Reconfigurable Computing Reconfigurable Computing Partitioning - - PowerPoint PPT Presentation

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Reconfigurable Computing

Reconfigurable Computing Reconfigurable Computing Partitioning Partitioning Chapter 5 Chapter 5

• Prof. Dr.
• Prof. Dr.-
• Ing. Jürgen Teich
• Ing. Jürgen Teich

Lehrstuhl für Hardware Lehrstuhl für Hardware-

• Software

Software-

• Co

Co-

• Design

Design

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Partitioning Partitioning -

• Motivation

Motivation

• A design implementation is often too big to allow an

implementation on a single FPGA.

• Possible solutions are:

Spatial partitioning: The design is partitioned into many

• FPGAs. Each partition block is implemented in one

single FPGA. All the FPGAs are used simultaneously. Temporal partitioning: The design is partitioned into blocks, each of which will be executed on one FPGA at a given time.

• We will give a short overview on spatial partitioning in the

first part of the chapter. Temporal partitioning algorithms will be considered in detail in the second part of the chapter.

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• Dataflow graph: A dataflow or sequencing graph or task

graph G =(V,E) is a directed acyclic graph in which: each node vi in V represents a task with execution time di An edge e =(u, v) represents a data dependency between the nodes u and v.

• Scheduling and ordering relation: Given a DFG G =(V,E) with

a precedence relation among the nodes

A schedule is a function s: V → N. A schedule defines for each node, the time at which the node will be executed on the reconfigurable device. – A schedule is feasible iff ∀(u,v) ∈ V: s(u) ≤ s(v) We define an ordering relation ≤ induced by any schedule s as follows: u ≤ v ↔ s(u) ≤ s(v)

Partitioning Partitioning – – definitions definitions

s: V

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The relation ≤ can be extended to sets as follows: (A ≤ B) ↔ ∀a ∈ A, b ∈ B: either a is not in relation with b or a ≤ b.

• Partition: Given a DFG G=(V,E) and a set R={R1, R2, ...., Rk}
• f reconfigurable devices. A partition P of a graph G toward

R is its division into some disjoint subsets P1, P2,,…,Pr : ∀Pi ∃Rj: S(Pi) ≤ S(Rj) ∧ T(Pi) ≤ T(Rj) where S(X) = size of X and T(X) = # terminals of X

• A partition is called spatial iff (pij=1 iff Pi will be implemented

in Rj) |{ Pi ∈ P: pij = 1}| =1 ∀Rj ∈ R

• A partition is temporal iff ∃Rj ∈ R: |{Pi ∈ P: pij = 1}| >1
• If all the devices in R are of the same type, then the partition

is said to be uniform.

• If |R|=1,we have a single device partition

Partitioning Partitioning – – definitions definitions

s: V

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Spatial partitioning Spatial partitioning

5

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Spatial Spatial partitioning partitioning -

• Problem

Problem

• Partitioning Constraints: Each FPGA

is characterized by:

The size, i.e., the number of LUTs, FFs available The terminals, i.e., the number of I/O pins available on the device A partition is valid iff: for a block B produced by the partition, we have:

S(B) <= S(device) where S(X) = size of X T(B) <= T(device) where T(X) = # terminals of X a f d b c e a c b f d e

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Spatial Spatial partitioning partitioning -

• Problem

Problem

• Objectives: The following objectives

are possible:

Minimize the number of cut nets Minimize the number of produced blocks Minimize the delay

• Difficult problem due to all the

constraints which are not always compatible.

• Solution approaches:

Use of heuristics for automatic partitioning Manual intervention

a f d b c e a c b f d e

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Spatial partitioning Spatial partitioning – – Approaches Approaches -

• Hierarchical

Hierarchical

• Motivation:

Reduce the problem complexity Keep the global view during partitioning Improve the final result in terms of number

• f devices.

