In this question I'm not asking how to do it but HOW IS IT DONE.
I'm trying (as an excersise) implement simple map and although I do not have problems with implementing links and they behavior (how to find next place to insert new link etc.) I'm stuck with the problem how to implement iterating over a map. When you think about it and look at std::map this map is able to return begin and end iterator. How? Especially end?
If map is a tree how can you say which branch of this map is an end? I just do not understand it. An how to iterate over a map? Starting from the top of the tree and then what? Go and list everything on the left? But those nodes on the left have also links to the right. I really don't know. I will be really glad if someone could explain it to me or give me a link so I could read about it.
A map is implemented using a binary search tree. To meet the complexity requirements it has to be a self-balancing tree, so a red-black tree is usually used, but that doesn't affect how you iterate over the tree.
To read the elements out of a binary search tree in order from least to greatest, you need to perform an in-order traversal of the tree. The recursive implementation is quite simple but isn't really practical for use in an iterator (the iterator would have to maintain a stack internally, which would make it relatively expensive to copy).
You can implement an iterative in-order traversal. This is an implementation taken from a library of tree containers I wrote a while ago. NodePointerT is a pointer to a node, where the node has left_, right_, and parent_ pointers of type NodePointerT.
// Gets the next node in an in-order traversal of the tree; returns null
// when the in-order traversal has ended
template <typename NodePointerT>
NodePointerT next_inorder_node(NodePointerT n)
{
if (!n) { return n; }
// If the node has a right child, we traverse the link to that child
// then traverse as far to the left as we can:
if (n->right_)
{
n = n->right_;
while (n->left_) { n = n->left_; }
}
// If the node is the left node of its parent, the next node is its
// parent node:
else if (n->parent_ && n == n->parent_->left_)
{
n = n->parent_;
}
// Otherwise, this node is the furthest right in its subtree; we
// traverse up through its parents until we find a parent that was a
// left child of a node. The next node is that node's parent. If
// we have reached the end, this will set node to null:
else
{
while (n->parent_ && n == n->parent_->right_) { n = n->parent_; }
n = n->parent_;
}
return n;
}
To find the first node for the begin iterator, you need to find the leftmost node in the tree. Starting at the root node, follow the left child pointer until you encounter a node that has no left child: this is the first node.
For an end iterator, you can set the node pointer to point to the root node or to the last node in the tree and then keep a flag in the iterator indicating that it is an end iterator (is_end_ or something like that).
The representation of your map's iterator is totally up to you. I think it should suffice to use a single wrapped pointer to a node. E.g.:
template <typename T>
struct mymapiterator
{
typename mymap<T>::node * n;
};
Or something similar. Now, mymap::begin() could return such instance of the iterator that n would point to the leftmost node. mymap::end() could return instance with n pointing to root probably or some other special node from which it is still possible to get back to rightmost node so that it could satisfy bidirectional iteration from end iterator.
The operation of moving between the nodes (operators++() and operator--(), etc.) are about traversing the tree from smaller to bigger values or vice versa. Operation that you probably have already implemented during insertion operation implementation.
For sorting purposes, a map behaves like a sorted key/value container (a.k.a. a dictionary); you can think of it as a sorted collection of key/value pairs, and this is exactly what you get when you query for an iterator. Observe:
map<string, int> my_map;
my_map["Hello"] = 1;
my_map["world"] = 2;
for (map<string, int>::const_iterator i = my_map.begin(); i != my_map.end(); ++i)
cout << i->first << ": " << i->second << endl;
Just like any other iterator type, the map iterator behaves like a pointer to a collection element, and for map, this is a std::pair, where first maps to the key and second maps to the value.
std::map uses a binary search internally when you call its find() method or use operator[], but you shouldn't ever need to access the tree representation directly.
One big trick you may be missing is that the end() iterator does not need to point to anything. It can be NULL or any other special value.
The ++ operator sets an iterator to the same special value when it goes past the end of the map. Then everything works.
To implement ++ you might need to keep next/prev pointers in each node, or you could walk back up the tree to find the next node by comparing the node you just left to the parent's right-most node to see if you need to walk to that parent's node, etc.
