I have been asked recently in a job interview to develop an algorithm that can determine whether a linked list is cyclical. As it's a linked list, we don't know its size. It's a doubly-linked list with each node having 'next' and 'previous' pointers. A node can be connected to any other node or it can be connected to itself.
The only solution that I came up at that time was to pick a node and check it with all the nodes of the linked list. The interviewer obviously didn't like the idea as it is not an optimal solution. What would be a better approach?
What you are looking for is a cycle-finding algorithm. The algorithm Joel refers to is called either the 'tortoise and hare' algorithm or Floyd's cycle finding algorithm. I prefer the second because it sounds like it would make a good D&D spell.
Wikpedia overview of cycle finding algorithms, with sample code
The general solution is to have 2 pointers moving at different rates. They will eventually be equal if some portion of the list is circular. Something along the lines of this:
function boolean hasLoop(Node startNode){
Node slowNode = startNode;
Node fastNode1 = startNode;
Node fastNode2 = startNode;
while (slowNode && fastNode1 = fastNode2.next() && fastNode2 = fastNode1.next()){
if (slowNode == fastNode1 || slowNode == fastNode2)
return true;
slowNode = slowNode.next();
}
return false;
}
Blatantly stolen from here: http://ostermiller.org/find_loop_singly_linked_list.html
Keep a hash of pointer values. Every time you visit a node, hash its pointer and store it. If you ever visit one that already has been stored you know that your list is circular.
This is an O(n) algorithm if your hash table is constant.
Another option is that since the list is doubly linked, you can traverse the list and check if the next pointers previous is always the current node or null and not the head. The idea here is that a loop must either encompass the entire list or look something like this:
- -*- \
\ \
\---
At Node * there are 2 incoming links only one of which can be the previous.
Something like:
bool hasCycle(Node head){
if( head->next == head ) return true;
Node current = head -> next;
while( current != null && current->next != null ) {
if( current == head || current->next->prev != current )
return true;
current = current->next;
}
return false; // since I've reached the end there can't be a cycle.
}
You can handle a general complete circular list like this: Loop through the linked list via the first element until you reach the end of the list or until you get back to the first element.
But if you want to handle the case where a portion of the list is circular then you need to also move ahead your first pointer periodically.
Start with two pointers pointing at the same element. Walk one pointer through the list, following the next pointers. The other walks the list following the previous pointers. If the two pointers meet, then the list is circular. If you find an element with a previous or next pointer set to NULL, then you know the list is not circular.
[Edit the question and subject has been reworded to clarify that we're checking for cycles in a doubly linked list, not checking if a doubly linked list is merely circular, so parts of this post may be irrelevant.]
Its a doubly link list with each node
having 'next' and 'previous' pointers.
Doubly-linked lists are commonly implemented with the head and tail of the list pointing to NULL to indicate where they end.
[Edit] As pointed out, this only checks if the list is circular as a whole, not if it has cycles in it, but that was the wording of the original question.
If the list is circular, tail->next == head and/or head->prev == tail. If you don't have access to both the tail and head node and only have one of those but not both, then it should suffice to simply check if head->prev != NULL or tail->next != NULL.
If this isn't a sufficient answer because we're only given some random node [and looking for cycles anywhere in the list], then all you have to do is take this random node and keep traversing the list until you reach a node that matches (in which case it is circular) or a null pointer (in which case it's not).
However, this is essentially the same thing as the answer you already provided which the interviewer didn't like. I'm quite certain that without some magical hack, there is no way to detect a cycle in a linked list, provided a random node, without a linear complexity algorithm.
[Edit] My mind has switched gears now with the focus on detecting cycles in a list as opposed to determining if a linked list is circular.
If we have a case like:
1<->2<->3<->[2]
The only way I can see that we can detect cycles is to keep track of all the elements we traversed so far and look for any match along the way.
Of course this could be cheap. If we're allowed to modify the list nodes, we could keep a simply traversed flag with each node that we set as we're doing this. If we encounter a node with this flag already set, then we've found a cycle. However, this wouldn't work well for parallelism.
