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/*
* (C) Copyright 2014-2016, by Dimitrios Michail
*
* JHeaps Library
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.jheaps.tree;
import java.io.Serializable;
import java.lang.reflect.Array;
import java.util.Comparator;
import java.util.NoSuchElementException;
import org.jheaps.AddressableHeap;
import org.jheaps.MergeableAddressableHeap;
import org.jheaps.annotations.ConstantTime;
import org.jheaps.annotations.LogarithmicTime;
/**
* Simple Fibonacci heaps. The heap is sorted according to the
* {@linkplain Comparable natural ordering} of its keys, or by a
* {@link Comparator} provided at heap creation time, depending on which
* constructor is used.
*
*
* This variant of the Fibonacci heaps is described in detail in the following
* paper:
*
* - Haim Kaplan, Robert E. Tarjan, Uri Zwick, Fibonacci Heaps Revisited,
* arXiv 1407.5750, 2014.
*
*
*
* This implementation provides amortized O(1) time for operations that do not
* involve deleting an element such as {@code insert}, and {@code decreaseKey}.
* Operation {@code findMin} is worst-case O(1). Operations {@code deleteMin}
* and {@code delete} are amortized O(log(n)). The operation {@code meld} is
* also amortized O(1).
*
*
* All the above bounds, however, assume that the user does not perform
* cascading melds on heaps such as:
*
*
* d.meld(e);
* c.meld(d);
* b.meld(c);
* a.meld(b);
*
*
* The above scenario, although efficiently supported by using union-find with
* path compression, invalidates the claimed bounds.
*
*
* Note that the ordering maintained by this heap, like any heap, and whether or
* not an explicit comparator is provided, must be consistent with
* {@code equals} if this heap is to correctly implement the
* {@code AddressableHeap} interface. (See {@code Comparable} or
* {@code Comparator} for a precise definition of consistent with
* equals.) This is so because the {@code AddressableHeap} interface is
* defined in terms of the {@code equals} operation, but this heap performs all
* key comparisons using its {@code compareTo} (or {@code compare}) method, so
* two keys that are deemed equal by this method are, from the standpoint of the
* Fibonacci heap, equal. The behavior of a heap is well-defined even
* if its ordering is inconsistent with {@code equals}; it just fails to obey
* the general contract of the {@code AddressableHeap} interface.
*
*
* Note that this implementation is not synchronized. If
* multiple threads access a heap concurrently, and at least one of the threads
* modifies the heap structurally, it must be synchronized externally.
* (A structural modification is any operation that adds or deletes one or more
* elements or changing the key of some element.) This is typically accomplished
* by synchronizing on some object that naturally encapsulates the heap.
*
* @param
* the type of keys maintained by this heap
* @param
* the type of values maintained by this heap
*
* @author Dimitrios Michail
*
*/
public class SimpleFibonacciHeap implements MergeableAddressableHeap, Serializable {
private final static long serialVersionUID = 1;
/**
* Size of consolidation auxiliary array. Computed for number of elements
* equal to {@link Long#MAX_VALUE}.
*/
private final static int AUX_CONSOLIDATE_ARRAY_SIZE = 91;
/**
* The comparator used to maintain order in this heap, or null if it uses
* the natural ordering of its keys.
*
* @serial
*/
private final Comparator comparator;
/**
* The root
*/
private Node root;
/**
* Size of the heap
*/
private long size;
/**
* Auxiliary array for consolidation
*/
private Node[] aux;
/**
* Used to reference the current heap or some other heap in case of melding,
* so that handles remain valid even after a meld, without having to iterate
* over them.
*
* In order to avoid maintaining a full-fledged union-find data structure,
* we disallow a heap to be used in melding more than once. We use however,
* path-compression in case of cascading melds, that it, a handle moves from
* one heap to another and then another.
*/
protected SimpleFibonacciHeap other;
/**
* Constructs a new, empty heap, using the natural ordering of its keys. All
* keys inserted into the heap must implement the {@link Comparable}
* interface. Furthermore, all such keys must be mutually
* comparable: {@code k1.compareTo(k2)} must not throw a
* {@code ClassCastException} for any keys {@code k1} and {@code k2} in the
* heap. If the user attempts to put a key into the heap that violates this
* constraint (for example, the user attempts to put a string key into a
* heap whose keys are integers), the {@code insert(Object key)} call will
* throw a {@code ClassCastException}.
