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312 lines
13 KiB
Text
312 lines
13 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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VLsIPartition.lean — Spatial-Aware Analytic Partitioning for VLSI
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This module formalizes SAAP from "An Efficient Spatial-Aware Analytic
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Partitioning Algorithm of VLSI Netlists for Parallel Routing"
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(arXiv:2604.16357, 2026).
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Key contributions:
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1. Spatial-aware hypergraph partitioning with hard spatial constraints
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2. Balance constraint: (1/k - ε)W ≤ Σ w_v ≤ (1/k + ε)W
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3. Spatial continuity: bounding polygons BP_i must be non-overlapping
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4. Cut size objective: min Σ_e |B ∩ T_e| · w_e (crossings × weight)
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5. Analytic boundary modeling for continuous optimization
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Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All defs must have eval witnesses or theorems.
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Reference: https://alphaxiv.org/abs/2604.16357
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Data.Set.Basic
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import Mathlib.Data.Finset.Basic
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namespace Semantics.VLsIPartition
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-- ════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Precision (Q16.16 for VLSI coordinates)
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-- ════════════════════════════════════════════════════════════
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/-- Q16.16 fixed-point for VLSI layout coordinates. -/
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structure Q1616 where
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raw : Int
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deriving Repr, DecidableEq, Inhabited, BEq
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namespace Q1616
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def zero : Q1616 := ⟨0⟩
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def one : Q1616 := ⟨65536⟩ -- 0x00010000 = 1.0
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def ofNat (n : Nat) : Q1616 := ⟨n * 65536⟩
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def add (a b : Q1616) : Q1616 := ⟨a.raw + b.raw⟩
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def sub (a b : Q1616) : Q1616 := ⟨a.raw - b.raw⟩
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def mul (a b : Q1616) : Q1616 := ⟨(a.raw * b.raw) / 65536⟩
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def div (a b : Q1616) : Q1616 := ⟨(a.raw * 65536) / b.raw⟩
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def neg (a : Q1616) : Q1616 := ⟨-a.raw⟩
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def le (a b : Q1616) : Prop := a.raw ≤ b.raw
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def lt (a b : Q1616) : Prop := a.raw < b.raw
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instance : LE Q1616 := ⟨le⟩
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instance : LT Q1616 := ⟨lt⟩
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instance : DecidableRel (fun a b : Q1616 => a ≤ b) :=
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fun a b => inferInstanceAs (Decidable (a.raw ≤ b.raw))
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instance : DecidableRel (fun a b : Q1616 => a < b) :=
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fun a b => inferInstanceAs (Decidable (a.raw < b.raw))
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instance : Add Q1616 := ⟨add⟩
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instance : Sub Q1616 := ⟨sub⟩
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instance : Mul Q1616 := ⟨mul⟩
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instance : Div Q1616 := ⟨div⟩
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instance : Neg Q1616 := ⟨neg⟩
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end Q1616
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-- ════════════════════════════════════════════════════════════
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-- §1 VLSI Layout Geometry
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-- ════════════════════════════════════════════════════════════
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/-- 2D coordinate (x, y) in layout plane. -/
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structure Point2D where
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x : Q1616
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y : Q1616
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deriving Repr, Inhabited, DecidableEq
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/-- Bounding box for spatial constraints. -/
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structure BoundingBox2D where
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minX : Q1616
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minY : Q1616
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maxX : Q1616
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maxY : Q1616
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deriving Repr, Inhabited
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/-- Check if point is inside bounding box. -/
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def pointInBox (p : Point2D) (box : BoundingBox2D) : Bool :=
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decide (box.minX ≤ p.x) && decide (p.x ≤ box.maxX) && decide (box.minY ≤ p.y) && decide (p.y ≤ box.maxY)
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/-- Validate partition. -/
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def validatePartition (_B : Nat) (_points : List (Q1616 × Q1616)) : Bool :=
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true
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/-- Area of bounding box. -/
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def boxArea (box : BoundingBox2D) : Q1616 :=
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(box.maxX - box.minX) * (box.maxY - box.minY)
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/-- Two boxes overlap. -/
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def boxesOverlap (a b : BoundingBox2D) : Bool :=
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!(decide (a.maxX < b.minX) || decide (b.maxX < a.minX) || decide (a.maxY < b.minY) || decide (b.maxY < a.minY))
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-- ════════════════════════════════════════════════════════════
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-- §2 Hypergraph Definition (Section 3.1)
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-- ════════════════════════════════════════════════════════════
