Constructing Augmented Multimode Compactors - IEEE Xplore

26th IEEE VLSI Test Symposium

Constructing Augmented Multimode Compactors Emil Gizdarski Synopsys Inc., 700 East Middlefield Road, Mountain View, CA 94043 E-mail: [email protected]

However, any effort to improve X-masking properties of the compactors while increasing output compaction ratio is a challenging task that requires an effective trade off between observability and the amount of the control data to be achieved. The discussion hereafter will be focused on improving the X-masking properties of the compactors. Space compactors are combinational circuits typically built of XOR gates that compact test response coming out from n scan chains in one shift cycle {0,1,X}n into a compacted test response observable at m outputs {0,1,X}m, where n > m and X denotes the unknown state. Accordingly, a space compactor can be represented as a (m x n) parity matrix of linear code with m rows and n columns where rows (columns) correspond to the outputs (inputs) of a space compactor [3]. The number of the 1’s in each column of the parity matrix of the code (or code weight) is usually fixed and determines an unique set of k outputs of the space compactor for observation of each scan chain. The ability of space compactors to tolerate unknown states in the test responses was addressed by the Xcompact technique [4] and X-codes [5]. We say that a space compactor can tolerate k-1 unknown states if a single error in test responses can be observed at the compactor outputs despite the presence of any k-1 unknown states in the same shift cycle. In general, improving X-masking properties has a negative impact on the compaction ratio n/m of the space compactors. In this sense, the analysis in [5] provides an estimation of the compaction ratio of the space compactors able to tolerate a certain number of unknown states rather than providing a feasible procedure for constructing compactor logic. In [6,7], a class of convolutional compactors that combine time and space compaction techniques is presented. The convolutional compactors convert test responses coming from n scan chains and a finite number of s shift cycles into a compacted test response observable at m outputs where n > m. The parity matrix of the convolutional compactors has a fixed weight k and certain properties to enable compaction across

Abstract In this paper, a new space compactor, called an augmented multimode compactor, is presented. Accordingly, scan chains are separated into groups using t orthogonal partitions. The augmented multimode compactor has three modes such that all scan chains, a group of scan chains and an intersection of two groups of scan chains is selected for compression. Respectively, 1, kt-1 and any number of unknown states per shift-out cycle can be tolerated in these modes where k is the number of the compactor outputs assigned for observation of each scan chain. Simulation results demonstrate the efficiency of the proposed principles for constructing fully X-tolerant compactors. In the range of 0 to 10 percent of unknown states in test responses, the proposed scheme achieved the same or up to 3 times better observability than the fully X-tolerant combinational compactor.

1. Introduction A common approach for reducing the output side test data volume is to generate a compacted test response or signature, rather than unloading the entire test response [2]. The signatures can be generated by space (combinational) compactors, and finite or infinite time (sequential) compactors. However, to avoid loss in fault coverage, these compactors should be designed so that the errors due to faults in the circuit-under-test (CUT) should propagate through the compactor to the output(s). Error propagation to the outputs of the compactor can be masked due to either error masking or Xmasking. Error masking (or aliasing) occurs when multiple errors cancel each other in the compactor and cause the signature to be correct despite the presence of errors. X-masking occurs when unknown states captured in the scan cell enter the compactor and mask errors generated by fault effect propagation to other scan cells. Most well-known output compactor schemes have good error masking properties [2-12].

