http://www.technicalypto.com/2010/04/trie-data-structure-part-5-complexity.html

Now that we’ve seen the basic operations on how to work with a TRIE, we shall now see the space and time complexities in order to get a real feel of how good a TRIE data structure is. Lets take the two important operations INSERT and SEARCH to measure the complexity.

INSERT operation first. Lets always take into account the worst case timing first and later convince ourselves of the practical timings. For every Node in the TRIE we had something called as Collection where the Collection can be either a Set or a List. If we choose Set, the order of whatever operation we perform over that will be in O(1) time, whereas if we use a LinkedList the number of comparisons at worst will be 26 (the number of alphabets). So for moving from one node to another, there will be at least 26 comparisons will be required at each step.

Having these in mind, for inserting a word of length ‘k’ we need (k * 26) comparisons. By Applying the Big O notation it becomes O(k) which will be again O(1). Thus insert operations are performed in constant time irrespective of the length of the input string (this might look lik an understatement, but if we make the length of the input string a worst case maximum, this sentence holds true).

Same holds true for the search operation as well. The search operation exactly performs the way the insert does and its order is O(k*26) = O(1).

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