Handbook of algorithms for physical design automation part 14 ppsx

Definition 3 Given a hypergraph, GV, E, nets that have vertices in multiple blocks belong to the cutset, Ec, of the hypergraph... Consequently, vertices may be assigned positions along t

For example, a net consisting of four vertices is represented by an equivalent six-edge complete graph To cut off one vertex from the rest requires cutting three edges, so the weight should be 1/3 (for a total edge weight of 1) However, to cut off two vertices from the rest requires cutting four edges (each with weight 1/4) Because some of the edges assigned weights of 1/3 and 1/4 may be the same, this weighting scheme is inconsistent

Lengauer [Len90] proves that no matter what weighting scheme is selected, there will always exist an exact graph bipartition with a deviation of
(√|e|) from the cost of cutting a single net.

Additionally, Ihler et al [IWW93] conjecture that a clique graph model is the best in terms of deviation from the true cost of cutting one net

In the generic clique model, a net on|e| vertices induces a complete graph where each edge has

weight

wi =|e| − 11 This weighting scheme arises from linear placements into fixed slots separated by a unit distance The denominator indicates the minimum total wirelength used to connect the|e| vertices Vannelli

and Hadley [VH90] propose the following metric that guarantees the weight of edges cut under a

k-way partitioning has an upper bound of 1.

wi = |e| 1

k

 |e|

k



Huang [HK97] proposes a weight of

wi =|e|(|e| − 1)4 that distributes the weight of one net evenly across two edges and gives an expected cut weight of 1 The following weighting scheme distributes the edge weight evenly across|e| − 1 edges:

w i = 2

|e|

In Ref [AY95], the authors use a variant of Huang’s metric:

|e|(|e| − 1)

2|e|− 2

2|e|

7.1.2 PARTITIONING ANDCLUSTERINGMETRICS

In this section, we give some definitions relevant to hypergraph partitioning and use them to describe metrics used in hypergraph partitioning

Definition 3 Given a hypergraph, G(V, E), nets that have vertices in multiple blocks belong to

the cutset, Ec, of the hypergraph Given k blocks, the cutset between the ith pair of blocks where

i= 1 · · ·

k

2



is denoted by Ec

i

C1 ∪ C2 ∪ · · · ∪ C k = V where C i ∩ C j = ∅ and α|V| ≤ |C i | ≤ β|V| for 1 ≤ i < j ≤ k and

0≤ α, β ≤ 1.

Definition 5 The weight of the ith block is denoted by w(C i) Usually, it is equal to the number

of vertices in the ith block, |C i|

construct a clustered hypergraph H(V, E) such that for every e ∈ E, there is a hyperedge e ∈ E

with e= {C|∃υ ∈ e ∩ C}.

Mincut partitioning: This metric counts the number of nets running between pairs of blocks

min f (V, k) =k

i=1|Ec

i| s.t α|V| ≤ w(Ci ) ≤ β|V|

For example, if a net spans three blocks then it would be counted three times in the objective [Alp96]

A slightly different objective is one that counts the number of entire nets that are cut (this is formally

called the netcut) These objectives are identical if the number of blocks is two Typically,α = 0.45

andβ = 0.55 for bipartitioning For k-way partitioning, some authors favor the following constraints:

|V|

αk ≤ w(C i ) ≤ α|V|

k

whereα > 1 [KK99].

Min-ratiocut bipartitioning: The ratiocut metric, rc, is used in Refs [WC91], [RDJ94] and others

as a way of incorporating balance constraints into the objective The objective is

min f (V, 2) = rc= |Ec|

w (C1)w(C2)

where the numerator indicates the netcut For a given netcut, this metric is minimized when the two blocks are of equal size However, as Alpert and Kahng [AK95] point out, the weakness in this metric

is that rcis very sensitive to change in|Ec| and relatively unaffected by changes in w(C1) or w(C2).

Thus, given a small enough netcut, it is possible to obtain a minimal ratio cut even if the block sizes are uneven

In the analytic bipartitioning technique described in Ref [RDJ94], vertices are assigned to

positions along the x-axis, simultaneously Block assignments are then derived from the coordinates

in some fashion Thus, it is not possible to move individual vertices from one block to another as

is the case with iterative-based partitioners Consequently, vertices may be assigned positions along

the x-axis that do not satisfy even fairly loose balance constraints in which a block is allowed to have

between 45 and 55 percent of the cells (or total cell area)

Min-ratiocut k-way partitioning: Chan et al [CSZ94] generalize the ratiocut metric for k blocks to

k



i=1

|Ec

i|

w (Ci) ≤

k



i=1

λi

whereλi is the ith smallest eigenvalue of the Laplacian matrix of G (V, E).

