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Atenţie! Aceasta este o versiune veche a paginii, scrisă la 2013-03-26 13:21:54.
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Numbers everyone 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 a few algorithms and data structure examples for each complexity class.
| maximum n | complexity | algorithms | data structures |
|---|---|---|---|
| 8 | nn | brute force, cartesian product | |
| 11 | n! | brute force, backtracking, next_permutation | |
| 13-17 | 3n | dynamic programming with exponential states | hash_map (to store the states) |
| 15 - 20 | n2 2n | dynamic programming with exponential states | bitsets, hash_map |
| 15 - 24 | 2n | subset enumeration, brute force, dynamic programming with exponential states | |
| 25 - 40 | 3n/2, 2n/2 | meet in the middle | hash tables (for set intersection) |
| 30-50 | n4, n5, n6 | ||
| 300-500 | n3 | all pairs shortest paths, largest sum submatrix, matrix multiplication, matrix chain multiplication, network flow | |
| 1000 - 10,000 | n2 | largest empty rectangle, Dijkstra, Prim (on dense graphs) | |
| 50,000 | n1.585, n sqrt n | Karatsuba, square root trick | two level tree |
| 100,000 | n log2 n | divide and conquer | 2d range trees |
| 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 |
| 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 hash |
| 1,000,000,000 | log n, sqrt n | binary search, ternary search, fast exponentiation, euclid algorithm |
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