二分查找
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#!/usr/bin/env python3
"""
Pure Python implementations of binary search algorithms
For doctests run the following command:
python3 -m doctest -v binary_search.py
For manual testing run:
python3 binary_search.py
"""
from __future__ import annotations
import bisect
def bisect_left(
sorted_collection: list[int], item: int, lo: int = 0, hi: int = -1
) -> int:
"""
Locates the first element in a sorted array that is larger or equal to a given
value.
It has the same interface as
https://docs.pythonlang.cn/3/library/bisect.html#bisect.bisect_left .
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item to bisect
:param lo: lowest index to consider (as in sorted_collection[lo:hi])
:param hi: past the highest index to consider (as in sorted_collection[lo:hi])
:return: index i such that all values in sorted_collection[lo:i] are < item and all
values in sorted_collection[i:hi] are >= item.
Examples:
>>> bisect_left([0, 5, 7, 10, 15], 0)
0
>>> bisect_left([0, 5, 7, 10, 15], 6)
2
>>> bisect_left([0, 5, 7, 10, 15], 20)
5
>>> bisect_left([0, 5, 7, 10, 15], 15, 1, 3)
3
>>> bisect_left([0, 5, 7, 10, 15], 6, 2)
2
"""
if hi < 0:
hi = len(sorted_collection)
while lo < hi:
mid = lo + (hi - lo) // 2
if sorted_collection[mid] < item:
lo = mid + 1
else:
hi = mid
return lo
def bisect_right(
sorted_collection: list[int], item: int, lo: int = 0, hi: int = -1
) -> int:
"""
Locates the first element in a sorted array that is larger than a given value.
It has the same interface as
https://docs.pythonlang.cn/3/library/bisect.html#bisect.bisect_right .
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item to bisect
:param lo: lowest index to consider (as in sorted_collection[lo:hi])
:param hi: past the highest index to consider (as in sorted_collection[lo:hi])
:return: index i such that all values in sorted_collection[lo:i] are <= item and
all values in sorted_collection[i:hi] are > item.
Examples:
>>> bisect_right([0, 5, 7, 10, 15], 0)
1
>>> bisect_right([0, 5, 7, 10, 15], 15)
5
>>> bisect_right([0, 5, 7, 10, 15], 6)
2
>>> bisect_right([0, 5, 7, 10, 15], 15, 1, 3)
3
>>> bisect_right([0, 5, 7, 10, 15], 6, 2)
2
"""
if hi < 0:
hi = len(sorted_collection)
while lo < hi:
mid = lo + (hi - lo) // 2
if sorted_collection[mid] <= item:
lo = mid + 1
else:
hi = mid
return lo
def insort_left(
sorted_collection: list[int], item: int, lo: int = 0, hi: int = -1
) -> None:
"""
Inserts a given value into a sorted array before other values with the same value.
It has the same interface as
https://docs.pythonlang.cn/3/library/bisect.html#bisect.insort_left .
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item to insert
:param lo: lowest index to consider (as in sorted_collection[lo:hi])
:param hi: past the highest index to consider (as in sorted_collection[lo:hi])
Examples:
>>> sorted_collection = [0, 5, 7, 10, 15]
>>> insort_left(sorted_collection, 6)
>>> sorted_collection
[0, 5, 6, 7, 10, 15]
>>> sorted_collection = [(0, 0), (5, 5), (7, 7), (10, 10), (15, 15)]
>>> item = (5, 5)
>>> insort_left(sorted_collection, item)
>>> sorted_collection
[(0, 0), (5, 5), (5, 5), (7, 7), (10, 10), (15, 15)]
>>> item is sorted_collection[1]
True
>>> item is sorted_collection[2]
False
>>> sorted_collection = [0, 5, 7, 10, 15]
>>> insort_left(sorted_collection, 20)
>>> sorted_collection
[0, 5, 7, 10, 15, 20]
>>> sorted_collection = [0, 5, 7, 10, 15]
>>> insort_left(sorted_collection, 15, 1, 3)
>>> sorted_collection
[0, 5, 7, 15, 10, 15]
"""
sorted_collection.insert(bisect_left(sorted_collection, item, lo, hi), item)
def insort_right(
sorted_collection: list[int], item: int, lo: int = 0, hi: int = -1
) -> None:
"""
Inserts a given value into a sorted array after other values with the same value.