1093-0167/08 $25.00 © 2008 IEEE DOI 10.1109/VTS.2008.40

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multiple shift cycles. As a result, the convolutional compactors achieve much higher compaction ratios n/m than space compactors using a single scan channel (or output). A modular compactor [8] is an infinite time compactor using multiple circular shift registers. Different schemes have been introduced to further increase the capabilities of the compactors to tolerate more than k-1 unknown states. In [9], i-Compact eliminates unknown states in the test responses, considering that the compacted test responses may have more than one correct state that requires a special ATE. In [10], the compaction scheme resolved this limitation using on-chip X-filtering. In [11], the block compactor uses programmable connections between scan chains and the inputs of the compactor. In [12], the authors showed that space compactors with multiple weights in the parity check matrix can improve the X-masking properties and compaction ratio, and reduce hardware overhead when the distribution of unknown states is not uniform. Significant improvement in the X-tolerant capabilities of the compactors was achieved based on masking and selection using control per test, control per shift or combined control [13-19]. Most of the presented schemes are compatible with certain types of decompressors and compressors. Also, most of these schemes are fully X-tolerant but they assume and work well when the X-density is relatively low. In this paper, the discussion is focused on the capabilities of the compactors to tolerate a large number of Xs in the test responses. The paper is organized as follows. In Section 2, the motivation and main idea is presented. In Section 3, procedures for constructing augmented multimode compactors are described. In Section 4, an analysis of the proposed compactor is given in the presence of a large number of Xs followed by the conclusions in Section 5.

space compactor, selector and control logic and has three modes: M0, M1 and M2. The augmented space compactor may have multiple output mappings constructed by partitioning the outputs of the space compactor into k non-overlapping groups of n outputs. Each output mapping is associated with a particular mode and defines a unique set of k outputs (one from every group) for observation of each scan chain so that the X-masking properties are maximized with respect to this mode. The selector is specified by t orthogonal partitions such that each partition divides all scan chains into n non-overlapping groups of n2 scan chains. The control logic has a few inputs and nt outputs to allow an independent selection of both each group of scan chains and an intersection of any two groups of scan chains. Accordingly, in mode M0, all scan chains are selected for compression and one X per shift-out cycle can be tolerated. In mode M1, n2 scan chains associated with a group of scan chains and a partition are selected for compression, and up to kt-1 Xs per shift-out cycle can be tolerated. In mode M2, n scan chains in the intersection of two groups of scan chains are selected for compression, and any number of Xs per shift-out cycle can be tolerated.

G1

...

G nt -1

...

selector

...

G0

n3

n3

Augmented space compactor

kn

nt control logic select output mapping

Figure 1: Augmented multimode compactor

3. Construction procedures In this section, procedures for constructing augmented multimode compactors based on self orthogonal 2D array codes are presented. Array codes are composite single parity check codes with a simple structure and low decoding complexity. In particular, the Smith procedure presented in [1] is adapted for constructing fully X-tolerant space compactors.

2. Motivation and main idea Assuming that the number and location of Xs in a CUT cannot be effectively predicted, it is a challenging task to establish an effective trade-off between the compaction ratio, the X-tolerant properties, the observability of the compactor and the control data volume. Clearly, improving X-tolerant properties will improve fault coverage of the modeled defects but may have a negative impact on compression ratio. An increase in observability, meanwhile, could improve test coverage of the unmodeled defects but may have a negative impact on the control data volume. A block scheme of a fully X-tolerant compactor, called an augmented multimode compactor, is given in Figure 1. Accordingly, it consists of an augmented

Smith procedure [1]: A p × p information bit array, p a prime, has rows and columns labeled 0,1,..,r,..,p-1 and 0,1,..,c,..,p-1, respectively. For each information bit, d1 parity check bits Di(r,c)=(r+ic) mod p are calculated where 0≤i≤d-2 and d≤p+1. Procedure 1(p,k): For p prime, let p3 scan chains be viewed as a p × p × p information bit array and (x,y,z) be a scan chain in row x, column y and block z. Output

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mappings are specified by observing scan chain (x,y,z) at k outputs that belong to k non-overlapping groups of p outputs, D0, D1, ..,Dk-1, in a way that: (i) For modes M1 & M2: Di(x,y,z)=(z+iy) mod p where 0≤i≤k-1 and k≤p. (ii) For mode M0: D0(x,y,z)=x and Dj(x,y,z)=(z+iy) mod p where 1≤i≤k-1 and k≤p.

