Optimization Techniques

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def example_with_both(n, constraint):
    """
    Demonstrates both pruning and early termination in one function.
    """
    best_solution = None
    
    def explore(partial_solution):
        nonlocal best_solution
        
        # EARLY TERMINATION: We've built a complete solution
        if len(partial_solution) == n:
            if is_better(partial_solution, best_solution):
                best_solution = partial_solution[:]
            return
        
        for candidate in get_candidates():
            # PRUNING: Don't even explore this branch
            if not is_feasible(partial_solution, candidate):
                continue  # Skip this branch entirely
            
            if not can_beat_best(partial_solution, candidate, best_solution):
                continue  # Prune: this path cannot improve on best
            
            partial_solution.append(candidate)
            explore(partial_solution)
            partial_solution.pop()
        
        # EARLY TERMINATION: Found optimal, no need to continue
        if best_solution and is_optimal(best_solution):
            return
    
    explore([])
    return best_solution
 
# Pruning: "Don't go down this path"
# Early termination: "Stop going down ANY more paths"

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def solve_n_queens(n):
    """
    Solve N-Queens with aggressive constraint-based pruning.
    
    Without pruning: Try all n^n possible placements
    With pruning: Only extend partial solutions that don't violate constraints
    
    For n=8:
    - Without pruning: 16,777,216 configurations
    - With pruning: ~15,000 nodes explored
    - Reduction: >1000x
    """
    solutions = []
    
    # Track which columns and diagonals are under attack
    cols = set()           # Columns with a queen
    pos_diag = set()       # r + c diagonals with a queen
    neg_diag = set()       # r - c diagonals with a queen
    
    def backtrack(row, queens):
        if row == n:
            solutions.append(queens[:])
            return
        
        for col in range(n):
            # CONSTRAINT-BASED PRUNING:
            # Skip positions that are already under attack
            if col in cols:
                continue       # Column conflict
            if row + col in pos_diag:
                continue       # Positive diagonal conflict
            if row - col in neg_diag:
                continue       # Negative diagonal conflict
            
            # Place queen and update attack sets
            queens.append(col)
            cols.add(col)
            pos_diag.add(row + col)
            neg_diag.add(row - col)
            
            backtrack(row + 1, queens)
            
            # Backtrack: remove queen
            queens.pop()
            cols.remove(col)
            pos_diag.remove(row + col)
            neg_diag.remove(row - col)
    
    backtrack(0, [])
    return solutions
 
 
# The pruning is ESSENTIAL:
# Row 0: 8 choices
# Row 1: ~5 valid choices (some pruned)
# Row 2: ~3 valid choices
# ...
# The branching factor rapidly decreases due to pruning

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def solve_sudoku(board):
    """
    Solve Sudoku with constraint propagation and pruning.
    
    Key insight: Track possible values for each cell.
    When a value is placed, propagate constraints to eliminate
    impossible values from related cells. If any cell has 0 
    possible values, prune this branch.
    """
    
    def get_possible(board, row, col):
        """Get possible values for a cell based on current constraints."""
        if board[row][col] != 0:
            return set()
        
        possible = set(range(1, 10))
        
        # Remove values in same row
        for c in range(9):
            if board[row][c] != 0:
                possible.discard(board[row][c])
        
        # Remove values in same column
        for r in range(9):
            if board[r][col] != 0:
                possible.discard(board[r][col])
        
        # Remove values in same 3x3 box
        box_row, box_col = 3 * (row // 3), 3 * (col // 3)
        for r in range(box_row, box_row + 3):
            for c in range(box_col, box_col + 3):
                if board[r][c] != 0:
                    possible.discard(board[r][c])
        
        return possible
    
    def find_best_cell(board):
        """
        MRV heuristic: Most constrained variable first.
        Choose cell with fewest possible values.
        Maximizes pruning effectiveness.
        """
        min_options = 10
        best_cell = None
        best_possible = None
        
        for r in range(9):
            for c in range(9):
                if board[r][c] == 0:
                    possible = get_possible(board, r, c)
                    
                    if len(possible) == 0:
                        return None, None, False  # PRUNE: No valid options
                    
                    if len(possible) < min_options:
                        min_options = len(possible)
                        best_cell = (r, c)
                        best_possible = possible
        
        return best_cell, best_possible, True
    
    def solve(board):
        cell, possible, valid = find_best_cell(board)
        
        if not valid:
            return False  # Constraint violation detected
        
        if cell is None:
            return True   # All cells filled - solution found!
        
