Linear vs Binary Search Algorithms

Linear and binary search are two prominent searching algorithms. Linear search works well for a small set but fails when the set becomes too large. Binary search works well in most cases, especially when the set is large.

Searching in a small list

Let's say we have a list of 7 items

first7Primes = [2, 3, 5, 7, 11, 13, 17]

We want to check if number 5 is in the list. We would utilize both linear and binary search

Linear Search

With linear search, we go through the list sequentially until we find what we are looking for. So, it'd be similar to this

first7Primes = [2, 3, 5, 7, 11, 13, 17]

def linearSearch(arr, item):
    for i in arr:
        if (i == item):
            return True
    return False

print(linearSearch(first7Primes , 5)) # True
print(linearSearch(first7Primes , 10)) # False

if we were dealing with the first 1000 primes, the time complexity would quickly rise because of the straight-line graph linear search plots. Below is an image of the straight-line linear search plots vs that of binary search.

Even if you don't understand BigO notation, you can clearly see that linear search keeps going straight up at approximately 45 degrees as the number of search terms increase.

Binary Search

With binary search, we use the mean of the highest and lowest numbers of the considered set. The list has to be ordered for binary-search to work.

How it works

Binary search repeatedly divides the array into halves and computes the average until it lands on the item being search. Let's say we are looking for number 32 in the first 50 whole numbers. We would work through this in pseudocode

// our set 
first50Numbers = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]

floor = a function that takes the whole number part of a number

// starting values
minIndex = 0
maxIndex = 49

// iteration 1
roundedAverage = floor((0 + 49)/2) = 24
roundedAverageValue = 25
is 25 < 30? yes, so we ignore anything less than 25. 
minIndex = roundedAverage  + 1 = 25


// iteration 2: minIndex is now 25 instead of 0; maxIndex remains 49
roundedAverage = floor((25 + 49)/2) = 37
roundedAverageValue  = 38
is 38 > 32? yes, so we ignore anything greater than 38. 
maxIndex = roundedAverage  + 1 = 38

// iteration 3: minIndex still remains 25; maxIndex is now 38
roundedAverage = floor((25 + 38)/2) = 31
roundedAverageValue  = 32
is 32 > 32? no, it is equal, so we return True since we've found the number.

The pseudocode is a simplified representation.

comparing linear and binary

In this array of 50 items. Linear search would perform a maximum of 50 'guesses' while binary search won't perform more than 6 (log to base 2 of 50). In our pseudocode above, we only needed 3

Here's the code for binary search. We are still using our array of first 7 primes.

from math import floor

first7Primes = [2, 3, 5, 7, 11, 13, 17]

def binarySearch(arr, targetValue):
    array = sorted(arr)
    min = 0
    max = len(array) - 1

    while True:
        average = floor((min + max)/2)
        if (max < min):
            return False
        if (array[average] == targetValue):
            return True
        if (array[average] < targetValue):
            min = average + 1
        else:
            max = average - 1

print(binarySearch(first7Primes, 5)) # True
print(binarySearch(first7Primes, 10)) # False

Comparing the efficiency of linear and binary search

To compare the efficiency of both algorithms, let's time them. Run the script in the repl below.

%[https://repl.it/@Vicradon/linear-vs-binary-search]

From the repl above, we see that linear wins over binary when the set is small but loses when the set is large.

Why is any of this important?

The speed of your software depends on the execution speed of your algorithms. To deliver high-quality software, you have to know the right algorithm for the job. You don't want to keep your users waiting for 10 seconds when the wait time could be less.

Further reading

1. Educative.io free course on algorithms
2. RealPython article on binary search

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Credits

1. Linear-Binary search comparison image from QsStudy