Анализ циклов и вложенных циклов

Один цикл: O(n)

# O(n): body runs n times
def count_ops_linear(n):
    ops = 0
    for i in range(n):
        ops += 1     # constant work
    return ops

print(count_ops_linear(100))  # 100

# Still O(n): step=2 halves count but same class
def count_ops_half(n):
    ops = 0
    for i in range(0, n, 2):
        ops += 1
    return ops

print(count_ops_half(100))    # 50  => O(n)

Вложенные циклы: O(n²) и выше

def count_pairs(n):
    ops = 0
    for i in range(n):          # n iterations
        for j in range(n):      # n iterations each
            ops += 1
    return ops

print(count_pairs(10))   # 100 = 10^2
print(count_pairs(100))  # 10000 = 100^2
# Doubling n quadruples ops: classic O(n^2)

Треугольный цикл: O(n²/2) = O(n²)

def count_unique_pairs(n):
    ops = 0
    for i in range(n):          # n iterations
        for j in range(i+1, n): # n-1, n-2, ..., 0
            ops += 1
    return ops

print(count_unique_pairs(10))  # 45 = 10*9/2
print(count_unique_pairs(100)) # 4950
# Still O(n^2) -- constant factor 1/2 dropped

Цикл с сокращающимся диапазоном: O(log n)

def count_log_ops(n):
    ops = 0
    i = n
    while i >= 1:
        ops += 1
        i //= 2   # halve each iteration
    return ops

import math
for n in [8, 16, 64, 1024]:
    ops = count_log_ops(n)
    print(f'n={n}, ops={ops}, log2={int(math.log2(n))}')
# ops tracks log2(n) closely

Вложенный цикл с сокращающимся внутренним диапазоном: O(n log n)

import math

def count_n_log_n(n):
    ops = 0
    for i in range(n):    # n iterations
        j = n
        while j >= 1:     # log n iterations
            ops += 1
            j //= 2
    return ops

for n in [8, 32, 128]:
    ops = count_n_log_n(n)
    predicted = int(n * math.log2(n))
    print(f'n={n}: actual={ops}, n*log2(n)~={predicted}')

Зависимые внутренние циклы

# Inner loop runs i times: total = 0+1+2+...+(n-1) = n(n-1)/2 => O(n^2)
def sum_inner_i(n):
    ops = 0
    for i in range(n):
        for j in range(i):   # runs 0,1,2,...,n-1 times
            ops += 1
    return ops

print(sum_inner_i(10))  # 45 = 10*9/2  => O(n^2)

# Inner loop runs n/i times (i doubles): sum ≈ n*log n => O(n log n)
def sum_inner_n_over_i(n):
    ops = 0
    i = 1
    while i <= n:
        for j in range(n // i):
            ops += 1
        i *= 2
    return ops
print(sum_inner_n_over_i(64))  # ~ 64*6 = 384

Пошаговый анализ пузырьковой сортировки

def bubble_sort(arr):
    n = len(arr)
    comparisons = 0
    for i in range(n):
        swapped = False
        for j in range(0, n - i - 1):
            comparisons += 1
            if arr[j] > arr[j+1]:
                arr[j], arr[j+1] = arr[j+1], arr[j]
                swapped = True
        if not swapped:  # early exit if sorted
            break
    return comparisons

arr = list(range(10, 0, -1))  # worst case: reversed
ops = bubble_sort(arr)
print(f'Sorted: {arr}')
print(f'Comparisons: {ops}')  # 45 = 10*9/2

Циклы по строкам и подстрокам

# O(n^2): string concat in loop
def build_bad(n):
    s = ''
    for i in range(n):
        s += str(i)  # copies s each time!
    return s

# O(n): join is a single pass
def build_good(n):
    parts = []
    for i in range(n):
        parts.append(str(i))
    return ''.join(parts)

print(build_good(10))  # '0123456789'

Несколько параметров входных данных

# O(m + n): two independent loops
def independent(m, n):
    a = sum(range(m))  # O(m)
    b = sum(range(n))  # O(n)
    return a + b       # total O(m + n)

# O(m * n): nested
def nested(m, n):
    count = 0
    for i in range(m):     # O(m)
        for j in range(n): # O(n) each
            count += 1
    return count  # O(m * n)

print(independent(5, 10))  # 10 + 45 = 55
print(nested(5, 10))       # 50

Цикл внутри цикла и последовательные вызовы

# Naive string matching: O(n*m)
def naive_search(text, pattern):
    n, m = len(text), len(pattern)
    matches = []
    for i in range(n - m + 1):  # O(n)
        if text[i:i+m] == pattern:  # O(m) comparison + O(m) slice
            matches.append(i)
    return matches
# Total: O(n*m)

print(naive_search('abcabcabc', 'abc'))  # [0, 3, 6]

Практика: определяем сложность с первого взгляда

# What is the complexity of this function?
def mystery(nums):
    result = []
    for i in range(len(nums)):          # O(n)
        for j in range(i, len(nums)):   # O(n) worst
            if sum(nums[i:j+1]) == 0:   # O(n) slice + sum!
                result.append((i, j))
    return result
# Answer: O(n^3)  -- three nested n-proportional ops
# Outer O(n) x inner O(n) x sum/slice O(n) = O(n^3)

Быстрая проверка

Итоги урока