Lec5

Graph Models for Global Routing: Grid Graph • Each cell is represented by a vertex. • Two vertices are joined by an edge if the corresponding cells are adjacent to each other. • The occupied cells are represented as filled circles, whereas the others are as clear circles.

d a

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Graph Model: Channel Intersection Graph • Channels are represented as edges. • Channel intersections are represented as vertices. • Edge weight represents channel capacity. • Extended channel intersection graph: terminals are also represented as vertices.

channel intersection graph extended channel intersection graph

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Global-Routing Problem • Given a netlist N={N1 , N2 , . . . , Nn}, a routing graph G = (V, E), find a Steiner Pn tree Ti for each net Ni, 1 ≤ i ≤ n, such that U (ej ) ≤ c(ej ), ∀ej ∈ E and i=1 L(Ti ) is minimized, where – c(ej ): capacity of edge ej ; – xij = 1 if ej is in Ti; xij = 0 otherwise; Pn – U (ej ) = i=1 xij : # of wires that pass through the channel corresponding to edge ej ; – L(Ti): total wirelength of Steiner tree Ti. • For high-performance, the maximum wirelength (maxni=1 L(Ti)) is minimized (or the longest path between two points in Ti is minimized).

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Global Routing in different Design Styles global routing

full custom flexible channels

standard cell

gate array

flexible channels fixed channels

most general problem

fixed feedthroughs

FPGA fixed routing tracks switchbox constraints

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Global Routing in Standard Cell • Objective – Minimize total channel height. – Assignment of feedthrough: Placement? Global routing? • For high performance, – Minimize the maximum wire length. – Minimize the maximum path length.

feedthroughs

failed net

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Global Routing in Gate Array • Objective – Guarantee 100% routability. • For high performance, – Minimize the maximum wire length. – Minimize the maximum path length.

2 tracks

failed connection Each channel has a capacity of 2 tracks.

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Global Routing in FPGA • Objective – Guarantee 100% routability. – Consider switch-module architectural constraints. • For performance-driven routing, – Minimize # of switches used. – Minimize the maximum wire length. – Minimize the maximum path length.

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s switch module

failed connection Each channel has a capacity of 2 tracks.

Classification of Global-Routing Algorithm • Sequential approach: Assigns priority to nets; routes one net at a time based on its priority (net ordering?). • Concurrent approach: All nets are considered at the same time (complexity?) global−routing algorithm

sequential approach

two−terminal

line−search

Lee

maze

Hadlock

multi−terminal

concurrent approach

hierarchical integer programming

Steiner−tree based

Soukup 8

Global-Routing: Maze Routing • Routing channels may be modelled by a weighted undirected graph called channel connectivity graph. • Node ↔ channel; edge ↔ two adjacent channels; capacity: (width, length) 7

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route A−A’ via 5−6−7

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route B−B’ via 5−2−3−6−9−10−7

updated channel graph

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route B−B’ via 5−6−7

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maze routing for nets A and B 9

Global Routing by Integer Programming • • • • •

Suppose that for each net i, there are ni possible trees ti1 , ti2 , . . . , tini to route the net. Constraint I: For each net i, only one tree tij will be selected. Constraint II: The capacity of each cell boundary ci is not exceeded. Minimize the total tree cost. Question: Feasible for practical problem sizes? – Key: Hierarchical approach! an routing instance 2

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trees of net 1

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C4 C1 C3

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C1

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C4 = 2

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C1 = 2

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C4

a feasible routing

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grid graph

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trees of net 2

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C1 C4 C3

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An Integer-Programming Example Boundary B1 B2 B3 B4