Scaled cost: Another metric that combines the usual minimum cut objective with block size

constraints is

f (V, k) = 1

|V|(k − 1)

k



i=1

|Ec

i|

w (Ci)

This metric is used in Refs [AK93,AK94,AK96,KK98,AKY99]

e

FIGURE 7.3 Absorption metric.

Absorption: The absorption metric measures the sum of nets, as a fraction, that are absorbed by

blocks [SS95]

max

k



i=1

 |e ∩ C i| − 1

|e| − 1

At the two extremes, nets that have one vertex in a block C add 0 to the absorption metric; nets that have all vertices in a block C add 1 to the absorption metric In Figure 7.3, |e ∩ C| = 2, |e∩C|−1

|e|−1 =1

3

In addition to the usual balance constraints, partitioning formulations may include constraints for vertices that are assigned to a specific block The presence of fixed terminals adds some convexity

to this otherwise highly nonconvex problem, making it computationally less expensive [EAV99] In Ref [ACKM00], the authors point out that the presence of fixed vertices makes the problem trivial

in the sense that only one or two passes of an iterative improvement engine are required to approach

a good solution

7.2 MOVE-BASED PARTITIONING METHODS

In this section, we will outline the most significant developments in the field of iterative improvement-based partitioning Iterative improvement forms the basis of multilevel partitioning, which represents the state of the art as far as partitioning is concerned

7.2.1 KERNIGHAN–LINHEURISTIC

Kernighan and Lin’s work was the earliest attempt at moving away from exhaustive search in determining the optimal netcut subject to balance constraints In Ref [KL70], they propose a

O (|V|2log|V|) heuristic for graph bipartitioning based on exchanging pairs of vertices with the highest gain between two blocks, C1and C2 They define the gain of a pair of vertices as the number

of edges by which the netcut decreases if vertices x and y are exchanged between blocks Assuming

a ijare entries of a graph adjacency matrix, the gain is given by the formula

g (vx , v y) =

⎛

⎝a xj−

vj∈C1

a xj

⎞

⎠ +

⎛

⎝a yj−

vj∈C2

a yj

⎞

⎠ − 2a xy

The terms in parentheses count the number of vertices that have edges entirely within one block minus the number of vertices that have edges connecting vertices in the complementary block

TABLE 7.1 Gain Computations

A B |AB A | + |OAB A | − |IA| − |OA|

B A |IAB A | + |OAB A | − |IA| − |OA|

The procedure works as follows: vertices are initially divided into two sets, C1and C2 A gain is computed for all pairs of vertices(vi , v j) with vi ∈ C1and v j ∈ C2; the pair of vertices,(vx , v y), with the highest gain is selected for exchange; v x and v yare then removed from the list of exchange candidates

The gains for all pairs of vertices, v i ∈ (C1 − {v x }) and v j ∈ (C2− {v y }), are recomputed and the pairing of vertices with the highest gain are exchanged The process continues until g (vx , v y) = 0,

at which point, the algorithm will have found a local minimum The algorithm can be repeated to improve upon the current local minimum Kernighan and Lin observe that two to four passes are necessary to obtain a locally optimal solution

In Ref [SK72], Schweikert and Kernighan introduce a model that deals with hypergraphs directly They point out that the major flaw of the (clique) graph model is that it exaggerates the importance

of nets with more than two connections After bipartitioning, vertices connecting large nets tend to end up in the same block, whereas vertices connected to two point nets end up in different blocks They combine their hypergraph model in the Kernighan–Lin partitioning heuristic and obtain much better results on circuit partitioning problems

7.2.2 FIDUCCIA–MATTHEYSESHEURISTIC

The Fiduccia–Mattheyses (FM) [FM82] method is a linear-time,[O(|P|)] per pass, hypergraph

bipartitioning heuristic Its impressive runtime is due to a clever way of determining which vertex to move based on its gain and to an efficient data structure called a bucket list

Borrowing the terminology and notation from Ref [KN91], a critical net is one that is connected

to a single vertex in one of the blocks (so that the removal of that vertex removes the net from that block) A net state is a combination of subnetworks that contain a net as well as the subnetworks in

which the net is critical Let A indicate that a net is entirely within block A; let A Aindicate that a net

in block A is critical to block A Let the prefix I or O indicate that the net is an input or output to the vertex, thus IAB A indicates the input net has vertices in blocks A and B but that is critical to block A.

