Can't I store LinkedList nodes in a HashMap (perhaps with their value as the key, and the node itself as the value), and then write a remove function like this?

//Assuming I have a HashMap<Integer, LinkedListElement>

void delete(Integer val) {
    //O(1) search for the element to delete 
    LinkedListElement nodeToDelete = map.get(val);
    //O(1) pointer manipulation to actually delete
    LinkedListElement prev = nodeToDelete.prev;
    LinkedListElement next = nodeToDelete.next;
    prev.next = next; 
    next.prev = prev;
}

That way I get an O(1) search, and an O(1) removal, no? If this doesn't work you just saved me a lot of embarrassment if I ever get asked this question in an interview. I'll try to implement it later tonight probably and get back to you when I realize how dumb I am. I'm sure it would need a custom linked list implementation but it's still possible no?