It has the same interface as
https://docs.pythonlang.cn/3/library/bisect.html#bisect.insort_right .
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item to insert
:param lo: lowest index to consider (as in sorted_collection[lo:hi])
:param hi: past the highest index to consider (as in sorted_collection[lo:hi])
Examples:
>>> sorted_collection = [0, 5, 7, 10, 15]
>>> insort_right(sorted_collection, 6)
>>> sorted_collection
[0, 5, 6, 7, 10, 15]
>>> sorted_collection = [(0, 0), (5, 5), (7, 7), (10, 10), (15, 15)]
>>> item = (5, 5)
>>> insort_right(sorted_collection, item)
>>> sorted_collection
[(0, 0), (5, 5), (5, 5), (7, 7), (10, 10), (15, 15)]
>>> item is sorted_collection[1]
False
>>> item is sorted_collection[2]
True
>>> sorted_collection = [0, 5, 7, 10, 15]
>>> insort_right(sorted_collection, 20)
>>> sorted_collection
[0, 5, 7, 10, 15, 20]
>>> sorted_collection = [0, 5, 7, 10, 15]
>>> insort_right(sorted_collection, 15, 1, 3)
>>> sorted_collection
[0, 5, 7, 15, 10, 15]
"""
sorted_collection.insert(bisect_right(sorted_collection, item, lo, hi), item)
def binary_search(sorted_collection: list[int], item: int) -> int:
"""Pure implementation of a binary search algorithm in Python
Be careful collection must be ascending sorted otherwise, the result will be
unpredictable
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item value to search
:return: index of the found item or -1 if the item is not found
Examples:
>>> binary_search([0, 5, 7, 10, 15], 0)
0
>>> binary_search([0, 5, 7, 10, 15], 15)
4
>>> binary_search([0, 5, 7, 10, 15], 5)
1
>>> binary_search([0, 5, 7, 10, 15], 6)
-1
"""
if list(sorted_collection) != sorted(sorted_collection):
raise ValueError("sorted_collection must be sorted in ascending order")
left = 0
right = len(sorted_collection) - 1
while left <= right:
midpoint = left + (right - left) // 2
current_item = sorted_collection[midpoint]
if current_item == item:
return midpoint
elif item < current_item:
right = midpoint - 1
else:
left = midpoint + 1
return -1
def binary_search_std_lib(sorted_collection: list[int], item: int) -> int:
"""Pure implementation of a binary search algorithm in Python using stdlib
Be careful collection must be ascending sorted otherwise, the result will be
unpredictable
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item value to search
:return: index of the found item or -1 if the item is not found
Examples:
>>> binary_search_std_lib([0, 5, 7, 10, 15], 0)
0
>>> binary_search_std_lib([0, 5, 7, 10, 15], 15)
4
>>> binary_search_std_lib([0, 5, 7, 10, 15], 5)
1
>>> binary_search_std_lib([0, 5, 7, 10, 15], 6)
-1
"""
if list(sorted_collection) != sorted(sorted_collection):
raise ValueError("sorted_collection must be sorted in ascending order")
index = bisect.bisect_left(sorted_collection, item)
if index != len(sorted_collection) and sorted_collection[index] == item:
return index
return -1
def binary_search_by_recursion(
sorted_collection: list[int], item: int, left: int = 0, right: int = -1
) -> int:
"""Pure implementation of a binary search algorithm in Python by recursion
Be careful collection must be ascending sorted otherwise, the result will be
unpredictable
First recursion should be started with left=0 and right=(len(sorted_collection)-1)
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item value to search
:return: index of the found item or -1 if the item is not found
Examples:
>>> binary_search_by_recursion([0, 5, 7, 10, 15], 0, 0, 4)
0
>>> binary_search_by_recursion([0, 5, 7, 10, 15], 15, 0, 4)
4
>>> binary_search_by_recursion([0, 5, 7, 10, 15], 5, 0, 4)
1
>>> binary_search_by_recursion([0, 5, 7, 10, 15], 6, 0, 4)
-1
"""
if right < 0:
