Your n/2 + n/3 + n/5 + … n/97 is not O(n), because the number of terms is not constant. [Edit after your edit: O(n2) is too loose an upper bound.] A loose upper-bound is n(1+1/2+1/3+1/4+1/5+1/6+…1/n) (sum of reciprocals of all numbers up to n), which is O(n log n): see Harmonic number. A more … Read more
Hash tables are O(1) average and amortized case complexity, however it suffers from O(n) worst case time complexity. [And I think this is where your confusion is] Hash tables suffer from O(n) worst time complexity due to two reasons: If too many elements were hashed into the same key: looking inside this key may take … Read more
For completeness, difflib in the standard-library provides loads of sequence-comparison utilities. For instance find_longest_match which finds the longest common substring when used on strings. Example use: from difflib import SequenceMatcher string1 = “apple pie available” string2 = “come have some apple pies” match = SequenceMatcher(None, string1, string2).find_longest_match() print(match) # -> Match(a=0, b=15, size=9) print(string1[match.a:match.a + … Read more
I have to agree that it’s pretty weird the first time you see an O(log n) algorithm… where on earth does that logarithm come from? However, it turns out that there’s several different ways that you can get a log term to show up in big-O notation. Here are a few: Repeatedly dividing by a … Read more
Dijkstra’s shortest path algorithm is O(ElogV) where: V is the number of vertices E is the total number of edges Your analysis is correct, but your symbols have different meanings! You say the algorithm is O(VElogV) where: V is the number of vertices E is the maximum number of edges attached to a single node. … Read more
New answer As of update 6 within Java 7’s lifetime, the behaviour of substring changed to create a copy – so every String refers to a char[] which is not shared with any other object, as far as I’m aware. So at that point, substring() became an O(n) operation where n is the numbers in … Read more
One trick for analyzing the time complexity of Euclid’s algorithm is to follow what happens over two iterations: a’, b’ := a % b, b % (a % b) Now a and b will both decrease, instead of only one, which makes the analysis easier. You can divide it into cases: Tiny A: 2a <= … Read more
O(log log n) terms can show up in a variety of different places, but there are typically two main routes that will arrive at this runtime. Shrinking by a Square Root As mentioned in the answer to the linked question, a common way for an algorithm to have time complexity O(log n) is for that … Read more
NOTE: While this answer was correct in 2012, engines use very different internal representations for both objects and arrays today. This answer may or may not be true. In contrast to most languages, which implement arrays with, well, arrays, in Javascript Arrays are objects, and values are stored in a hashtable, just like regular object … Read more
To understand the difference between polynomial time and pseudopolynomial time, we need to start off by formalizing what “polynomial time” means. The common intuition for polynomial time is “time O(nk) for some k.” For example, selection sort runs in time O(n2), which is polynomial time, while brute-force solving TSP takes time O(n · n!), which … Read more