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<HTML><HEAD> <TITLE>Data Structures and Algorithms: Heaps</TITLE> <META name="description" content="Data Structures and Algorithms Course Notes, PLDS210 University of Western Australia"> <META name="keywords" content="data structures,algorithms,abstract data types, heap, complete tree"> </HEAD> <BODY BGCOLOR="#ffffff"> <TABLE BGCOLOR="#00c0f0" WIDTH="100%" CELLSPACING=0 CELLPADDING=0> <TR BGCOLOR="#00f0f0"><TD ALIGN=right> <FONT FACE=helvetica SIZE=+1><I>Data Structures and Algorithms</I></FONT> </TD></TR> <TR><TD><FONT FACE=helvetica SIZE=+2><B>6.2 Heaps</B></FONT> </TD></TR> </TABLE> <P> Heaps are based on the notion of a <FONT COLOR="#fa0000"><B>complete tree</B></FONT>, for which we gave an informal definition earlier. <P> Formally: <P> A binary tree is <FONT COLOR="#fa0000"><B>completely full</B></FONT> if it is of height, <I>h</I>, and has 2<SUP><I>h</I>+1</SUP>-1 nodes. <P> <A NAME=complete_tree> A binary tree of height, <I>h</I>, is <FONT COLOR="#fa0000"><B>complete</B></FONT> <I>iff</I> <OL TYPE="a"> <LI> it is empty <I>or</I> <LI> its left subtree is complete of height <I>h</I>-1 and its right subtree is completely full of height <I>h</I>-2 <I>or</I> <LI> its left subtree is completely full of height <I>h</I>-1 and its right subtree is complete of height <I>h</I>-1. </OL> A complete tree is filled from the left: <UL> <LI> all the leaves are on <UL> <LI> the same level <I>or</I> <LI> two adjacent ones <I>and</I> </UL> <LI> all nodes at the lowest level are as far to the left as possible. </UL> <H3>Heaps</H3> A binary tree has the <FONT COLOR="#fa0000"><B>heap property</B></FONT> <I>iff</I> <OL TYPE="a"> <LI> it is empty <I>or</I> <LI> the key in the root is larger than that in either child and both subtrees have the heap property. </OL> A heap can be used as a priority queue: the highest priority item is at the root and is trivially extracted. But if the root is deleted, we are left with two sub-trees and we must <I>efficiently</I> re-create a single tree with the heap property. <BLOCKQUOTE> The value of the heap structure is that we can both extract the highest priority item and insert a new one in <B>O(log<I>n</I>)</B> time. </BLOCKQUOTE> <P> How do we do this? <P> <TABLE> <TR><TD>Let's start with this heap. <P> A deletion will remove the T<BR> at the root. </TD> <TD><IMG SRC="fig/heap.gif"></TD></TR> </TABLE> <TABLE> <TR> <TD> To work out how we're going to maintain the heap property, use the fact that a complete tree is filled from the left. So that the position which must become empty is the one occupied by the M. <P> Put it in the vacant root position. </TD> <TD><IMG SRC="fig/heap1.gif"></TD></TR> </TABLE> <TABLE> <TR> <TD> This has violated the condition that the root must be greater than each of its children. <P> So interchange the M with the larger of its children. <P> </TD> <TD><IMG SRC="fig/heap2.gif"></TD></TR> </TABLE> <TABLE> <TR> <TD> The left subtree has now lost the heap property. <P> So again interchange the M with the larger of its children. <P> </TD> <TD><IMG SRC="fig/heap3.gif"></TD></TR> </TABLE> This tree is now a heap again, so we're finished. <P> We need to make at most <I>h</I> interchanges of a root of a subtree with one of its children to fully restore the heap property. Thus deletion from a heap is <B>O(<I>h</I>)</B> or <B>O(log<I>n</I>)</B>. <H3>Addition to a heap</H3> <TABLE> <TR> <TD> To add an item to a heap, we follow the reverse procedure. <P> Place it in the next leaf position and move it <B>up</B>. <P> Again, we