Don't forget that the iterators to a map should stay valid during insert/erase operations (as long as you didn't erase the iterator's current node).
Related
I'm implementing LRUCache, where in unordered_map I store an iterator to list. When I move the most "fresh" element to the head, I need to iterator not changed.
I need to swap exactly nodes, not values in nodes. I'm finding the way to do it.
I tried to do it with std::iter_swap, but it's just implemented as std::swap(*it_first, *it_second)
std::list<std::string> list;
list.emplace_back("first");
list.emplace_back("second");
auto it_first = list.begin();
auto it_second = ++list.begin();
std::iter_swap(it_first, it_second);
assert(list.begin() == it_second);
I need to swap two nodes to passed assert.
splice looks like it can do this with something like:
list.splice(it_first, list, it_second);
That says "Splice in it_second from myself (list, the second argument), before the first node in myself". The method guarantees that "the iterators to moved elements remain valid, but now refer into *this, not into other.", which implies the raw nodes themselves are moved.
I use the following method to traverse* a binary tree of 300 000 levels:
Node* find(int v){
if(value==v)
return this;
else if(right && value<v)
return right->find(v);
else if(left && value>v)
return left->find(v);
}
However I get a segmentation fault due to stack overflow.
Any ideas on how to traverse the deep tree without the overhead of recursive function calls?
*
By "traverse" I mean "search for a node with given value", not full tree traversal.
Yes! For a 300 000 level tree avoid recursion. Traverse your tree and find the value iteratively using a loop.
Binary Search Tree representation
25 // Level 1
20 36 // Level 2
10 22 30 40 // Level 3
.. .. .. .. .. .. ..
.. .. .. .. .. .. .. .. // Level n
Just to clarify the problem further. Your tree has a depth of n = 300.000 levels. Thus, in the worst case scenario a Binary Search Tree (BST) will have to visit ALL of the tree's nodes. This is bad news because that worst case has an algorithmic O(n) time complexity. Such a tree can have:
2ˆ300.000 nodes = 9.9701e+90308 nodes (approximately).
9.9701e+90308 nodes is an exponentially massive number of nodes to visit. With these numbers it becomes so clear why the call stack overflows.
Solution (iterative way):
I'm assuming your Node class/struct declaration is a classic standard integer BST one. Then you could adapt it and it will work:
struct Node {
int data;
Node* right;
Node* left;
};
Node* find(int v) {
Node* temp = root; // temp Node* value copy to not mess up tree structure by changing the root
while (temp != nullptr) {
if (temp->data == v) {
return temp;
}
if (v > temp->data) {
temp = temp->right;
}
else {
temp = temp->left;
}
}
return nullptr;
}
Taking this iterative approach avoids recursion, hence saving you the hassle of having to recursively find the value in a tree so large with your program call stack.
A simple loop where you have a variable of type Node* which you set to the next node, then loop again ...
Don't forget the case that the value you are searching for does not exist!
You could implement the recursion by not using the call stack but a user-defined stack or something similar; this could be done via the existing stack template. The approach would be to have a while loop which iterates until the stack is empty; as the existing implementaion uses depth-first search, elimination of the recursive calls can be found here.
When the tree that you have is a Binary Search Tree, and all you want to do is search for a node in it that has a specific value, then things are simple: no recursion is necessary, you can do it using a simple loop as others have pointed out.
In the more general case of having a tree which is not necessarily a Binary Search Tree, and wanting to perform a full traversal of it, the simplest way is using recursion, but as you already understand, if the tree is very deep, then recursion will not work.
So, in order to avoid recursion, you have to implement a stack on the C++ heap. You need to declare a new StackElement class that will contain one member for each local variable that your original recursive function had, and one member for each parameter that your original recursive function accepted. (You might be able to get away with fewer member variables, you can worry about that after you have gotten your code to work.)
You can store instances of StackElement in a stack collection, or you can simply have each one of them contain a pointer to its parent, thus fully implementing the stack by yourself.
So, instead of your function recursively calling itself, it will simply consist of a loop. Your function enters the loop with the current StackElement being initialized with information about the root node of your tree. Its parent pointer will be null, which is another way of saying that the stack will be empty.
