Lecture2: Search 1

Search Problems

• state space \(S\)
• initial state \(s_0\)
• Action \(A(s)\)
• Transition model \(Result(s,a)\)
• goal test \(G(s)\)
• Action cost \(c(s,a,s')\)

Uninformed Search

Uniform Cost Search Properties

Strategy: expand lowest \(g(n)\) := cost from root to \(n\)

Frontier is a priority queue sorted by \(g(n)\)

Complete when \(\epsilon > 0\)

Informed Search

Heuristic Search

A heuristic is - A function that estimates how close a state is to a goal - Example: Manhattan distance, Euclidean distance for pathing

A* Tree Search

Strategy: Combining UCS and Greedy Search

• Sorted by \(f(n) = g(n) + h(n)\)
  • \(g(n)\): uniform cost by path cost
  • \(h(n)\): greedy function

When Should A* Terminate

Only stop when we dequeue a goal

Optimality

Admissible Heuristics: A heuristic \(h\) is admissible if:

\[ 0 \leq h(n) \leq h^*(n) \]

Optimality of A^* search

Assume:

• \(A\) is an optimal goal node, \(B\) is a suboptimal goal node
• \(h\) is admissible

Claim: \(A\) will exit the frontier before \(B\)

Efficiency

\(A^*\) explores all state \(s\) satisfying

\[ g(s) \leq g(s_{goal}) - h(s) \]

\(A^*\) is more efficient than UCS.

\(A^*\) Graph Search

Admissibility:

Consistency: heuristic "arc" \(\leq\) actual cost for each arc

\[ h(n) - h(n') \leq c(n, a, n') \]