Respecting the design hierarchy facilitates debugging Performance improvement

• Approach:

Apply an algorithm for clustering a flat netlist (creates red envelopes) Flatten the hierarchy except created (red) clusters Partition this flat netlist (reduced problem size)

Hierarchical spatial partitioning

B

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Spatial partitioning Spatial partitioning – – Approaches Approaches -

• Hierarchical

Hierarchical

• Removing all non-valid blocks

may produce a big amount of glue logic in the final problem

• Some non-valid blocks may be

partitioned separately by applying a divide-and-conquer strategy

• ST quality is used to determine

how good a partition block is: ST = S/T (S=Size, T=Terminal) defines the ratio size/terminal

• Poor ST-quality: Blocks having

many connections with other hierarchy blocks

Removing hierarchy is preferable

Flattening the hierarchy

Small size, big I/O pin number, poor ST-quality Partitioning

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Spatial partitioning Spatial partitioning – – Approaches Approaches -

• Hierarchical

Hierarchical

• Good ST-quality: Blocks having few

connections with other hierarchy blocks

Splitting is preferable

• Average ST-quality: calculated recursively

in a bottom-up fashion (for a global view)

• Device ST-quality: ST(D).

Device filling is good when the ST-quality of the assigned block is larger or equal to the device quality.

Big size, small I/O pin number, good ST-quality Splitting

ST < ST(D) Leaf block Remove Split Split Remove Split Split Remove Split Remove ST >= ST(D) and ST >= average ST ST >= ST(D) and ST < average ST Non leaf block with big amount of glue logic Non leaf block with small amount of glue logic

ST-quality Blocks

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Spatial partitioning Spatial partitioning – – User intervention User intervention

• Fully automatic partitioning never

satisfies designers

• User intervention may lead to more

efficient results

• A mixture of manual and automatic

strategies istherefore common

• User intervention:

Assignment of hierarchy blocks to devices Hierarchy modification Manual guidance of the automatic partitioning Invoking automatic partitioning on selected blocks (splitting)

Top B C D F E G A H FPGA 1 FPGA 2 FPGA 3

Top B C D F E G A H Pre-assignment of blocks to FPGAs Top B C F E G A

Flattening

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Spatial partitioning Spatial partitioning – – User intervention User intervention

Top B C D F E G A H Top B F E G A

Ungrouping

Top B C D F E G A H C

Splitting

Top B C D F E G A H

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Spatial partitioning Spatial partitioning – – Timing Timing – – Block replication Block replication

Critical path optimization

10 ns 20 ns 30 ns 10 ns 30 ns 70 ns 20 ns

Reducing the number of I/O pins

10 ns 20 ns 30 ns 10 ns 30 ns 70 ns

B1 B3 B2 B1 B3 B2 B2 B2

10 ns 20 ns 30 ns 10 ns 30 ns 50 ns

B1 B1 B2 B3 B2

10 ns 20 ns 30 ns 10 ns 30 ns 50 ns

B1 B3 B2 B2 B1

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Temporal partitioning Temporal partitioning

14

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Temporal partitioning Temporal partitioning – – Problem definition Problem definition

• Temporal partitioning:

We consider a single device temporal partitioning of a DFG G=(V,E) for a device R A temporal partition can also be defined as an ordered partition of G with the constraints imposed by R. With the ordering relation imposed on the partition, we reduce the solution space to only those partitions which can be scheduled on the device for execution. Therefore, cycles are not allowed in the dataflow graph. Otherwise, the resulting partition may not be schedulable on the device

a f d b c e a c b f d e

cycle

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Temporal partitioning Temporal partitioning -

• Problem

Problem

• Goal:

Computation and scheduling of a Configuration graph

• In a configuration graph,

Nodes are partitions or bitstreams Edges reflect the precedence in a given DFG The partition blocks communicate by means

• f inter-configuration registers usually

mapped into the processor address space The configuration sequence is controlled by a host processor On configuration, save register values. This requires a given amount of memory After reconfiguration, copy values back

A configuration graph

P1 P2 P3 P4 P5 Inter-configuration registers

IO Register IO Register IO Register IO Register

Processor

Bus Block

IO Register IO Register IO Register IO Register

FPGA

FPGA register mapping into the address spaces of the processor

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Temporal partitioning Temporal partitioning -

• Problem

Problem

• Objectives:

Minimize the number of interconnections. This is one

• f the most important objective since it will minimize

the amount of exchanged data the amount of memory for temporally storing the data

Minimize the number of produced blocks Minimize the overal computation delay

• Quality of the result: Provides a mean to

measure how good an algorithm performs

Connectivity of a graph G=(V,E): con(G) = 2*|E|/(|V|2 - |V|) Quality of Partitioning P = {P1,…,Pn}: Average connectivity over P High (low) quality means algorithm performs well (poor).

