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

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

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!

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

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)

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

Locus

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

AB-region

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

Horizontal Visibility Graph (HVG)

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.

Visibility graph of a placement

The overall flow and so on …

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 )

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?

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

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

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

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

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

Thank you! Q & A

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

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

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

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