require <B>O(<I>h</I>)</B> or <B>O(log<I>n</I>)</B> exchanges. </TD> <TD><IMG SRC="fig/heap4.gif"></TD></TR> </TABLE> <HR> <H3>Storage of complete trees</H3> The properties of a complete tree lead to a very efficient storage mechanism using <I>n</I> sequential locations in an array. <TABLE> <TR> <TD><IMG SRC="fig/heap5.gif"></TD> <TD> If we number the nodes from 1 at the root and place: <UL> <LI>the left child of node <I>k</I> at position 2<I>k</I> <LI>the right child of node <I>k</I> at position 2<I>k</I>+1 </UL> <P> Then the 'fill from the left' nature of the complete tree ensures that the heap can be stored in consecutive locations in an array. </TD> </TR> </TABLE> <TABLE> <TR> <TD> Viewed as an array, we can see that the <I>n</I>th node is always in index position <I>n</I>. </TD> <TD><IMG SRC="fig/heap6.gif"></TD> </TR> </TABLE> <P> The code for extracting the highest priority item from a heap is, naturally, recursive. Once we've extracted the root (highest priority) item and swapped the last item into its place, we simply call <TT><FONT COLOR=green>MoveDown</FONT></TT> recursively until we get to the bottom of the tree. <P> <A HREF=source/heap_delete.c TARGET=heap_delete.c>Click here to load heap_delete.c</A> <P> Note the macros <TT><FONT COLOR=green>LEFT</FONT></TT> and <TT><FONT COLOR=green>RIGHT</FONT></TT> which simply encode the relation between the index of a node and its left and right children. Similarly the <TT><FONT COLOR=green>EMPTY</FONT></TT> macro encodes the rule for determining whether a sub-tree is empty or not. <P> Inserting into a heap follows a similar strategy, except that we use a <TT><FONT COLOR=green>MoveUp</FONT></TT> function to move the newly added item to its correct place. (For the <TT><FONT COLOR=green>MoveUp</FONT></TT> function, a further macro which defines the <TT><FONT COLOR=green>PARENT</FONT></TT> of a node would normally be added.) <P> Heaps provide us with a method of sorting, known as <A HREF=heapsort.html>heapsort</A>. However, we will examine and analyse the simplest method of sorting first. <P> <A NAME=p_queue_anim> <H3>Animation</H3> In the animation, note that both the array representation (used in the implementation of the algorithm) and the (logical) tree representation are shown. This is to demonstrate how the tree is restructured to make a heap again after every insertion or deletion. <P> <TABLE BGCOLOR="#00f0f0" WIDTH="100%"> <TR><TD > <FONT COLOR=blue FACE=helvetica> <B>Priority Queue Animation</B><BR> This animation was written by Woi Ang.</FONT></TD> <TD ALIGN=center> <TABLE BORDER=0> <TR><TD> <applet CODEBASE="Java/p_queue" code = "AlgAnimApp.class" width = 200 height = 35> <param name = "filename" value = "AlgThread.java"> <param name = "buttonname" value = "Run the Animation"> <param name = "algname" value = "Priority Queue"> </applet> </TD> </TR> </TABLE> </TD> <TD><FONT FACE=helvetica>Please email comments to:<BR> <img src="john_email.gif"> </TR> </TABLE> <P> <TABLE WIDTH="100%" BGCOLOR="#00c0f0"> <TR><TD><H3>Key terms</H3></TD></TR></TABLE> <TABLE><TR><TD> <DL> <DT><FONT COLOR="#fa0000"><B>Complete Tree</B></FONT> <DD>A balanced tree in which the distance from the root to any leaf is either <I><B>h</B></I> or <I><B>h</B></I>-1. </DL> </TD></TR></TABLE> <P> <TABLE CELLPADDING=5 WIDTH="100%" BGCOLOR="#00f0f0" CELLSPACING=0> <TR><TD WIDTH="50%"> Continue on to <A HREF="sorting.html">Sorting</A></TD> <TD>Back to the <A HREF="ds_ToC.html">Table of Contents</A> </TD></TR></TABLE> <SMALL> © <img src="john_email.gif">, 1998 </SMALL> </BODY> </HTML>