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A Theoretical Study on Wire Length Estimation Algorithms for Placement with Opaque Blocks Tan Yan*, Shuting Li Yasuhiro Takashima, Hiroshi Murata The University of Kitakyushu * Now with University of Illinois at Urbana-Champaign

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Motivation Opaque blocks makes HPWL inexact Because of IP blocks, analog blocks, memory module… Lead to timing violation, unroutable nets…

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Motivationcontd Exact wire length estimation for Block Placement the obstacle-avoiding shortest path length Time complexity: O(n)? O(n 2) ? O(nlogn)?... Time complexity is almost the same as HPWL! Already proposed in Computational Geometry However Not well-known in CAD community Need interpretation to be applicable to CAD!

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Our Contribution We restate the results in [P.J.de Rezende 85] & [M.J.Atallah 91] Simplify the discussion (with Block Placement notions) CAD-oriented language Tailor the theory to fit into Physical Design background

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Problem Formulation Input: Block location Pin location (on block boundaries) ABLR relations * (obtainable from Sequence Pair, etc) Output: Rectilinear block-avoiding shortest path length for every 2-pin net = Minimal Wire Length (MWL)

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Assumption 2-pin net s on S, t on T S T S is left-to T y s y t S T s t

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Locus

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Theorem 1 MWL = HPWL RU locus of s goes below or through t Proof omitted

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AB-region

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Lemma 2 There exists an MWL routing inside the AB-region

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Horizontal Visibility Graph (HVG)

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MWL = shortest path length Only linear number of edges, but still captures MWL! Lemma 4: There exists a path (s,t) on the visibility graph that corresponds to an MWL routing.

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Visibility graph of a placement

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The overall flow and so on …

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Time complexity M = # of blocks, N = # of nets Building visibility graph: O(M logM) Estimating one net: O(M) Total: O(M logM + NM) Shortest path on channel graph takes O(NM 2 )

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Use LUT to enhance the speed No path between two vertices? (a 2 b 2 ) Need to judge whether RU locus above t ? How to find out A & B promptly?

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Two lemmas: Lemma 5: Two vertices s and t on visibility graph. If there is no path between them, then MWL = HPWL Lemma 6: If t is above ss RU locus and there exists a shortest path between them, then its length = HPWL. MWL(a,b) = HPWL ShortestPath(c,d) = MWL (c,d) = HPWL

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Theorem 3 The MWL of any two vertices on the visibility graph can be obtained by shortest path algorithm: Shortest path exists, MWL = path length Otherwise, MWL = HPWL

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How it works MWL = shortest path length No path! MWL = HPWL And so on… Lookup table

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Time complexity Building LUT: O(M 2 ) Estimating one net: O(1) Total: O(M 2 +N) Almost the same as HPWL!

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Future works Integration of routing congestion Extension to handle multi-pin nets Application to global router Experimental study

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Thank you! Q & A

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Proof of Theorem 1 MWL = HPWL RU locus of s goes below or through t

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Proof of Lemma 2 There exists an MWL routing completely inside AB-region

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Proof of Lemma 4 There exists a path p from s to t on HVG that corresponds to an MWL routing.

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Proof of Lemma 6 If t is above ss RU locus and there exists a shortest path between them, then its length = HPWL.

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