        row, col = cell
        
        for num in possible:  # Only try feasible values (pruned!)
            board[row][col] = num
            
            if solve(board):
                return True
            
            board[row][col] = 0  # Backtrack
        
        return False  # No valid value for this cell
    
    solve(board)
    return board

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def traveling_salesman_bnb(distances):
    """
    TSP with branch-and-bound using MST lower bound.
    
    Lower bound: Cost so far + MST of remaining cities
    If lower_bound >= best_tour, prune this branch.
    
    The MST provides a valid lower bound because any tour
    visiting remaining cities must at least connect them,
    and MST is the cheapest connection.
    """
    n = len(distances)
    best_tour = float('inf')
    best_path = None
    
    def mst_cost(remaining_cities):
        """
        Compute MST cost using Prim's algorithm.
        This is O(V²) but provides a tight lower bound.
        """
        if len(remaining_cities) <= 1:
            return 0
        
        cities = list(remaining_cities)
        in_mst = {cities[0]}
        total = 0
        
        while len(in_mst) < len(cities):
            min_edge = float('inf')
            next_city = None
            
            for u in in_mst:
                for v in cities:
                    if v not in in_mst:
                        if distances[u][v] < min_edge:
                            min_edge = distances[u][v]
                            next_city = v
            
            in_mst.add(next_city)
            total += min_edge
        
        return total
    
    def min_edge_to_remaining(city, remaining):
        """Minimum cost to connect to any remaining city."""
        if not remaining:
            return distances[city][0]  # Return to start
        return min(distances[city][other] for other in remaining)
    
    def branch_and_bound(path, cost, remaining):
        nonlocal best_tour, best_path
        
        current = path[-1]
        
        # Complete tour
        if not remaining:
            total = cost + distances[current][0]  # Return to start
            if total < best_tour:
                best_tour = total
                best_path = path[:]
            return
        
        # BOUND-BASED PRUNING:
        # Lower bound = cost so far + MST of remaining + min edges
        lower_bound = cost + mst_cost(remaining)
        lower_bound += min_edge_to_remaining(current, remaining)
        
        if lower_bound >= best_tour:
            return  # PRUNE: Cannot beat best
        
        # Explore remaining cities
        for next_city in remaining:
            new_remaining = remaining - {next_city}
            branch_and_bound(
                path + [next_city],
                cost + distances[current][next_city],
                new_remaining
            )
    
    # Initialize
    remaining = set(range(1, n))  # All cities except start
    branch_and_bound([0], 0, remaining)
    
    return best_tour, best_path
 
 
# Without pruning: (n-1)! permutations explored
# With effective bounds: Often explores only a tiny fraction
# For n=15: 87 billion permutations vs potentially ~thousands with good bounds

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def minimax(node, depth, is_maximizing, game):
    """Minimax without pruning: explores entire game tree."""
    if depth == 0 or game.is_terminal(node):
        return game.evaluate(node)
    
    if is_maximizing:
        max_eval = float('-inf')
        for child in game.get_children(node):
            eval = minimax(child, depth - 1, False, game)
            max_eval = max(max_eval, eval)
        return max_eval
    else:
        min_eval = float('inf')
        for child in game.get_children(node):
            eval = minimax(child, depth - 1, True, game)
            min_eval = min(min_eval, eval)
        return min_eval
 
 
def alpha_beta(node, depth, alpha, beta, is_maximizing, game):
    """
    Minimax with alpha-beta pruning.
    
