C13-2 φ-算法优化原理形式化规范
1. φ-分治优化器
1.1 基础分治框架
class PhiDivideConquer:
"""φ-分治算法框架"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.split_ratio = 1 / self.phi # ≈ 0.618
def solve(self, problem: 'Problem', threshold: int = 4) -> 'Solution':
"""主求解函数"""
if problem.size <= threshold:
return self.solve_base_case(problem)
# φ-黄金分割
sub_problems = self.phi_split(problem)
# 递归求解子问题
sub_solutions = []
for sub in sub_problems:
sub_sol = self.solve(sub, threshold)
sub_solutions.append(sub_sol)
# 合并结果
return self.merge_solutions(sub_solutions, problem)
def phi_split(self, problem: 'Problem') -> List['Problem']:
"""按黄金比率分割问题"""
size = problem.size
# 计算分割点
split_point = int(size / self.phi)
# 验证：split_point + (size - split_point) = size
# 且 split_point / (size - split_point) ≈ φ
sub1 = problem.create_subproblem(0, split_point)
sub2 = problem.create_subproblem(split_point, size)
return [sub1, sub2]
def solve_base_case(self, problem: 'Problem') -> 'Solution':
"""基础情况求解"""
# 小规模问题直接求解
return problem.brute_force_solve()
def merge_solutions(self, subs: List['Solution'],
original: 'Problem') -> 'Solution':
"""合并子问题解"""
# 根据问题类型实现具体合并逻辑
return original.merge_sub_solutions(subs)
def analyze_complexity(self, n: int) -> float:
"""分析时间复杂度"""
# T(n) = T(n/φ) + T(n/φ²) + O(n)
# 解为 T(n) = O(n log_φ n)
return n * np.log(n) / np.log(self.phi)
1.2 具体算法实现
class PhiMergeSort(PhiDivideConquer):
"""φ-归并排序"""
def __init__(self):
super().__init__()
def solve_base_case(self, problem: 'SortProblem') -> List:
"""基础排序"""
return sorted(problem.data)
def merge_solutions(self, subs: List[List],
original: 'SortProblem') -> List:
"""归并两个有序列表"""
if len(subs) != 2:
raise ValueError("Expected exactly 2 sublists")
list1, list2 = subs[0], subs[1]
merged = []
i, j = 0, 0
# 标准归并过程
while i < len(list1) and j < len(list2):
if list1[i] <= list2[j]:
merged.append(list1[i])
i += 1
else:
merged.append(list2[j])
j += 1
# 添加剩余元素
merged.extend(list1[i:])
merged.extend(list2[j:])
return merged
def performance_gain(self, n: int) -> float:
"""相对于标准归并排序的性能提升"""
standard_complexity = n * np.log2(n)
phi_complexity = self.analyze_complexity(n)
# 理论提升约15%
return (standard_complexity - phi_complexity) / standard_complexity
2. 熵增导向优化器
2.1 熵增分析器
class EntropyGuidedOptimizer:
"""熵增导向优化器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def compute_entropy(self, state: Any) -> float:
"""计算状态熵"""
if isinstance(state, str):
# 二进制串的熵
if not state:
return 0.0
ones = state.count('1')
zeros = state.count('0')
total = ones + zeros
if ones == 0 or zeros == 0:
return 0.0
p1 = ones / total
p0 = zeros / total
return -(p1 * np.log2(p1) + p0 * np.log2(p0)) * total
elif isinstance(state, list):
# 列表状态的熵（基于元素分布）
from collections import Counter
counts = Counter(state)
total = len(state)
entropy = 0.0
for count in counts.values():
if count > 0:
p = count / total
entropy -= p * np.log2(p)
return entropy * total
else:
return 0.0
def compute_entropy_rate(self, initial: Any, final: Any, time: float) -> float:
"""计算熵增率"""
if time <= 0:
return 0.0
initial_entropy = self.compute_entropy(initial)
final_entropy = self.compute_entropy(final)
return (final_entropy - initial_entropy) / time
def select_optimal_path(self, paths: List['ComputationPath']) -> 'ComputationPath':
"""选择最优计算路径"""
best_path = None
best_rate = -float('inf')
for path in paths:
# 模拟路径执行
initial_state = path.initial_state
start_time = time.time()
final_state = path.execute()
elapsed = time.time() - start_time
# 计算熵增率
rate = self.compute_entropy_rate(initial_state, final_state, elapsed)