*/
@ConstantTime
public SimpleFibonacciHeap() {
this(null);
}
/**
* Constructs a new, empty heap, ordered according to the given comparator.
* All keys inserted into the heap must be mutually comparable by
* the given comparator: {@code comparator.compare(k1,
* k2)} must not throw a {@code ClassCastException} for any keys {@code k1}
* and {@code k2} in the heap. If the user attempts to put a key into the
* heap that violates this constraint, the {@code insert(Object key)} call
* will throw a {@code ClassCastException}.
*
* @param comparator
* the comparator that will be used to order this heap. If
* {@code null}, the {@linkplain Comparable natural ordering} of
* the keys will be used.
*/
@ConstantTime
@SuppressWarnings("unchecked")
public SimpleFibonacciHeap(Comparator comparator) {
this.root = null;
this.comparator = comparator;
this.size = 0;
this.aux = (Node[]) Array.newInstance(Node.class, AUX_CONSOLIDATE_ARRAY_SIZE);
this.other = this;
}
/**
* {@inheritDoc}
*
* @throws IllegalStateException
* if the heap has already been used in the right hand side of a
* meld
*/
@Override
@SuppressWarnings("unchecked")
@ConstantTime(amortized = true)
public AddressableHeap.Handle insert(K key, V value) {
if (other != this) {
throw new IllegalStateException("A heap cannot be used after a meld");
}
if (key == null) {
throw new NullPointerException("Null keys not permitted");
}
Node n = new Node(this, key, value);
if (root == null) {
root = n;
} else {
if (comparator == null) {
if (((Comparable) n.key).compareTo(root.key) < 0) {
root = link(root, n);
} else {
link(n, root);
}
} else {
if (comparator.compare(n.key, root.key) < 0) {
root = link(root, n);
} else {
link(n, root);
}
}
}
size++;
return n;
}
/**
* {@inheritDoc}
*
* @throws IllegalStateException
* if the heap has already been used in the right hand side of a
* meld
*/
@Override
@ConstantTime(amortized = true)
public AddressableHeap.Handle insert(K key) {
return insert(key, null);
}
/**
* {@inheritDoc}
*/
@Override
@ConstantTime(amortized = true)
public AddressableHeap.Handle findMin() {
if (size == 0) {
throw new NoSuchElementException();
}
return root;
}
/**
* {@inheritDoc}
*/
@Override
@LogarithmicTime(amortized = true)
public AddressableHeap.Handle deleteMin() {
if (comparator == null) {
return comparableDeleteMin();
} else {
return comparatorDeleteMin();
}
}
/**
* {@inheritDoc}
*/
@Override
@ConstantTime
public boolean isEmpty() {
return size == 0;
}
/**
* {@inheritDoc}
*/
@Override
@ConstantTime
public long size() {
return size;
}
/**
* {@inheritDoc}
*/
@Override
public Comparator comparator() {
return comparator;
}
/**
* {@inheritDoc}
*/
@Override
@ConstantTime
public void clear() {
root = null;
size = 0;
}
/**
* {@inheritDoc}
*/
@Override
@SuppressWarnings("unchecked")
@ConstantTime(amortized = true)
public void meld(MergeableAddressableHeap other) {
SimpleFibonacciHeap h = (SimpleFibonacciHeap) other;
// check same comparator
if (comparator != null) {
if (h.comparator == null || !h.comparator.equals(comparator)) {
throw new IllegalArgumentException("Cannot meld heaps using different comparators!");
}
} else if (h.comparator != null) {
throw new IllegalArgumentException("Cannot meld heaps using different comparators!");
}
if (h.other != h) {
throw new IllegalStateException("A heap cannot be used after a meld.");
}
// meld
if (root == null) {
root = h.root;
} else if (h.root != null) {
if (comparator == null) {
if (((Comparable) h.root.key).compareTo(root.key) < 0) {
root = link(root, h.root);
} else {
link(h.root, root);
}
} else {
if (comparator.compare(h.root.key, root.key) < 0) {
root = link(root, h.root);
} else {
link(h.root, root);
}
}
}
size += h.size;
// clear other
h.size = 0;
h.root = null;
// take ownership
h.other = this;
}
// --------------------------------------------------------------------
static class Node implements AddressableHeap.Handle, Serializable {
private final static long serialVersionUID = 1;
/*
* We maintain explicitly the belonging heap, instead of using an inner
* class due to possible cascading melding.
