en>Rjwilmsi: /* References */Added 4 dois to journal cites using AWB (10104)

2014-05-08T17:02:35Z

References: Added 4 dois to journal cites using AWB (10104)

en>Mgarraha: /* Example */ corrected m_1

2013-12-23T22:30:31Z

Example: corrected m_1

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			In [[computer science]], [[Disjoint-set data structure\|Union Find]] is an [[algorithm]] for doing certain operations on sets. This page is about proof of ''O''(log<sup>*</sup>''n'') [[Amortized analysis\|amortized time]] <ref name="name for reference">[[Raimund Seidel]], Micha Sharir. "Top-down analysis of path compression", SIAM J. Computing 34(3):515–525, 2005</ref> of [[Disjoint-set data structure\|Union Find]]<ref>{{cite journal \|last1=Tarjan \|first1=Robert Endre \|year=1975 \|title=Efficiency of a Good But Not Linear Set Union Algorithm \|journal=Journal of the ACM \|volume=22 \|issue=2 \|pages=215–225 \|url=http://portal.acm.org/citation.cfm?id=321884 \|doi=10.1145/321879.321884 }}</ref><ref>{{cite journal \|last1=Hopcroft \|first1=J. E. \|last2=Ullman \|first2=J. D. \|year=1973 \|title=Set Merging Algorithms \|journal=SIAM Journal on Computing \|volume=2 \|issue=4 \|pages=294–303 \|doi=10.1137/0202024}}</ref><ref>[[Robert E. Tarjan]] and [[Jan van Leeuwen]]. Worst-case analysis of set union algorithms. Journal of the ACM, 31(2):245–281, 1984.</ref>

			Statement: If ''m'' operations, either [[Disjoint-set data structure#Disjoint-set forests\|Union]] or [[Disjoint-set data structure#Disjoint-set forests\|Find]], are applied to ''n'' elements, the total run time is ''O''(''m'' log<sup></sup>''n''), where log<sup></sup> is the [[iterated logarithm]].

			==Proof==

			{{anchor\|increasing rank lemma}}Lemma 1: As the [[Disjoint-set data structure#Disjoint-set forests\|find function]] follows the path along to the root, the rank of node it encounters is strictly increasing

			:Proof: claim that as Find and Union operations are applied to the data set, this fact remains true over time. Initially when each node is the root of its own tree, it's trivially true. The only case when the rank of a node might be changed is when the [[Disjoint-set data structure#Disjoint-set forests\|Union by Rank]] operation is applied. In this case, a tree with smaller rank will be attached to a tree with greater rank, rather than vice versa. And during the find operation, all nodes visited along the path will be attached to the root, which has larger rank than its children, so this operation won't change this fact either.

			{{anchor\|min subtree size lemma}}Lemma 2: A node ''u'' which is root of a subtree with rank ''r'' has at least 2<sup>''r''</sup> nodes.

			:Proof: Initially when each node is the root of its own tree, it's trivially true. Assume that a node ''u'' with rank ''r'' has at least 2<sup>''r''</sup> nodes. Then when two tree with rank ''r'' [[Disjoint-set data structure#Disjoint-set forests\|Unions by Rank]] and form a tree with rank ''r'' + 1, the new node has at least 2<sup>''r''</sup> + 2<sup>''r''</sup> = 2<sup>''r'' + 1</sup> nodes.

			[[File:ProofOflogstarnRank.jpg\|center\|alt=Alternative text\|Caption text]]

			Lemma 3: The maximum number of nodes of rank ''r'' is at most ''n''/2<sup>''r''</sup>.

			:Proof: From [[#min subtree size lemma\|lemma 2]], we know that a node ''u'' which is root of a subtree with rank ''r'' has at least 2<sup>''r''</sup> nodes. We will get the maximum number of nodes of rank ''r'' when each node with rank ''r'' is the root of a tree that has exactly 2<sup>''r''</sup> nodes. In this case, the number of nodes of rank ''r'' is ''n''/2<sup>''r''</sup>

			For convenience, we define "bucket" here: a bucket is a set that contains vertices with particular ranks.

