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# Jump Search

Continuing our recent dive into search algorithms, we will now be covering the Jump Search algorithm. Similar to Binary Search the Jump Search algorithm is a searching algorithm for pre-sorted arrays.

The basic idea is to check fewer elements than Linear Search by jumping ahead by a fixed increment and backtracking when a larger value than the one searched for is found, running a Linear Search from there to find if the value exists. This may sound strange at first but by the end of this article, all will be clear.

This algorithm saves time and is thus faster when compared to a Linear Search since we do not look at all elements in the collection. It is however slower than Binary Search since the Jump Search algorithm runs with a best case Big O of `O(√n)`.

Jump search has one advantage over Binary Search however and that is that we only go backwards once whereas Binary Search can jump back as many as `O(Log n)` times. This being the case, in situations where Binary Search seems to be running slow due to backward lookups, Jump Search might be able to help you in such a scenario to speed things up.

## Tests

``````from main import jumpSearch;

def test_strings():
assert jumpSearch(["Dave", "James"], "Dave") == 0;
assert jumpSearch(["Dave", "James"], "abc") == -1;

def test_integers():
assert jumpSearch([1,2,3,4], 3) == 2;
assert jumpSearch([1,2,3,4], 5) == -1;

def test_floats():
assert jumpSearch([3.14], 3.14) == 0;
assert jumpSearch([3.141], 3.14) == -1;

def test_booleans():
assert jumpSearch([False, True], True) == 1;
assert jumpSearch([False, True], False) == 0;

def test_lists():
assert jumpSearch([[3.14]], [3.14]) == 0;
assert jumpSearch([[3.141]], [3.14]) == -1;

def test_nested_lists():
assert jumpSearch([[3.14, ]], [3.14, ]) == 0;
assert jumpSearch([[3.141]], [3.14]) == -1;

def test_sets():
assert jumpSearch([set()], set()) == 0;
assert jumpSearch([set()], set([1,2])) == -1;

def test_tuples():
assert jumpSearch([tuple()], tuple()) == 0;
assert jumpSearch([tuple()], tuple([1, 2])) == -1;
``````

You may notice that these are similar to the tests for Linear Search and Binary Search that we wrote in previous articles and this is on purpose.

The difference in tests comes, as with binary search, in the comparison operators we are using. In our case the tests for `dict` were removed and the test for `boolean` vs `None` have also been removed. These are not possible to have due to these not being able to compare to one another using the less than (`<`) operator which is used in the Jump Search algorithm implementation.

You can run and explore through the tests via the repl below:

## Implementation

``````from math import sqrt;
from typing import List, TypeVar;
from operator import eq as deep_equals, ge as greater_than_or_equal, ne as not_equal, lt as less_than;

T = TypeVar('T')

def linearSearch(collection: List[T], value: T) -> int:
for index, item in enumerate(collection):
if deep_equals(item, value): return index;
return -1;

def jumpSearch(collection: List[T], value: T) -> int:
n = len(collection);
step = sqrt(n);
prev = 0;

while less_than(collection[int(min(step, n) - 1)], value):
prev = step;
step *= 2;
if greater_than_or_equal(prev, n): return -1;

foundIndex = linearSearch(collection[int(prev):], value);
if not_equal(foundIndex, -1): foundIndex += int(prev);
return foundIndex;
``````

Note 1: Since Jump Search utilises Linear Search, we will use the implementation we built previously in this article series for that.

For the `jumpSearch` function, we require a `collection` and a `value` to be provided and will return an `int` (integer) representing the index of the `value` if it is found and `-1` if it is not. The `value` itself is what we will be looking for in the `collection`.

Note 2: The `collection` should be a `List` of items matching the same type as the `value` we are searching for and thus, if the `value` is of type `str` (string) then we expect the `collection` to be a `List[str]` for example.

Whence we enter the `jumpSearch` function body, we declare 3 variables:

1. The variable `n` to represent the amount of items in the `collection`
2. The variable `step` to be the value of how far we should jump each time, this is defined as the square root of `n`
3. The variable `prev` to represent the last index visited when the last `step` was taken

From here we run a `while` loop as long as the current item at `int(min(step, n) - 1)` is less than the `value` we are searching for. You may be wondering why we use `int(min(step, n) - 1)`. Let's break it down from the inside out. Let's use the first iteration of the `while` loop for a Pseudocode example:

``````collection = [1, 2, 3];
value = 2;
n = len(collection); # 3
step = sqrt(n); # 1.73205080757
prev = 0; # 0
nextIndex = min(step, n) - 1; # 0.73205080757
indexAsInteger = int(nextIndex) # 0
itemAtIndex = collection[indexAsInteger]; # 1
condition = less_than(itemAtIndex, value); # True
while condition is True: # now we run the loop since the condition is True
``````

That is the essential breakdown of the `while` loops condition and hopefully that makes sense as to why we run `int(min(step, n) - 1)` in the implementation now.

Inside the loop we set the `prev` to equal the current `step` and then add the `step` to itself for the next iteration of the `while` loop. During the loop execution we check that the `prev` variable is not greater than or equal to the length of the `collection` which we stored in the variable `n`. If it is, we haven't found the `value` in the `collection` and return `-1` to represent that it wasn't found.

If we exit the loop and didn't return `-1`, we must have the `value` still in the `collection` and so we then run the `linearSearch` function with a new array created from a slice of the `collection` running from the `prev` step index to the end of the collection and pass in the `value` we are looking for also. If the `linearSearch` run doesn't return `-1`, we have found the `value` and all we need to do is add the `prev` step index value to the `foundIndex`.

Note 3: We do this because if `-1` wasn't returned and thus the `value` was found. If you remember we took a slice of the `collection` beginning from the `prev` step to the end of the `collection` for the `linearSearch` to run over and thus if the value was found by `linearSearch` at index `1` then it would be in the `collection` at index `prev + 1`.

This works well for us since we return the `foundIndex` either way and so if the `value` is not found we will just return the `-1` anyway.

## Conclusions

Jump Search is useful in places where Binary Search is potentially slow due to backwards lookups. It is faster than a regular Linear Search and tries to maximise efficiency by using jumps through the sorted collection.

I hope you found some value in this article and if you have any questions or comments, feel free to write them below!

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