14 numbers every developer should know

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14 numbers every developer should know

Cosmin Negruseri

26 martie 2013

Jeff Dean , a famous Google engineer, popularized a list of latency numbers everyone should

know . The list is a great resource for designin g 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 higherlog n, sqrt n

binary search, ternary search, fast

exponentiation, euclid algorithm

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10,000,000n, 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 hash

deque

1,000 ,000 n lo g nsorting, divide and conquer, sweep line,

Kruskal, Dijkstra

segment trees, range

trees, heaps, treaps,

binary indexed trees,

suffix arrays

100,000 n log2n divide and conquer 2d range trees

50,000n1.585, n

sqrt nKaratsuba, square root trick two level tree

1000 - 10,000 n2largest 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 middlehash tables (for set

intersection)

15 - 24 2nsubset enumeration, brute force, dynamic

programming with exponential states

15 - 20 n22ndynamic programming with exponential

statesbitsets, hash_map

13-17 3ndynamic 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 millio ns of ed ges given that Di jkstra i mplementatio ns h ave m l og n time co mplexity (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?


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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.

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