There is a solution proposed here [which I stole from another answer] called "Floyd's Cycle-Finding Algorithm". Let's take a look at it (modified to make it a little easier for me to read).
function boolean hasLoop(Node startNode)
{
Node fastNode2 = startNode;
Node fastNode1 = startNode;
Node slowNode = startNode;
while ( slowNode && (fastNode1 = fastNode2.next()) && (fastNode2 = fastNode1.next()) )
{
if (slowNode == fastNode1 || slowNode == fastNode2)
return true;
slowNode = slowNode.next();
}
return false;
}
It basically involves using 3 iterators instead of 1. We can look at a case like: 1->2->3->4->5->6->[2] case:
First we start at [1] with a fast iterator to [2] and another at [3] or [1, 2, 3]. We stop when the first iterator matches either of the two second iterators.
We proceed with [2, 4, 5] (the first fast iterator traverses the next node of the second fast iterator, and the second fast iterator traverses the next node of the first fast iterator after that). Then [3, 6, 2], and finally [4, 3, 4].
Yay, we've found a match, and have thus determined the list to contain a cycle in 4 iterations.
Assuming that someone says "Here a pointer to a member of a list. Is it a member of a circular list?" then you could examine all reachable members in one direction of the list for pointers to the one node that you were given a pointer to in their pointer which should point away from you. If you decide to go in the next direction then you look for next pointers that are equal to the pointer you were first given. If you choose to go in the prev direction then you look for prev pointers that equal the pointer that you were first given. If you reach a NULL pointer in either direction then you have found the end and know that it is not circular.
You could extend this by going in both directions at the same time and seeing if you bump into yourself, but it gets more complicated and it really doesn't save you anything. Even if you implemented this with 2 threads on a multi-core machine you'd be dealing with shared volatile memory comparisons, which would kill performance.
Alternately, if you can mark each node in the list you could try to determine if there was a cycle by looking for your mark while you searched for the end. If you found your mark in a node you would know that you had been there before. If you found an end before you found one of your marks you would know it wasn't circular. This would not work of another thread were trying to do this at the same time, though, because you would get your marks mixed up, but the other implementation wouldn't work if other threads were reordering the list at the same time as the test.
What you need is Floyd's cycle-finding algorithm. You can also think of finding the the intersection point of the cycle as homework.
Here is a clean approach to test if a linked list has cycles in it (if it's cyclical) based on Floyd's algorithm:
int HasCycle(Node* head)
{
Node *p1 = head;
Node *p2 = head;
while (p1 && p2) {
p1 = p1->next;
p2 = p2->next->next;
if (p1 == p2)
return 1;
}
return 0;
}
The idea is to use two pointers, both starting from head, that advance on different speeds. If they meet each other, that's our clue that there is a cycle in our list, if not, the list is cycle-less.
It is unbelievable how wide can complicated solutions spread.
Here's an absolute minimum required for finding whether a linked list is circular:
bool is_circular(node* head)
{
node* p = head;
while (p != nullptr) {
p = p->next;
if (p == head)
return true;
}
return false;
}
Related
Im trying to use the insertion sort method in order to sort nodes from a LinkedList. I've adjusted the code so many times but I can't quite seem to get it, keep getting different types of results none which are sorted.
Heres the code:
Node* sort_list(Node* head)
{
Node* node_ptr = NULL;
for(Node* i = head->next; i->next != NULL; i = i->next){
if (i->key < head->key) {
node_ptr = i;
head = head->next;
}
}
return node_ptr;
}
This is a homework problem so instead of outright writing a code, I will first point out where you went wrong.
In an insertion sort like algorithm, obviously there needs to be some kind of swapping that needs to be done between elements that are out of place (that is need to be inserted). Hence start with thinking about how you can swap two elements of the array. Pay special attention to the cases when one is head or one is tail.
Your implemented code doesn't have any trace of pointer swaps so this is where you are wrong.
Next you must think about the cases when we need to sort. In this case, it is rather simple. If the current element and the next are in sorted order (assuming ascending order, current < next). Then nothing needs to be done but simply make the next one the current.
Then you can obviously infer that violation of this case is when you need to swap the elements. After the swap (with proper attention to where the pointers were and will be after sorting), repeat the process till you hit the null wall.
P.S : This is a possible duplicate of another SO question.