The gain equations are computed with respect to vertices and are given in Table 7.1

In Figure 7.4, if we move vertex u from A to B, the gain is computed in the following way.

|AB A | = 2 because nets 1 and 2 are critical to A, |OAB A | = 2 refers to nets 1 and 2 as well, |IA| = 3

Net 1

Net 2

v u

FIGURE 7.4 Example of gain computations.

because there are three inputs to u in A, and |OA| = 2 because there are two outputs to u in A The

gain is given by|AB A | + |OAB A | − |IA| − |OA| = 2 + 2 − 3 − 2 = −1, which implies moving vertex

u will result in the netcut increasing by 1 On the other hand, moving vertex v from B to A implies

|IAB A | = 1 because v has one input that is critical to A, |OAB A | = 0 because v has no outputs critical

to A, |IA| = 0 because v has no inputs in A, and |OA| = 0 because v has no outputs in A, for a total

gain of 1

The gains are maintained in a[−|P|max· · · |P|max] bucket array whose ith entry contains a doubly linked list of free vertices with gains currently equal to i The maximum gain, |P|max, is obtained when a vertex of degree|P|max(i.e., a vertex that is incident on|P|maxnets) is moved across the block boundary and all of its incident nets are removed from the cutset (Figure 7.5)

The free vertices in the bucket list are linked to vertices in the main vertex array so that during a gain update, vertices are removed from the bucket list in constant time (per vertex) Superior results are obtained if vertices are removed and inserted from the bucket list using a last-in-first-out (LIFO) scheme (over first-in-first-out [FIFO] or random schemes) [HHK97] The authors of Ref [HHK97] speculate that a LIFO implementation is better because vertices that naturally block together will tend to be listed sequentially in the netlist Care must be exercised to compute gains correctly for situations where a cell has two inputs on the same net

7.2.3 IMPROVEMENTS ON THEFIDUCCIA–MATTHEYSESHEURISTIC

This section discusses a few noteworthy improvements to the original FM implementation that have helped the acceptance of FM as the most popular partitioning technique

The first improvement to FM is Krishnamurthy’s look-ahead scheme [Kri84], in which a vertex belonging to a multivertex net, which is in the cutset, is considered for a move Moving this vertex may not necessarily remove the net from the cutset in the current pass, but may do so in a future pass

Kirshnamurthy’s method calculates a gain vector consisting of a sequence of r gain values, which are likely to result in r moves from the current move The rth level gain counts the reduced netcut after r

moves If there are ties in the current gain value, gain vectors are calculated for those configurations;

ties at the ith move are broken by looking at the possible gains at the (i + 1)st iteration.

Sanchis [San89] extends the FM concept to deal with multiway partitioning Her method incor-porates Fiduccia and Mattheyses’ gain bucket structure (modified for multiway partitioning) with Krishnamurthy’s look-ahead scheme One pass of the algorithm consists of examining moves that

result in the highest gain among all k (k −1) bucket lists The vertex with the highest gain that satisfies the balance criterion is then moved After a move, all k (k − 1) bucket lists are updated.

Cell

Maximum gain

−Pmax

Pmax

FIGURE 7.5 Bucket-list data structure used in FM iterative improvement partitioning algorithm.

Cell 1

Cell 2

Cell 1

Cell 2

FIGURE 7.6 On the left side, the netcut is 2; on the right side, the inverter has been replicated so the netcut

is only 1

An extension to the FM algorithm works by replicating cells across blocks to reduce the netcut as

in Figure 7.6 [KN91] Replication is useful in the context of field programmable gate array (FPGA) partitioning where there are hard limits on the number of I/O resources The concept of replication

is depicted in Figure 7.6

The principal modification to the FM algorithm is the insertion of gain equations that model cell replication Again, using the notation from Ref [KN91] and the previous section, the gains are given

in Table 7.2, where the last four lines are due to vertex replication or unreplication Notice that there

is very little change to the algorithm, thus the asymptotic complexity is equivalent to that of FM Dutt and Deng [DD96] observe that iterative improvement engines such as FM or look-ahead do not identify blocks adequately, and consequently, miss locally optimal solutions with respect to the netcut They point out that in FM, the total gain of a vertex is composed of the sum of an initial gain component and an updated gain component They propose to make the decision regarding which subsequent vertices to move based on the updated gain component exclusively

