14 numbers every developer should know

Jeff Dean , a famous Google engineer, popularized a list of latency 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(n⁵) one.

In the coding contest world, competitors think about these tradeoffs all the time. No wonder, there's a set of numbers every algorithm designer should know.

The table below shows the limits that can be reached in a few seconds by algorithms of different complexities, n being the input size. I've added some algorithms and data structure examples for each complexity class.

maximum n	complexity	algorithms	data structures
1,000,000,000 and higher	log n, sqrt n	binary search, ternary search, fast exponentiation, euclid algorithm
10,000,000	n, n log log n, n log* n	set intersection, Eratosthenes sieve, radix sort, KMP, topological sort, Euler tour, strongly connected components, 2sat	disjoint sets, tries, hash_map, rolling hashdeque
1,000,000	n log n	sorting, divide and conquer, sweep line, Kruskal, Dijkstra	segment trees, range trees, heaps, treaps, binary indexed trees, suffix arrays
100,000	n log2 n	divide and conquer	2d range trees
50,000	n1.585, n sqrt n	Karatsuba, square root trick	two level tree
1000 - 10,000	n2	largest empty rectangle, Dijkstra, Prim (on dense graphs)
300-500	n3	all pairs shortest paths, largest sum submatrix, naive matrix multiplication, matrix chain multiplication, gaussian elimination, network flow
30-50	n4, n5, n6
25 - 40	3n/2, 2n/2	meet in the middle	hash tables (for set intersection)
15 - 24	2n	subset enumeration, brute force, dynamic programming with exponential states
15 - 20	n2 2n	dynamic programming with exponential states	bitsets, hash_map
13-17	3n	dynamic programming with exponential states	hash_map (to store the states)
11	n!	brute force, backtracking, next_permutation
8	nn	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.

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 behave for this problem.