4 5 1 2 8 7 9 1 3 6

Quality = 0.2

1 3 4 5 6 2 8 7 9 1

Quality = 0.45

1 2 3 4 5 6 8 7 9 1

Connectivity = 0.24

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Temporal partitioning vs Scheduling Temporal partitioning vs Scheduling

• Scheduling: Given is a DFG and an architecture which is a

set of resources

Compute the starting time of each node on a given resource

• Temporal partitioning: Given is a DFG and a reconfigurable

device

The starting time of each node is the starting time of the partition to which it belongs Compute the starting time of each node on the device

• Solution approaches:

List scheduling Integer Linear Programming Network Flow Spectral method

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Unconstrained Scheduling Unconstrained Scheduling

• ASAP (as soon as possible)

Defines the earliest starting time for each node in the DFG Computes the minimal latency (lower bound)

• ALAP (as late as possible)

Defines the latest starting time for each node in the DFG according to a given latency

• The mobility of a node is the difference between the

ALAP-starting time and ASAP-starting time

Mobility is 0 node is on a critical path

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ASAP ASAP-

• Example

Example

Unconstrained scheduling with optimal latency: L = 4

Zeit 4

* +

• <

Zeit 0 Zeit 3 Zeit 4

* * * * * +

• Time 1

Time 2 Time 3 Zeit 3 Time 4 Time 0

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ASAP ASAP-

• Algorithm

Algorithm

ASAP(G(V,E),d) { FOREACH (vi without predecessor)

• s(vi) := 0;

REPEAT { choose a node vi whose predecessors are all planned; s(vi) := maxj:(vj,vi)∈E {s(vj)+ dj}; } UNTIL (all nodes vi are planned); RETURN s }

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ALAP ALAP-

• Example

Example

Unconstrained scheduling with optimal latency: L = 4

* +

• <

Zeit 1 Zeit 3 Zeit 4

* * * * * +

• Zeit 4

Time 1 Time 2 Time 3 Time 4 Time 0

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ALAP ALAP-

• Algorithm

Algorithm

ALAP(G(V,E),d, L) { FOREACH (vi without successor) s(vi) := L - di; REPEAT { Choose a node vi whose successors are all planned; s(vi) := minj:(vi,vj)∈E {s(vj)} - di; } UNTIL (all nodes vi are planned); RETURN s }

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Mobility Mobility

* * 1 1

Zeit 0 Zeit 1 Zeit 2 Zeit 3 Zeit 4

* * + < * + *

• *

*

• 2

2 2 2 * + + <

Time 1 Time 2 Time 3 Time 4 Time 0

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Constrained scheduling Constrained scheduling

• Extended ASAP, ALAP

Compute ASAP or ALAP Assign the tasks earlier (ALAP) or later (ASAP) such that the resource constraints are always fulfilled by construction

• Listscheduling

A list L of ready to run tasks is created Tasks are placed in L in decreasing priority order At a given step, the task with highest priority is assigned to the free resource. Criteria can be: number of successors, mobility, connectivity, etc.

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Extended ASAP, ALAP Extended ASAP, ALAP

* +

• <

* * * * * +

• 2 Multiplier, 2 ALUs (+, −, <)

Time 0 Time 1 Time 2 Time 3 Time 4

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Constrained scheduling Constrained scheduling

* +

• <

* * * * * +

• 3

3 2 2 1 1 1 1

Criterion: number of successors Resource: 1 multiplier, 1 ALU (+, −, <)

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Constrained scheduling Constrained scheduling

Time 0 Time 1 Time 2 Time 3 Time 4 Time 5 Time 6 Time 7

* +

• <

* * * * + *

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Temporal partitioning vs constrained scheduling Temporal partitioning vs constrained scheduling

• List Scheduling (LS) for partitioning
• 1. Construct a list L of all nodes with priorities
• 2. Create a new empty partition Pact

2.1 Remove a node from the list and place it in the partition 2.2 If size(Pact) <= size(R) and T(Pact) <= T(R) goto 2.1, else goto 2.3 2.3 If empty(list), stop else goto 2.