(i) Let an error and kt-1 unknown states exist in one shift cycle and the error and each unknown state under consideration belong to different classes of scan chains. In this case, each unknown state can be associated with at most one partition such that the scan chains containing the error and this unknown state belong to the same group of scan chains. As a result, at least one partition should exist where the scan chain containing the error can be selected for compression in a group of scan chains containing less than k-1 unknown states. Since the augmented space compactor can tolerate k-1 unknown states per shift-out cycle with respect to each partition, the error can be observed using this partition. Therefore AMC(p,k,t) can tolerate tk-1 unknown states in this case. (ii) Let an error and an unknown state exist in one shift cycle in scan chains that belong to the same class of scan chains. In this case, the scan chains under consideration don’t share any common output and the error under consideration is still observable at k outputs of the compactor. Therefore AMC(p,k,t) can still tolerate k-1 unknown states with respect to each partition and all t partitions are still available. (End)

Procedure 2(p,k,t): For p prime, let p3 scan chains be viewed as a p × p × p information bit array and (x,y,z) be a scan chain in row x, column y and block z. Let a cluster (y,z) be a set of p scan chains in column y and block z. A selector function is specified by t orthogonal partitions, Pi(x,y,z)=(x + (p-i-1)y) mod p, where 0≤i≤t1and t≤p. Lemma 1: For p prime and k 3, let an augmented space compactor is constructed using output mappings specified by Procedure 1. Then one and k-1 Xs per shift-out cycle can be tolerated in mode M0 and mode M1, respectively. Proof: In mode M0, each scan chain is observable at a unique set of k outputs. Therefore one X per shift-out cycle can be tolerated in this mode. In mode M1, each chain is observable at k outputs and any two scan chains in one group belong to different clusters and have at most one common output. Therefore k-1 Xs can be tolerated in this mode. (End)

d22 d21 d20 d12 d11 d10 d02

Lemma 2: For p prime, let an AMC(p,k,t) be constructed using output mappings and a selector function specified by Procedures 1 and 2, respectively. Then AMC(p,k,t) can tolerate kt-1 unknown states per shift-out cycle when a group of p2 scan chains is selected for compression, mode M1, and any number of unknown states when p scan chains in the intersection of two groups are selected for compression, mode M2, where k≤p and t≤p.

M1

d01 d00

M0

G00 G01 G02 G10 G11 G12 G20 G21 G22

Partition P 0 0 1 2 2 0 1 1 2 0 0 1 2 2 0 1 1 2 0 0 1 2 2 0 1 1 2 0 Partition P 1 0 1 2 1 2 0 2 0 1 0 1 2 1 2 0 2 0 1 0 1 2 1 2 0 2 0 1 Partition P 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2

Proof: For p prime, a selector function Pi(x,y,z)=(x + (p-i-1)y) mod p for 0≤i≤t-1 defines t orthogonal partitions of scan chains such that any two scan chains in one block belong to the same group in at most one partition. Also, each chain has exactly p-1 other chains in different blocks belonging to the same group for all partitions which are observable at non-overlapping sets of k outputs of the compactor. In other words, by intersecting two groups, the scan chains are divided into p2 classes of p scan chains such that each class of scan chains is observable at non-overlapping sets of k outputs of the compactor. As a result, in mode M2, Xmasking between the scan chains selected for compression is not possible. To prove Lemma 2 for mode M1, two cases need to be considered:

C0

C1

C2

Block B 0

C3

C4

Block B 1

C5

C6

C7

C8

Block B 2

Figure 2: AMC(p,k,t) where p=3, k=3 and t=3 Figure 2 illustrates the proposed procedures for constructing AMC(p,k,t) when p=3, k=3 and t=3. Accordingly, the AMC(p,k,t) consists of an augmented space compactor constructed based on Procedure 1 and a selector constructed based on Procedure 2. Let p3 scan chains attached to the selector inputs be labeled from 0 to 26 starting from left to right in Figure 2. Case A: In mode M1, an error in scan chain 0 is observable at outputs {d00,d10,d20}. The AMC(p,k,t) can