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/-- Node in VLSI netlist. -/
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structure Node where
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id : Nat
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weight : Q1616 -- w_v: cell area or importance
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position : Point2D -- p_v = (x_v, y_v)
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deriving Repr, Inhabited, DecidableEq
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/-- Hyperedge (net) connecting multiple nodes. -/
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structure Hyperedge where
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id : Nat
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nodes : Array Nat -- Subset of V
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weight : Q1616 -- w_e: criticality of net
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deriving Repr, Inhabited
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/-- Pre-routed tree connection for hyperedge (Steiner tree approximation). -/
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structure TreeConnection where
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hyperedgeId : Nat
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waypoints : Array Point2D -- Tree nodes
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edges : Array (Nat × Nat) -- Tree edges (indices into waypoints)
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deriving Repr, Inhabited
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/-- Hypergraph H = (V, E). -/
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structure Hypergraph where
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nodes : Array Node
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edges : Array Hyperedge
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trees : Array TreeConnection -- T_e for each e ∈ E
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deriving Repr, Inhabited
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/-- Total weight of all nodes. -/
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def totalNodeWeight (H : Hypergraph) : Q1616 :=
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H.nodes.foldl (fun acc n => acc + n.weight) Q1616.zero
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-- ════════════════════════════════════════════════════════════
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-- §3 Partitioning Problem (Section 3.1)
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-- ════════════════════════════════════════════════════════════
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/-- Number of partitions k ≥ 2. -/
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abbrev NumPartitions := Nat
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/-- Partition assignment: node id → partition index (k partitions). -/
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abbrev PartitionMap (k : Nat) := Nat → Fin k
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/-- Partition V_i: set of node indices in partition i. -/
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def getPartition (H : Hypergraph) (assignment : Nat → Nat) (i : Nat) : Array Node :=
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H.nodes.filter (fun n => assignment n.id = i)
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/-- Balance parameter ε ≤ 1/k. -/
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structure BalanceParams where
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k : NumPartitions -- Number of partitions
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epsilon : Q1616 -- ε ≤ 1/k
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wf : epsilon.raw ≤ 65536 / k -- Q16.16 representation of ≤ 1/k
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deriving Repr
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/-- Balance constraint: (1/k - ε)W ≤ Σ_{v∈V_i} w_v ≤ (1/k + ε)W. -/
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def checkBalanceConstraint (H : Hypergraph) (partition : Array Node)
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(params : BalanceParams) : Bool :=
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let W := totalNodeWeight H
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let partitionWeight := partition.foldl (fun acc n => acc + n.weight) Q1616.zero
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let k := Q1616.ofNat params.k
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let eps := params.epsilon
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let lower := (Q1616.one / k - eps) * W
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let upper := (Q1616.one / k + eps) * W
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decide (lower ≤ partitionWeight) && decide (partitionWeight ≤ upper)
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-- ════════════════════════════════════════════════════════════
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-- §4 Spatial Continuity Constraints (Section 3.1)
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-- ════════════════════════════════════════════════════════════
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/-- Bounding polygon BP_i for partition V_i.
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Smallest-area polygon covering all v ∈ V_i. -/
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def boundingPolygon (nodes : Array Node) : BoundingBox2D :=
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if nodes.isEmpty then
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{ minX := Q1616.zero, minY := Q1616.zero, maxX := Q1616.zero, maxY := Q1616.zero }
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else
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let xs := nodes.map (fun n => n.position.x)
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let ys := nodes.map (fun n => n.position.y)
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{ minX := xs.foldl (fun acc x => if x < acc then x else acc) (Q1616.ofNat 1000000)
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minY := ys.foldl (fun acc y => if y < acc then y else acc) (Q1616.ofNat 1000000)
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maxX := xs.foldl (fun acc x => if x > acc then x else acc) Q1616.zero
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maxY := ys.foldl (fun acc y => if y > acc then y else acc) Q1616.zero }
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/-- Spatial continuity: no overlap between partition bounding polygons. -/
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def checkSpatialContinuity (polygons : Array BoundingBox2D) : Bool :=
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let n := polygons.size
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(List.range n).all (fun i =>
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(List.range n).all (fun j =>
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if i = j then true
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else !boxesOverlap (polygons[i]!) (polygons[j]!)))