I am reading about list traversals in Algorithms book by RobertSedwick. Function definitions are shown below. It is mentioned that it is possible to have traverse and remove functions can have iterative counter parts, but traverseR cannot have. My question why traverseR cannot have iterative counter part? Is it that if recursive call is not end of function i.e., like in traverse then we cannot have iterative, Is my understanding right?
Thanks for your time and help.
void traverse(link h, void visit(link))
{
if (h == 0) return;
visit(h);
traverse(h->next, visit);
}
void traverseR(link h, void visit(link))
{
if (h == 0) return;
traverseR(h->next, visit);
visit(h);
}
void remove(link& x, Item v)
{
while (x != 0 && x->item == v)
{ link t = x; x = x->next; delete t; }
if (x != 0) remove(x->next, v);
}
traverseR uses the call stack to store pointers to all the nodes of the list, so that they can be accessed in reverse order as the call stack unwinds.
In order to do this without a call stack (i.e. non-recursively), you'll need some other stack-like data structure to store these pointers in.
The other functions simply work on the current node and move on, with no need to store anything for use after the recursive function call returns. This means that the tail recursion can be replaced with a loop (either by modifying the code or, depending on the compiler, letting it determine that that's possible and make the transformation itself).
Assuming that the list is single-linked, it is not possible to visit it iteratively in the backward order because there's no pointer from a node to a previous node.
What the recursive implementation of traverseR essentially does is that it implicitly reverses the list and visits it in the forward order.
You could write and iterative version of traverseR using a stack: in a loop iterate from one node to another, pushing the nodes on the stack. When you get to the end of the list then, in another loop, pop and visit the nodes you visited.
But his is basically what the recursive version does.
It is possible to traverse a singly linked list in reverse order with only O(1) extra space -- i.e., without a stack of previously visited nodes. It is, however, a little tricky, and not at all thread safe.
The trick to this is to traverse the list from beginning to end, reversing it in place as you do so, then traverse it back to the beginning, reversing it again on the way back through.
Since it is a linked list, reversing it in place is fairly straightforward: as you get to a node, save the current value of its next pointer, and overwrite that with the address of the previous node in the list (see the code for more detail):
void traverseR(node *list, void (*visit)(node *)) {
node *prev = nullptr;
node *curr = list;
node *next;
if (!curr)
return;
// Traverse forwards, reversing list in-place as we go.
do {
next = curr->next;
curr->next = prev;
prev = curr;
curr = next;
} while (curr->next);
// fix up so we have a fully reversed list
curr->next = prev;
prev = nullptr;
// Traverse the reversed list, visiting each node and reversing again
do {
visit(curr);
next = curr->next;
curr->next = prev;
prev = curr;
curr = next;
} while (curr->next);
}
Like almost anything dealing with linked lists, I feel obliged to add that (at least IMO) they should almost always be treated as a purely intellectual exercise. Using them in real code is usually a net loss. You typically end up with code that's slow, fragile, and hard to understand, as well as typically wasting quite a bit of memory (unless the data you store in each node is pretty big, the pointer can often use as much space as the data itself).
My question why traverseR cannot have iterative counter part? Is it that if recursive call is not end of function i.e., like in traverse then we cannot have iterative, Is my understanding right?
Correct. The functions traverse and remove end with a call to themselves. They are tail recursive functions. The call in traverseR to itself is not at the end of the function; traverseR is not tail recursive.
Recursion in general has an expense of creating and later destroying stack frames. This expense can be completely avoided with tail recursive functions by changing the recursion into iteration. Most compilers recognize tail recursive functions and convert the recursion to iteration.
It is possible to write an iterative version of traverseR depending on what you mean by iterative. If you are limited so a single traversal through the list, it is not possible. But if you can sacrifice a lot processing time it can be done. It does use less memory in the classic speed vs. memory trade-off.
void traverseRI(link h, void visit(link))
{
if (h == 0) return;
link last = 0;
while (last != h)
{
link test = h;
while (test->next != last)
{
test = test->next;
}
visit(test);
last = test;
}
}
Our instructor has shown has several examples of functions that process linked lists (Show All Items, Delete At, Insert As Head, Insert As Tail..)
Now, in those example I noticed that he was using different approaches of traversal.
In some instances he would use
while(head !=0)
{
head=head->link;
}
In other instances he uses to move from node to node.
while(head->link !=0)
{
head=head->link;
}
This is confusing to me. Is there a reason to use one over the other for certain operations ?
