#問題
二分查找
list.index()無法應對大規模數據的查詢，需要用其它方法解決，這里談的就是二分查找
#思路說明
在查找方面，python中有list.index()的方法。例如：
>>> a=[2,4,1,9,3] #list可以是無序，也可以是有序
>>> a.index(4) #找到后返回該值在list中的位置
1
>>> a.index(5) #如果沒有該值，則報錯
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: 5 is not in list
這是python中基本的查找方法，雖然簡單，但是，如果由於其時間復雜度為O(n)，對於大規模的查詢恐怕是不足以勝任的。二分查找就是一種替代方法。
二分查找的對象是：有序數組。這點特別需要注意。要把數組排好序先。怎么排序，可以參看我這里多篇排序問題的文章。
基本步驟：
1. 從數組的中間元素開始，如果中間元素正好是要查找的元素，則搜素過程結束；
2. 如果某一特定元素大於或者小於中間元素，則在數組大於或小於中間元素的那一半中查找，而且跟開始一樣從中間元素開始比較。
3. 如果在某一步驟數組為空，則代表找不到。
這種搜索算法每一次比較都使搜索范圍縮小一半。時間復雜度：O(logn)
# 遞歸
def binary_search(lst, value, lo, hi):
if lo > hi:
return -1
half = (lo + hi)/2
if lst[half] == value:
return half
elif lst[half] > value:
return binary_search(lst, value, lo, half-1)
else:
return binary_search(lst, value, half+1, hi)
# 循環
def binary_search_while(lst, value):
lo, hi = 0, len(lst)-1
while lo <= hi:
half = (lo + hi)/2
if lst[half] > value:
hi = half - 1
elif lst[half] < value:
lo = half + 1
else:
return half
return -1
對於python，不能忽視其強大的標准庫。經查閱，發現標准庫中就有一個模塊，名為：bisect。
- 模塊接受排序后的列表。
- 本模塊同樣適用於長列表項。因為它就是用二分查找方法實現的，有興趣可以看其源碼（源碼是一個很好的二分查找算法的例子，特別是很好地解決了邊界條件極端的問題.)
- 關於bisect模塊的更多內容，可以參看[官方文檔](https://docs.python.org/2/library/bisect.html)
"""Bisection algorithms."""
def insort_right(a, x, lo=0, hi=None):
"""Insert item x in list a, and keep it sorted assuming a is sorted.
If x is already in a, insert it to the right of the rightmost x.
Optional args lo (default 0) and hi (default len(a)) bound the
slice of a to be searched.
"""
if lo < 0:
raise ValueError('lo must be non-negative')
if hi is None:
hi = len(a)
while lo < hi:
mid = (lo+hi)//2
if x < a[mid]: hi = mid
else: lo = mid+1
a.insert(lo, x)
insort = insort_right # backward compatibility
def bisect_right(a, x, lo=0, hi=None):
"""Return the index where to insert item x in list a, assuming a is sorted.
The return value i is such that all e in a[:i] have e <= x, and all e in
a[i:] have e > x. So if x already appears in the list, a.insert(x) will
insert just after the rightmost x already there.
Optional args lo (default 0) and hi (default len(a)) bound the
slice of a to be searched.
"""
if lo < 0:
raise ValueError('lo must be non-negative')
if hi is None:
hi = len(a)
while lo < hi:
mid = (lo+hi)//2
if x < a[mid]: hi = mid
else: lo = mid+1
return lo
bisect = bisect_right # backward compatibility
def insort_left(a, x, lo=0, hi=None):
"""Insert item x in list a, and keep it sorted assuming a is sorted.
If x is already in a, insert it to the left of the leftmost x.
Optional args lo (default 0) and hi (default len(a)) bound the
slice of a to be searched.
"""
if lo < 0:
raise ValueError('lo must be non-negative')
if hi is None:
hi = len(a)
while lo < hi:
mid = (lo+hi)//2
if a[mid] < x: lo = mid+1
else: hi = mid
a.insert(lo, x)
def bisect_left(a, x, lo=0, hi=None):
"""Return the index where to insert item x in list a, assuming a is sorted.
The return value i is such that all e in a[:i] have e < x, and all e in
a[i:] have e >= x. So if x already appears in the list, a.insert(x) will
insert just before the leftmost x already there.
Optional args lo (default 0) and hi (default len(a)) bound the
slice of a to be searched.
"""
if lo < 0:
raise ValueError('lo must be non-negative')
if hi is None:
hi = len(a)
while lo < hi:
mid = (lo+hi)//2
if a[mid] < x: lo = mid+1
else: hi = mid
return lo
# Overwrite above definitions with a fast C implementation
try:
from _bisect import *
except ImportError:
pass
參考資料：https://github.com/qiwsir/algorithm