Week 4 Lecture Notes Part 1 Blind Search cont. and Informed Search

Transcription

1 Week 4 Lecture Notes Part 1 Blind Search cont. and Informed Search Created by Nicholas Collins (s ) and Nicholas Mayer (s ) Admin Assignment 1 is due 9 September, not 4 September as it says in the course profile. Students can work in groups of up to 3, but it is already too late to register, as registrations closed Thursday night. The close in registrations is to ensure no one join s their friend the day before the assignment is due without doing any work. Groups do not have a larger assignment, but have to be able to work together. Cheating on the assignment will not be tolerated. Main Topics Blind Search (continued) Uniform Cost Search algorithm Informed Search Greedy Best First Search algorithm A* Search algorithm Last Week Formulating a problem in terms of search: Find a sequence of actions to move from the initial state to the goal state Uses a graph, which is seen as a tuple of vertices and edges General structure of a search algorithm: Put initial vertex in a container of states to be expanded Loop: Select a vertex 'v' from the container If 'v' is the goal, return Otherwise, expand 'v' (put results of successor(v) in container) successor(v), a function that: Semester

2 Takes a vertex 'v' as input Outputs the set of vertices immediately visitable from 'v', i.e. the vertices from the edges coming out from 'v' Search algorithm performance can be measured by: Completeness Complete If a solution exists, it will be found Optimality Optimal Path found is path with lowest possible cost Cost is defined by a cost function or by number of steps Complexity Time and memory space consumed Measured with asymptotic big-o notation, which should be familiar to students from a previous course Blind search algorithms include: Breadth First Search (BFS) Depth First Search (DFS) Iterative Deepening DFS Uniform Cost Search At each step, look in the container and expand the node with the lowest cost from the root node (out of the nodes currently in the container) Example: Expansion Order 1 Container Goal Found 1 82D g(n) = 0 2 Taxi Rank g(n) = 50 3 Great Court g(n) = 70 6 Bldg g(n) = Theatre g(n) = Union (via GC) g(n) = Union (via Thr) g(n) = G (via GC,Un) g(n) = G (via B50) g(n) = G (via Th,Un) g(n) = 190 Semester

3 Search stops only when the goal node is expanded, not when it is first discovered/added to the container This ensures that the final solution has the minimum possible cost The first time the goal is discovered may not be the optimal solution (e.g. if the route to the goal is several very low cost paths or one very high cost path) When a location is found in multiple paths (e.g. the Union building), a new element is added to the container for each time the location is found Implementation: Uses a Priority Queue (with cost as priority) as the node container Follows general search algorithm structure Properties: Complete? If b is finite and all edges have cost > ε, with epsilon being a small number > 0 Optimal? If all edges have positive cost Complexity? Informed Search Uses some information from the structure of the problem Decompose cost function into two parts: g(n) = cost from root to node n h(n) = estimated cost based on heuristics from n to goal Cost f(n) = some function of g(n), h(n) [e.g. g(n) + h(n)] Node to expand is selected based on f(n) In informed search, f(n) must have some dependence on h(n) If f(n) depends only on g(n) then the algorithm is a Uniform Cost Search (not an informed search) Informed search is generally faster, but is difficult to analyse as performance is highly dependent on the heuristics used Commonly used informed search algorithms include: Greedy Best First Search Semester

4 A* Search Blind Search Informed Search Greedy Best First Search Expand fringe node with the lowest estimated cost to the goal f(n) = h(n) Select node with lowest h(n) (of the nodes in the container) for expansion Ignores the cost from the root to the current node g(n) ignored Example: 1 2 Expansion Order 4 3 Goal Found Container 1 82D h(n) = 80 Taxi Rank h(n) = 70 2 Great Court h(n) = 60 3 Union (via GC) h(n) = 20 Theatre h(n) = G (via Un) h(n) = 0 Semester

5 Properties: Complete? If branching factor and maximum depth are finite, and there are no loops in the state graph such that re-visited states are avoided Optimal? No Complexity? Highly dependent on heuristic A* Search Select fringe node to expand based on estimated cost from the root to the goal via the node g(n) = cost from root to node n h(n) = estimated cost from node n to goal based on heuristic f(n) = g(n) + h(n) Expand node with lowest f(n) of the nodes in the container Example: 1 Expansion Order 3 4 Container Goal Found 1 82D f(n) = g+h = 0+80 f(n) = 80 2 Taxi Rank = f(n) = Great Court = f(n) = 150 Bldg 50 = f(n) = Union(via GC) = f(n) = Theatre = f(n) = 130 Union(via Th) = f(n) = G(via GC,Un) = f(n) = 150 Semester

6 Admissible heuristics Heuristics which never overestimate the cost of a goal h(n) <= h*(n) where h*(n) is the true cost of the goal Properties: Complete? If all edges have cost > ε, with epsilon being a small number > 0 Optimal? If all edges have cost > ε and heuristic is admissible Complexity? Highly dependent on heuristic Sketches of proofs of completeness and optimality of A* search were shown in the lecture, but were not explained in great detail due to time constraints (refer to lecture slides) Consistent Heuristics More restrictive than an admissible heuristic, so consistent admissible h(n) <= c(n, a, n') + h(n') Checks that if you have a heuristic from n to goal, that heuristic must be <= the cost to move from that node n to another node n' and the heuristic from the other node n' to the goal, with a being the action to get to n' Equivalent to the triangle inequality If h(n) is a consistent heuristic, then: f(n) is a non-decreasing function When A* search expands a node n, the path to n is optimal Note that a consistent heuristic may not always be desirable Semester