In every place where the recursive version of your function was calling itself, your new function will be allocating a new instance of StackElement, initializing it, and repeating the loop using this new instance as the current element.
In every place where the recursive version of your function was returning, your new function will be releasing the current StackElement, popping the one that was sitting on the top of the stack, making it the new current element, and repeating the loop.
When you find the node you were looking for, you simply break from the loop.
Alternatively, if the node of your existing tree supports a) a link to its "parent" node and b) user data (where you can store a "visited" flag) then you don't need to implement your own stack, you can just traverse the tree in-place: in each iteration of your loop you first check if the current node is the node you were looking for; if not, then you enumerate through children until you find one which has not been visited yet, and then you visit it; when you reach a leaf, or a node whose children have all been visited, then you back-track by following the link to the parent. Also, if you have the freedom to destroy the tree as you are traversing it, then you do not even need the concept of "user data": once you are done with a child node, you free it and make it null.
Well, it can be made tail recursive at the cost of a single additional local variable and a few comparisons:
Node* find(int v){
if(value==v)
return this;
else if(!right && value<v)
return NULL;
else if(!left && value>v)
return NULL;
else {
Node *tmp = NULL;
if(value<v)
tmp = right;
else if(value>v)
tmp = left;
return tmp->find(v);
}
}
Walking through a binary tree is a recursive process, where you'll keep walking until you find that the node you're at currently points nowhere.
It is that you need an appropriate base condition. Something which looks like:
if (treeNode == NULL)
return NULL;
In general, traversing a tree is accomplished this way (in C):
void traverse(treeNode *pTree){
if (pTree==0)
return;
printf("%d\n",pTree->nodeData);
traverse(pTree->leftChild);
traverse(pTree->rightChild);
}
I would like to create an iterator over the binary tree so as to be able to use range-based for loop. I understand I ought to implement the begin() and end() function first.
Begin should probably point to the root. According to the specification, however, the end() functions returns "the element following the last valid element". Which element (node) is that? Would it not be illegal to point to some "invalid" place?
The other thing is the operator++. What is the best way to return "next" element in tree? I just need some advice to begin with this programming.
I would like to expand/augment my question*. What if I wanted to iterate over a tree with an arbitrary arity? Let each node have a vector of children and let begin() point to the "real" root. I would probably have to implement a queue (for breadth-first) inside the iterator class to store the unique_ptr's to nodes, right? Then, when the queue is empty I would know that I have passed all nodes and thus should return TreeIterator(nullptr) when oprator++() is called. Does it make sense? I want it as simple as possible and only forward iteration.
*Or should I create a new thread?
Where your begin() should point to pretty much depends on the order in which you want to traverse your tree. Using the root may be sensible, e.g., for a breadth first walk of the tree. The end() doesn't really sit on a tree node: this position is accessed and indicates that the end of the sequence is reached. Whether it indicates anything related to the tree somewhat depends on what sort of iteration you want to support: when supporting only forward iteration it can just indicate the end. When also supporting bidirectional iteration, it needs to know how to find the node right before the end.
In any case, the place pointed to isn't really accessed and you need a suitable indicator. For a forward iteration only iterator end() could just return an iterator pointing to null and when you move on from the last node you just set the iterator's pointer to null as well: equality comparing the two pointers would yield true, indicating that you have reached the end. When wanting to support bidirectional iteration you'll need some sort of link record which can be used to navigate to the previous node but which doesn't need to store a value.
The ordered associated containers (std::map<K, V>, std:set<V>, etc.) are internally implemented as some sort of tree (e.g., a Red/Black-tree). The begin() iterator starts with the left-most node and the end() iterator refers to the position after the right-most node. The operator++() just finds the next node to the right of the current:
if the iterator sits on a node without a right child node, it walks along the chain of parents until it finds a parent reaching its child via the left branch of the tree
if it sits on a node with a right child node it walks to the child and then down the sequence of left children of this child (if any) to find the left-most child in the right subtree.
Obviously, if you don't walk your tree from left to right but rather, e.g., from top to bottom, you'll need a different algorithm. The easiest approach for me is to draw a tree on a piece of paper and see how to get to the next node.
