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- 6

## What will I learn?

## Statement

## Goal

**Huffman Coding**is an algorithm for doing data compression and it forms the basic idea behind file compression. Instead of allowing every character to occupy 8 bits in a file, we use

**variable-length encoding**to assign each symbol a unique binary code according to the frequency of the character in the file, without any ambiguities.

**To put this into perspective:**Suppose a file contains a string “aabacdeade”, where frequency of characters

`a`,

`b`,

`c`,

`d`and

`e`is 4, 1, 1, 2 and 2 respectively. We assign binary codes to each character as follows:

a --> 00 b --> 010 c --> 011 d--> 10 e--> 11

The process of encoding can be divided into two parts:

**Part 1: Building a Huffman tree**

First, assume all of the characters as individual trees with frequency as their weight. Now, we use a greedy approach to find the two trees with the smallest weights. Then, join them to create a new tree with the sum of those two as its weight and repeat this process until we have a single tree remaining.

For the above example:

Step 1: [a] [d] [e] # --> Here: a = 4, d = 2, e = 2, (bc) = 2

/ \

[b] [c]

Step 2: [a] # # --> Here: a = 4, (bc) = 2, (de) = 4

/ \ / \

[b] [c] [d] [e]

Step 3: # # --> Here: (de) = 4, (a(bc)) = 6

/ \ / \

[a] # [d] [e]

/ \

[b] [c]

Step 4: # --> Here: ((de)(a(bc))) = 10

/ \

/ \

/ \

# #

/ \ / \

[a] # [d] [e]

/ \

[b] [c]

**Part 2: Assigning binary codes to each symbol by traversing Huffman tree**

Generally, bit ‘0’ represents the left child and bit ‘1’

represents the right child

#

0 / \ 1

/ \

/ \

/ \

# #

0 / \ 1 0 / \ 1

/ \ / \

[a] # [d] [e]

0 / \ 1

/ \

[b] [c]

Thus by going through the tree, we will come up with

a = 00, b = 010, c = 011, d = 10, e = 11

**Test Case 1**

n = 5

frequencies = 4 1 1 2 2

bit length for each test case in order = 2 3 3 2 2 [see above for clarification]

total bit count = 4 * 2 + 1 * 3 + 1* 3 + 2 * 2 + 2 * 2 = 22

**0utput :**22

**Here are some links to understand it further :**

https://www.geeksforgeeks.org/huffman-coding-greedy-algo-3/

https://www.studytonight.com/data-structures/huffman-coding

Input

**Line 1:**A single integer

`N`representing the number of characters

**Line 2:**An ordered sequence of

`frequency`values separated by space, where

1 2 3 ... N represents char1 = 1, char2 = 2, char3 = 3 ... charN = N

Output

A single value representing the

**least number of bits**used to store the complete file.Constraints

1 ≤

`N`≤ 3000Example

Input

5 4 1 1 2 2

Output

22

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