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tolerate kt-1=8 unknown states in this mode. A distribution of 9 unknown states for masking the error in scan chain 0 at outputs d00, d10 and d20 is defined by k pairs of clusters {C1,C2}, {C5,C7} and {C4,C8}, respectively. Let us select 9 scan chains in clusters C1, C5 and C8. If all these scan chains contain unknown states in this shift-out cycle then the error in scan chain 0 is masked with respect to all outputs assigned for observation of this scan chain. By excluding just one of these scan chains, for example scan chain 26 in cluster C8, the error under consideration can be observed at output d20 using partition P0. Case B: In mode M2, scan chains 0, 9, and 18 in the intersection of groups G00 and G10 can be selected for compression. Any number of unknown states can be tolerated in this mode since these scan chains are observable at non-overlapping sets of k outputs of the compactor, {d00,d10,d20}, {d01,d11,d21} and {d02,d12,d22}, respectively.

of this cluster. As a result, any two clusters share at most one common output. Next, the rows in the first table that correspond to clusters in the same position of all blocks must have different numbers in all columns. As a result, scan chains in the intersection of two groups of scan chains are observable at nonoverlapping sets of k outputs. Next, partitions P0, P1 and P2 for each block are specified by columns 4, 3 and 2 in the first table. As a result, scan chains in one cluster belong to different groups for all partitions and any two scan chains within one block belong to the same group in at most one partition. The remaining four tables in Figure 3 specify the output mapping for mode M0. In this mode, all scan chains are selected for compression. Accordingly, scan chains in one cluster have one unique output (first column) and share three common outputs (second, third and fourth columns). Also, any two scan chains share at most k-1common outputs.

Corollary 1: For n 3, an AMC(n,k,t) that is able to tolerate 1, kt-1 and any number of unknown states per shift-out cycle in modes M0, M1 and M2 respectively can be constructed if h mutually orthogonal Latin squares of order n exist where k≤h+2≤n and t≤h+1.

Corollary 2: For n 3, an AMC(n,k,t) that is able to tolerate 1, t(k-1) and any number of unknown states per shift-out cycle in modes M0, M1 and M2 respectively can be constructed using a single output mapping if h mutually orthogonal Latin squares of order n exist where k≤h+1, t≤h and k+t≤h+2.

Horizontal square 0 1 2 3

0 1 2 3

0 1 2 3

Vertical square

0 1 2 3

0 0 0 0

1 1 1 1

2 2 2 2

Output mapping and selector function for modes M1 & M2 C0 B0

C1 C2 C3 C5

B1

C4 C7 C6

B2

C10 C11 C8 C9 C15

B3

C14 C13 C12

2 Latin squares of order 4

3 3 3 3

0 2 3 1

1 3 2 0

2 0 1 3

3 1 0 2

0 3 1 2

1 2 0 3

2 1 3 0

3 0 2 1

Output mapping for mode M0

D0 D1 D2 D3

D0 D1 D2 D3

D0 D1 D2 D3

D0 D1 D2 D3

D0 D1 D2 D3

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3

0 1 2 3 2 3 0 1 3 2 1 0 1 0 3 2

0 1 2 3 3 2 1 0 1 0 3 2 2 3 0 1

For Corollary 2, the worst case scenario exists when an error and one unknown state belong to the same class of scan chains. In this case, the scan chains under consideration share a common output in group D0. As a result, the error under consideration is observable at k1 compactor outputs. Therefore, the AMC(p,k,t) can tolerate k-2 unknown states with respect to each partition and all t partitions are still available.

