A heap is a partially sorted binary tree. Although a heap is not completely in order, it conforms to a sorting principle: every node has a value less (for the sake of simplicity, we will assume that all orderings are from least to greatest) than either of its children. Additionally, a heap is a "complete tree" -- a complete tree is one in which there are no gaps between leaves. For instance, a tree with a root node that has only one child must have its child as the left node. More precisely, a complete tree is one that has every level filled in before adding a node to the next level, and one that has the nodes in a given level filled in from left to right, with no breaks.
Why use a
heap?
A heap can be thought of as a priority queue; the most
important node will always be at the top, and when removed, its replacement will
be the most important. This can be useful when coding algorithms that require
certain things to processed in a complete order, but when you don‘t want to
perform a full sort or need to know anything about the rest of the nodes. For
instance, a well-known algorithm for finding the shortest distance between nodes
in a graph, Dijkstra‘s Algorithm, can be optimized by using a priority
queue.
Heaps can also be used to sort data. A heap sort is
O(nlogn) efficiency, though it is not the fastest possible sorting algorithm.
Check out thistutorial
heap sort for more information related to heap
sort.
How
do you implement a heap?
Although the concept of a heap is
simple, the actual implementation can appear tricky. How do you remove the root
node and still ensure that it is eventually replaced by the correct node? How do
you add a new node to a heap and ensure that it is moved into the proper
spot?
The answers to these questions are more straight
forward than meets the eye, but to understand the process, let‘s first take a
look at two operations that are used for adding and removing nodes from a heap:
upheaping and downheaping.
Upheap:
The upheap process is used to add a node to a heap. When you upheap a node, you
compare its value to its parent node; if its value is less than its parent node,
then you switch the two nodes and continue the process. Otherwise the condition
is met that the parent node is less than the child node, and so you can stop the
process. Once you find a parent node that is less than the node being upheaped,
you know that the heap is correct--the node being upheaped is greater than its
parent, and its parent is greater than its own parent, all the way up to the
root.
Downheap: The downheap process
is similar to the upheaping process. When you downheap a node, you compare its
value with its two children. If the node is less than both of its children, it
remains in place; otherwise, if it is greater than one or both of its children,
then you switch it with the child of lowest value, thereby ensuring that of the
three nodes being compared, the new parent node is lowest. Of course, you cannot
be assured that the node being downheaped is in its proper position -- it may be
greater than one or both of its new children; the downheap process must be
repeated until the node is less than both of its
children. 
When
you add a new node to a heap, you add it to the rightmost unoccupied leaf on the
lowest level. Then you upheap that node until it has reached its proper
position. In this way, the heap‘s order is maintained and the heap remains a
complete tree.
Removing
the root node from a heap is almost as simple: when you take the node out of the
tree, you replace it with "last" node in the tree: the node on the last level
and rightmost on that level.
Once the top node has been
replaced, you downheap the node that was moved until it reaches its proper
position. As usual, the result will be a proper heap, as it will be complete,
and even if the node in the last position happens to be the greatest node in the
entire heap, it will do no worse than end up back where it
started.
Efficiency
of a heap
Whenever you work with a heap, most of the time
taken by the algorithm will be in upheaping and downheaping. As it happens, the
maximum number of levels of a complete tree is log(n)+1, where n is the number
of nodes in the tree. Because upheap or downheap moves an element from one level
to another, the order of adding to or removing from a heap is O(logn), as you
can make switches only log(n) times, or one less time than the number of levels
in the tree (consider that a two level tree can have only one
switch).
reference from :http://www.cprogramming.com/tutorial/computersciencetheory/heap.html
What is a heap?--reference,布布扣,bubuko.com
原文：http://www.cnblogs.com/davidwang456/p/3708618.html