The second variant will cause a segfault if head is initially NULL.
Other than that, the first variant will iterate N times (where N is the number of items in the list). The second variant will only iterate N-1 times.
The first variant will leave "head" pointing to a "null" value after traversal. The second variant assumes head must be pointing to a good (non-NULL) head value to start with, and will leave head pointing to an element with a null link. Thus the second variant is useful for finding the final element of the list, and the first variant is useful for counting the number of items in the list.
In the first case, he is covering the case where the list might be initially empty (head = nil). You would normally do any processing internal to the loop before the
head = head->link
line.
In the second case, presumably he knows the list is not initially empty. In this case you would normally do any processing after the
head = head->link
line, although you could, if there was a reason, do some before as well. Of course it is also possible that this is not a conscious decision, since professors are people too ;-)
The second example has two problems actually. Use the first, always.
The first problem is as Oli Charlesworth said, in that it will cause a null pointer dereference (segmentation fault) if the loop is entered with head being NULL.
The second problem is that any code between the top of the loop and the head=head->link; statement will not occur on the last node in the linked list. So if this update statement is at the end of the loop (that's the usual way of doing things), then the last node will be completely bypassed. So if your code was this:
while(head->link !=0)
{
dostufftoNode(head);
head=head->link;
}
Then the dostufftoNode() function would be called for every node EXCEPT the last one.
while(head !=0)
{
head=head->link;
}
This will
check if head is not null
set head to head->link
go to one
this will iterate a total of n times
while(head->link !=0)
{
head=head->link;
}
This will
check if head->link is not null
set head to head->link
go to one
this will iterate a total of n-1 times
Here's the node definition:
struct node{
int data;
stuct node * left;
struct node * right;
};
What I am trying to do is list all the nodes that point to an ancestor node. After posting the wrong solution and taking advice from the answers, my new solution is:
Recursively go through the binary tree. Add the current node to an array of nodes and then check if the children of the current node point to any of the previous ancestor nodes.
The default case is the node being NULL. If that happens the function returns.
How it is supposed to work:
Adds the node to the array
Checks if the left child is NULL.
If not, it compares the child to each of the previous nodes.
If it finds a fault, it reports it.
If not, it calls the function with the child as the argument.
Repeat until finished.
(Does same for rhs of binary tree)
Questions:
Is an array the best thing to store
the nodes?
Does this work? for (i = 0; i < sizeof(arrOfNodes) / sizeof(node); i++)
Because the function is recursive,
the array and the array index can't
be initialized inside the function
(or can they be?) so should they be
global?
Would it be better to have two arrays?
(one for the LHS and one for the
RHS)
The code:
void findFault(node * root){
if (root == NULL){
return;
}
arrOfNodes[index++] == root; // array of nodes
if (root->left != NULL){
for (i = 0; i < sizeof(arrOfNodes) / sizeof(node); i++){
if (ar[i] == root->left){
printf("%d", root->left);
return;
}
}
findFault(root->left);
} else return;
if (root->right != NULL){
for (i = 0; i < sizeof(ar) / sizeof(node); i++){
if (ar[i] == root->right){
printf("%d", root->right);
return;
}
}
findFault(root->right);
} else return;
}
I don't know about recursion, but this:
if (&root->left->left == &root){
is wrong in more ways that I can possibly describe, but anyway here are three issues:
Why are you taking the address of root?
Why don't you test that the first left pointer is null?
You could simply use a std::map, but learning how to implement a binary tree is a good idea too.
This is an incorrect solution to the problem. Neil Butterworth already noted on your code, I'll note on the algorithm.
Your algorithm only checks a very specific case - whether a grandchild node points to its grandparent. What you should do is collect the parents along the way to a node and see that a node's child isn't one of its parents.
There are many ways to do this. One is to add a counter to your node struct and set all nodes' counters to zero before you begin traversing the tree. Whenever you reach a node you make sure the counter is zero and then increase it by one. This means that if you see a child whose counter isn't zero, you've already visited it and therefore the tree isn't valid.
