): d=12 IDS = 3,644,035 nodes
A*(h1) = 227 nodes A*(h2) = 73 nodes
d=24 IDS = too many nodesA*(h1) = 39,135 nodes A*(h2) = 1,641 nodes
![Page 35: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/35.jpg)
35
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
![Page 36: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/36.jpg)
36
Local search algorithms
In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution
State space = set of "complete" configurations
Find configuration satisfying constraints, e.g., n-queens
In such cases, we can use local search algorithms keep a single "current" state, try to improve it
![Page 37: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/37.jpg)
37
Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal
![Page 38: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/38.jpg)
38
Hill-climbing search
"Like climbing Everest in thick fog with amnesia"
![Page 39: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/39.jpg)
39
Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima
![Page 40: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/40.jpg)
40
Hill-climbing search: 8-queens problem
h = number of pairs of queens that are attacking each other, either directly or indirectly
h = 17 for the above state
![Page 41: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/41.jpg)
41
Hill-climbing search: 8-queens problem
• A local minimum with h = 1
![Page 42: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/42.jpg)
42
Simulated annealing search
Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency
![Page 43: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/43.jpg)
43
Properties of simulated annealing search
One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1
Widely used in VLSI layout, airline scheduling, etc
![Page 44: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/44.jpg)
44
Local beam search
Keep track of k states rather than just one
Start with k randomly generated states
At each iteration, all the successors of all k states are generated
If any one is a goal state, stop; else select the k best successors from the complete list and repeat.
![Page 45: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/45.jpg)
45
Genetic algorithms
A successor state is generated by combining two parent states
Start with k randomly generated states (population)
A state is represented as a string over a finite alphabet (often a string of 0s and 1s)
Evaluation function (fitness function). Higher values for better states.
Produce the next generation of states by selection, crossover, and mutation
![Page 46: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/46.jpg)
46
Genetic algorithms
Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28)
24/(24+23+20+11) = 31%23/(24+23+20+11) = 29% etc
![Page 47: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/47.jpg)
47
Genetic algorithms
![Page 48: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/48.jpg)
Online Searching
An online search agent operates by interleaving computation and action. First, it takes action, then it observes the environment and computes the next action.
However, there is a penalty for sitting around and computing too long. This is why we need efficient code.
Usually used in exploration situations.
![Page 49: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/49.jpg)
Offline Search vs. Online Search
Offline Search Knows the “map” of the situation Basically finds the shortest path knowing the whole
layout of the situation Works like a GPS navigation system
Online Search: Doesn’t know the “map” of the situation Has to explore and find out where to go, then
determine the shortest path
![Page 50: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/50.jpg)
Search Patterns
Often, online search agents search using a depth-first search pattern
The search pattern must include whether or not the state space is safely explorable.
Random walk method works, but not very well. It takes exponentially many steps to find the goal.
Hill climbing search is by default an online search method, but only keeps one state in memory and can’t go back. Depth-first is more efficient.
![Page 51: CHAPTER 4 Informed search algorithms 1. Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing](https://reader035.vdocuments.mx/reader035/viewer/2022081418/56649e005503460f94aea318/html5/thumbnails/51.jpg)
References
http://theory.stanford.edu/~amitp/GameProgramming/Heuristics.html
http://buildnewgames.com/astar/http://
www.autonlab.org/tutorials/hillclimb02.pdfhttp://
www.autonlab.org/tutorials/search16.pdf