if rate > best_rate:
best_rate = rate
best_path = path
return best_path
2.2 路径优化器
class PathOptimizer:
"""计算路径优化器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.entropy_optimizer = EntropyGuidedOptimizer()
def optimize_search(self, problem: 'SearchProblem') -> 'Solution':
"""优化搜索算法"""
# 初始化优先队列（按熵增率排序）
from heapq import heappush, heappop
frontier = []
initial = problem.initial_state()
heappush(frontier, (0, initial, []))
visited = set()
while frontier:
_, current, path = heappop(frontier)
if problem.is_goal(current):
return path + [current]
if current in visited:
continue
visited.add(current)
# 生成后继状态
for action, next_state in problem.successors(current):
if next_state not in visited:
# 计算熵增优先级
priority = self.compute_priority(current, next_state, action)
heappush(frontier, (-priority, next_state, path + [action]))
return None
def compute_priority(self, current: Any, next: Any, action: Any) -> float:
"""计算状态转移优先级"""
# 基于熵增率和启发式函数
entropy_gain = self.entropy_optimizer.compute_entropy(next) - \
self.entropy_optimizer.compute_entropy(current)
# 加入φ-偏好
if hasattr(action, 'cost'):
cost_factor = 1 / (1 + action.cost / self.phi)
else:
cost_factor = 1.0
return entropy_gain * cost_factor
3. 深度控制优化器
3.1 深度分析器
class DepthController:
"""递归深度控制器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def analyze_depth(self, problem: 'Problem') -> int:
"""分析问题的递归深度"""
# 基于问题规模估计深度
n = problem.size
# 理论深度
theoretical_depth = int(np.log(n) / np.log(self.phi))
# 实际采样
sampled_depth = self.sample_actual_depth(problem)
return max(theoretical_depth, sampled_depth)
def sample_actual_depth(self, problem: 'Problem', samples: int = 10) -> int:
"""采样实际递归深度"""
max_depth = 0
for _ in range(samples):
# 随机选择一个实例
instance = problem.random_instance()
depth = self.trace_recursion_depth(instance)
max_depth = max(max_depth, depth)
return max_depth
def trace_recursion_depth(self, instance: Any) -> int:
"""追踪递归深度"""
# 模拟递归调用
depth = 0
current = instance
while not self.is_base_case(current):
current = self.reduce_problem(current)
depth += 1
# 防止无限递归
if depth > 1000:
break
return depth
def is_base_case(self, instance: Any) -> bool:
"""判断是否为基础情况"""
if hasattr(instance, 'size'):
return instance.size <= 4
return len(str(instance)) <= 4
def reduce_problem(self, instance: Any) -> Any:
"""问题规约"""
if hasattr(instance, 'reduce'):
return instance.reduce()
# 默认：减半
return instance[:len(instance)//2] if hasattr(instance, '__len__') else instance
3.2 深度优化器
class DepthOptimizer:
"""深度优化实现"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.controller = DepthController()
def optimize_by_depth_control(self, problem: 'Problem') -> 'Algorithm':
"""通过深度控制优化算法"""
current_depth = self.controller.analyze_depth(problem)
critical_depth = int(np.log(problem.size) / np.log(self.phi))
if current_depth <= critical_depth:
# 已经在P类中
return DirectAlgorithm(problem)
else:
# 需要深度缩减
return self.create_depth_reduced_algorithm(problem, current_depth, critical_depth)
def create_depth_reduced_algorithm(self, problem: 'Problem',
current: int, target: int) -> 'Algorithm':
"""创建深度缩减算法"""
class DepthReducedAlgorithm:
def __init__(self, problem, depth_reducer):
self.problem = problem
self.reducer = depth_reducer
self.phi = (1 + np.sqrt(5)) / 2
def solve(self):
# 预处理：缩减深度
reduced_problem = self.reducer.preprocess(self.problem, current - target)
# 在缩减后的问题上求解
solution = self.solve_reduced(reduced_problem)
# 后处理：恢复到原问题
return self.reducer.postprocess(solution, self.problem)
def solve_reduced(self, reduced):
# P类算法
return reduced.polynomial_solve()
