XOR(^) is a binary operation called exclusive OR and works as

1^0 = 1  
0^1 = 1  
0^0 = 0
1^1 = 0  

XOR by 1 can work like a toggle switch that turns 1 to 0 or 0 to 1.

Another interesting thing to note is

x^0 = x 
x^x = 0

Usage:

Problem 1: Given a number . Flip all bits in its binary representation.

Solution 1: ^ considering is bit integer.

Problem 2: Given two numbers and . Swap and without using airthmetic operator and without using third variable.

Solution 2:

= ^

= ^

= ^

Caching or Indexing is a technique used to store counts of values which lie in a small range.

Example:
You are given a million numbers in a range, say ; now, if ordering is not important, a good way to store these value is to create an array of size 1000 and simply update counts of each index.

This has many benefits in simple array-based problems and string problems.

A more advanced topic is hashing, where large values of can be hashed into small index values with a collision probability which, if below a certain value, can be used in practice.

Usage:

• Such mapping of elements from larger domain to comparatively smaller domain is useful when we need to compare elements only with or symbol. For example Coordinate Compression is useful in case of Longest Increasing Subsequence Problem and some other problems if