If you haven't implemented a data structure using your own iterators I'd recommend trying things out on a simple sequential data structure, e.g., a list: There it is pretty obvious how to reach the next node and when the end is reached. Once the general iteration principle is clear, creating a tree is just a matter of getting the navigation right.
Look at SGI implementation of RBTree (this is the base for std::set/std::map... containers).
http://www.sgi.com/tech/stl/stl_tree.h
You will see that begin() is the leftmost node.
You will see that end() is a special "empty" node header which is the super root - I mean a real root (preset only if the tree is not empty) is its child node.
operator ++ is to go to right child and then find this child leftmost node.
If such child does not exist - we go to left parent of the rightmost parent node. As in this example (red lines are skip move, blue ones ends are the given steps of iteration):
The code copied from SGI:
void _M_increment()
{
if (_M_node->_M_right != 0) {
_M_node = _M_node->_M_right;
while (_M_node->_M_left != 0)
_M_node = _M_node->_M_left;
}
else {
_Base_ptr __y = _M_node->_M_parent;
while (_M_node == __y->_M_right) {
_M_node = __y;
__y = __y->_M_parent;
}
if (_M_node->_M_right != __y)
_M_node = __y;
}
}
When a tree is empty - begin() is leftmost of header - so it is header itself - end() is also header - so begin() == end(). Remember - your iteration scheme must match this condition begin() == end() for empty trees.
This seems to be very smart iteration scheme.
Of course you can define you own scheme - but the lesson learned is to have special node for end() purpose.
An iterator for a tree is going to be more than just a pointer, although it will likely contain a pointer:
struct TreeIterator {
TreeIterator(TreeNode *node_ptr) : node_ptr(node_ptr) { }
TreeIterator &operator++();
TreeIterator operator++(int);
bool operator==(const TreeIterator &) const;
private:
TreeNode *node_ptr;
};
TreeIterator begin(const Tree &tree) { ... }
TreeIterator end(const Tree &tree) { ... }
You can make your end() function return something special, like TreeIterator(nullptr)
What your begin iterator points to will depend on the type of traversal that you want. If you are doing breadth-first traversal, then starting at the root makes sense.
I have implement a link-based BST (binary search tree) in C++ for one of my assignment. I have written my whole class and everything works good, but my assignment asks me to plot the run-times for:
a. A sorted list of 50000, 75000, and 100000 items
b. A random list of 50000, 75000, and 100000 items
That's fine, I can insert the numbers but it also asks me to call the FindHeight() and CountLeaves() methods on the tree. My problem is that I've implemented the two functions using recursion. Since I have a such a big list of numbers I'm getting getting a stackoverflow exception.
Here's my class definition:
template <class TItem>
class BinarySearchTree
{
public:
struct BinarySearchTreeNode
{
public:
TItem Data;
BinarySearchTreeNode* LeftChild;
BinarySearchTreeNode* RightChild;
};
BinarySearchTreeNode* RootNode;
BinarySearchTree();
~BinarySearchTree();
void InsertItem(TItem);
void PrintTree();
void PrintTree(BinarySearchTreeNode*);
void DeleteTree();
void DeleteTree(BinarySearchTreeNode*&);
int CountLeaves();
int CountLeaves(BinarySearchTreeNode*);
int FindHeight();
int FindHeight(BinarySearchTreeNode*);
int SingleParents();
int SingleParents(BinarySearchTreeNode*);
TItem FindMin();
TItem FindMin(BinarySearchTreeNode*);
TItem FindMax();
TItem FindMax(BinarySearchTreeNode*);
};
FindHeight() Implementation
template <class TItem>
int BinarySearchTree<TItem>::FindHeight()
{
return FindHeight(RootNode);
}
template <class TItem>
int BinarySearchTree<TItem>::FindHeight(BinarySearchTreeNode* Node)
{
if(Node == NULL)
return 0;
return 1 + max(FindHeight(Node->LeftChild), FindHeight(Node->RightChild));
}
CountLeaves() implementation
template <class TItem>
int BinarySearchTree<TItem>::CountLeaves()
{
return CountLeaves(RootNode);
}
template <class TItem>
int BinarySearchTree<TItem>::CountLeaves(BinarySearchTreeNode* Node)
{
if(Node == NULL)
return 0;
else if(Node->LeftChild == NULL && Node->RightChild == NULL)
return 1;
else
return CountLeaves(Node->LeftChild) + CountLeaves(Node->RightChild);
}
I tried to think of how I can implement the two methods without recursion but I'm completely stumped. Anyone have any ideas?
