# Hashtable with doubly linked lists

algorithmhashtable

Introduction to Algorithms (CLRS) states that a hash table using doubly linked lists is able to delete items more quickly than one with singly linked lists. Can anybody tell me what is the advantage of using doubly linked lists instead of single linked list for deletion in Hashtable implementation?

#### Best Solution

The confusion here is due to the notation in CLRS. To be consistent with the true question, I use the CLRS notation in this answer.

We use the hash table to store key-value pairs. The value portion is not mentioned in the CLRS pseudocode, while the key portion is defined as `k`.

In my copy of CLR (I am working off of the first edition here), the routines listed for hashes with chaining are insert, search, and delete (with more verbose names in the book). The insert and delete routines take argument `x`, which is the linked list element associated with key `key[x]`. The search routine takes argument `k`, which is the key portion of a key-value pair. I believe the confusion is that you have interpreted the delete routine as taking a key, rather than a linked list element.

Since `x` is a linked list element, having it alone is sufficient to do an O(1) deletion from the linked list in the `h(key[x])` slot of the hash table, if it is a doubly-linked list. If, however, it is a singly-linked list, having `x` is not sufficient. In that case, you need to start at the head of the linked list in slot `h(key[x])` of the table and traverse the list until you finally hit `x` to get its predecessor. Only when you have the predecessor of `x` can the deletion be done, which is why the book states the singly-linked case leads to the same running times for search and delete.

Although CLRS says that you can do the deletion in O(1) time, assuming a doubly-linked list, it also requires you have `x` when calling delete. The point is this: they defined the search routine to return an element `x`. That search is not constant time for an arbitrary key `k`. Once you get `x` from the search routine, you avoid incurring the cost of another search in the call to delete when using doubly-linked lists.
The pseudocode routines are lower level than you would use if presenting a hash table interface to a user. For instance, a delete routine that takes a key `k` as an argument is missing. If that delete is exposed to the user, you would probably just stick to singly-linked lists and have a special version of search to find the `x` associated with `k` and its predecessor element all at once.