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# prob01 - Problem Solving 1 Show that in a heap of length n...

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Problem Solving 1

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1 Show that in a heap of length , the number of nodes rooted at which the subtree has height is # ( ) . 2 h n h n node h + =
. 2 / ) 0 ( # Thus, nodes. internal of # 1 leaves of # nodes internal of # Hence, child. one with node internal one most at is there e, binary tre complete nearly in that Note leaves. of # ) 0 ( # , 0 For 2 ) ( # : induction) (by Show 1 n node node h n h node h = + = = = +

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= = + = + + + 2 1 1 2 2 2 ) 1 ( # Then . 2 ) ( # Assume h h h n n h node n h node
Planar Graph Separator partition. for this separator the called is . in endpoint one and in endpoint one with edges no are there and vertices, ) O( has vertices, 3 2 most at has and of each such that , and , , sets three into of vertices the of partition a exists there , ( graph planar vertex - any In S B A n S n/ B A B S A G V,E) G = n Tarjan-Lipton Theorem (1979)

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Application graphs. planar on problems solving for algorithms conquer and divide efficient designing in use of be can ions decomposit Separator
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