/// <inheritdoc cref="IHeap{TKey,TValue}.Update"/>
public override void Update(BinomialHeapNode <TKey, TValue> node, TKey key)
{
if (node == null)
{
throw new ArgumentNullException(nameof(node));
}
var relation = this.Comparer.Compare(key, node.Key);
node.Key = key;
// If the new value is considered equal to the previous value, there is no need
// to fix the heap property, because it is already preserved.
if (relation < 0)
{
this.MoveUp(node);
}
else if (relation > 0)
{
// If one of the roots is moved down (or it is not moved, but its value is
// modified) then there is a chance that the top should be updated as well.
var isRootBeingUpdated = node.Parent == null;
this.MoveDown(node);
if (isRootBeingUpdated)
{
this.UpdateTop();
}
}
}
/// <summary>
/// Breaks a tree represented by the specified node apart and inserts all its
/// subtrees to the list of binomial trees.
/// </summary>
/// <param name="node">The root of the tree to break.</param>
private void BreakTreeAndInsertChildren(BinomialHeapNode <TKey, TValue> node)
{
var current = node.Left;
while (current != null)
{
var next = current.Right;
this.MakeRoot(current);
current = next;
}
}
/// <inheritdoc cref="IHeap{TKey,TValue}.Add"/>
public override BinomialHeapNode <TKey, TValue> Add(TKey key, TValue value)
{
// Create a one-node tree for the specified item and perform
// the usual merging of heaps.
var tree = new BinomialHeapNode <TKey, TValue>(key, value);
this.MakeRoot(tree);
++this.count;
return(tree);
}
/// <summary>
/// Merges two trees of equal rank and returns the root of the resulting heap.
/// </summary>
private BinomialHeapNode <TKey, TValue> Merge(BinomialHeapNode <TKey, TValue> a, BinomialHeapNode <TKey, TValue> b)
{
// If both binomial heaps are non-empty, the merge function returns
// a new heap where the smallest root of the two heaps is the root of
// the new combined heap and adds the other heap to the list of its children.
BinomialHeapNode <TKey, TValue> parent, child;
if (this.Comparer.Compare(a.Key, b.Key) < 0)
{
parent = a;
child = b;
}
else
{
parent = b;
child = a;
}
// The smallest root will be the new leftmost child of the largest root.
// Thus, make a parent-child relation.
child.Right = parent.Left;
if (parent.Left != null)
{
parent.Left.Parent = child;
}
child.Parent = parent;
parent.Left = child;
++parent.Rank;
// Adjust the top if we improved it.
if (this.top == child)
{
this.top = parent;
}
return(parent);
}
/// <summary>
/// Moves the node down in the tree to restore heap order.
/// </summary>
/// <param name="node">The node to be moved down.</param>
private void MoveDown(BinomialHeapNode <TKey, TValue> node)
{
// We want to move one node down in a binomial tree. However,
// what we have is a binary tree. We need to go to the first child
// and then iterate all the way right (to check all siblings). Our goal
// is to find the child that should be extracted first from the heap.
// Repeat the process until the heap property is restored.
// Check if the current node should really be extracted first, or maybe
// one of its children should be extracted earlier.
var topChild = node.Left;
// In case the current node has no children, there is nothing more to do.
// It cannot be moved down any further.
if (topChild == null)
{
return;
}
// There are some children, so browse them and find the top one.
for (var child = topChild.Right; child != null; child = child.Right)
{
if (this.Comparer.Compare(child.Key, topChild.Key) < 0)
{
topChild = child;
}
}
// In case no child needs to be extracted earlier than the current node,
// there is nothing more to do - the right spot was found.
if (this.Comparer.Compare(node.Key, topChild.Key) <= 0)
{
return;
}
// Move the top child up by one node and now investigate the
// node that was considered to be the top child (recursive).
this.SwapWithParent(topChild, node);
this.MoveDown(node);
}
/// <summary>
/// Tries to insert a tree into a collection of binomial trees. If the operation
/// fails, it returns a merged tree.
/// </summary>
/// <param name="node">The root of the tree to insert.</param>
/// <param name="merged">The result of merging trees if the appropriate slot is occupied.</param>
private bool TryInsert(BinomialHeapNode <TKey, TValue> node, out BinomialHeapNode <TKey, TValue> merged)
{
// If the appropriate slot for our tree is empty, simply insert the root
// and return. However, if there is already a tree with the same rank,
// join the two and pass it as the out parameter.
var rank = node.Rank;
var root = this.roots[rank];
if (root == null)
{
this.roots[rank] = node;
merged = null;
return(true);
}
this.roots[rank] = null;
merged = this.Merge(root, node);
return(false);
}
/// <summary>
/// Iterates through the collection of binomial trees and updates the top node.
/// </summary>
private void UpdateTop()
{
foreach (var root in this.roots)
{
if (root == null)
{
continue;
}
if (this.top == null)
{
this.top = root;
continue;
}
if (this.Comparer.Compare(root.Key, this.top.Key) < 0)
{
this.top = root;
}
}
}
/// <summary>
/// Moves the node up in the tree to restore heap order.
