Numbers everyone should know

14 numbers every developer should know

'Jeff Dean':http://research.google.com/people/jeff/ , a famous Google engineer, popularized a list of latency 'numbers everyone should know':http://highscalability.com/numbers-everyone-should-know. The list is a great resource for designing large scale infrastructure systems.

*sidenote* 'Jeff Dean':http://research.google.com/people/jeff/ , a famous Google engineer, popularized a list of latency 'numbers everyone should know':http://highscalability.com/numbers-everyone-should-know. The list is a great resource for designing large scale infrastructure systems.

Algorithms and their complexity often occur in critical parts of computer systems, but I find that few engineers have a good understanding of how a O(n!) algorithm compares to a O(n^5^) one.

| 100,000 | n log^2^ n| divide and conquer | 2d range trees |
| 50,000 | n^1.585^, n sqrt n| Karatsuba, 'square root trick':blog/square-root-trick |two level tree |
| 1000 - 10,000 | n^2^ | largest empty rectangle, Dijkstra, Prim (on dense graphs) |   |

| 300-500 | n^3^ | all pairs shortest paths, largest sum submatrix,  matrix multiplication, matrix chain multiplication, network flow |   |

| 300-500 | n^3^ | all pairs shortest paths, largest sum submatrix, naive matrix multiplication, matrix chain multiplication, gaussian elimination, network flow |   |

| 30-50 | n^4^, n^5^, n^6^ |   |   |
| 25 - 40 | 3^n/2^, 2^n/2^ | 'meet in the middle':blog/meet-in-the-middle | hash tables (for set intersection) |
| 15 - 24 | 2^n^ | subset enumeration, brute force, dynamic programming with exponential states |   |

| 11 | n! | brute force, backtracking, next_permutation |   |
| 8 |  n^n^ | brute force, cartesian product |   |

These numbers aren't very precise, they assume in memory operations and some varying constant factors, but they do give a good starting point in your search for a solution that fits your problem and your data size.

These numbers aren't very precise, they assume in memory operations and some varying constant factors, but they do give a good starting point in your search for a solution that fits your problem and your data size.
 
Let's go through an example.
 
Suppose you work for a GPS company and your project is to improve their directions feature. In school you've learned about using Dijkstra's algorithm to find the shortest path between two nodes in a graph. Knowing these numbers you will understand that it will take seconds to process a graph with millions of edges given that Dijkstra implementations have m log n time complexity (where m is the number of edges and n the number of nodes).
 
Now you face a few questions:
 
How fast do you want your code to be? seconds? hundreds of milliseconds?
 
A response on the web feels fast if it takes less then 500 milliseconds. So let's pick half a second.
 
How big is the graph? Do you want to solve the problem for a city, a coutry or even a continent?
 
Each is about a magnitude larger than the other and will be solved by different approaches. Let's say we want to solve the problem for the whole of Europe.
 
Here are sizes for a few input sets:
 
|_. input |_. Europe |_. USA/CAN |_. USA (Tiger) |
| #nodes | 18 029 721 | 18 741 705 | 24 278 285 |
| #directed edges | 42 199 587 | 47 244 849 | 58 213 192 |
| #road categories | 13 | 13 | 4 |
 
Since we chose half a second to be our execution time and the size of our problem to be about 40 million edges it's clear from our table that m log n is too slow. So pure Dijkstra won't do. We need to look at how other algorithms like A star search or one based on 'Highway hierarchies':http://algo2.iti.kit.edu/schultes/hwy/esa06HwyHierarchies.pdf behave for this problem.

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