In Ref [CL98], the authors propose a k-way FM-based partitioning method that does not rely

on recursive subdivision of the solution space Up until that point, a k-way partitioning solution

meant partitioning into two blocks, then four, and so on in a recursive fashion The problem with this approach is that vertices can only be moved between the two blocks within the current partitioning level, so the partitioning solution is fairly localized, as is illustrated on the left in Figure 7.7 In their approach, two blocks form a pair when the cutsize between them is maximum or minimum during the last several passes FM-type partitioning is then performed on vertices within the two selected blocks as on the right in Figure 7.7

7.2.4 SIMULATEDANNEALING

In the late 1980s, simulated annealing emerged as a viable means to solve difficult combinatorial problems The nomenclature comes from the process of crystal growth, called annealing A material

is initially heated to molten state If it is cooled slowly enough, the molecules gradually fall into a state of minimal energy and the material assumes a beautiful crystalline shape The mathematical analogy is that state corresponds to a feasible solution, energy corresponds to solution cost, and

TABLE 7.2 More Detailed set of Gain Computations Previous State Next State Gain Equation

A B |AB A | + |OAB A | − |IA| − |OA|

B A |IAB A | + |OAB A | − |IA| − |OA|

AB A |IAB B | − |OAB A | − |OAB|

AB B |IAB A | − |OAB B | − |OAB|

FIGURE 7.7 Recursive versus pairwise k–way partitioning.

minimal energy corresponds to an optimal solution The principal advantage of simulated annealing over other methods is its ability to accept moves that will increase the cost function, initially with

a reasonably high probability, but with decreasing probability as the temperature decreases In this way, it is possible to climb out of local optima The details of the simulated annealing algorithm are given in Ref [KGV83]

Simulated annealing is applied to the problem of graph partitioning in Ref [JAMS89] Any

partition of vertices into two sets, C1and C2, is a valid solution, where C1and C2are not necessarily the same size The algorithm attempts to even out partition sizes by moving the vertex that results in the least increase in cutsize from the larger set to the smaller set using the cost function

cost(C1, C2) = |{{u, v} ∈ E : u ∈ C1 and v ∈ C2}| + α(|C1| − |C2|)2

The temperature is embedded in theα parameter, thus at higher temperatures, imbalanced block

sizes are penalized according to the square of the difference in block sizes As the temperature decreases, block sizes become more balanced Simulated annealing tends to produce smaller netcuts than iterative methods, albeit with much greater runtimes [JAMS89]

7.3 MATHEMATICAL PARTITIONING FORMULATIONS

Analytical partitioning methods use equation solving techniques to assign vertices to one of the two blocks so that the number of edges with endpoints in both blocks is minimized In this section, we

use the notation C1and C2to denote the sets of vertices in each block In Ref [CGT95], the authors use spectral bipartitioning to bipartition the vertices about the median of the entries in the second eigenvector of the Laplacian matrix In Ref [AK94], the authors use a space-filling curve traversal

of the space spanned by a small set of eigenvectors to determine where to split the ordering induced

by the set of eigenvectors This section discusses analytical approaches to partitioning in more detail The formulations in this section use the definitions given below

edge connecting vertices i and j, the adjacency matrix, A, is defined as

aij=



wij

Note that usually, wij is set to 1.

We denote the ith eigenvalue of A by αi

Definition 8 Given an adjacency matrix of a graph on |V| vertices, the diagonal degree matrix,

D, is defined as

dii=

|V|



j=1

aij

Definition 9 Given a graph on |V| vertices, define the corresponding |V| × |V| Laplacian matrix,

L= D − A.

7.3.1 QUADRATICPROGRAMMINGFORMULATION

Given a graph G (V, E), vertices u ∈ V and v ∈ V, let xv = 1 if vertex v belongs to block 1 and

xv = −1 if vertex v belongs to block 2 We wish to minimize the number of edges with endpoints

in both blocks Because x v = ±1, this is equivalent to minimizing the one-dimensional (integer) distance between all pairs of connected vertices [HK91]

(u,v)∈E

The nonzero pattern in the summand results in the matrix formulation:

min

x (PT

x)T(PT

x) = min xT

PPTx = min xT

where P is the|V| × |E| node-arc incidence matrix defined as [NW88]

pv,e=

⎧

⎪

⎪

+1 if vertex v is the head of edge e

−1 if vertex v is the tail of edge e

0 otherwise

and where L is the Laplacian matrix of the graph To accommodate nonunit weights on the edges, one forms the product PWPT, where W is a diagonal matrix with the nonzero entries representing the weights of edges [GM00] We list the properties of L here.

Property 1 L is a symmetric, positive semi definite matrix.

