Given a sorted array `arr`

, **Array Bisection Algorithm** (a.k.a. Bisection Method, Binary Search Method) enables us to find an insertion point `i`

for a new element `val`

such that `arr[i-1] < val <= arr[i]`

(or, `arr[i] < val <= arr[i+1]`

).

### Problem

Consider we want to insert a number `10`

to a sorted array `[2, 4, 8, 16, 32]`

. Here, an insertion point should be `i=3`

as `arr[2] < 10 <= arr[3]`

.

A naive method is sequentially walking through the elements until we hit the condition:

```
def search(arr, val):
"""
>>> search([2, 4, 8, 16, 32], 10)
3
"""
if val < arr[0]:
return 0
for i in range(1, len(arr)):
if arr[i-1] < val and val <= arr[i]:
return i
return len(arr)
```

The time complexity of this approach is $O(N)$, and larger arrays take more time to complete the operation.

### Intuition

As an optimized way to solve the problem, binary search finds out an insertion point in $O(\log N)$ time complexity.

The basic idea of the method is to repeatedly split `arr`

into two chunks, first-half `arr[:mid]`

and last-half `arr[mid+1:]`

of the elements, until a dividing point `mid`

reaches the target value `val`

.

A GIF image below from penjee.com illustrates how it works in comparison with the naive method:

### Implementation

Although Python implements the algorithm as a standard library `bisect`

, let's try to implement it from scratch.

Starting from `lo=0`

and `hi=len(arr)-1`

, what we have to do is to keep narrowing down a focused range while maintaining `arr[lo] < val <= arr[hi]`

.

```
def bisect(arr, val):
"""Bisection algorithm
Return an index of an ascending-ordered array `arr` where `val` can be inserted. A returned index `i` indicates a potential insertion point, and
`arr[i:]` must come after `val` once inserted.
>>> bisect([2, 4, 8, 16, 32], 1)
0
>>> bisect([2, 4, 8, 16, 32], 4)
1
>>> bisect([2, 4, 8, 16, 32], 3)
1
>>> bisect([2, 4, 8, 16, 32], 10)
3
>>> bisect([2, 4, 8, 16, 32], 64)
5
"""
if len(arr) == 0:
return 0
if val < arr[0]:
return 0
if arr[-1] < val:
return len(arr)
lo, hi = 0, len(arr) - 1
while lo < hi:
if val == arr[lo]:
return lo
elif val == arr[hi]:
return hi
mid = (lo + hi) // 2
if val == arr[mid]:
return mid
elif val < arr[mid]:
hi = mid
else:
lo = mid + 1
return lo
```

In the case of looking for a position where `10`

fits in `[2, 4, 8, 16, 32]`

, the method updates `lo`

and `hi`

as follows.

First, all elements from head to tail are considered:

```
2 4 8 16 32
^ ^ ^
L M H
```

Next, the method realizes the first-half of the array elements is smaller than `10`

, and hence they are ignored so that the following process can focus only on the second half:

```
2 4 8 16 32
^ ^
M H
L
```

Finally, `arr[2] < 10 <= arr[3]`

is confirmed, and `3`

is returned as a potential insertion point:

```
2 4 8 16 32
^
H
M
L
```

### Application

The bisection method can widely be applicable for searching a certain data point from historical records.

In real-world applications, it's safe to say that historical records arrive in the order of timestamp, and hence a target array is typically pre-ordered when we search something from there.

To give an example, assume you have a Tweet database for each user:

```
class User(object):
def __init__(self):
self.tweets = []
def tweet(self, datetime, text):
self.tweets.append((datetime, text))
```

The database sequentially stores a new tweet as soon as it's posted:

```
user = User()
user.tweet(20100101, 'Hello, world.')
# ...
user.tweet(20201201, 'I am hungry.')
user.tweet(20201231, 'Sleepy...')
user.tweet(202101015, 'Happy New Year!')
# ....
```

A question here could be *"What was the last tweet in 2020?"*

If we use `bisect`

, an answer to the query can be easily and efficiently found by searching an insertion point for `(20210101, '')`

:

```
def last_before(timestamp, arr):
"""
arr[i] := (timestamp, value)
"""
pos = bisect(arr, (timestamp, ''))
if pos == 0:
return ''
if pos == len(arr):
return arr[-1][1]
if arr[pos][0] == timestamp:
return arr[pos][1]
return arr[pos-1][1]
last_before(20210101, user.tweets) # => "Sleepy..."
```

Even if a target list is not pre-sorted, growing an array while maintaining its order is not hard when we leverage heap (sorted dictionary/queue, to be more precise). It only takes $O(\log N)$ for insertion.

The method itself is simple, but the efficient searching technique could accelerate a lot of real-life applications we can think of.

#### Share

#### Support (Thank you!)

Note that, as an Amazon Associate, I earn from qualifying purchases on amazon.ca.

#### See also

- 2021-04-04
- The Essence of Supply Chain Management
- 2017-01-21
- FluRS: A Python Library for Online Item Recommendation
- 2017-01-14
- Recommendation.jl: Building Recommender Systems in Julia

Last updated: 2022-09-02

#### Author: Takuya Kitazawa

**Takuya Kitazawa** is a freelance software developer, minimalistic traveler, ultralight hiker & runner, and craft beer enthusiast. While my area of specialty is in data & AI ethics and machine learning productization, I have worked full-stack throughout the career e.g., as a frontend/backend engineer, OSS developer, technical evangelist, solution architect, data scientist, and product manager. You can find what I am doing lately at my "**now**" page, and your inquiry is always welcome at **[email protected]**, including comments on my blog posts.

#### Disclaimer

- Opinions are my own and do not represent the views of organizations I am/was belonging to.
- I am doing my best to ensure the accuracy and fair use of the information. However, there might be some errors or biased subjective statements because the main purpose of this blog is to jot down my personal thoughts as soon as possible before conducting an extensive investigation. Visitors understand the limitations and rely on any information at their own risk.
- That said, if there is any issue with the content, please contact me so I can take the necessary action.