Outline: Giải quyết bài toán và tìm kiếm(Problem solving and search, problem solving agents)( g p g g )

Hình thành bài toán (Problem formulation)• Xác định các thành phần của bài toán (problem formulation)• Lược đồ chung để giải bài toán (general-solving procedure)• Đánh giá một giải thuật tìm kiếm

Uninformed (blink) search (Tìm không có thông tin phản hồi, hoặctìm kiếm mù)• Search strategies: depth-first, breadth-first (Các chiến lược tìm kiếm: theo chiềug p , ( ợ

sâu, theo chiều rộng)

Informed search (Tìm kiếm dựa trên các hàm đánh giá, heuristics)• Search strategies: best first search• Search strategies: best-first search• Heuristic functions (Các hàm heuristic)

Uniform searchGreedy searchGreedy searchA* search

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Xem lại giải thuật uninformed searchg

Function General-Search(problem, Queue/Stack) returns a solution, or failureQueue/Stack make-queue(make-node(initial-state[problem]));Queue/Stack make-queue(make-node(initial-state[problem]));father(initial-state[problem]) = empty;while (1)

if Queue/Stack is empty then return failure;if Queue/Stack is empty then return failure;node = pop(Queue/Stack) ;if test(node,Goal[problem]) then return path(node,father);expand-nodes adjacent-nodes(node, Operators[problem]);p j ( , p [p ]);push(Queue/Stack, expand-nodes );foreach ex-node in expand-nodes

father(ex-node) = node;end

Có thể sử dụng cấu trúc dữ liệu khác Queue/Stack tốt hơn không?

Hàm đánh giá – heuristicg

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Best-first search

Idea:Sử dụng hàm đánh giá trong quá trình phát triển cây tìm kiếm, hàm đánh giá ước lượng “độ gần” của mỗi đỉnh

á ớ á í ứ à ỉ ể â ìtrạng thái với trạng thái đích. Tức là chỉ triển khai cây tìm kiếm theo nhánh có triển vọng đi đến đích nhanh nhất.

Implementation:Thay vì dùng Queue/Stack, dùng mảng có sắp xếp theo hàm đ h iđánh giá

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Cài đặt chi tiết giải thuật Best-first search

Function General-Search(problem, ordered-array,h) returns a solution, or failureordered-array make-queue(make-node(initial-state[problem]));ordered-array make-queue(make-node(initial-state[problem]));father(initial-state[problem]) = empty;while (1)

if ordered-array is empty then return failure;if ordered array is empty then return failure;node = pop(ordered-array) ; // node with max/min hif test(node,Goal[problem]) then return path(node,father);expand-nodes adjacent-nodes(node, Operators[problem]);p j ( , p [p ]);push(ordered-array, expand-nodes ,h);foreach ex-node in expand-nodes

father(ex-node) = node;Lấy các đỉnh kề với node, nhưng chưa xuất hiện trong

end

Function push(ordered-array, expand-nodes ,h);

ordered-array

Chèn các nodes trong expand-nodes vào ordered-array sao cho mảng ordered-array sắp theo thứ tự tăng/giảmtheo hàm h

Các loại hàm đánh giá - hạ g

Hàm h(n) – chi phí từ trạng thái đầu đến n uniform search

Hàm h(n) – Ước lượng chi phí từ n đến trạng thái đích greedy search

Hàm h(n) = f(n) + g(n)Trong đó:

f(n) – chi phí từ trạng thái đầu đến ng(n) – Ước lượng chi phí từ n đến trạng thái đích

và g(n) ≤ g*(n), g*(n) là chi phí thực sự từ n đến trạng thái đích

A* search…

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Ví dụ: Romania with step costs in kmụ p

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Uniform-cost search

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Uniform-cost search

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Uniform-cost search

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Properties of uniform-cost searchp

• Completeness: Yes, if step cost ≥ ε >0• Time complexity: # nodes with g ≤ cost of optimal solution, ≤ O(b d)• Space complexity: # nodes with g ≤ cost of optimal solution, ≤ O(b d)• Optimality: Yes as long as path cost never decreases• Optimality: Yes, as long as path cost never decreases

g(n) is the path cost to node nRemember:

b = branching factorb = branching factord = depth of least-cost solution

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Greedy searchy

Estimation function:

h(n) = estimate of cost from n to goal (heuristic)

For example:For example:

hSLD(n) = straight-line distance from n to Bucharest

Greedy search expands first the node that appears to be closest to the goal, according to h(n).