Proof Using Equations 7.1 and 7.2, we have

xTLx=1

2

|V|



i=1

|V|



j=1

(xi − x j)2≥ 0, ∀ x i j

By Property 1, the eigenvalues are all real and nonnegative.

Property 2 The sum of elements in each row of L equal 0

Proof Recall that L= D − A and that d ii=|V|

j=1a ij, thus

|V|



j=1

ij = d ii−

|V|



j=1

aij=

|V|



j=1

aij−

|V|



j=1

aij= 0

The graph partitioning formulation includes block assignment constraints on the vertices [Hal70, PSL90]:

min

s.t xv= ±1 Because of its discrete nature, this problem is very difficult to solve exactly The discrete constraints

can be modeled by n ith order constraints [TK91] In practice, the integer constraints are

approx-imated by first- and second-order constraints only The second-order constraints in Equation 7.4 spread vertices about the median The first-order constraints in Equation 7.5 dictate that there are an

approximately equal number of vertices on both sides of the median For convenience, we define e

as the vector of all ones Thus, the optimization problem is

min

This formulation essentially replaces the solution space consisting of the vertices of the ±1 unit hypercube with the points on the surface of the Euclidian unit sphere

is the eigenvector corresponding to the second smallest eigenvalue of L.

u2is formally known as the Fiedler vector [Fie73] The components of the Fiedler vector that are negative valued represent the coordinates of vertices in the first block; components of the second eigenvector that are greater than or equal to 0 represent the coordinates of vertices in the second block The effect is that the eigenvector components of strongly (weakly) connected vertices are close (far away), thus strongly connected vertices are more likely to be assigned to the same block Because it minimizes the distance between pairs of vertices, the technique we have just described can be used as a one-dimensional placement of vertices [TK91] Unfortunately, there is no guarantee that the optimal solution obtained by the continuous optimization problem closely approximates the discrete optimum [CGT95]

7.3.1.1 Lower Bounds on the Cutset Size

The Laplacian matrix used in the bipartitioning quadratic programming formulation is, in fact, the discretized version of the Laplace operator from partial differential equation (PDE) theory If the PDE is solved exactly, one can obtain theoretical bounds on the number of edges cut for a line graph

fixed at both ends The canonical graph used to obtain lower bounds is a line graph of length L with

tethered endpoints, as in Figure 7.8 We represent the string by|V| = n weighted masses connected

by n + 1 pieces of string such that each piece is ∆x = L

n units long

u1

u3

u2

u4 u5

FIGURE 7.8 Line graph.

The bipartitioning problem for the line graph has an exact solution, which is given by the second

eigenvalue of L,λ2 =2aπ

L and the second eigenvector of L is given by

u2 =

sin

2π(x)

n+ 1 sin

2π(2x)

n+ 1 · · · sin



2π[(n − 1)x]

n+ 1

 sin

2π(nx)

n+ 1

when n = 2 (i.e., the case with bipartitioning), the string vibrates such that the u-coordinate at the midpoint is always 0 The area to the left of the midpoint (thus, half of the vertices) has u > 0 and the area to the right of the midpoint has u < 0 (the other half of vertices).

Some of the earliest theoretical developments in eigenvector bipartitioning were concerned with finding lower bounds on the size of the cutset First, we give two definitions:

Definition 10 A block assignment matrix, X, is defined as

xis=



1 if vertex i is in block s

0 otherwise

A related matrix describes whether two vertices are in the same block

, is defined as

b ij=



1 if vertex i is in the same block as vertex j

0 otherwise

The eigenvalues of B are{β1,β2, , βk, 0, , 0} where k indicates the desired number of blocks.

Donath and Hoffman [DH73] provide lower bounds on the number of edges cut:

|Ec| ≥ −1

2

k



j=1

λj βj

whereλj is an eigenvalue of the adjacency matrix plus a diagonal matrix U, such that|V|

i=1uii =

−|V|

j=1

|V|

i=1aij

Barnes [Bar82] restated k-way graph partitioning in terms of finding a block assignment matrix,

B, so that the distance (in a two-norm sense) between B and the adjacency matrix, A, is as small as

possible The rationale is that if vertices i and j are adjacent (i.e., a ij = 1), then they should end up

in the same block (i.e., b ij= 1) He shows that

Hagen and Kahng [HK92] proved that

|Ec| ≥|C1| · |C2|

|V| λ2

which agrees with Boppana’s [Bop87] bounds of

|Ec| ≥ |V|

4 λ2

when|V1| = |C2| = |V| Other bounds are found in Ref [FRW92]