Calculating Memory Capacity

Computer Science tests students’ ability to calculate the information volume of a message. This task is considered a higher difficulty level and requires 3 minutes to complete.

How to calculate the memory capacity for storing identifiers. A simple, step-by-step explanation.

Question:

When registering in the computer system, each object is assigned an identifier consisting of 317 characters. The following characters are used to generate it:

• decimal digits (0–9), 10 characters in total;
• a special alphabet containing 4090 different symbols.

In the database, each identifier is stored using the same minimum number of bytes. Character-based encoding is used, where all characters are encoded with the same minimum possible number of bits.

Required: Determine the amount of memory (in megabytes) required to store 262,144 such identifiers.

Key features of the solution:

• Counting bits per symbol: It is necessary to take into account the size of the alphabet and determine the minimum number of bits to encode one symbol.
• Converting to bytes and megabytes: It is important to round values ​​correctly when converting from bits to bytes and then to megabytes.
• Rounding: All memory capacity calculations are rounded to the nearest whole megabyte.

This approach not only tests knowledge of information theory, but also requires attention to detail when performing arithmetic operations.

Step-by-step solution to the problem

1. Determining the cardinality of the alphabet. Each identifier symbol can be either:

• Decimal digit (0 to 9), which gives 10 characters;
• A special symbol from an alphabet of 4090 characters.

Thus, the total number of possible symbols (the size of the alphabet):

10 + 4090 = 4100 characters.

2. Determining the number of bits to encode one symbol. To encode each symbol, it is necessary to select a number of bits that will allow for the unambiguous representation of all 4100 possible variants.

The amount of information is measured in bits, and each bit can take one of two values: 0 or 1. This means that n bits can encode 2 ⁿ.

To find out how many bits are required to encode 4100 symbols, you need to find a number n such that: 2 ⁿ ≥ 4100.

Let’s calculate: log ² 4100 ≈ 12.0014. We round the result up: n = 13 bits.

Let’s check:

• 2¹² = 4096, which is less than 4100 = 12 bits are not enough;
• 2¹³ = 8192, which is greater than 4100 =13 bits are enough.

Thus, to encode one symbol , 13 bits are required , since 12 bits are not enough, and 13 is already more than enough.

3. Calculating the number of bits for a single identifier. One identifier consists of 317 characters. Each character is encoded in 13 bits, meaning:

317 × 13 bits = 4121 bits.

Let’s convert bits to bytes (since 1 byte = 8 bits):

4121 bits ÷ 8 = 515.125 bytes.

Since it is not possible to allocate a fractional number of bytes, we round up to an integer: 516 bytes.

4. Calculate the total memory required for all identifiers. A total of 262,144 identifiers need to be stored. Each one takes up 516 bytes, meaning:

516 bytes × 262,144 = 135,266,304 bytes.

5. Convert bytes to megabytes. To convert bytes to megabytes, divide by 1024 × 1024:

135,266,304 bytes ÷ 1024 ÷ 1024 = 129 MB.

Answer: 129 MB 

Solution in Python

import math

# Problem parameters
num_digits = 10
num_special = 4090
alphabet_size = num_digits + num_special
length = 317
num_identifiers = 262144

# Bits per character (minimum possible integer number of bits to represent alphabet_size)
# 2^i >= alphabet_size => i = ceil(log2(alphabet_size))
bits_per_char = math.ceil(math.log2(alphabet_size))

# Total bits per identifier
bits_per_identifier = length * bits_per_char

# Bytes per identifier (minimum possible integer number of bytes)
# Since 1 byte = 8 bits, bytes = ceil(bits_per_identifier / 8)
bytes_per_identifier = math.ceil(bits_per_identifier / 8)

# Total bytes for all identifiers
total_bytes = bytes_per_identifier * num_identifiers

# Megabytes (usually 1 MB = 1024 * 1024 bytes = 2^20 bytes, or sometimes 10^6. Let's check both or 2^20)
mb_binary = total_bytes / (1024 * 1024)
mb_decimal = total_bytes / 1000000

print(f"Alphabet size: {alphabet_size}")
print(f"Bits per char: {bits_per_char}")
print(f"Bits per identifier: {bits_per_identifier}")
print(f"Bytes per identifier: {bytes_per_identifier}")
print(f"Total bytes: {total_bytes}")
print(f"MB (binary, 2^20): {mb_binary}")
print(f"MB (decimal, 10^6): {mb_decimal}")

# Verification of steps
print(f"2^12 = {2**12}")
print(f"2^13 = {2**13}")
print(f"4121 / 8 = {4121 / 8}")
print(f"516 * 262144 / (1024 * 1024) = {516 * 262144 / (1024**2)}")

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