Finding all names matching a query: how to use a suffix tree?

Question

Question : You have a smartphone and you opened the contact app. You want to search a contact. let's say manmohan. but you don't remember his full name. you only remember mohan so you started typing. the moment you type 'm' contact app will start searching for contact which has letter 'm' available. suppose you have stored names in your contact list ("manmohan", "manoj", "raghav","dinesh", "aman") now contact will show manmohan,manoj and aman as a result. Now the next character you type is 'o' (till now you have typed "mo" ) now the result should be "manmohan". How would you implement such data structure?

My approach was applying KMP as you look for pattern "m" then "mo" in all available contact. then display the string which has the match. But interviewer said it's not efficient. ( I couldn't think of any better approach. ) Before leaving he said there is an algorithm which will help. if you know it you can solve it. I couldn't do it. (before leaving I asked about that standard algorithm. Interviewer said : suffix tree). can anyone explain please how is it better ? or which is the best algorithm to implement this data structure.

Answer 1

The problem you're trying to solve essentially boils down to the following: given a fixed collection of strings and a string that only changes via appends, how do you efficiently find all strings that contain that pattern as a substring?

There's a neat little result on strings that's often useful for taking on problems that involve substring searching: a string P is a substring of a string T if and only if P is a prefix of at least one suffix of T. (Do you see why?)

So imagine that you take every name in your word bank and construct a trie of all the suffixes of all the words in that bank. Now, given the pattern string P to search for, walk down the trie, reading characters of P. If you fall off the trie, then the string P must not be a substring of any of the name bank (otherwise, it would have been a prefix of at least one suffix of one of the strings in T). Otherwise, you're at some trie node. Then all of the suffixes in the subtree rooted at the node you're currently visiting correspond to all of the matches of your substring in all of the names in T, which you can find by DFS-ing the subtrie and recording all the suffixes you find.

A suffix tree is essentially a time- and space-efficient data structure for representing a trie of all the suffixes of a collection of strings. It can be built in time proportional to the number of total characters in T (though the algorithms for doing so are famously hard to intuit and code up) and is designed so that you can find all matches of the text string in question rooted at a given node in time O(k), where k is the number of matches.

To recap, the core idea here is to make a trie of all the suffixes of the strings in T and then to walk down it using the pattern P. For time and space efficiency, you'd do this with a suffix tree rather than a suffix trie.