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<H1>[MUD-Dev] Re: Red Black Tree ?</H1>
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<LI><em>To</em>: <A HREF="mailto:mud-dev#kanga,nu">mud-dev#kanga,nu</A></LI>
<LI><em>Subject</em>: [MUD-Dev] Re: Red Black Tree ? </LI>
<LI><em>From</em>: "T. Alexander Popiel" <<A HREF="mailto:popiel#snugharbor,com">popiel#snugharbor,com</A>></LI>
<LI><em>Date</em>: Fri, 09 Oct 1998 11:06:39 -0600</LI>
<LI><em>cc</em>: <A HREF="mailto:popiel#snugharbor,com">popiel#snugharbor,com</A></LI>
<LI><em>Reply-To</em>: <A HREF="mailto:mud-dev#kanga,nu">mud-dev#kanga,nu</A></LI>
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<PRE>
In message: <<A HREF="msg00135.html">361E2BAC.E9C9C8FE#mediacom,it</A>>
Valerio Santinelli <tanis#mediacom,it> writes:
>
>Can anybody explain me what's a red black tree or just point me to a page
>with some docs about it ?
There are several pages on the web of fairly uniformly low quality
(just search for 'red-black tree'). Here's the beginning of what
_Introduction to Algorithms_ by Cormen, Leiserson, and Rivest has
to say on the subject:
# A red-black tree is a binary search tree with one extra bit of storage per
# node: its color, which can be either RED or BLACK. By constraining the
# way nodes can be colored on any path from the root to a leaf, red-black
# trees ensure that no such path is more than twice as long as any other,
# so that the tree is approximately balanced.
#
# Each node of the tree now contains the fields color, key, left,
# right, and p. If a chils or the parent of a node does not exist,
# the corresponding pointer field of the node contains the value NIL.
# We shall regard these NIL's as being pointers to external nodes (leaves)
# of the binary search tree and the normal, key-bearing nodes as being
# internal nodes of the tree.
#
# A binary search tree is a red-black tree if it satisfies the following
# red-black properties:
#
# 1. Every node is either red or black.
# 2. Every leaf (NIL) is black.
# 3. If a node is red, then both its children are black.
# 4. Every simple path from a node to a descendant leaf contains the same
# number of black nodes.
#
# We call the number of black nodes on any path from, but not including, a
# node x to a leaf the black-height of the node, denoted bh(x). By property
# 4, the notion of black-height is well defined, since all descending
# paths from the node have the same number of black nodes. We define the
# black-height of a red-black tree to be the black-height of its root.
#
# The following lemma shows why red-black trees make good search trees.
#
# Lemma 14.1: A red-black tree with n internal nodes has height at most
# 2lg(n+1).
[ Proof omitted ]
# An immediate consequence of this lemma is that the dynamic-set operations
# Search, Minimum, Maximum, Successor, and Predecessor can be implemented
# in O(lg n) time on red-black trees, since they can be made to run in
# O(h) time on a search tree of height H (as shown in Chapter 13) and any
# red-black tree of n nodes is a search tree with height O(lg n). Although
# the the algorithms Tree-Insert and Tree-Delete form Chapter 13 run in O(lg
# n) time when given a red-black tree as input, they do not directly support
# the dynamic-set operations Insert and Delete, since they do not guarantee
# that the modified binary search tree will be a red-black tree. We shall
# see in Sections 14.3 and 14.4, however, that these two operations can
# indeed be supported in O(lg n) time.
Enjoy.
- Alex
</PRE>
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<ul><li>Thread context:
<BLOCKQUOTE><UL>
<LI><STRONG>[MUD-Dev] Re: Laws of Online World Design</STRONG>, <EM>(continued)</EM>
<ul compact>
<ul compact>
<LI><strong><A NAME="00188" HREF="msg00188.html">[MUD-Dev] Re: Laws of Online World Design</A></strong>,
J C Lawrence <a href="mailto:claw#under,engr.sgi.com">claw#under,engr.sgi.com</a>, Tue 13 Oct 1998, 22:14 GMT
</LI>
</ul>
<LI><strong><A NAME="00162" HREF="msg00162.html">[MUD-Dev] Re: Laws of Online World Design</A></strong>,
J C Lawrence <a href="mailto:claw#under,engr.sgi.com">claw#under,engr.sgi.com</a>, Tue 13 Oct 1998, 02:45 GMT
</LI>
<LI><strong><A NAME="00173" HREF="msg00173.html">[MUD-Dev] Re: Laws of Online World Design</A></strong>,
Hans-Henrik Staerfeldt <a href="mailto:hhs#cbs,dtu.dk">hhs#cbs,dtu.dk</a>, Tue 13 Oct 1998, 11:28 GMT
</LI>
</ul>
</LI>
<LI><strong><A NAME="00135" HREF="msg00135.html">[MUD-Dev] Re: Red Black Tree ?</A></strong>,
Valerio Santinelli <a href="mailto:tanis#mediacom,it">tanis#mediacom,it</a>, Fri 09 Oct 1998, 15:29 GMT
<UL>
<LI><strong><A NAME="00136" HREF="msg00136.html">[MUD-Dev] Re: Red Black Tree ?</A></strong>,
T. Alexander Popiel <a href="mailto:popiel#snugharbor,com">popiel#snugharbor,com</a>, Fri 09 Oct 1998, 18:08 GMT
</LI>
<LI><strong><A NAME="00141" HREF="msg00141.html">[MUD-Dev] Re: Red Black Tree ?</A></strong>,
Caliban Tiresias Darklock <a href="mailto:caliban#darklock,com">caliban#darklock,com</a>, Fri 09 Oct 1998, 21:10 GMT
<UL>
<LI><strong><A NAME="00142" HREF="msg00142.html">[MUD-Dev] Re: Red Black Tree ?</A></strong>,
T. Alexander Popiel <a href="mailto:popiel#snugharbor,com">popiel#snugharbor,com</a>, Fri 09 Oct 1998, 21:42 GMT
<UL>
<LI><strong><A NAME="00143" HREF="msg00143.html">[MUD-Dev] Re: Red Black Tree ?</A></strong>,
Caliban Tiresias Darklock <a href="mailto:caliban#darklock,com">caliban#darklock,com</a>, Fri 09 Oct 1998, 23:06 GMT
<UL>
<LI><strong><A NAME="00144" HREF="msg00144.html">[MUD-Dev] Re: Red Black Tree ?</A></strong>,
Ben Greear <a href="mailto:greear#cyberhighway,net">greear#cyberhighway,net</a>, Sat 10 Oct 1998, 06:55 GMT
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