/// </summary>
/// <param name="node">The node to be moved up.</param>
private void MoveUp(BinomialHeapNode <TKey, TValue> node)
{
// Exchange the element with its parent, and possibly also with its
// grandparent, and so on, until the heap property is no longer violated.
// Each binomial tree has height at most log n, so this takes O(log n) time.
while (node.Parent != null)
{
var parent = node.Parent;
var current = node;
// Remember we want to go up in a binomial tree having a binary tree.
// For that reason, we need to find a parent in a binary tree. Thus,
// go up until you reach the leftmost sibling. From there, go up once more.
// You'll reach the parent in the binomial tree.
while (parent.Right == current)
{
current = current.Parent;
parent = current.Parent;
}
if (this.Comparer.Compare(node.Key, parent.Key) < 0)
{
this.SwapWithParent(node, parent);
}
else
{
break;
}
}
// If we got to the very top of our tree, there is a chance that the global
// top (for the collection of binomial trees) should be updated as well.
if (this.Comparer.Compare(node.Key, this.top.Key) < 0)
{
this.top = node;
}
}
/// <inheritdoc cref="IHeap{TKey,TValue}.Remove"/>
public override TValue Remove(BinomialHeapNode <TKey, TValue> node)
{
if (node == null)
{
throw new ArgumentNullException(nameof(node));
}
// To delete a node from the heap we will move it up to the top and
// then simply pop it. Instead of assigning some 'hacky' values like
// negative infinity, we will temporarily use a fake comparer that always
// says the first parameter is smaller.
var result = node.Value;
this.SwapRealAndFakeComparer();
this.MoveUp(node);
this.SwapRealAndFakeComparer();
this.Pop();
return(result);
}
/// <summary>
/// Makes an arbitrary node a root.
/// </summary>
/// <param name="node">Node to be made a root.</param>
private void MakeRoot(BinomialHeapNode <TKey, TValue> node)
{
// Roots have no parents and siblings, so adjust the pointers accordingly.
node.Parent = null;
node.Right = null;
// Since we insert a new root to out collection of binomial trees,
// there is a chance that the global top should be updated as well.
if (this.top == null || this.Comparer.Compare(node.Key, this.top.Key) < 0)
{
this.top = node;
}
// Now, as long as the appropriate slot for our tree is occupied by another
// tree with the same rank, join the two and continue the process.
while (this.TryInsert(node, out node) == false)
{
}
}
/// <inheritdoc cref="IHeap{TKey,TValue}.Pop"/>
public override BinomialHeapNode <TKey, TValue> Pop()
{
// Get the root with the top value and remove it from the collection
// of binomial trees.
var result = this.Peek();
var toRemove = this.top;
this.top = null;
this.roots[toRemove.Rank] = null;
// Now we should break the tree represented by the node 'toRemove' apart
// and insert back all its subtrees to our list of binomial trees.
this.BreakTreeAndInsertChildren(toRemove);
// The old top node is ready to be garbage collected (unless the user
// owns a reference to it). Adjust the number of elements and update the top.
--this.count;
this.UpdateTop();
return(result);
}
/// <summary>
/// Swaps a node with its parent.
/// </summary>
private void SwapWithParent(BinomialHeapNode <TKey, TValue> node, BinomialHeapNode <TKey, TValue> parent)
{
// Remember that binomial trees are represented here as binary trees.
// This makes thinking about swapping much easier.
var np = node.Parent;
var nl = node.Left;
var nr = node.Right;
var pp = parent.Parent;
var pl = parent.Left;
var pr = parent.Right;
// Let's start from adjusting the ranks. Basic swapping.
var tmp = node.Rank;
node.Rank = parent.Rank;
parent.Rank = tmp;
// For the node and its parent we will now adjust their
// left and right pointers.
parent.Left = nl;
if (nl != null)
{
nl.Parent = parent;
}
parent.Right = nr;
if (nr != null)
{
nr.Parent = parent;
}
node.Right = pr;
if (pr != null)
{
pr.Parent = node;
}
// To adjust the left pointer of the node we need to consider two cases,
// depending on whether our 'node' in the binary tree is the left child of
// the 'parent' (equivalently, it is also the first child of the 'parent'
// in the corresponding binomial tree then).
if (node == pl)
{
// 'node' is indeed the left child of the 'parent', so swapping is simple.
parent.Parent = node;
node.Left = parent;
}
else
{
// 'node' is not the left child of the 'parent', so we also need to adjust
// 'pl' and 'np', which are some other nodes.
node.Left = pl;
if (pl != null)
{
pl.Parent = node;
}
parent.Parent = np;
np.Right = parent;
}
// Now we're left with adjusting the grandparent of our node.
node.Parent = pp;
if (pp == null)
{
// Our node's parent was a root, so right now the node is a new root.
// We need to update the array of roots then. The top *could* be updated
// here, but this is done elsewhere as an optimization.
this.roots[node.Rank] = node;
}
else
{
// Our node is not a new root.
if (pp.Left == parent)
{
pp.Left = node;
}
else
{
pp.Right = node;
}
}
}