right = len(sorted_collection) - 1
if list(sorted_collection) != sorted(sorted_collection):
raise ValueError("sorted_collection must be sorted in ascending order")
if right < left:
return -1
midpoint = left + (right - left) // 2
if sorted_collection[midpoint] == item:
return midpoint
elif sorted_collection[midpoint] > item:
return binary_search_by_recursion(sorted_collection, item, left, midpoint - 1)
else:
return binary_search_by_recursion(sorted_collection, item, midpoint + 1, right)
def exponential_search(sorted_collection: list[int], item: int) -> int:
"""Pure implementation of an exponential search algorithm in Python
Resources used:
https://en.wikipedia.org/wiki/Exponential_search
Be careful collection must be ascending sorted otherwise, result will be
unpredictable
:param sorted_collection: some ascending sorted collection with comparable items
:param item: item value to search
:return: index of the found item or -1 if the item is not found
the order of this algorithm is O(lg I) where I is index position of item if exist
Examples:
>>> exponential_search([0, 5, 7, 10, 15], 0)
0
>>> exponential_search([0, 5, 7, 10, 15], 15)
4
>>> exponential_search([0, 5, 7, 10, 15], 5)
1
>>> exponential_search([0, 5, 7, 10, 15], 6)
-1
"""
if list(sorted_collection) != sorted(sorted_collection):
raise ValueError("sorted_collection must be sorted in ascending order")
bound = 1
while bound < len(sorted_collection) and sorted_collection[bound] < item:
bound *= 2
left = bound // 2
right = min(bound, len(sorted_collection) - 1)
last_result = binary_search_by_recursion(
sorted_collection=sorted_collection, item=item, left=left, right=right
)
if last_result is None:
return -1
return last_result
searches = ( # Fastest to slowest...
binary_search_std_lib,
binary_search,
exponential_search,
binary_search_by_recursion,
)
if __name__ == "__main__":
import doctest
import timeit
doctest.testmod()
for search in searches:
name = f"{search.__name__:>26}"
print(f"{name}: {search([0, 5, 7, 10, 15], 10) = }") # type: ignore[operator]
print("\nBenchmarks...")
setup = "collection = range(1000)"
for search in searches:
name = search.__name__
print(
f"{name:>26}:",
timeit.timeit(
f"{name}(collection, 500)", setup=setup, number=5_000, globals=globals()
),
)
user_input = input("\nEnter numbers separated by comma: ").strip()
collection = sorted(int(item) for item in user_input.split(","))
target = int(input("Enter a single number to be found in the list: "))
result = binary_search(sorted_collection=collection, item=target)
if result == -1:
print(f"{target} was not found in {collection}.")
else:
print(f"{target} was found at position {result} of {collection}.")
关于此算法
问题陈述
给定一个包含 n 个元素的已排序数组，编写一个函数来搜索给定元素（目标）的索引。
方法
- 通过重复将数组分成两半来搜索数组。
- 最初考虑实际数组并选择中间索引处的元素。
- 保持较低索引（即 0）和较高索引（即数组长度）。
- 如果它等于目标元素，则返回索引。
- 否则，如果它大于目标元素，则仅考虑数组的左半部分。（较低索引 = 0，较高 = 中间 - 1）
- 否则，如果它小于目标元素，则仅考虑数组的右半部分。（较低索引 = 中间 + 1，较高 = 数组长度）
- 如果在数组中找不到目标元素，则返回 -(插入索引 + 1)（如果较低索引大于或等于较高索引）。一些更简单的实现如果找不到元素则只返回 -1。必须添加 1 的偏移量，因为插入索引可能是 0（搜索值可能小于数组中的所有元素）。由于索引从 0 开始，因此必须将其与目标元素具有索引 0 的情况区分开来。
时间复杂度
O(log n) 最坏情况
O(1) 最佳情况（如果初始数组的中间元素是目标元素）
空间复杂度
O(1) 对于迭代方法
O(1) 对于递归方法，*如果使用了尾调用优化*，否则由于递归调用栈，O(log n)
示例
arr = [1,2,3,4,5,6,7]
target = 2
Initially the element at middle index is 4 which is greater than 2. Therefore we search the left half of the
array i.e. [1,2,3].
Here we find the middle element equal to target element so we return its index i.e. 1
target = 9
A simple Binary Search implementation may return -1 as 9 is not present in the array. A more complex one would return the index at which 9 would have to be inserted, which would be `-8` (last position in the array (7) plus one (7+1), negated)`.