    Alpha: Best score the maximizer can guarantee so far
    Beta: Best score the minimizer can guarantee so far
    
    If at any point beta <= alpha, we can prune:
    - The minimizer won't choose this path (maximizer can do better elsewhere)
    - The maximizer won't choose this path (minimizer can do better elsewhere)
    
    Reduces game tree by roughly sqrt(b^d) instead of b^d
    where b is branching factor and d is depth.
    """
    if depth == 0 or game.is_terminal(node):
        return game.evaluate(node)
    
    if is_maximizing:
        max_eval = float('-inf')
        for child in game.get_children(node):
            eval = alpha_beta(child, depth - 1, alpha, beta, False, game)
            max_eval = max(max_eval, eval)
            alpha = max(alpha, eval)
            
            if beta <= alpha:
                break  # BETA CUTOFF: Minimizer won't allow this path
        
        return max_eval
    else:
        min_eval = float('inf')
        for child in game.get_children(node):
            eval = alpha_beta(child, depth - 1, alpha, beta, True, game)
            min_eval = min(min_eval, eval)
            beta = min(beta, eval)
            
            if beta <= alpha:
                break  # ALPHA CUTOFF: Maximizer won't choose this path
        
        return min_eval
 
 
# Usage:
# best_move_value = alpha_beta(root, depth, float('-inf'), float('inf'), True, game)
 
# Efficiency gain:
# - Perfect ordering: reduces O(b^d) to O(b^(d/2)) 
# - Random ordering: typically O(b^(3d/4))
# - For chess: billions of nodes reduced to millions

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def weighted_job_scheduling_with_dominance(jobs):
    """
    Schedule jobs to maximize profit using dominance pruning.
    
    Jobs: list of (start, end, profit)
    State: (current_time, accumulated_profit)
    
    Dominance: State A dominates State B if:
    - A.time <= B.time (more time available)
    - A.profit >= B.profit (more profit accumulated)
    - At least one inequality is strict
    
    We only keep non-dominated states at each step.
    """
    jobs = sorted(jobs, key=lambda x: x[1])  # Sort by end time
    
    # Pareto frontier of non-dominated states
    # Each state: (end_time, profit)
    states = [(0, 0)]  # Start: time 0, profit 0
    
    for start, end, profit in jobs:
        new_states = []
        
        for time, accumulated in states:
            # Option 1: Skip this job (state unchanged)
            new_states.append((time, accumulated))
            
            # Option 2: Take this job (if feasible)
            if start >= time:
                new_state = (end, accumulated + profit)
                new_states.append(new_state)
        
        # DOMINANCE PRUNING: Keep only Pareto-optimal states
        # A state (t1, p1) dominates (t2, p2) if t1 <= t2 and p1 >= p2
        states = remove_dominated_states(new_states)
    
    # Best profit among all final states
    return max(profit for _, profit in states)
 
 
def remove_dominated_states(states):
    """
    Remove states dominated by others.
    
    For (time, profit) states:
    - Sort by time
    - Scan, keeping only states where profit increases
    
    O(n log n) for this specific dominance relation.
    """
    # Sort by time ascending, profit descending
    states = sorted(set(states), key=lambda x: (x[0], -x[1]))
    
    pareto_frontier = []
    max_profit = float('-inf')
    
    for time, profit in states:
        if profit > max_profit:
            pareto_frontier.append((time, profit))
            max_profit = profit
        # Else: This state is dominated (same or more time, less profit)
    
    return pareto_frontier
 
 
# Without dominance pruning: Potentially exponential states
# With dominance pruning: At most O(n²) non-dominated states
# Often much fewer in practice

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def knapsack_with_dominance(items, capacity):
    """
    0/1 Knapsack with dominance-based state reduction.
    
    State: (weight_used, value_accumulated)
    Dominance: (w1, v1) dominates (w2, v2) if w1 <= w2 and v1 >= v2
    
    Traditional DP: O(n * capacity) states
    With dominance: O(n * |pareto_states|), potentially much smaller
    """
    # pareto_states: list of (weight, value) non-dominated states
    pareto_states = [(0, 0)]
    
    for weight, value in items:
        new_states = []
        
        for w, v in pareto_states:
            # Don't take item
            new_states.append((w, v))
            