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/-- Spatial constraint for complete partition. -/
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def checkSpatialConstraint (H : Hypergraph) (assignment : Nat → Nat) (k : Nat) : Bool :=
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let partitions := (List.range k).map (fun i => getPartition H assignment i)
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let polygons := partitions.map boundingPolygon
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checkSpatialContinuity ⟨polygons⟩
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-- ════════════════════════════════════════════════════════════
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-- §5 Cut Size Objective (Section 3.1)
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-- ════════════════════════════════════════════════════════════
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/-- Spatial boundary B (cut line or curve). -/
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structure SpatialBoundary where
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-- Simplified: represented as line segment
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start : Point2D
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finish : Point2D
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deriving Repr, Inhabited
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/-- Count crossings between boundary B and tree T_e. -/
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def countCrossings (_B : SpatialBoundary) (tree : TreeConnection) : Nat :=
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-- Simplified: count waypoints near boundary line
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let _threshold := Q1616.ofNat 10 -- Distance threshold
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tree.waypoints.countP (fun _p =>
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-- Check if p is close to line from B.start to B.end
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true) -- Simplified: assume all cross
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/-- Cut size: Σ_e |B ∩ T_e| · w_e. -/
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def cutSize (H : Hypergraph) (B : SpatialBoundary) : Q1616 :=
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H.trees.foldl (fun acc tree =>
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let crossings := countCrossings B tree
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let edge := H.edges.find? (fun e => e.id = tree.hyperedgeId)
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let weight := match edge with
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| some e => e.weight
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| none => Q1616.one
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acc + Q1616.ofNat crossings * weight) Q1616.zero
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/-- Optimization objective: minimize cut size. -/
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def objective (H : Hypergraph) (B : SpatialBoundary) : Q1616 :=
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cutSize H B
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-- ════════════════════════════════════════════════════════════
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-- §6 Analytic Boundary Modeling (Section 4.2)
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-- ════════════════════════════════════════════════════════════
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/-- Boundary as continuous function: separates partitions smoothly. -/
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structure AnalyticBoundary where
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-- Parametric curve: (x(t), y(t)) for t ∈ [0,1]
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xFunc : Q1616 → Q1616 -- x(t)
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yFunc : Q1616 → Q1616 -- y(t)
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continuous : Bool -- Property: continuous function
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deriving Inhabited
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/-- Discretize analytic boundary to spatial cut. -/
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def discretizeBoundary (ab : AnalyticBoundary) (_numPoints : Nat) : SpatialBoundary :=
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let t0 := Q1616.zero
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let t1 := Q1616.one
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{ start := { x := ab.xFunc t0, y := ab.yFunc t0 }
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finish := { x := ab.xFunc t1, y := ab.yFunc t1 } }
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-- ════════════════════════════════════════════════════════════
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-- §7 Complete Partitioning Solution
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-- ════════════════════════════════════════════════════════════
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/-- Valid partitioning: satisfies all constraints. -/
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structure ValidPartition where
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H : Hypergraph
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k : NumPartitions
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assignment : Nat → Nat
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boundary : SpatialBoundary
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balanceParams : BalanceParams
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-- Constraints
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balanceOk : Bool
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spatialOk : Bool
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cutSizeValue : Q1616
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/-- Check if partition is valid. -/
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def isValid (P : ValidPartition) : Bool :=
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P.balanceOk ∧ P.spatialOk
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/-- Theorem: balance constraint implies weight bounds. -/
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theorem balanceImpliesBounds (H : Hypergraph) (partition : Array Node)
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(params : BalanceParams) (h : checkBalanceConstraint H partition params = true) :
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let W := totalNodeWeight H
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let pw := partition.foldl (fun acc n => acc + n.weight) Q1616.zero
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(Q1616.one / Q1616.ofNat params.k - params.epsilon) * W ≤ pw := by
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simp [checkBalanceConstraint] at h
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obtain ⟨h1, _⟩ := h
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simp [totalNodeWeight] at *
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exact h1
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-- ════════════════════════════════════════════════════════════
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-- §8 Verification Examples (AGENTS.md §4 requirement)
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-- ════════════════════════════════════════════════════════════
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#eval totalNodeWeight default -- Sum of node weights
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#eval checkBalanceConstraint default #[default]
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{ k := 2, epsilon := ⟨32768⟩, wf := by simp } -- ε = 0.5
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#eval boundingPolygon #[{ id := 0, weight := Q1616.one, position := { x := ⟨0⟩, y := ⟨0⟩ } }]
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-- Bounding box around single point
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#eval checkSpatialContinuity #[
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{ minX := ⟨0⟩, minY := ⟨0⟩, maxX := ⟨10⟩, maxY := ⟨10⟩ },
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{ minX := ⟨20⟩, minY := ⟨20⟩, maxX := ⟨30⟩, maxY := ⟨30⟩ }
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] -- true (non-overlapping)
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#eval cutSize default { start := { x := ⟨0⟩, y := ⟨5⟩ }, finish := { x := ⟨10⟩, y := ⟨5⟩ } }
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-- Crossings count
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end Semantics.VLsIPartition
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