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Temporal partitioning vs constrained scheduling Temporal partitioning vs constrained scheduling

* +

• <

* * * * * +

• 3

3 2 2 1 1 1 1

Criterion: number of successors size(FPGA) = 250, size (mult) = 100, size(add) = size(sub) = 20, size(comp) = 10

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P2 P1

Temporal partitioning vs constrained scheduling Temporal partitioning vs constrained scheduling

+ < * * * * P3

• *
• *

+ * +

• <

* * * * * +

• 3

3 2 2 1 1 1 1

Connectivity: c(P1) = 1/6, c(P2) = 1/3, c(P3) = 2/6 Quality: 5/6

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P2 P1

Temporal partitioning vs constrained scheduling Temporal partitioning vs constrained scheduling

+ < * * * P3 *

• *
• *

+

• <

* * * * * +

• 3

3 2 2 1 1 1 1

Connectivity: c(P1) = 2/10, c(P2) = 2/3, c(P3) = 2/3 Quality: 1.2 Connectivity is better

* +

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Temporal partitioning Temporal partitioning – – List scheduling List scheduling – – list list construction construction

• ASAP

Place the currently processed node in the list if all its predecessors are already in the list. This corresponds to: Assigning level-number to nodes Scheduling the nodes for execution according to the level number

• Drawback

“Levelization”: Nodes are assigned to partitions

• nly on the basis of their level-number

(increase data exchange)

• Advantage

Fast (linear run-time) Local optimization possible

+ / * * +

• *
• /

Level 0 Level 1 Level 2 Level 3

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Temporal partitioning Temporal partitioning – – List scheduling List scheduling -

• Improvement

Improvement

• Local optimization by configuration switching (Bobda)
• If two consecutive partitions P1 and P2 share a common set
• f operators, then:

We implement the minimal set of operators needed for the two partitions. We use signals multiplexing to switch from one partition to the next

• ne.
• Drawbacks: More resources are needed to implement the

signal switching

• Advantages:

Reconfiguration time is reduced Device operation is not interrupted

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Temporal partitioning Temporal partitioning – – List scheduling List scheduling – – config config switching switching

a b c d e j i h

A d d S u b A d d M ul t A d d S u b M ul t

f b c j h

A d d A d d M u lt

g

S u b

a i d f e Configuration 2 Configuration 1 Inter configuration register g

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Temporal partitioning Temporal partitioning – – List scheduling List scheduling-

• config

config switching switching

a b c d e j i h

A d d S u b A d d M ul t A d d S u b M ul t

f b c j h

A d d A d d M u lt

g

S u b

a i d f e Configuration 2 Configuration 1 Inter configuration register g

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Temporal partitioning Temporal partitioning – – List scheduling List scheduling-

• config

config switching switching

a b c d e j i h

A d d S u b A d d M ul t A d d S u b M ul t

f b c j h

A d d A d d M u lt

g

S u b

a i d f e Configuration 2 Configuration 1 Inter configuration register g

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Temporal partitioning Temporal partitioning – – List scheduling List scheduling -

• Improvement

Improvement

• Improved List Scheduling algorithm
• 1. Generate the list of nodes node_list
• 2. Build a first partition P1
• 3. While (!node_list.empty( ))

4. build a new partition P2 5. If union(P1, P2) fits on the device, then implement configuration switching with P1 and P2 6. else set P1 = P2 and goto 3

• 7. Exit

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Temporal partitioning Temporal partitioning – – ILP ILP

• With the ILP (Integer Linear Programming),

the temporal partitioning constraints are formulated as inequalities. The system of inequalities is then solved using an ILP-solver.

