From: Martin Mares <mj@ucw.cz>
Date: Mon, 21 Jan 2008 14:01:34 +0000 (+0100)
Subject: Disctractors and hedgehogs.
X-Git-Tag: printed~282
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=342b251ea0bbeffd8c49ea36f4c73d845a93a71b;p=saga.git

Disctractors and hedgehogs.
---

diff --git a/mst.tex b/mst.tex
index 2925414..e8ec543 100644
--- a/mst.tex
+++ b/mst.tex
@@ -144,7 +144,7 @@ $w(f)\ge w(e')$ due to the choice of $e'$ and all other edges on~$C$ are lighter
 than~$e'$ as $e'$ was not $T$-light.
 \qed
 
-\thm\id{mstthm}%
+\thmn{Minimality by order}\id{mstthm}%
 A~spanning tree~$T$ is minimum iff there is no $T$-light edge.
 
 \proof
@@ -191,6 +191,28 @@ explicitly state so.
 When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
 its unique minimum spanning tree.
 
+Another useful consequence is that whenever two graphs are isomorphic and the
+isomorphism preserves weight order, the isomorphism applies to their MST's
+as well:
+
+\defn
+A~\df{monotone isomorphism} of two weighted graphs $G_1=(V_1,E_1,w_1)$ and
+$G_2=(V_2,E_2,w_2)$ is a bijection $\pi: V_1\rightarrow V_2$ such that
+for each $u,v\in V_1: uv\in E_1 \Leftrightarrow \pi(u)\pi(v)\in E_2$ and
+for each $e,f\in E_1: w_1(e)<w_1(f) \Leftrightarrow w_2(\pi[e]) < w_2(\pi[f])$.
+
+\lemman{MST of isomorphic graphs}\id{mstiso}%
+Let~$G_1$ and $G_2$ be two weighted graphs with unique edge weights and $\pi$
+their monotone isomorphism. Then $\mst(G_2) = \pi[\mst(G_1)]$.
+
+\proof
+The isomorphism~$\pi$ maps spanning trees onto spanning trees and it preserves
+the relation of covering. Since it is monotone, it preserves the property of
+being a light edge (an~edge $e\in E(G_1)$ is $T$-light $\Leftrightarrow$
+the edge $\pi[e]\in E(G_2)$ is~$f[T]$-light). Therefore by Theorem~\ref{mstthm}, $T$
+is the MST of~$G_1$ if and only if $\pi[T]$ is the MST of~$G_2$.
+\qed
+
 \section{The Red-Blue Meta-Algorithm}
 
 Most MST algorithms can be described as special cases of the following procedure
@@ -589,8 +611,9 @@ which of course works in multigraphs as well.)
 In the previous algorithm, the role of the mapping~$\pi^{-1}$ is of course played by the edge labels~$\ell$.
 
 Finally, we will show a family of graphs where the $\O(m\log n)$ bound on time complexity
-is tight. The graphs do not have unique weights, so the edge comparison oracle will break
-ties by comparing the edges by their arbitrary identifiers as usually.
+is tight. The graphs do not have unique weights, but they are constructed in a way that
+the algorithm never compares two edges with the same weight. Therefore, when two such
+graphs are monotonely isomorphic (see~\ref{mstiso}), the algorithm processes them in the same way.
 
 \defn
 A~\df{distractor of order~$k$,} denoted by~$D_k$, is a path on $n=2^k$~vertices $v_1,\ldots,v_n$
@@ -598,26 +621,26 @@ where each edge $v_iv_{i+1}$ has its weight equal to the number of trailing zero
 representation of the number~$i$. The vertex $v_1$ is called a~\df{base} of the distractor.
 
 \rem
-Alternatively, we can define~$D_k$ recursively: $D_0$ is a single vertex, $D_k$ consists
+Alternatively, we can use a recursive definition: $D_0$ is a single vertex, $D_k$ consists
 of two disjoint copies of $D_{k-1}$ joined by an edge of weight~$k$.
 