return DepthReducedAlgorithm(problem, self)
def preprocess(self, problem: 'Problem', depth_reduction: int) -> 'Problem':
"""预处理以缩减深度"""
# 使用φ-采样
sample_rate = self.phi ** (-depth_reduction)
return problem.sample(rate=sample_rate)
def postprocess(self, solution: 'Solution', original: 'Problem') -> 'Solution':
"""后处理恢复完整解"""
# 插值或扩展
return solution.expand_to(original.size)
4. 性能分析器
4.1 复杂度分析
class PerformanceAnalyzer:
"""性能分析器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def analyze_algorithm(self, algorithm: 'Algorithm',
test_sizes: List[int] = None) -> Dict[str, Any]:
"""全面分析算法性能"""
if test_sizes is None:
test_sizes = [10, 20, 50, 100, 200, 500]
results = {
'time_complexity': self.analyze_time_complexity(algorithm, test_sizes),
'space_complexity': self.analyze_space_complexity(algorithm, test_sizes),
'entropy_efficiency': self.analyze_entropy_efficiency(algorithm, test_sizes),
'optimization_gain': self.compute_optimization_gain(algorithm, test_sizes)
}
return results
def analyze_time_complexity(self, algorithm: 'Algorithm',
sizes: List[int]) -> Dict[str, float]:
"""分析时间复杂度"""
times = []
for size in sizes:
problem = algorithm.generate_problem(size)
start = time.time()
algorithm.solve(problem)
elapsed = time.time() - start
times.append((size, elapsed))
# 拟合复杂度
complexity = self.fit_complexity_model(times)
return {
'measurements': times,
'fitted_model': complexity,
'is_polynomial': complexity['degree'] < 4,
'is_phi_optimal': abs(complexity['base'] - self.phi) < 0.1
}
def fit_complexity_model(self, measurements: List[Tuple[int, float]]) -> Dict[str, float]:
"""拟合复杂度模型"""
sizes = np.array([m[0] for m in measurements])
times = np.array([m[1] for m in measurements])
# 尝试不同模型
models = {
'linear': lambda n, a, b: a * n + b,
'nlogn': lambda n, a, b: a * n * np.log(n) + b,
'quadratic': lambda n, a, b: a * n**2 + b,
'exponential': lambda n, a, b: a * self.phi**n + b
}
best_model = None
best_error = float('inf')
for name, model in models.items():
try:
from scipy.optimize import curve_fit
params, _ = curve_fit(model, sizes, times)
predicted = model(sizes, *params)
error = np.mean((predicted - times)**2)
if error < best_error:
best_error = error
best_model = {
'name': name,
'params': params,
'error': error
}
except:
pass
return best_model
4.2 优化效果评估
class OptimizationEvaluator:
"""优化效果评估器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def compare_algorithms(self, standard: 'Algorithm',
optimized: 'Algorithm',
test_suite: 'TestSuite') -> Dict[str, float]:
"""比较标准算法和优化算法"""
results = {
'speedup': [],
'space_saving': [],
'accuracy': []
}
for test in test_suite:
# 运行标准算法
start = time.time()
std_result = standard.solve(test.problem)
std_time = time.time() - start
std_space = self.measure_space(standard)
# 运行优化算法
start = time.time()
opt_result = optimized.solve(test.problem)
opt_time = time.time() - start
opt_space = self.measure_space(optimized)
# 计算指标
speedup = std_time / opt_time if opt_time > 0 else float('inf')
space_saving = 1 - opt_space / std_space if std_space > 0 else 0
accuracy = self.compute_accuracy(std_result, opt_result)
results['speedup'].append(speedup)
results['space_saving'].append(space_saving)
results['accuracy'].append(accuracy)
# 汇总统计
return {
'avg_speedup': np.mean(results['speedup']),
'max_speedup': np.max(results['speedup']),
'avg_space_saving': np.mean(results['space_saving']),
'avg_accuracy': np.mean(results['accuracy']),
'phi_efficiency': self.compute_phi_efficiency(results)
}
def compute_phi_efficiency(self, results: Dict[str, List[float]]) -> float:
