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# L03 - 6.889 Lecture 3 Planar Separators Christian Sommer...

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6.889 — Lecture 3: Planar Separators Christian Sommer September 13, 2011 We shall prove theorems of the following ﬂavor (see textbook/papers for precise statements and proofs). Thm. For any planar graph G = ( V,E ) on n = | V | vertices and for any 1 weight function w : V R + , we can partition V into A,B,S V such that [ α –balanced] w ( A ) ,w ( B ) α · w ( V ) for some α (0 , 1) [separation] no edge between any a A and b B ( A × B E = ) [small separator] | S | ≤ f ( n ) [efﬁcient] A,B,S can be found in linear time. Trees what if G is a (binary) tree? can do 1 / 2 –balanced partition with | S | = 1 ? only 2 / 3 –balanced! with one edge in separator? only 3 / 4 –balanced for binary trees 1 / 3 1 / 3 1 / 3 1 / 4 1 / 4 1 / 4 1 / 4 Grids What happens for a grid on n vertices (say square: n × n )? | S | ≤ n 1 / 2 –balanced n 2 / 3 –balanced < n but Θ( n ) cut out a diagonal and remain 2 / 3 –balanced, s vertices separate s 2 / 2 vertices from the rest, n/ 3 s 2 / 2 . O ( n ) is “right order of magnitude” today’s lecture: can generalize to all planar graphs Beyond Extensions to bounded-genus and minor-free graphs to be discussed in Lecture 5 General Graphs Is there a separator theorem that works for any graph? No (complete graph)! For any sparse graph? No (expander graphs)! 1 almost — need individual weights (1 - α ) w ( V ) 1

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We shall prove two versions of the main theorem. Both proofs use the following lemma (a weighted version). Lemma. For any planar graph G = ( V,E ) with a spanning tree of radius d rooted at r V , we can partition V into A,B,S V such that [balanced] | A | , | B | ≤ 3 n/ 4 [separation] no edge between any a A and b B ( A × B E = ) [separator size] | S | ≤ 2 d + 1 [efﬁcient] A,B,S can be found in linear time. Proof (sketch). Let T be the spanning tree of depth d rooted at r . Triangulate G . Recall interdigitating trees from Lecture 2. Let T * be the dual tree in the triangulated version of G . Every non-tree edge e deﬁnes a fundamental cycle C ( e ) . Since T has depth d , we have | C ( e ) | ≤ 2 d + 1 . assign appropriate weights to faces. then ﬁnd edge separator in interdigitating tree! ( T * has degree 3 ) Problem Set: how to efﬁciently ﬁnd the best edge e (how to compute w ( ext ( C ( e ))) ,w ( int ( C ( e ))) for each edge e , where ext ( C ) , int ( C ) denote the exterior and interior of a cycle C , respectively)
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L03 - 6.889 Lecture 3 Planar Separators Christian Sommer...

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