⭐ A* Search

1. 🎯 Learning Objectives

• Trace A* with f(N)=g(N)+h(N) showing OPEN/CLOSED tables
• Prove A* admissibility: if h(N) ≤ true cost, A* finds optimal path
• Explain the dominance relation among heuristics
• Compare A* with Best First Search and Branch & Bound
• Identify when A* degenerates into BFS or DFS

2. 📖 Core Content

3.1 Intuition: Combining Cost-So-Far with Estimated-Remaining

• $g(N)$: Actual cost from start to N (known)
• $h(N)$: Estimated cost from N to goal (heuristic)
• $f(N)$: Estimated total cost through N

3.2 The A* Algorithm

text
AStar(initial_state, goal_test, move_gen, heuristic):
    OPEN = priority_queue()  // Ordered by f(N) = g(N) + h(N)
    OPEN.add(initial_state, h(initial_state))
    g[initial_state] = 0
    CLOSED = {}
    nodePairs = {}
    while OPEN is not empty:
        N = OPEN.extract_min()
        if goal_test(N):
            return reconstruct_path(N, nodePairs)
        CLOSED.add(N)
        for each child in move_gen(N):
            tentative_g = g[N] + cost(N, child)
            if child in CLOSED and tentative_g >= g[child]:
                continue  // Already have a better path
            if child not in OPEN or tentative_g < g[child]:
                g[child] = tentative_g
                f[child] = tentative_g + h(child)
                OPEN.add(child, f[child])
                nodePairs[child] = N
    return NO_SOLUTION

3.3 Worked Example: A* Trace

3.4 Admissibility

1. Assume A* returns a suboptimal path with cost $C > C^*$ (optimal)
2. At the point where A* terminates, there must be some node N on the optimal path in OPEN
3. $f(N) = g(N) + h(N) \leq g(N) + h^*(N) = C^*$ (by admissibility)
4. $f(N) \leq C^* < C = f(G)$ (where G is the goal A* returned)
5. So A* would have expanded N before G — contradiction

3.5 Dominance

• It provides tighter bounds
• It prunes more nodes
• It expands fewer states
• A* with $h_1$ never expands more nodes than A* with $h_2$

3.6 A* Properties

3.7 When A* Degrades

• h(N) = 0 for all N: A* becomes BFS (or Dijkstra for weighted graphs)
• h(N) very accurate: A* expands almost only nodes on the optimal path
• h(N) > h(N)* (inadmissible): A* may return suboptimal solutions
• h(N) very inaccurate: A* may expand almost as many nodes as BFS

4. 📝 Practice Questions

Q1: Show that the Manhattan distance heuristic is admissible for the 8-puzzle.

Answer: Manhattan distance sums the horizontal and vertical distances each tile must travel. Each move moves one tile one step, reducing Manhattan by at most 1. Therefore, the true minimum moves ≥ Manhattan distance. Hence h(N) ≤ h*(N) — admissible. Q2: If h(N) is admissible but not consistent, what could go wrong?

Answer: A* may need to re-expand nodes (move from CLOSED back to OPEN) when a better path is found. This increases time but does not affect optimality. With a consistent heuristic, the first expansion of a node is always optimal. Q3: Does a dominating heuristic always reduce A's runtime?*

Answer: A dominating heuristic prunes more nodes (reducing expansions) but takes more time per node to compute. There is a trade-off between heuristic accuracy and computational cost. Join Discord PreviousAdmissibility & ConsistencyNextTSP Branch & Bound