			We create some buckets and put vertices into buckets according to their ranks. That is, vertices with rank 0 goes into the zeroth bucket, vertices with rank 1 goes into the first bucket, vertices with rank 2 and 3 goes into the second bucket, and vertices with ranks ''B'' to 2<sup>''B''</sup> − 1 go into the ''B''th bucket.

			[[File:Proof of O(log*n) Union Find.jpg\|center\|alt=Alternative text\|Caption text]]
			We can make two observations about the buckets.

			# {{anchor\|max buckets}}The total number of buckets is at most log<sup>*</sup>''n''
			#:Proof: When we go from one bucket to the next, we add one more two to the power, that is, the next bucket to [''B'', 2<sup>''B''</sup> − 1] will be [2<sup>''B''</sup>, 2<sup>2<sup>''B''</sup></sup> − 1]
			#{{anchor\|max bucket size}}The maximum number of elements in bucket [''B'', 2<sup>''B''</sup> – 1] is at most 2''n''/2<sup>''B''</sup>
			#:Proof: The maximum number of elements in bucket [''B'', 2<sup>''B''</sup> – 1] is at most ''n''/2<sup>''B''</sup> + ''n''/2<sup>''B''+1</sup> + ''n''/2<sup>''B''+2</sup> + … + ''n''/2<sup>2^''B'' – 1</sup> ≤ 2''n''/2<sup>''B''</sup>

			Let ''F'' represent the list of "find" operations performed, and let

			<math>T_1 = \sum_F\text{(link to the root)}</math>

			<math>T_2 = \sum_F\text{(number of links traversed where the buckets are different)}</math>

			<math>T_3 = \sum_F\text{(number of links traversed where the buckets are the same).}</math>

			Then the total cost of ''m'' finds is ''T'' = ''T''<sub>1</sub> + ''T''<sub>2</sub> + ''T''<sub>3</sub>

			Since each find operation makes exactly one traversal that leads to a root, we have ''T''<sub>1</sub> = ''O''(''m'').

			Also, from the bound above on the number of buckets, we have ''T''<sub>2</sub> = ''O''(''m''log<sup>*</sup>''n'').

			For T<sub>3</sub>, suppose we are traversing from ''u'' to ''v'', where ''u'' and ''v'' have rank in the bucket [''B'', 2<sup>''B''</sup> − 1]. From [[#increasing rank lemma\|lemma 1]], we know that the number of times we traversed a link (''u'',''v'') where ''u'' and ''v'' were in the same bucket is at most 2<sup>''B''</sup> − 1 − ''B'', which is at most 2<sup>''B''</sup>.

			Therefore, <math>T_3=\sum_{[B, 2^B - 1]} \sum_{u} 2^B</math>

			From Observations [[#max buckets\|1]] and [[#max bucket size\|2]], we can conclude that <math>T_3\le\sum_{B} 2^B\frac{2n}{2^B}\le 2n\log^*n.</math>

			Therefore, ''T'' = ''T''<sub>1</sub> + ''T''<sub>2</sub> + ''T''<sub>3</sub> = ''O''(''m'' log<sup>*</sup>''n'').

			== References ==
			<references />

			[[Category:Search algorithms]]

en>Sun Creator: /* Derivation */Typo fixing and checking, typos fixed: ,, → , (2) using AWB (8097)

2012-08-08T17:17:47Z

Derivation: Typo fixing and checking, typos fixed: ,, → , (2) using AWB (8097)

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Combined Linear Congruential Generator - Revision history

en>Rjwilmsi: /* References */Added 4 dois to journal cites using AWB (10104)

en>Mgarraha: /* Example */ corrected m_1

en>Sun Creator: /* Derivation */Typo fixing and checking, typos fixed: ,, → , (2) using AWB (8097)