Recursion on a tree with 100,000 nodes should not be a problem if it is balanced. The depth would only be maybe 17, which would not use very much stack in the implementations shown. (log2(100,000) = 16.61). So it seems that maybe the code that is building the tree is not balancing it correctly.
I found this page very enlightening because it talks about the mechanics of converting a function that uses recursion to one that uses iteration.
It has examples showing code as well.
May be you need to calculate this while doing the insert. Store the heights of nodes, i.e add an integer field like height in the Node object. Also have counters height and leaves for the tree. When you insert a node, if its parent is (was) a leaf, the leaf count doesnt change, but if not, increase leaf count by 1. Also the height of the new node is parent's height + 1, hence if that is greater than the current height of the tree, then update it. Its a homework, so i wont help with the actual code
Balance your tree occasionally. If your tree is getting stackoverflow on FindHeight(), that means your tree is way unbalanced. If the tree is balanced it should only have a depth of about 20 nodes for 100000 elements.
The easiest (but fairly slow) way of re-balancing unbalanced binary tree is to allocate an array of TItem big enough to hold all of the data in the tree, insert all of your data into it in sorted order, and delete all of the nodes. Then rebuild the tree from the array recursively. The root is the node in the middle. root->left is the middle of the left half, root->right is the middle of the right half. Repeat recursively. This is the easiest way to rebalance, but it is slowish and takes lots of memory temporarily. On the other hand, you only have to do this when you detect that the tree is very unbalanced, (depth on insert is more than 100).
The other (better) option is to balance during inserts. The most intuitive way to do this is to keep track of how many nodes are beneath the current node. If the right child has more than twice as many "child" nodes as the left child, "rotate" left. And vice-versa. There's instrcutions on how to do tree rotates all over the internet. This makes inserts slightly slower, but then you don't have occassional massive stalls that the first option creates. On the other hand, you have to constantly update all of the "children" counts as you do the rotates, which isn't trivial.
In order to count the leaves without recursion, use the concept of an iterator like the STL uses for the RB-tree underlying std::set and std::map ... Create a begin() and end() function for you tree that indentifies the ordered first and last node (in this case the left-most node and then the right-most node). Then create a function called
BinarySearchTreeNode* increment(const BinarySearchTreeNode* current_node)
that for a given current_node, will return a pointer to the next node in the tree. Keep in mind for this implementation to work, you will need an extra parent pointer in your node type to aid in the iteration process.
Your algorithm for increment() would look something like the following:
Check to see if there is a right-child to the current node.
If there is a right-child, use a while-loop to find the left-most node of that right subtree. This will be the "next" node. Otherwise go to step #3.
If there is no right-child on the current node, then check to see if the current node is the left-child of its parent node.
If step #3 is true, then the "next" node is the parent node, so you can stop at this point, otherwise go the next step.
If the step #3 was false, then the current node is the right-child of the parent. Thus you will need to keep moving up to the next parent node using a while loop until you come across a node that is a left-child of its parent node. The parent of this left-child node will then be the "next" node, and you can stop.
Finally, if step #5 returns you to the root, then the current node is the last node in the tree, and the iterator has reached the end of the tree.
Finally you'll need a bool leaf(const BinarySearchTreeNode* current_node) function that will test whether a given node is a leaf node. Thus you counter function can simply iterate though the tree and find all the leaf nodes, returning a final count once it's done.
If you want to measure the maximum depth of an unbalanced tree without recursion, you will, in your tree's insert() function, need to keep track of the depth that a node was inserted at. This can simply be a variable in your node type that is set when the node is inserted in the tree. You can then iterate through the three, and find the maximum depth of a leaf-node.
BTW, the complexity of this method is unfortunately going to be O(N) ... nowhere near as nice as O(log N).
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);
}