⛰️ Hill Climbing

1. 🎯 Learning Objectives

• Trace Hill Climbing on a simple state landscape
• Explain the difference between Hill Climbing and Best First Search (no OPEN list)
• Identify local optima, plateaus, and ridges
• Describe variants: Stochastic Hill Climbing, Random Restart Hill Climbing
• Explain why Hill Climbing uses O(1) space

2. 📋 Prerequisites

3. 📖 Core Content

3.1 Intuition: Always Go Uphill

3.2 The Hill Climbing Algorithm

text
HillClimbing(initial_state, goal_test, move_gen, heuristic):
    current = initial_state
    while True:
        if goal_test(current):
            return current  // Found goal
        neighbors = move_gen(current)
        if neighbors is empty:
            return current  // Stuck at local optimum
        // Find the best neighbor (lowest h-value for minimization)
        best = neighbor with minimum heuristic value
        if h(best) >= h(current):
            return current  // Stuck — no improvement
        current = best  // Move to the best neighbor

3.3 Worked Example: Hill Climbing on 8-Puzzle

pseudo
2 8 3
1 6 4
7 0 5

pseudo
1 2 3
8 0 4
7 6 5

• Move blank UP: [[2, 0, 3], [1, 6, 4], [7, 8, 5]] → Compute h = ? Let's trace carefully:
• State after blank up: [2, 0, 3], [1, 6, 4], [7, 8, 5] Tile positions: 1(1,0), 2(0,0), 3(0,2), 4(1,2), 5(2,2), 6(1,1), 7(2,0), 8(1,1? wait 8 is at (1,1)) Manhattan: 1→goal(0,0): |1-0|+|0-0|=1; 2→goal(0,1): |0-0|+|0-1|=1; 3→goal(0,2): 0; 4→goal(1,2): 0; 5→goal(2,2): 0; 6→goal(2,1): |1-2|+|1-1|=1; 7→goal(2,0): 0; 8→goal(1,0): |1-1|+|1-0|=1 Total = 4
• Move blank LEFT: [2, 8, 3], [1, 6, 4], [0, 7, 5] Manhattan ≈ 5 (similar to start)
• Move blank RIGHT: [2, 8, 3], [1, 6, 4], [7, 5, 0] Manhattan ≈ 6 Best neighbor: Blank UP (h=4) which is < h(current)=5. Move to it. Step 3: Current = [2, 0, 3], [1, 6, 4], [7, 8, 5]. h = 4. Continue until goal reached or stuck.

3.4 Hill Climbing vs. Best First Search

3.5 Local Optima and How Hill Climbing Gets Stuck

1. Local optima: A state that is better than all its neighbors but not the global optimum
2. Plateaus: All neighbors have the same heuristic value
3. Ridges: A sequence of states where each is better than the previous, but no single move improves

3.6 Variants of Hill Climbing

text
RandomRestartHillClimbing(num_restarts, move_gen, heuristic):
    best_solution = null
    best_value = infinity
    for i = 1 to num_restarts:
        initial = random_state()
        solution = HillClimbing(initial, move_gen, heuristic)
        if value(solution) < best_value:
            best_value = value(solution)
            best_solution = solution
    return best_solution

4. 📐 Key Formulas / Concepts

5. ⚠️ Common Pitfalls

Pitfall 1: Confusing Hill Climbing with Best First

Pitfall 2: Expecting Optimal Solutions

Pitfall 3: Applying Hill Climbing to Discontinuous Landscapes

6. 📝 Practice Questions

Q1: Trace Hill Climbing on a problem where start has h=10, neighbors have h=[12, 8, 15, 9].

Answer: Hill Climbing picks the neighbor with h=8 (lowest). If that state's neighbors all have h ≥ 8, Hill Climbing stops there — it has found a local optimum (h=8) even though the global optimum might have h=0. Q2: Why doesn't Hill Climbing use an OPEN list?

Answer: Hill Climbing is a local search — it only keeps the current state and tries to improve it. It has no need to store alternative paths because it never backtracks. This makes it memory-efficient (O(1)) but also means it can get stuck in local optima. Q3: How does Random Restart Hill Climbing improve over standard Hill Climbing?

Answer: Random Restart runs Hill Climbing from multiple random starting positions. If one run gets stuck in a local optimum, another run from a different start might reach the global optimum. It provides probabilistic completeness — as the number of restarts increases, the probability of finding the global optimum approaches 1. Join Discord PreviousHeuristic FunctionsNextLocal Search Overview