*/
SimpleFibonacciHeap heap;
K key;
V value;
Node parent; // parent
Node child; // any child
Node next; // younger sibling
Node prev; // older sibling
int rank; // node rank
boolean mark; // marked or not
Node(SimpleFibonacciHeap heap, K key, V value) {
this.heap = heap;
this.key = key;
this.value = value;
this.parent = null;
this.child = null;
this.next = this;
this.prev = this;
this.rank = 0;
this.mark = false;
}
/**
* {@inheritDoc}
*/
@Override
public K getKey() {
return key;
}
/**
* {@inheritDoc}
*/
@Override
public V getValue() {
return value;
}
/**
* {@inheritDoc}
*/
@Override
public void setValue(V value) {
this.value = value;
}
/**
* {@inheritDoc}
*/
@Override
@ConstantTime(amortized = true)
public void decreaseKey(K newKey) {
SimpleFibonacciHeap h = getOwner();
if (h.comparator == null) {
h.comparableDecreaseKey(this, newKey);
} else {
h.comparatorDecreaseKey(this, newKey);
}
}
/**
* {@inheritDoc}
*/
@Override
@LogarithmicTime(amortized = true)
public void delete() {
if (this.next == null) {
throw new IllegalArgumentException("Invalid handle!");
}
SimpleFibonacciHeap h = getOwner();
h.forceDecreaseKeyToMinimum(this);
h.deleteMin();
}
/*
* Get the owner heap of the handle. This is union-find with
* path-compression between heaps.
*/
SimpleFibonacciHeap getOwner() {
if (heap.other != heap) {
// find root
SimpleFibonacciHeap root = heap;
while (root != root.other) {
root = root.other;
}
// path-compression
SimpleFibonacciHeap cur = heap;
while (cur.other != root) {
SimpleFibonacciHeap next = cur.other;
cur.other = root;
cur = next;
}
heap = root;
}
return heap;
}
}
/*
* Decrease the key of a node.
*/
@SuppressWarnings("unchecked")
private void comparableDecreaseKey(Node n, K newKey) {
int c = ((Comparable) newKey).compareTo(n.key);
if (c > 0) {
throw new IllegalArgumentException("Keys can only be decreased!");
}
n.key = newKey;
if (c == 0) {
return;
}
if (n.next == null) {
throw new IllegalArgumentException("Invalid handle!");
}
// if not root and heap order violation
Node y = n.parent;
if (y != null && ((Comparable) n.key).compareTo(y.key) < 0) {
cut(n, y);
root.mark = false;
cascadingRankChange(y);
if (((Comparable) n.key).compareTo(root.key) < 0) {
root = link(root, n);
} else {
link(n, root);
}
}
}
/*
* Decrease the key of a node.
*/
private void comparatorDecreaseKey(Node n, K newKey) {
int c = comparator.compare(newKey, n.key);
if (c > 0) {
throw new IllegalArgumentException("Keys can only be decreased!");
}
n.key = newKey;
if (c == 0) {
return;
}
if (n.next == null) {
throw new IllegalArgumentException("Invalid handle!");
}
// if not root and heap order violation
Node y = n.parent;
if (y != null && comparator.compare(n.key, y.key) < 0) {
cut(n, y);
root.mark = false;
cascadingRankChange(y);
if (comparator.compare(n.key, root.key) < 0) {
root = link(root, n);
} else {
link(n, root);
}
}
}
/*
* Decrease the key of a node to the minimum. Helper function for performing
* a delete operation. Does not change the node's actual key, but behaves as
* the key is the minimum key in the heap.
