Back-of-the-Envelope Calculations

Essential reference numbers and calculations for system design interviews

You might be asked to do some estimates by hand during system design interviews. For example, you might need to determine how long it will take to generate 100 image thumbnails from disk or how much memory a data structure will take.

Memorize the Powers of Two table and Latency numbers - they’re essential for quick calculations during interviews.

Powers of Two Table

Use this table for quick approximations of storage and memory calculations:

Power           Exact Value         Approx Value        Bytes
---------------------------------------------------------------
7                             128
8                             256
10                           1024   1 thousand           1 KB
16                         65,536                       64 KB
20                      1,048,576   1 million            1 MB
30                  1,073,741,824   1 billion            1 GB
32                  4,294,967,296                        4 GB
40              1,099,511,627,776   1 trillion           1 TB

Source: Powers of two - Wikipedia

Latency Numbers Every Programmer Should Know

These numbers are critical for understanding system performance:

Latency Comparison Numbers
--------------------------
L1 cache reference                           0.5 ns
Branch mispredict                            5   ns
L2 cache reference                           7   ns                      14x L1 cache
Mutex lock/unlock                           25   ns
Main memory reference                      100   ns                      20x L2 cache, 200x L1 cache
Compress 1K bytes with Zippy            10,000   ns       10 us
Send 1 KB bytes over 1 Gbps network     10,000   ns       10 us
Read 4 KB randomly from SSD*           150,000   ns      150 us          ~1GB/sec SSD
Read 1 MB sequentially from memory     250,000   ns      250 us
Round trip within same datacenter      500,000   ns      500 us
Read 1 MB sequentially from SSD*     1,000,000   ns    1,000 us    1 ms  ~1GB/sec SSD, 4X memory
HDD seek                            10,000,000   ns   10,000 us   10 ms  20x datacenter roundtrip
Read 1 MB sequentially from 1 Gbps  10,000,000   ns   10,000 us   10 ms  40x memory, 10X SSD
Read 1 MB sequentially from HDD     30,000,000   ns   30,000 us   30 ms 120x memory, 30X SSD
Send packet CA->Netherlands->CA    150,000,000   ns  150,000 us  150 ms

Notes
-----
1 ns = 10^-9 seconds
1 us = 10^-6 seconds = 1,000 ns
1 ms = 10^-3 seconds = 1,000 us = 1,000,000 ns

Handy Metrics

Based on the numbers above:

• Read sequentially from HDD at 30 MB/s
• Read sequentially from 1 Gbps Ethernet at 100 MB/s
• Read sequentially from SSD at 1 GB/s
• Read sequentially from main memory at 4 GB/s
• 6-7 world-wide round trips per second
• 2,000 round trips per second within a data center

Latency Numbers Visualized

Using These Numbers

When performing back-of-the-envelope calculations:

1. Start with the constraint: Understand the scale (users, requests, data size)
2. Use approximations: Round to powers of 2 for easier math
3. Work through the math step by step: Show your work to the interviewer
4. Validate your assumptions: Check if your numbers make sense
5. Consider trade-offs: Use these numbers to justify design decisions

It’s okay to round aggressively during calculations. The goal is to get within the right order of magnitude, not to be perfectly precise.

Additional Resources

• Use back of the envelope calculations
• Latency numbers every programmer should know - Gist 1
• Latency numbers every programmer should know - Gist 2
• Designs, lessons, and advice from building large distributed systems
• Software Engineering Advice from Building Large-Scale Distributed Systems