-\FIXME{Picture of a distractor and of a hedgehog.}
+\figure{distractor.eps}{\epsfxsize}{A~distractor $D_3$ and its evolution (bold edges are contracted)}
 
 \lemma
 A~single iteration of the contractive algorithm reduces~$D_k$ to a graph isomorphic with~$D_{k-1}$.
 
 \proof
 Each vertex~$v$ of~$D_k$ is incident with a single edge of weight~1. The algorithm therefore
-selects all weight~1 edges and contract them. This produces a graph which is
+selects all weight~1 edges and contracts them. This produces a graph which is
 exactly $D_{k-1}$ with all weights increased by~1, which does not change the relative order of edges.
 \qed
 
-\FIXME{Define isomorphism of weighted graphs.}
-
 \defn
 A~\df{hedgehog}~$H_{a,k}$ is a graph consisting of $a$~distractors $D_k^1,\ldots,D_k^a$ of order~$k$
 together with edges of a complete graph on the bases of the distractors. These additional edges
-have weight $k+1$, i.e., they are heavier than the edges of the distractors.
+have unique weights larger than~$k$, i.e., they are heavier than the edges of the distractors.
+
+\figure{hedgehog.eps}{\epsfxsize}{A~hedgehog $H_{5,2}$ (quills bent to fit in the picture)}
 
 \lemma
 A~single iteration of the contractive algorithm reduces~$H_{a,k}$ to a graph isomorphic with $H_{a,k-1}$.
@@ -626,7 +649,7 @@ A~single iteration of the contractive algorithm reduces~$H_{a,k}$ to a graph iso
 Each vertex is incident with an edge of some distractor, so the algorithm does not select
 any edge of the complete graph. Contraction therefore reduces each distractor to a smaller
 distractor (modulo an additive factor in weight) and leaves the complete graph intact.
-This is exactly $H_{a,k-1}$ (again modulo a~shift of weights).
+This is monotonely isomorphic to $H_{a,k-1}$.
 \qed
 
 \thmn{Lower bound for Contractive Bor\o{u}vka}%
@@ -689,5 +712,6 @@ See Theorem 6.1 in \cite{nesetril:minors}, which also lists some other equivalen
 % mention matroids
 % sorted weights
 % \O(...) as a set?
+% impedance mismatch in terminology: contraction of G along e vs. contraction of e.
 