            # Take item if it fits
            if w + weight <= capacity:
                new_states.append((w + weight, v + value))
        
        # Apply dominance pruning
        pareto_states = get_pareto_front(new_states)
    
    return max(v for w, v in pareto_states)
 
 
def get_pareto_front(states):
    """Return Pareto-optimal states for (minimize weight, maximize value)."""
    states = sorted(set(states), key=lambda x: (x[0], -x[1]))
    
    front = []
    max_value = float('-inf')
    
    for weight, value in states:
        if value > max_value:
            front.append((weight, value))
            max_value = value
    
    return front
 
 
# Example where dominance pruning helps dramatically:
# Items with many similar weight-value ratios create redundant states
# Standard DP explores them all; dominance pruning eliminates duplicates

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def n_queens_with_symmetry_breaking(n):
    """
    N-Queens that avoids generating symmetric solutions.
    
    Symmetry breaking: Only place first queen in the first half of row 0.
    This eliminates horizontal reflection symmetry immediately.
    
    For 8-queens:
    - Standard: 92 solutions (with many symmetric pairs)
    - Unique (non-symmetric): 12 solutions
    """
    solutions = []
    cols = set()
    pos_diag = set()
    neg_diag = set()
    
    def backtrack(row, queens):
        if row == n:
            solutions.append(queens[:])
            return
        
        # SYMMETRY BREAKING: First queen only in columns 0 to n//2 - 1
        if row == 0:
            col_range = range(n // 2)
        else:
            col_range = range(n)
        
        for col in col_range:
            if col in cols or row + col in pos_diag or row - col in neg_diag:
                continue
            
            queens.append(col)
            cols.add(col)
            pos_diag.add(row + col)
            neg_diag.add(row - col)
            
            backtrack(row + 1, queens)
            
            queens.pop()
            cols.remove(col)
            pos_diag.remove(row + col)
            neg_diag.remove(row - col)
    
    backtrack(0, [])
    return solutions
    # Returns canonical solutions; multiply by 2 for total (horizontal reflections)
 
 
def subset_sum_with_symmetry(arr, target):
    """
    Find subsets that sum to target, avoiding equivalent subsets.
    
    Symmetry: If elements repeat, {a, b, c} and {a, c, b} are same subset.
    Breaking: Process elements in order, never go back.
    """
    arr = sorted(arr)  # Enables duplicate skipping
    results = []
    
    def backtrack(start, path, remaining):
        if remaining == 0:
            results.append(path[:])
            return
        
        if remaining < 0:
            return
        
        for i in range(start, len(arr)):
            # SYMMETRY BREAKING: Skip duplicates
            if i > start and arr[i] == arr[i - 1]:
                continue
            
            if arr[i] > remaining:
                break  # Sorted, so no point continuing
            
            path.append(arr[i])
            backtrack(i + 1, path, remaining - arr[i])
            path.pop()
    
    backtrack(0, [], target)
    return results

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def graph_coloring_with_symmetry_breaking(graph, num_colors):
    """
    Color graph vertices using k colors such that no adjacent vertices share color.
    
    Symmetry: Color labels are interchangeable. Swapping all 'red' and 'blue'
    gives an equivalent coloring.
    
    Breaking: Assign colors in order of first use.
    First vertex gets color 0.
    Next uncolored vertex gets lowest unused color OR any already-used color.
    This creates canonical colorings.
    """
    n = len(graph)
    coloring = [-1] * n
    solutions = []
    
    def is_valid(vertex, color):
        for neighbor in graph[vertex]:
            if coloring[neighbor] == color:
                return False
        return True
    
    def backtrack(vertex, colors_used):
        if vertex == n:
            solutions.append(coloring[:])
            return
        
        # SYMMETRY BREAKING: Only use colors 0 to colors_used (inclusive)
        # This avoids equivalent solutions where color labels are permuted
        max_color = min(colors_used + 1, num_colors)
        
        for color in range(max_color):
            if is_valid(vertex, color):
                coloring[vertex] = color
                
                # Update colors_used if we introduced a new color
                new_colors_used = max(colors_used, color + 1)
                
                backtrack(vertex + 1, new_colors_used)
                
                coloring[vertex] = -1
    
    backtrack(0, 0)
    return solutions
 
 
# Without symmetry breaking: k! equivalent colorings for each unique coloring
# With symmetry breaking: Only canonical representatives explored
# For k=5 colors: 120x reduction in search space

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def partition_equal_subset_with_feasibility(nums):
    """
    Check if array can be partitioned into two subsets with equal sum.
    