C0

C1 B0 C2

C3

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

0 0 C5 0 0 1 1 C4 1 1 B1 2 2 C7 2 2 3 3 C6 3 3

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

2 2 2 2 3 3 3 3 0 0 0 0 1 1 1 1

3 3 C10 3 3 2 2 C11 2 2 B2 1 1 C8 1 1 0 0 C9 0 0

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0

1 1 C15 1 1 0 0 C14 0 0 B3 3 3 C13 3 3 2 2 C12 2 2

0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

1 1 1 1 0 0 0 0 3 3 3 3 2 2 2 2

2 2 2 2 3 3 3 3 0 0 0 0 1 1 1 1

4. Simulation experiment In this section, the properties of the proposed compactor are analyzed. The analysis was focused on the observability since this property had the highest impact on the number of test patterns and the test quality in the presence of a large number of Xs. More formally, the AMC(n,k,t) constructed based on Corollary 2 (using a simple output mapping) was compared with the fully X-tolerant compactor [19]. Accordingly, both compactors had the same number of inputs and outputs, and used control per shift. In a full observability mode, referenced here as a mode M0, each scan chain is observable at the same set of k outputs for both compactors. In X-tolerant modes, referenced here as modes M1 and M2, the fully X-tolerant compactor had a direct observation of each scan chain by dividing scan chains into g=n2/k groups of nk scan chains.

P2 P1 P0

Figure 3: AMC(n,k,t) where n=4, k=4 and t=3 Figure 3 illustrates a generic procedure for constructing AMC(n,k,t) based on Corollary 1 where n=4, k=4 and t=3. In this case, horizontal, vertical and 2 Latin squares of order 4 are used (shown in the top). The first table specifies the output mapping and partitions for modes M1 and M2. In this case, each Latin square is represented by a column in the first table. Also, each column is associated with a group of n outputs of the augmented space compactor: D0, D1, D2 and D3. Each row in the first table corresponds to a cluster of n scan chains and defines a unique set of k outputs (one from each group) assigned for observation

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Accordingly, one control bit was required to enable/disable the full observability mode and log 2 g control bits were required to select a particular group of scan chains in the X-tolerant mode. Table 1 summarizes the number of control bits required per shift cycle for both compactors.

configuration, mode M1 had dominant usage when the X-density was higher than 1.3 percent. 1.2 1 OBS 0.8

Table 1: Control bits and observed cells

0.6

Mode

0.4

M0 M1 M2

X-tolerant compactor [19] Control bits Observed cells 1 n3 kn 1+log2(n2/k) kn 1+log2(n2/k)

Proposed AMC(n,k,t) Control bits Observed cells 1 n3 n2 2+log2nt n 2+log2n2

M0 M1 M2

0.2 0 0

2

4

6

8 10 X-density, %

12

14

16

Figure 4: Observability and probabilities of modes of AMC(11,3,2)

When t=2, both compactors were constrained to use the same number of control bits for modes M0 and M1. As a result, when n=5 and n=9, the number of the available partitions of the AMC(n,k,2) for some scan chains was restricted to 1. In this sense, a difference in the control data volume of both compactors might exist if mode M2 was intensively used. Also, both compactors had the same number of XOR gates since they used the same output mapping. In this sense, a difference in the area overhead (hardware and wire) might exist in the selector and the control logic. In the selector, the fully X-tolerant compactor and the AMC(n,k,t) required two 2-input AND gates and one (t+1)-input AND gate per scan chain, respectively. In the control logic, the area overhead depends on the total number of groups of scan chains. As a result, when t=2, the AMC(n,k,2) had lower area overhead than the fully X-tolerant compactor. A simulation approach was used to estimate the observability and the control data volume for both compactors. One thousand experiments were run for each configuration and X-density in the specified range. For each experiment, the positions of Xs were randomly generated and each cell (not in state X) was selected for compression in a mode providing the highest observability. Accordingly, if a particular cell could not be observed in mode M0, then a partition in mode M1 that maximizes the observability for this cell was selected. If there was no such partition then this cell was observed in mode M2. The number of control bits and observed cells was calculated as average numbers across all non-X cells and experiments. In this sense, the number of the observed cells in Table 1 defines the maximum number of observed cells for each mode. Figure 4 shows the observability and the probability of modes, M0, M1 and M2, of the AMC(11,3,2) when the X-density changes from 0 to 15 percent. For this