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Romania with step costs in kmp

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Properties of Greedy Searchp y

Complete?

Time?

Space?

Optimal?Optimal?

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Properties of Greedy Searchp y

Complete? No – can get stuck in loopse.g., Iasi > Neamt > Iasi > Neamt > …Complete in finite space with repeated-state checking.

Time? O(b^m) but a good heuristic can givedramatic improvementp

Space? O(b^m) – keeps all nodes in memory

Optimal? NoOptimal? No.

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A* search

Idea: avoid expanding paths that are already expensive

evaluation function: h(n) = f(n) + g(n), with:f(n) – cost so far to reach ng(n) – estimated cost to goal from nh(n) – estimated total cost of path through n to goal

A* search uses an admissible heuristic, that is,

h(n) ≤ h*(n) where h*(n) is the true cost from n.For example: hSLD(n) never overestimates actual road distance.

Theorem: A* search is optimal

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Optimality of A* (standard proof)p y ( p )

Suppose some suboptimal goal G2 has been generated and is in the L b d d d h hqueue. Let n be an unexpanded node on a shortest path to an

optimal goal G1.

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Optimality of A* (more useful proof)p y ( p )

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f-contoursf contours

Ho do the conto s look like hen h(n) 0?How do the contours look like when h(n) =0?

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Properties of A*p

Complete?

Time?Time?

Space?

Optimal?

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Properties of A*p

Complete? Yes, unless infinitely many nodes with f ≤ f(G)

Time? Exponential in [(relative error in h) x (length of solution)]Time? Exponential in [(relative error in h) x (length of solution)]

Space? Keeps all nodes in memory

Optimal? Yes – cannot expand fi+1 until fi is finished

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Admissible heuristics

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Admissible heuristics

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Relaxed Problem

Admissible heuristics can be derived from the exact solution cost of l d i f h bla relaxed version of the problem.

If the rules of the 8-puzzle are relaxed so that a tile can move h th h ( ) i th h t t l tianywhere, then h1(n) gives the shortest solution.

If the rules are relaxed so that a tile can move to any adjacent th h ( ) i th h t t l tisquare, then h2(n) gives the shortest solution.

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Iterative improvementp

In many optimization problems, path is irrelevant;

the goal state itself is the solution.

Then, state space = space of “complete” configurations.e , state space space o co p ete co gu at o s

Algorithm goal:- find optimal configuration (e.g., TSP), or,g g- find configuration satisfying constraints

(e.g., n-queens)

In such cases, can use iterative improvement algorithms: keep a single “current” state, and try to improve it.

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Iterative improvement example: n-queensp p q

Goal: Put n chess-game queens on an n x n board, with no two h l di lqueens on the same row, column, or diagonal.

fHere, goal state is initially unknown but is specified by constraints that it must satisfy.

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Summaryy

Best-first search = general search, where the minimum-cost nodes ( o ding to ome me e) e e p nded fi t(according to some measure) are expanded first.

Uniform search = best-first with the cost from initial state as a heuristic

Greedy search = best-first with the estimated cost to reach the goal as a heuristic measure.

generally faster than uninformed searchgenerally faster than uninformed searchnot optimalnot complete.

A* search = best-first with measure = path cost so far + estimated path cost to goal.

bi d t f if t d d hcombines advantages of uniform-cost and greedy searchescomplete, optimal and optimally efficientspace complexity still exponential

41Time complexity of heuristic algorithms depend on quality of heuristic function.