CS 430 Lecture 9 - uenics.evansville.eduuenics.evansville.edu/~hwang/s17-courses/cs430/... · Monday, January 30 CS 430 Artificial Intelligence - Lecture 9 5 Iterative-Deepening Depth-First

Monday, January 30 CS 430 Artificial Intelligence - Lecture 9 1

Lecture 9

Questions?


Outline

Chapter 3 - Solving Problems by Searching Uninformed Search Strategies

Depth-limited Search Iterative-deepening Search

Informed (Heuristic) Search Strategies Greedy Best-First Search A* Search


Recall: Depth-First Search (DFS)

Depth-first search expands the deepest node in the frontier. It searches down a path to the leaf nodes, then "backs up" to the next deepest unexplored node.

Frontier is implemented using a stack. Tree-Search is not complete, but Graph-Search

is complete. Time complexity is O(bm), but space complexity is O(bm). (Note: m the maximum depth, rather than d, depth of shallowest node.)


Depth-Limited Search (DLS)

Depth-limited search (DLS) alleviates the problem of infinite paths in DFS tree search by limiting the depth of the search.

All nodes at the depth limit l are treated as if they have no successors. When this happens a cutoff failure occurs for that branch.

Time complexity becomes O(bl) and space complexity becomes O(bl). Still not complete or optimal.


Iterative-DeepeningDepth-First Search (IDS)

Iterative-deepening search combines the characteristics of DFS and BFS. Call DLS with depths from 0 to .

Explores frontier like BFS, but with time and space complexity like DFS.


Iterative-DeepeningDepth-First Search (IDS)


Comparison of UninformedSearch Strategies

Criterion BFS UCS DFS DLS IDS

Complete? Yes Yes No No YesOptimal? Sometimes Yes No No SometimesTime O(bd) O(b1+C*/ε) O(bm) O(bl) O(bd)

Space O(bd) O(b1+C*/ε) O(bm) (O(bl) O(bd)

When using Tree-Search. Graph-Search makes DFS complete for finite

spaces and space and time complexity bounded by size of state space.


Informed Search Strategies

Previous search algorithms generated new states from problem definition data only.

If search uses problem-specific data beyond the problem definition can do an informed search much more efficiently.

General approach is a best-first strategy that bases selection on an evaluation function, traditionally called f(n), that is construed as a cost estimate.


Informed Search Strategies

Choice of f(n) determines the search strategy, and most best-first algorithms include a heuristic function, denoted h(n), where

h(n) = estimated cost of cheapest path from the state at node n to the goal state

For now, h(n) is an arbitrary, non-negative, problem specific function with constraint that h(n) = 0 when n is a goal node.


Greedy Best-First Search

Greedy best-first search expands the node that is believed to be the closest to the goal node by using f(n) = h(n).

E.g., for the Romanian map problem, can use the straight-line distance heuristic, denoted h

SLD.

Straight-line distanceto Bucharest


Example: Romanian Map

Choose Sibiu (253), then Faragas (176), then Bucharest (0), a goal node.

Start with Arad and expand to Sibiu (253), Timisoara (329), and Zerind (374).


Greedy Best-First Search

Note that the value of hSLD

cannot be computed

from the problem description.

Also, it takes experience to know that hSLD

is

correlated to actual road distances, and therefore a useful heuristic function.

Evaluation: Complete? Optimal? Time? Space?


Map of Romania


A* Search

Mostly widely known form of best-first search is called A* (pronounced "A-star") search. It uses

f(n) = g(n) + h(n)

where g(n) is the (actual) path cost to reach node n and h(n) is the estimated cost from node n to the goal, making f(n) the estimated cost of the cheapest solution through n.

Expand the node with the lowest f(n). Note if h(n) = 0 for all nodes, this is the same as uniform-cost search.



Choose Sibiu, then Rimnicu Vilcea (413=220+ 193), then Faragas (415 =239+176).

Start with Arad and expand to Sibiu (393=140+253), Timisoara (447=118+329), and Zerind (449=75+374).



Choose Pitesti, then Bucharest (418=418+0), a goal node.

Faragas expands to Sibiu and Bucharest, but next lowest node is Pitesti (417=317+100).


A* Search

Time? Space? Complete? Optimal?


Optimality of A* Search

For certain heuristic functions, A* search is both complete and optimal using Graph-Search.

First, h(n) must be an admissible heuristic, meaning that it never overestimates the cost to reach the goal.

E.g., hSLD

is admissible, since the straight-line

distance is the minimum distance between two points, it can never be shorter than the actual road distance between two cities.



Second, h(n) must be consistent, meaning that for every node n and every successor n' of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus the estimated cost of reaching the goal from n'. I.e.,

h(n) <= c(n, a, n') + h(n')

This a form of the triangle inequality: a side of a triangle cannot be longer than the sum of the other two sides.



Proof of A* optimality with consistent h(n): If h(n) is consistent, then the values of f(n)

along any path are non-decreasing. This follows directly from the definition of

consistency. Suppose n' is a successor to n, then g(n') = g(n) + c(n, a, n') for some action a, giving

f(n') = g(n') + h(n') = g(n) + c(n, a, n') + h(n') >= g(n) + h(n) = f(n)



Whenever A* selects a node n for expansion, the optimal path to that node has been found.

Prove this by contradiction: If this was not the case, there would have to be another frontier node n' on the optimal path from the start node to n by the graph separation property of Graph-Search. Because f is non-decreasing along any path, n' would have a lower f-cost than n and would have been selected first.



Thus the sequence of nodes expanded by A* is in non-decreasing order of f(n).

Hence the first goal node selected for expansion must be an optimal solution, because f(n) is the true cost (h(n) = 0) and all later nodes will be at least as expensive.



Then A* expands some nodes on the goal contour (f(n) = C*) before selecting a goal node.

Can draw this as contours in the search space. For example, in the Romanian map, the contour labeled 400 encloses all nodes that have f(n) <= 400.

A* expands all nodes with f(n) < C*



Note: A* expands no nodes with f(n) > C*, effectively pruning away parts of the search space without having to examine them.

Among algorithms that extend search paths from the root and use the same heuristic function, A* is optimally efficient, meaning that no other optimal algorithm is guaranteed to expand fewer nodes than A*.

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CS 430 Lecture 9 - uenics.evansville.eduuenics.evansville.edu/~hwang/s17-courses/cs430/... · Monday, January 30 CS 430 Artificial Intelligence - Lecture 9 5 Iterative-Deepening Depth-First