3.5 3

t=1

2.5

t=2

2

t=3 t=4

1.5

t=5 1 0.5 0 0

2

4

6

8 10 X-density, %

12

14

16

Figure 5: Relative observability of AMC(11,3,t) 3.5 3 n=5

2.5

n=7

2

n=9 1.5

n=11

1

n=13

0.5 0 0

2

4

6

8

10

12

14

16

X-density, %

Figure 6: Relative observability of AMC(n,3,2) 1.25 1.2 1.15

n=5

1.1

n=7 n=9

1.05

n=11 1

n=13

0.95 0.9 0

2

4

6

8

10

12

14

16

X-density, %

Figure 7: Control data volume of AMC(n,3,2) Figure 5 shows the relative observability of the AMC(11,3,t) with respect to the fully X-tolerant compactor when parameter t changes from 1 to 5. In this case, the AMC(11,3,t) consistently achieved the same or better observability than the fully X-tolerant compactor in the presented range. Also, the observability of the AMC(11,3,t) was between than

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08, Center for Reliable and High-Performance Computing, Univ. of Illinois at Urbana-Champaign, 2003.

2.5-3 times better than the observability of the fully Xtolerant compactor when the X-density was around 5 percent. Figure 6 shows the relative observability of the AMC(n,3,2) with respect to the fully X-tolerant compactor when the parameter n changes from 5 to 13. In this case, the AMC(n,3,2) consistently achieved the same or up to 3 times better observability than the fully X-tolerant compactor when the X-density was in the range between 0 and 14 percent. Figure 7 shows the relative control data volume of the AMC(n,3,2) with respect to the fully X-tolerant compactor when the parameter n changes from 5 to 13. In this case, the AMC(n,3,2) required up to 20% higher control data volume than the fully X-tolerant compactor when the X-density was around 15 percent.

[6] J. Rajski, J. Tyszer, C. Wang and S. M. Reddy, “Finite Memory Test Response Compactors for Embedded Test Applications,” in IEEE Transaction on CAD, vol.24, No 4, pp. 622-634, April 2005. [7] J. Rajski and J. Tyszer, "Synthesis of X-tolerant convolutional compactors," in Proc. VLSI Test Symposium, pp. 114-119, 2005. [8] W. Rajski and J. Rajski, “Modular compactor of

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5. Conclusions

[11] C. Wang, S.M. Reddy, I. Pomeranz, J. Rajski and J. Tyszer, "On compacting test response data containing unknown values," in Proc. International Conference on Computer Aided Design, pp. 855 - 862, 2003.

In this paper, a set of procedures for constructing fully X-tolerant space compactors have been presented. Accordingly, the proposed space compactor had a full observability mode and two X-tolerant modes such that scan chains were selected for compression using t orthogonal partitions. Also, each scan chain was observed at a unique set of k outputs of the compactor such that the X-masking properties were optimized with respect to each mode. In the range of the Xdensity between 0 and 10 percent, the proposed scheme had achieved the same or up to 3 times better observability than the fully X-tolerant compactor [19] with a negligible increase in the control data volume. As a result of the improved observability, a reduction in the number of test patterns and better test coverage of the unmodeled defects can be expected. The presented principles can be extended for constructing fully X-tolerant finite and infinite time compactors.

[12] T. Clouqueur, K. Zarrineh, K. K. Saluja and H. Fujiwara, “Design and analysis of multiple weight linear compactors of responses containing unknown values,” in Proc. International Test Conference, pp. 1099-1108, 2005. [13] M. Naruse, I. Porneranz, S. M. Reddy and S. Kundu, "On-chip compression of output responses with unknown values using LFSR reseeding," in Proc. International Test Conference, pp. 1060–1068, 2003. [14] V. Chickermane, B. Foutz and B. Keller, “Channel masking synthesis for efficient on-chip test compression”, in Proc. International Test Conference, pp. 452-461, 2004. [15] Y. Tang, H.-J. Wunderlich, H. Vranken, F. Hapke, M. Wittke, P. Engelke, I. Polian and B. Becker, “X-masking during logic BIST and its impact on defect coverage”, in Proc. International Test Conference, pp. 442-451, 2004. [16] M.C.-T. Chao, S. Wang, S. T. Chakradhar and K.-T. Cheng, “Response shaper: a novel technique to enhance unknown tolerance for output response compaction”, in Proc International Conf. Computer Aided Design, pp. 80 – 87, 2005.

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