t11 0 1 0 1

t12 1 0 1 1

t13 1 1 1 0

t21 1 0 1 0

t22 0 1 1 1

t23 1 1 0 1

t31 1 1 0 0

t32 0 0 1 1

• gi,j : cost of tree tij ⇒ g1,1 = 2, g1,2 = 3, g1,3 = 3, g2,1 = 2, g2,2 = 3, g2,3 = 3, g3,1 = 2, g3,2 = 2. Minimize 2x1,1 + 3x1,2 + 3x1,3 + 2x2,1 + 3x2,2 + 3x2,3 + 2x3,1 + 2x3,2 subject to x1,1 + x1,2 + x1,3 x2,1 + x2,2 + x2,3 x3,1 + x3,2 x1,2 + x1,3 + x2,1 + x2,3 + x3,1 x1,1 + x1,3 + x2,2 + x2,3 + x3,1 x1,2 + x1,3 + x2,1 + x2,2 + x3,2 x1,1 + x1,2 + x2,2 + x2,3 + x3,2 xi,j

= = = ≤ ≤ ≤ ≤ =

1 (Constraint 1 (Constraint 1 (Constraint 2 (Constraint 2 (Constraint 2 (Constraint 2 (Constraint 0, 1, 1 ≤ i, j ≤ 3

I : t1 ) I : t2 ) I : t3 ) II : B1) II : B2) II : B3) II : B4)

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Hierarchical Global Routing • Marek-Sadowska, “Router planner for custom chip design,” ICCAD, 1986. • At each level of the hierarchy, an attempt is made to minimize the cost of nets crossing cut lines. • At the lowest level of the hierarchy, the layout surface is divided into R × R grid regions with boundary capacity equal to C tracks. • Let Rl be the # of grid regions of a given cut line l; a cut line can be divided into M = RCl sections. • Global routing can be formulated as a linear assignment problem: – xi,j = 1 if net i is assigned to section j; xi,j = 0 otherwise. – Each net crosses the cut line exactly once: – Capacity constraint of each section:

PN

i=1

PM

j=1

xij = 1, 1 ≤ i ≤ N .

xij ≤ C, 1 ≤ j ≤ M .

– wij : cost of assigning net i to section j. Minimize

PN PM

i=1 Upper leftmost pin of net

Section

j=1

wij xij .

Bounding box Ideal sections to route this net

Cut line

Lower rightmost pin of net

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The Routing-Tree Problem • Problem: Given a set of pins of a net, interconnect the pins by a “routing tree.”

gate array

standard cell

building block

• Minimum Rectilinear Steiner Tree (MRST) Problem: Given n points in the plane, find a minimum-length tree of rectilinear edges which connects the points. • M RST (P ) = M ST (P ∪ S), where P and S are the sets of original points and Steiner points, respectively.

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Steiner points

minimum spanning tree MST

MRST

Theoretic Results for the MRST Problem • Hanan’s Thm: There exists an MRST with all Steiner points (set S) chosen from the intersection points of horizontal and vertical lines drawn points of P . – Hanan, “On Steiner’s problem with rectilinear distance,” SIAM J. Applied Math., 1966. • Hwang’s Theorem: For any point set P ,

Cost(M ST (P )) Cost(M RST (P ))

≤ 32 .

– Hwang, “On Steiner minimal tree with rectilinear distance,” SIAM J. Applied Math., 1976. • Best existing approximation algorithm: Performance bound

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by Foessmeier et al.

– Foessmeier et al, “Fast approximation algorithm for the rectilinear Steiner problem,” Wilhelm Schickard-Institut f¨ ur Informatik, TR WSI-93-14, 93. – Zelikovsky, “An 11 approximation algorithm for the network Steiner problem,” Al6 gorithmica., 1993. MRST

MST

Hanan grid

Cost(MST)/Cost(MRST) −> 3/2

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A Simple Performance Bound • Easy to show that

Cost(M ST (P )) Cost(M RST (P ))

≤ 2.

• Given any MRST T on point set P with Steiner point set S, construct a spanning tree T 0 on P as follows: 1. Select any point in T as a root. 2. Perform a depth-first traversal on the rooted tree T . 3. Construct T 0 based on the traversal. 2 7 1

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depth−first traversal every edge is visited twice

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Cost(T’) <= 2 Cost(T)

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