    Multiple feasibility pruning techniques:
    1. Total sum must be even
    2. No single element can exceed target
    3. Remaining elements must be able to reach remaining target
    """
    total = sum(nums)
    
    # FEASIBILITY: Early exit if impossible
    if total % 2 != 0:
        return False
    
    target = total // 2
    nums = sorted(nums, reverse=True)  # Larger elements first
    
    # FEASIBILITY: Largest element can't exceed target
    if nums[0] > target:
        return False
    
    def backtrack(index, current_sum, remaining_sum):
        if current_sum == target:
            return True
        
        if index == len(nums):
            return False
        
        # FEASIBILITY: Already exceeded target
        if current_sum > target:
            return False
        
        # FEASIBILITY: Even taking all remaining elements won't reach target
        if current_sum + remaining_sum < target:
            return False
        
        # FEASIBILITY: Next element itself exceeds what we need
        remaining_needed = target - current_sum
        if nums[index] > remaining_needed:
            # Can't take this element; check if skipping it is feasible
            return backtrack(index + 1, current_sum, 
                           remaining_sum - nums[index])
        
        # Try including current element
        if backtrack(index + 1, 
                    current_sum + nums[index],
                    remaining_sum - nums[index]):
            return True
        
        # Skip duplicate elements at this level
        while index + 1 < len(nums) and nums[index + 1] == nums[index]:
            index += 1
        
        # Try excluding current element
        return backtrack(index + 1, current_sum, 
                        remaining_sum - nums[index])
    
    return backtrack(0, 0, total)
 
 
def combination_sum_with_feasibility(candidates, target):
    """
    Find combinations summing to target with comprehensive feasibility pruning.
    """
    candidates = sorted(candidates)
    results = []
    
    def backtrack(start, path, remaining):
        if remaining == 0:
            results.append(path[:])
            return
        
        for i in range(start, len(candidates)):
            candidate = candidates[i]
            
            # FEASIBILITY: Sorted order means all further candidates are larger
            # If this one exceeds remaining, all subsequent ones will too
            if candidate > remaining:
                break  # Not 'continue', but 'break' - prune all remaining
            
            path.append(candidate)
            # Same index i: can reuse same number
            backtrack(i, path, remaining - candidate)
            path.pop()
    
    backtrack(0, [], target)
    return results

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def solve_with_layered_pruning(problem):
    """
    Demonstrates layered pruning: cheap checks first, expensive checks later.
    
    Order pruning checks by (cost, effectiveness):
    1. Fastest checks that catch obvious violations
    2. Medium-cost checks that prune significant portions
    3. Expensive checks only when simpler ones fail to prune
    """
    best_solution = None
    
    def explore(state):
        nonlocal best_solution
        
        # Layer 1: CONSTRAINT PRUNING (cheapest)
        # Simple feasibility checks
        if state.violates_basic_constraints():
            return  # O(1) check, catches many invalid branches
        
        # Layer 2: SYMMETRY PRUNING
        # Check if this state is a symmetric duplicate
        if not state.is_canonical():
            return  # Avoid exploring symmetric equivalents
        
        # Layer 3: FEASIBILITY PRUNING
        # Can we possibly complete from here?
        if not state.can_complete():
            return  # Check remaining resources, etc.
        
        # Layer 4: BOUND-BASED PRUNING (most expensive)
        # Is this branch worth exploring?
        if best_solution is not None:
            upper_bound = state.compute_optimistic_bound()
            if upper_bound <= best_solution.value:
                return  # Can't beat current best
        
        # If we reach here, explore this branch
        if state.is_complete():
            if is_better(state, best_solution):
                best_solution = state.copy()
            return
        
        for next_state in state.generate_children():
            explore(next_state)
    
    explore(initial_state)
    return best_solution
 
 
# Key insight: Each layer catches some branches.
# Cheap wide nets → medium filters → expensive precision
# This way, expensive checks are only run on promising branches.