Lecture 3: Solving Problems by Searching

Lecture 3: Solving Problems by Searching¶

AIMA Chapter 3 — 1 hour¶

Why Search Matters¶

• Route planning, puzzles, scheduling, games

• Same abstract problem: initial state → goal state via actions

• Theory: completeness, optimality, complexity

• Practice: state space size, heuristic design

Learning Objectives¶

• Formulate problems as search problems

• Understand uninformed search: BFS, DFS, UCS, iterative deepening

• Understand informed search: greedy best-first, A*

• Design and analyze heuristic functions

Problem-Solving Agents¶

• Goal: Reach a goal state

• Formulation: Initial state, actions, transition model, goal test, path cost

• Solution: Sequence of actions from initial to goal state

• Optimal solution: Lowest path cost

Example: Romania¶

• States: Cities

• Actions: Drive between adjacent cities

• Goal: Bucharest

• Path cost: Sum of edge costs (distances)

Search Trees and Graphs¶

• Search tree: Expand states, generate successors

• State space graph: States + edges (may have cycles)

• Explored set: States already expanded

• Frontier: States to expand (open list)

Measuring Performance¶

• Completeness: Does it find a solution if one exists?

• Optimality: Does it find the optimal solution?

• Time complexity: Number of nodes expanded

• Space complexity: Maximum nodes in memory

Breadth-First Search (BFS)¶

• Expand shallowest node first

• Frontier = FIFO queue

• Complete: Yes (if branching factor finite)

• Optimal: Yes (if step cost = 1)

• Time: O(b^d)

• Space: O(b^d)

Uniform-Cost Search (UCS)¶

• Expand lowest-cost node first

• Frontier = priority queue (by g(n))

• Complete: Yes

• Optimal: Yes (if costs non-negative)

• Equivalent to Dijkstra’s algorithm

Depth-First Search (DFS)¶

• Expand deepest node first

• Frontier = LIFO stack

• Complete: No (infinite depth)

• Optimal: No

• Time: O(b^m)

• Space: O(bm) — linear!

Depth-Limited Search¶

• DFS with depth limit ℓ

• Complete: No (if ℓ < d)

• Optimal: No

• Avoids infinite paths

Iterative Deepening Search¶

• Run depth-limited search with ℓ = 0, 1, 2, ...

• Complete: Yes

• Optimal: Yes (if step cost = 1)

• Time: O(b^d) — same as BFS

• Space: O(bd) — like DFS!

Bidirectional Search¶

• Search forward from start and backward from goal

• Meet in the middle

• Time: O(b^(d/2)) — much faster!

• Space: O(b^(d/2))

• Requires explicit goal states

Informed Search: Heuristics¶

• Heuristic h(n): Estimated cost from n to goal

• Admissible: h(n) ≤ actual cost (never overestimate)

• Consistent: h(n) ≤ c(n,a,n’) + h(n’) for successor n’

Greedy Best-First Search¶

• Expand node with lowest h(n)

• Not complete (can get stuck in loops)

• Not optimal

• Fast when heuristic is good

A* Search¶

• Expand node with lowest f(n) = g(n) + h(n)

• Complete: Yes (if finite branching)

• Optimal: Yes (if h admissible)

• Optimally efficient for given heuristic

A*: Optimality Proof¶

• A* expands nodes in order of f(n)

• When goal is expanded, its g(n) is optimal

• Admissible h ensures we never ignore a node on optimal path

Heuristic Design: Relaxed Problems¶

• Relaxation: Remove constraints from problem

• Cost of optimal solution to relaxed problem = admissible heuristic

• Example: 8-puzzle — ignore “blocking” → Manhattan distance

Heuristic Design: Subproblems¶

• Pattern databases: Store exact solution costs for subproblems

• Landmarks: Key states, precompute distances

• Learning: Learn heuristics from experience

Dominance and Consistency¶

• h₂ dominates h₁ if h₂(n) ≥ h₁(n) for all n

• Dominating heuristic → fewer nodes expanded

• Consistent heuristics are admissible

• A* with consistent h never re-expands nodes

Summary¶

*With unit step cost

References¶

• Russell & Norvig, AIMA 4e, Ch. 3

• Chapter PDF: chapters/chapter-03.pdf

• aima-python: search4e.ipynb

• Pseudocode: aima.cs.berkeley.edu/algorithms.pdf

Questions?¶

Next lecture: Search in Complex Environments (Chapter 4)