• The constraints usually considered are:

Uniqueness constraint Precedence (temporal order) constraint Memory constraint Resource constraint Latency constraint

• Notations: yvi = 1 ↔ v ∈ Pi

wuv = 1 ↔ u ∈ Pi ∧ v ∈ Pj ∧ Pi ≠ Pj

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Temporal partitioning Temporal partitioning – – ILP ILP

• Unique assignment constraint: Each task should be placed

in exactly one partition: ∀v ∈ V: ∑i=1,…,m yvi = 1

• Precedence constraint: for each edge (u,v) in the graph,

node u must be placed either in the same partition as v or in an earlier partition than that in which v is placed:

• Resource constraint: The sum of the resources needed to

implement the modules in one partition should not exceed the total amount of available resources:

Device area constraint Device terminal constraints

∑ ∑

= =

≤ ∈ ∀

m 1 i vi m 1 i ui

y i y i : E ) v , u (

) device ( S ) u ( s y : P P

V u ui i

≤ ∈ ∀

∑ ∈

) device ( T w ) device ( T w : P P

i i i i

P v , P u uv P v , P u uv i

≤ ∧ ≤ ∈ ∀

∑ ∑

∈ ∉ ∉ ∈

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Temporal partitioning Temporal partitioning – – Network Network-

• flow

flow-

• approach

approach

• Recursive bi-partitioning:

The goal at each step is the generation of a uni-directional bi-partition The goal at each step is to compute a bi- partition wich minimizes the edge-cut size between the two partition blocks. Network flow methods are used to compute the bi-partition with minimal edge-cut size. Directly applying the min-cut max-flow theorem may lead to non-unidirectional cuts. Therefore, the original G is first transformed into a new graph G' in which each cut will be unidirectional in an optimal solution.

Unidirectional recursive bipartitioning A bidirectional cut

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Temporal partitioning Temporal partitioning – – Network Network-

• flow

flow – – graph graph transformations transformations

• Two-terminal net transformation

Replace an edge (v1, v2) by two edges (v1, v2) with capacity 1 and (v2, v1) with infinite capacity

• Multi-terminal net transformation

For a multi-terminal net {v1, v2,…,vn}, introduce a dummy node v with no weight and a bridging edge (v1, v) with capacity 1. Introduce the edges (v, v2), .... (v, vn), each of which is assigned a capacity of 1. Introduce the edges (v2, v1), ..., (vn, v1), each

• f which is assigned an infinite capacity

Having computed a min-cut in the trans- formed graph G, a min-cut can be derived in G: for each node of G' assigned to a partition, its counterpart in G is assigned to the corresponding partition in G.

1 1

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Temporal partitioning Temporal partitioning – – Spectral method Spectral method -

• approach

approach

• Goal:

Increase the quality of a partitioning

• 2-Step Method:

Place the connected components of the graphs in the same area before partitioning Use a partitioning strategy to compute the configuration graph

• To solve Step 1:

Use the wire length model Place the component in an n-D vector space in such a way that the sum of the distances among the components is minimized (k-D spectral embedding)

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• 3

3-

• D spectral embedding

D spectral embedding

Given a graph G = (V, E), find the k placement vectors X1 = (X1i),…,Xk = (Xki), (i=1,…,|V|) which minimize:

j i 2 j k i k 2 j 2 i 2 n 1 i n 1 j 2 j 1 i 1

c ) ) x x ....( ) x x ( ) x x (( 2 1 Z − + − + − =

∑ ∑

= =

To avoid the trivial case where all components have the same position, the following constraints are additionally imposed:

1 X X .... X X X X

k T k 2 T 2 1 T 1

= = = =

Given a graph G = (V, E), a placement vector X1 = (x11,x12,..., X1|V|) is a vector in which the i-th entry describes the coordinate of the i-th node in

• ne dimension.

