Question

Given the following code, what is the complexity of 3. and how would I represent simple algorithms with the following complexities?

O(n²+n)
O(n²+2n)
O(logn)
O(nlogn)

var collection = new[] {1,2,3};
var collection2 = new[] {1,2,3};

//1.
//On
foreach(var i in c1)
{
}

//2.
//On²
foreach(var i in c1)
{
  foreach(var j in c1)
  {
  }
}

//3.
//O(nⁿ⁺ᵒ)?
foreach(var i in c1)
{
  foreach(var j in c2)
  {
  }
}

Answer 1

The outer foreach is executed n = |c1| times (where |x| is the size of c1), while the inner foreach is executed m = |c2| times. That’s O(n * m) times in total.

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how would I represent simple algorithms with the following complexities?

• O(n²+n)

This is the same as O(n^2). Something that takes O(n^2) time would be drinking a toast with every other person at a party, assuming that there’s always exactly two people in a toast, and only one person does the toasting at a time.

• O(n²+2n)

Same as above; the O(n^2) term dominates. Another example of an O(n^2) effort is planting trees in a square garden of length n, assuming it takes constant time to plant each tree, and that once you plant a tree other trees are excluded from its vicinity.

• O(logn)

An example of this would be finding a word in a dictionary by repeatedly picking the midpoint of the region of pages you need to search next. (In other words, a binary search.)

• O(nlogn)

Use the above algorithm, but now you have to find every word in the dictionary.