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Multi-Commodity Flow Based Routing. Set up ILP formulation for MCF routing Capacity of each edge in G is 2 Each edge in G becomes a pair of bi-directional arcs in F n 1 = { a,l }, n 2 = { i,c }, n 3 = { d,f }, n 4 = { k,d }, n 5 = { g,h }, n 6 = { b,k }. Flow Network.

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multi commodity flow based routing
Multi-Commodity Flow Based Routing
  • Set up ILP formulation for MCF routing
    • Capacity of each edge in G is 2
    • Each edge in G becomes a pair of bi-directional arcs in F
    • n1 = {a,l}, n2 = {i,c}, n3 = {d,f}, n4 = {k,d}, n5 = {g,h}, n6 = {b,k}

Practical Problems in VLSI Physical Design

flow network
Flow Network
  • Each arc has a cost based on its length
    • Let xek denote a binary variable for arc e w.r.t. net k
    • xek= 1means net k uses arc e in its route
    • Total number of x-variables: 16 × 2 × 6 = 192

Practical Problems in VLSI Physical Design

ilp objective function
ILP Objective Function
  • Minimize

Practical Problems in VLSI Physical Design

ilp demand constraint
ILP Demand Constraint
  • Utilize demand constant
    • zvk= 1 means node v is the source of net k (= −1 if sink)
    • Total number of z-constants: 12 × 6 = 72

Practical Problems in VLSI Physical Design

ilp demand constraint cont
ILP Demand Constraint (cont)
  • Node a: source of net n1

Practical Problems in VLSI Physical Design

ilp demand constraint cont1
ILP Demand Constraint (cont)
  • Node b: source of net n6

Practical Problems in VLSI Physical Design

ilp capacity constraint
ILP Capacity Constraint
  • Each edge in the routing graph allows 2 nets

Practical Problems in VLSI Physical Design

ilp solutions
ILP Solutions
  • Min-cost: 108 (= sum of WL), 22 non-zero variable

Practical Problems in VLSI Physical Design

ilp based mcf routing solution
ILP-based MCF Routing Solution
  • Net 6 is non-optimal
    • Due to congestion

Practical Problems in VLSI Physical Design

drawback of ilp based method
Drawback of ILP-based Method
  • ILP is non-scalable
    • Runtime quickly increases with bigger problem instances
  • Shragowitz and Keel presented a heuristic instead
    • Called MM (MiniMax) heuristic [1987]
    • Repeatedly perform shortest path computation and rip-up-and-reroute

Practical Problems in VLSI Physical Design

mm heuristic
MM Heuristic
  • Initial set up: shortest path computation
    • Ignore capacity, some paths are not unique

Practical Problems in VLSI Physical Design

first iteration of mm heuristic
First Iteration of MM Heuristic
  • Step 1
    • Capacity of channel c(e,f) and c(d,i) is violated
    • Max overflow M1 = 3 − 2 = 1 > 0, so we proceed
    • Notation: channel c(e,f) represents arc pair (e,f) and (f,e)

Practical Problems in VLSI Physical Design

first iteration of mm heuristic cont
First Iteration of MM Heuristic (cont)
  • Step 2
    • Set of channels with overflow of M1: J1 = {c(d,i), c(e,f)}
    • Set of channels with overflow of M1 and M1− 1 : J10 = {c(a,d), c(e,h), c(i,j), c(j,k), c(d,i), c(e,f)}
  • Step 3
    • Cost of J10 = {c(a,d), c(e,h), c(i,j), c(j,k), c(d,i), c(e,f)} is ∞

Practical Problems in VLSI Physical Design

first iteration of mm heuristic cont1
First Iteration of MM Heuristic (cont)
  • Step 4
    • Set of nets using channels in J1: K1 = {n1, n2, n3, n4, n5, n6}
    • Set of nets using channels in J10: K10 = K1

Practical Problems in VLSI Physical Design

first iteration of mm heuristic cont2
First Iteration of MM Heuristic (cont)
  • Step 5
    • Compute shortest paths for nets in K1 using new cost (= Step 3)
    • n1 & n6 have non-infinity cost, so we proceed

Practical Problems in VLSI Physical Design

first iteration of mm heuristic cont3
First Iteration of MM Heuristic (cont)
  • Step 6
    • Net with minimum wirelength increase between n1 & n6: k0 = n1

Practical Problems in VLSI Physical Design

first iteration of mm heuristic cont4
First Iteration of MM Heuristic (cont)
  • Step 7
    • Use new routing for n1
    • Wirelength didn’t change, but congestion improved

Practical Problems in VLSI Physical Design

second iteration of mm heuristic
Second Iteration of MM Heuristic
  • Details in the book
    • Use new routing for n3
    • Wirelength increased (due to detour in n3), but congestion improved

Practical Problems in VLSI Physical Design

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