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• 3

3-

• D spectral embedding

D spectral embedding

• To solve this optimization problem, introduce the Lagrange multipliers

and the Lagrangian:

• Taking the first partial derivative of L with respect to the vectors X1, X2,

…, Xk leads to the following equation

• The solutions are the eigenvectors X1, X2, …, Xk related to the

eigenvalues

k 2 1

,......, , α α α

Hall(1970) proved that with B being the laplacian (D - C) matrix of G

• Connection matrix C: C(i,j) = 1 if nodes i and j are connected and 0
• therwise (symmetric matrix).
• Degree matrix D: D(i,i) = sum of column i entries in C. D(i,j) = 0 else.

k T k 2 T 2 1 T 1

BX X ... BX X BX X z + + + =

) 1 X X ( ... ) 1 X X ( BX X ... BX X L

k T k 1 1 T 1 1 k T k 1 T 1

− − − − − + + = α α ) X BX ( 2 )... X BX ( 2 ) X BX ( 2

k k k 2 2 2 1 1 1

= − = − = − α α α

k 2 1

,......, , α α α

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• 3

3-

• D spectral embedding

D spectral embedding

                                            1 1

• 1
• 3

1

• 1
• 3

1

• 1
• 1
• 3

1

• 1
• 1
• 1
• 3

1

• 1
• 1
• 1
• 3

1

• 1
• 1
• 3

1

• 1
• 1

1

• 1

1

• 1

1

• 1

1

• 1

1

• 1

1

• 1

                    0.355147 0.241072

• 0.241072
• 0.225048

0.152761

• 0.0410593

                    0.0 0.325058

• 0.325058

0.0 0.0 0.0                     0.191984

• 0.0749483
• 0.0749481
• 0.191984

0.0749482 0.34188

0.438447 0.267949 0.112586

1 3 6

= = = α α α

1 3 2

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• Generation of Partitions

Generation of Partitions

Incremental generation of P0, P1, …,Pr Select components along the z-axis and put them in the actual partition block until the size of the block exceeds the size of the FPGA: Initially, At each step i, is partitioned into and This process is repeated until We have successive bipartitions

V P ~

0 =

∅ =

i

P ~

i

P

1 i

P ~

+

i

P ~ ) P P ~ P ~ , P (

i i 1 i i

− =

+

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• Generation of Partitions

Generation of Partitions -

• elimination of cycles

elimination of cycles

Because the spectral method works only on undirected graphs, the computed temporal partitioning may be not satisfying order constraints. Post-processing is required to eliminate cycles in the configuration graph P0 P1

Post-processing is done via a modified version of the Kernighan-Lin Algorithms

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• Generation of Partitions

Generation of Partitions -

• elimination of cycles

elimination of cycles

) v ( E ) v ( I ) v ( D

1 i ~ i

P P

+

− =

Cut = 3

i

P

1 i

P ~

+

Cut = 4

i

P

1 i

P ~

+

Iterative improvement using the Kernighan-Lin (KL) algorithm Improve the cut by moving a component v from one partition block to another block. Gain: Problem: How to define the cut and the gain for directed graphs?

Cut ?

1 i

P ~

+

i

P

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• Generation of Partitions

Generation of Partitions -

• elimination of cycles

elimination of cycles

Definitions:

∑

∈

=

P v | ) v , v ( ij i P

j j i

w ) v ( E

{ }

Q v P v | E ) v , v ( E

j i j i PQ

∈ ∧ ∈ ∈ =

i

P

1 i

P ~

+

) v ( E

i

P

) v ( E

1 i ~

P +

) v ( I

1 i ~

P +

) v ( I

i

P

i

P

1 i

P ~

+

) v ( E

i

P

) v ( I

i

P

) v ( I

1 i ~

P +

) v ( E

1 i ~

P +

∑

∈

=

P v | ) v , v ( ij i P

j i j

w ) v ( I

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Temporal partitioning Temporal partitioning – – Spectral approach Spectral approach -

• Generation of Partitions

Generation of Partitions -

• elimination of cycles

elimination of cycles

Goal:

• r

Use the KL algorithm on two instances of the same problem

In the first instance, we seek the following goal where the gain of moving a component v is In the second instance, we seek the following goal where the gain of moving a component v is

1 i ~ i P

P

E

+

=

i 1 i ~

P P

E

+

=

) v ( I ) v ( E ) v ( D

1 i ~ i

P P

+

− = ) v ( E ) v ( I ) v ( D

1 i ~ i

P P

+

− =