*/
private void forceDecreaseKeyToMinimum(Node n) {
Node y = n.parent;
if (y != null) {
cut(n, y);
root.mark = false;
cascadingRankChange(y);
root = link(root, n);
}
}
@SuppressWarnings("unchecked")
private AddressableHeap.Handle comparableDeleteMin() {
if (size == 0) {
throw new NoSuchElementException();
}
Node z = root;
Node x = root.child;
// clear fields of previous root
z.child = null;
z.next = null;
z.prev = null;
// simple case, no children
if (x == null) {
root = null;
size = 0;
return z;
}
// iterate over all children of root
int maxDegree = -1;
while (x != null) {
Node nextX = (x.next == x) ? null : x.next;
// clear parent
x.parent = null;
// remove from child list
x.prev.next = x.next;
x.next.prev = x.prev;
x.next = x;
x.prev = x;
int d = x.rank;
while (true) {
Node y = aux[d];
if (y == null) {
break;
}
// make sure x's key is smaller
if (((Comparable) y.key).compareTo(x.key) < 0) {
Node tmp = x;
x = y;
y = tmp;
}
// make y a child of x
link(y, x);
// make link fair by increasing rank
x.rank++;
aux[d] = null;
d++;
}
// store result
aux[d] = x;
// keep track of max degree
if (d > maxDegree) {
maxDegree = d;
}
// advance
x = nextX;
}
// recreate tree
int i = 0;
while (i <= maxDegree && aux[i] == null) {
i++;
}
root = aux[i];
aux[i] = null;
i++;
while (i <= maxDegree) {
Node n = aux[i];
if (n != null) {
if (((Comparable) n.key).compareTo(root.key) < 0) {
root = link(root, n);
} else {
link(n, root);
}
aux[i] = null;
}
i++;
}
// decrease size
size--;
return z;
}
private AddressableHeap.Handle comparatorDeleteMin() {
if (size == 0) {
throw new NoSuchElementException();
}
Node z = root;
Node x = root.child;
// clear fields of previous root
z.child = null;
z.next = null;
z.prev = null;
// simple case, no children
if (x == null) {
root = null;
size = 0;
return z;
}
// iterate over all children of root
int maxDegree = -1;
while (x != null) {
Node nextX = (x.next == x) ? null : x.next;
// clear parent
x.parent = null;
// remove from child list
x.prev.next = x.next;
x.next.prev = x.prev;
x.next = x;
x.prev = x;
int d = x.rank;
while (true) {
Node y = aux[d];
if (y == null) {
break;
}
// make sure x's key is smaller
if (comparator.compare(y.key, x.key) < 0) {
Node tmp = x;
x = y;
y = tmp;
}
// make y a child of x
link(y, x);
// make link fair by increasing rank
x.rank++;
aux[d] = null;
d++;
}
// store result
aux[d] = x;
// keep track of max degree
if (d > maxDegree) {
maxDegree = d;
}
// advance
x = nextX;
}
// recreate tree
int i = 0;
while (i <= maxDegree && aux[i] == null) {
i++;
}
root = aux[i];
aux[i] = null;
i++;
while (i <= maxDegree) {
Node n = aux[i];
if (n != null) {
if (comparator.compare(n.key, root.key) < 0) {
root = link(root, n);
} else {
link(n, root);
}
aux[i] = null;
}
i++;
}
// decrease size
size--;
return z;
}
/*
* (unfair) Link y as a child of x.
*/
private Node link(Node y, Node x) {
y.parent = x;
Node child = x.child;
if (child == null) {
x.child = y;
y.next = y;
y.prev = y;
} else {
y.prev = child;
y.next = child.next;
child.next = y;
y.next.prev = y;
}
return x;
}
/*
* Cut the link between x and its parent y.
*/
private void cut(Node x, Node y) {
// advance y child
y.child = x.next;
if (y.child == x) {
y.child = null;
}
// remove x from child list of y
x.prev.next = x.next;
x.next.prev = x.prev;
x.next = x;
x.prev = x;
x.parent = null;
// clear mark
x.mark = false;
}
/*
* Cascading rank change. Assumes that the root is unmarked.
*/
private void cascadingRankChange(Node y) {
while (y.mark == true) {
y.mark = false;
if (y.rank > 0) {
--y.rank;
}
y = y.parent;
}
y.mark = true;
if (y.rank > 0) {
--y.rank;
}
}
}