 \endpart
diff --git a/pic/distractor.eps b/pic/distractor.eps
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+%%EndResource
+%%BeginResource: font CMR7
+%!PS-AdobeFont-1.1: CMR7 1.0
+%%CreationDate: 1991 Aug 20 16:39:21
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR7) readonly def
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+/Weight (Medium) readonly def
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+/isFixedPitch false def
+end readonly def
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+dup 93 /bracketright put
+dup 94 /circumflex put
+dup 95 /dotaccent put
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+dup 123 /endash put
+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
+dup 160 /space put
+readonly def
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+%%BeginResource: font CMSY10
+%!PS-AdobeFont-1.1: CMSY10 1.0
+%%CreationDate: 1991 Aug 15 07:20:57
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMSY10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMSY10 def
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+dup 123 /paragraph put
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+
+%%EndResource
+%%BeginResource: font CMR10
+%!PS-AdobeFont-1.1: CMR10 1.00B
+%%CreationDate: 1992 Feb 19 19:54:52
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMR10 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 161 /Gamma put
+dup 162 /Delta put
+dup 163 /Theta put
+dup 164 /Lambda put
+dup 165 /Xi put
+dup 166 /Pi put
+dup 167 /Sigma put
+dup 168 /Upsilon put
+dup 169 /Phi put
+dup 170 /Psi put
+dup 173 /Omega put
+dup 174 /ff put
+dup 175 /fi put
+dup 176 /fl put
+dup 177 /ffi put
+dup 178 /ffl put
+dup 179 /dotlessi put
+dup 180 /dotlessj put
+dup 181 /grave put
+dup 182 /acute put
+dup 183 /caron put
+dup 184 /breve put
+dup 185 /macron put
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+/slj {setlinejoin} bind def
+/sdh {0 setdash} bind def
+end
+%%EndProcSet
+%%EndProlog
+%%BeginSetup
+%%BeginResource: font CMSY10
+%!PS-AdobeFont-1.1: CMSY10 1.0
+%%CreationDate: 1991 Aug 15 07:20:57
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMSY10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
+end readonly def
+/FontName /CMSY10 def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
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+dup 163 /multiply put
+dup 164 /asteriskmath put
+dup 165 /divide put
+dup 166 /diamondmath put
+dup 167 /plusminus put
+dup 168 /minusplus put
+dup 169 /circleplus put
+dup 170 /circleminus put
+dup 173 /circlemultiply put
+dup 174 /circledivide put
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+dup 176 /circlecopyrt put
+dup 177 /openbullet put
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+dup 179 /equivasymptotic put
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+dup 183 /lessequal put
+dup 184 /greaterequal put
+dup 185 /precedesequal put
+dup 186 /followsequal put
+dup 187 /similar put
+dup 188 /approxequal put
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+dup 192 /greatermuch put
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+dup 0 /minus put
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+dup 2 /multiply put
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+dup 23 /followsequal put
+dup 24 /similar put
+dup 25 /approxequal put
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+dup 30 /precedes put
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+dup 34 /arrowup put
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+dup 60 /Rfractur put
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+dup 62 /latticetop put
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+dup 86 /V put
+dup 87 /W put
+dup 88 /X put
+dup 89 /Y put
+dup 90 /Z put
+dup 91 /union put
+dup 92 /intersection put
+dup 93 /unionmulti put
+dup 94 /logicaland put
+dup 95 /logicalor put
+dup 96 /turnstileleft put
+dup 97 /turnstileright put
+dup 98 /floorleft put
+dup 99 /floorright put
+dup 100 /ceilingleft put
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+dup 104 /angbracketleft put
+dup 105 /angbracketright put
+dup 106 /bar put
+dup 107 /bardbl put
+dup 108 /arrowbothv put
+dup 109 /arrowdblbothv put
+dup 110 /backslash put
+dup 111 /wreathproduct put
+dup 112 /radical put
+dup 113 /coproduct put
+dup 114 /nabla put
+dup 115 /integral put
+dup 116 /unionsq put
+dup 117 /intersectionsq put
+dup 118 /subsetsqequal put
+dup 119 /supersetsqequal put
+dup 120 /section put
+dup 121 /dagger put
+dup 122 /daggerdbl put
+dup 123 /paragraph put
+dup 124 /club put
+dup 125 /diamond put
+dup 126 /heart put
+dup 127 /spade put
+dup 128 /arrowleft put
+readonly def
+/FontBBox{-29 -960 1116 775}readonly def
+/UniqueID 5000820 def
+currentdict end
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+%%EndResource
+%%BeginResource: font CMR10
+%!PS-AdobeFont-1.1: CMR10 1.00B
+%%CreationDate: 1992 Feb 19 19:54:52
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
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+/PaintType 0 def
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+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
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+dup 90 /Z put
+dup 91 /bracketleft put
+dup 92 /quotedblleft put
+dup 93 /bracketright put
+dup 94 /circumflex put
+dup 95 /dotaccent put
+dup 96 /quoteleft put
+dup 97 /a put
+dup 98 /b put
+dup 99 /c put
+dup 100 /d put
+dup 101 /e put
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+dup 103 /g put
+dup 104 /h put
+dup 105 /i put
+dup 106 /j put
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+dup 108 /l put
+dup 109 /m put
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+dup 114 /r put
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+dup 116 /t put
+dup 117 /u put
+dup 118 /v put
+dup 119 /w put
+dup 120 /x put
+dup 121 /y put
+dup 122 /z put
+dup 123 /endash put
+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
+dup 160 /space put
+readonly def
+/FontBBox{-251 -250 1009 969}readonly def
+/UniqueID 5000793 def
+currentdict end
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