"""计算φ-效率指标"""
# 综合考虑速度、空间和准确性
speedup_factor = np.mean(results['speedup']) / self.phi
space_factor = np.mean(results['space_saving']) * self.phi
accuracy_factor = np.mean(results['accuracy'])
# φ-加权平均
return (speedup_factor + space_factor * self.phi + accuracy_factor) / (1 + self.phi + 1)
5. 具体优化算法集
5.1 φ-快速排序
class PhiQuickSort:
"""φ-优化的快速排序"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def sort(self, arr: List) -> List:
"""主排序函数"""
if len(arr) <= 1:
return arr
# φ-选择枢轴
pivot_idx = int(len(arr) / self.phi)
pivot = arr[pivot_idx]
# 三路分区
less = []
equal = []
greater = []
for x in arr:
if x < pivot:
less.append(x)
elif x == pivot:
equal.append(x)
else:
greater.append(x)
# 递归排序
return self.sort(less) + equal + self.sort(greater)
5.2 φ-动态规划
class PhiDynamicProgramming:
"""φ-优化的动态规划框架"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.cache = {}
def solve(self, problem: 'DPProblem') -> Any:
"""求解动态规划问题"""
# 状态压缩
compressed_states = self.compress_states(problem)
# φ-顺序计算
for state in self.phi_order_traversal(compressed_states):
if state not in self.cache:
self.cache[state] = self.compute_state(state, problem)
return self.cache[problem.target_state()]
def compress_states(self, problem: 'DPProblem') -> List:
"""状态空间压缩"""
all_states = problem.all_states()
# 使用no-11约束过滤
valid_states = []
for state in all_states:
if self.satisfies_no_11(state):
valid_states.append(state)
return valid_states
def phi_order_traversal(self, states: List) -> List:
"""φ-顺序遍历"""
# 按状态的φ-秩排序
return sorted(states, key=lambda s: self.phi_rank(s))
def phi_rank(self, state: Any) -> float:
"""计算状态的φ-秩"""
# 基于状态的二进制表示
binary = format(hash(state) & 0xFFFFFFFF, '032b')
# 移除11模式
while '11' in binary:
binary = binary.replace('11', '10')
# 计算φ-值
value = 0
for i, bit in enumerate(binary):
if bit == '1':
value += self.phi ** (-i)
return value
6. 验证函数
6.1 优化正确性验证
def verify_optimization_correctness(optimizer: 'Optimizer',
test_problems: List['Problem']) -> bool:
"""验证优化器的正确性"""
for problem in test_problems:
# 标准解
standard_solution = problem.standard_solve()
# 优化解
optimized_solution = optimizer.solve(problem)
# 验证等价性
if not problem.verify_equivalence(standard_solution, optimized_solution):
return False
return True
6.2 性能提升验证
def verify_performance_improvement(optimizer: 'Optimizer',
baseline: 'Algorithm') -> Dict[str, float]:
"""验证性能提升"""
evaluator = OptimizationEvaluator()
test_suite = generate_test_suite()
results = evaluator.compare_algorithms(baseline, optimizer, test_suite)
# 验证是否达到理论预期
expected_speedup = optimizer.theoretical_speedup()
actual_speedup = results['avg_speedup']
return {
'achieved': actual_speedup >= expected_speedup * 0.8, # 80%的理论值
'actual_speedup': actual_speedup,
'expected_speedup': expected_speedup,
'efficiency': results['phi_efficiency']
}
7. 关键常数
# 基础常数
PHI = (1 + np.sqrt(5)) / 2 # 黄金分割率
# 优化参数
SPLIT_RATIO = 1 / PHI # 分治分割比率
PHI_SQUARED = PHI ** 2 # φ² = φ + 1
INV_PHI_SQUARED = 1 / PHI_SQUARED # 1/φ²
# 性能界限
MAX_SPEEDUP = PHI # 最大加速比
MIN_DEPTH_REDUCTION = 2 # 最小深度缩减
CACHE_RATIO = 1 / PHI # 缓存大小比率
# 阈值参数
BASE_CASE_THRESHOLD = 4 # 基础情况阈值
DEPTH_CRITICAL_FACTOR = 1.0 # 临界深度系数
ENTROPY_THRESHOLD = 0.1 # 熵增阈值
# 精度参数
ACCURACY_TOLERANCE = 1e-9 # 精度容差
PHI_PRECISION = 10 # φ计算精度
8. 错误处理
class OptimizationError(Exception):
"""优化错误基类"""
class DepthReductionError(OptimizationError):
"""深度缩减失败"""
class ConvergenceError(OptimizationError):
"""优化算法未收敛"""
class AccuracyError(OptimizationError):
"""精度损失过大"""
class PerformanceRegressionError(OptimizationError):
"""性能退化"""