Another way to accomplish this kind of check is to do a breadth-first sweep of the nodes, all the while keeping a vector of nodes you have visited already (which you can keep sorted by address). Each time you visit a node, assert it is not in the vector, then add it to the appropriate place to keep the visited list sorted.
The advantage to this kind of check is it can be performed without modifying the tree or node struct itself, though there is a bit of a performance penalty.
Notes:
An array would be a fine way to store the nodes. If you're avoiding STL (curious: why?) then you'll have to manage your own memory. Doable, but it's a brittle wheel to reinvent.
Your for loop check to get the size of the arrays will not work; if you use malloc/free or new/delete then you'll have to specify the size of the array you want beforehand; you should use that size instead of calculating it every time through the for loop.
The typical pattern for a recursive algorithm is to have an "outer" and "inner" function. The outer function is the one called by external code and does the initial setup, etc. The inner function is only called by the outser function, tends to have a more complicated parameter set (taking data set up by the outer function), and calls itself to perform the actual recursion.
You will need two arrays: one for the list of nodes you have visited, and one for the list of nodes you have yet to visit.
I don't know if the algorithm that generates the binary tree is able to propagate a fault other than node's left/right child.
Anyway, this is a corrected version for your code:
void findFault(node * root){
if (root == NULL){
return;
}
if (root->left == root){
printf("left: %d", root->data);
} else findFault(root->left);
if (root->right == root){
printf("right: %d", root->data);
} else findFault(root->right);
}
int LinkedList::DoStuff()
{
Node *Current = next_;
while ( Current != NULL )
{
Current = Current->next_;
length_++;
}
// At the last iteration we have reached the end/tail/last node
return length_;
}
there are no more nodes beyond the last. How can i traverse to the tail-end to the front-head?
Unless your linked list is a doubly-linked one, this is difficult to do. Recursion is one way, assuming you don't have lists so big that you'll run out of stack space, something like this (pseudo-code):
DoStuffBackwards (currNode) {
if (currNode != NULL) {
DoStuffBackwards (currNode->next);
// Process currNode here.
}
}
DoStuffBackwards (firstNode);
This works because you keep calling DoStuffBackwards() for the next node until you exhaust the list then, as you roll back up the recursion stack, you process each node.
If you just want to go backwards from last node to current node, than Pax's answer (using recursion) is your best bet, also see my version below. If your current node is not the head of your non-circular-singly-linked-list, and you want to go from current node to head node, it is impossible.
int LinkedList::DoStuff()
{
return DoStuffBackward(next_, 0);
}
int LinkedList::DoStuffBackward(Node* node, int n)
{
if (!node)
{
return n;
}
int len = DoStuffBackward(node->next_, n + 1);
std::cout << "doing stuff for node " << n << std::endl;
return len;
}
This has the smell of homework, so no code, but here's an overview of a solution that doesn't require recursion:
If you want to run through the list backward one option to relink the list to point backwards as you're traversing it to find the end. Then as you re-traverse the list (which visits the nodes in the reverse order from the original list) you repeat the relinking same as before and the list ends up in its original order.
This is simple in concept, but handling the pointers and links correctly (especially at the start and end of the list) can be a bit tricky.
Recursion can work, as can building an auxiliary data structure, such as an array with one entry for each element of the original list. If you want a solution for a single-threaded list without requiring O(n) extra storage, the best bet is to reverse the list in place as Michael suggests. I wrote an example for this, [but I'll leave it out given the concern about homework]. One caution about reversing the list: if there are any other data structures that hold pointers to the original list, and you might be accessing them during your traversal, they won't work if they need to access the list while it's reversed, and this might lead to data corruption if they try to modify the list.
Update: Ok, here's the (C++) routine to reverse a list in place. It hasn't been tested, though. And I'll note that if this is homework, the poster still needs to figure out how to use this routine correctly to get a complete answer.
Node *ReverseList(Node *head) {
// Reverse a single-threaded list in place, return new head
Node *prev=NULL;
Node *cur=head;
while (Node *next=cur->next_) {
cur->next_ = prev;
prev = cur;
cur = next;
}
cur->next_ = prev;
return cur;
}
push the list on a stack and then pop them off.
Is your linked list class doubly-linked or singly-linked? If there is no previous pointer inside each node, you can't traverse backwards.
I also suggest you post more code and take the time to make your question readable.