Average Placement Method with Common Centroid Constraints for Analog IC Layout Design Kunihiro Fujiyoshi, Keitaro Ue Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan
[email protected] [email protected] Abstract—To improve immunity against process gradients, a common centroid constraint, in which every pair of capacitors should be placed symmetrically with respect to a common center point, is widely used. Several methods to obtain a good placement satisfying the constraints were proposed. However, the distance between cells in a common centroid group may be redundantly big, and it degrades the immunity. In this paper, we propose a novel algorithm to place cells belonging to each common centroid group. The algorithm is simple, but it is proved to be effective. I. INTRODUCTION Recently, most integrated circuit chips are made from monolithic IC, which can be manufactured at low cost. In monolithic IC, the absolute error which is the proportion of the difference of actual value to the designed value of each element is large (several tens of percentages), but the maximum difference of the absolute error of devices on a chip, is small (a few percentages). So it is suitable to build circuits such as A/D, D/A converters and switched-capacitor filters. But if the accuracy of these circuits is required to be higher, such a small mismatch of capacitors or resistors still becomes a critical issue. The causes of the mismatch can be divided into two categories: systematic mismatch and random mismatch [1], [2]. One of the frequent systematic mismatch is caused by process gradients, whose effect can be represented by the following linear model [3]: Cxy = C + αx + βy, where Cxy is the capacitance of the unit capacitor in the coor- dinates x and y, C is the ideal value of the unit capacitance, and α and β are the process gradient components in x and y directions, respectively if the capacitors are in the close proximity. Therefore to improve immunity against process gradients, a common centroid constraint, in which every pair of capacitors should be placed symmetrically with respect to a common center point, is widely used. In order to obtain a placement which satisfies common centroid constraint, some methods are proposed [4], [5], [6], [7], [8], [15]. Strasser et al. proposed a method using B∗-tree [4], but this method is deterministic. C. W. Lin et al. proposed a method to obtain placement considering random mismatches [5], but this method can handle unit-capacitors only. Some methods using sequence-pair (seq-pair) [9], which is a topological representation of rectangular block placement, are proposed [6], [7], [8], [15]. Ma et al. proposed a method using Center-based Corner Block List [6], which is an exten- sion of CBL [10]. They regard each common centroid group as a rectangular super-block, and search whole placements by seq-pair, so area obtained by the method may be redundantly large. Also, since the cells in each common centroid group are placed along the boundary of the rectangular super-block, distance between the cells may be redundantly big and the immunity against process gradients degrades. Xiao et al. proposed a method which uses seq-pair to obtain placement satisfying common centroid constraint [7]. This method can handle all cells together by using seq-pair. But it has a defect that some blocks may be overlapped as shown in [8]. Ue et al. proposed another method which uses seq-pair to obtain placement satisfying common centroid constraint [8], [15]. Their algorithm determines relative coordinates of all cells in each common centroid group individually, and deter- mines the whole placement by using convex block packing algorithm [12] in O(n2) time, where n is the number of cells. The algorithm does not overlap any cells, but, the distance between cells in a common centroid group may be redundantly big, and it degrades the immunity. In this paper, we propose a novel decoding algorithm to place cells belonging to each common centroid group by applying average placement, where coordinates of each cell are set to the averages of bottom-left packing and top-right packing. In any obtained placements, it can be guaranteed that x-distance and y-distance between cells in every common centroid group is the minimum separately. The algorithm is very simple, but effective, and the effectiveness of the method is examined by experimental comparisons. II. PROBLEM DEFINITION A. Common centroid constraint A common centroid constraint gives a constraint, in which every pair of cells in the same group should be placed symmetrically with respect to the common center point of the group. These groups are called common centroid groups. A common centroid group consists of some pairs of cells and at most one self-symmetry cell, whose center conforms to the common center point of the group. In this paper, cells in each common centroid group should be placed as close as possible to the common center point since immunity against process gradients degrades if the distance between cells and the common center point is large. A common centroid group is represented by a set of cell pairs in parentheses and at most one self-symmetry cell such as {(a1, a2), bs, (c1, c2)}. B. Representation Our decoding method uses a sequence-pair which satisfies some constraints. These constraints will be defined later in this section. 1) Sequence-pair (seq-pair) [9]: A sequence-pair (seq- pair) is an ordered pair of Γ+ and Γ−, each of which is a permutation of names of n rectangles. For example, (Γ+; Γ−) = (abc; bac) is a seq-pair for rectangle set {a, b, c}. It is easily understood that the variety of the seq-pair for n rectangles is (n!)2. If rectangle a is the i’th element in Γ+, we denote Γ+(i) = a as well as Γ −1 + (a) = i. A similar notation is used also for Γ−. A seq-pair imposes a horizontal/vertical (H/V) constraint for every pair of rectangles as follows. For every rectangle pair {a, b}, rectangle a is left of rectangle b (equivalently, b is right of a) if Γ−1+ (a) < Γ −1 + (b) and Γ −1 − (a) < Γ −1 − (b). Similarly, rectangle a is below rectangle b (equivalently, b is above a) if Γ−1+ (a) > Γ −1 + (b) and Γ −1 − (a) < Γ −1 − (b). For example, seq-pair 978-1-4799-4132-2/14/$31.00 ©2014 IEEE 226 bsa1 a2 c1 c2 d1 d2 (a) a1 a2bs (b) r1 of G1 c1 c2 d1 d2 (c) r2 of G2 Fig. 1. (a) A packing satisfying common centroid constraints, common centroid group: G1 = {(a1, a2), bs}, G2 = {(c1, c2), (d1, d2)}, point- symmetric feasible seq-pair S=(c1d1a1bsa2d2c2; a1d2c2c1d1bsa2). Note that S satisfies Group Clustered Placement condition, but does not satisfy separate condition. (b) Cells in G1 are placed in the minimum bounding rectangle r1. (c) Cells in G2 are placed in the minimum bounding rectangle r2. (abc; bac) imposes a set of H/V constraints: {a is left of c, b is left of c, and b is below a}. From a given seq-pair, the packing to the bottom left corner based on H/V constraints imposed by it can be obtained in O(n2) time by using the H/V constraint graphs constructed faithfully with the H/V constraints. 2) Point-symmetric feasible sequence-pair: Suppose (ΓGi+; ΓGi−) is a sub-seq-pair obtained by extracting just the cells in common centroid group Gi from seq-pair S. Point-symmetric feasible seq-pair is defined as follows: Seq-pair S is point-symmetric feasible if ΓGi+ and ΓGi− satisfy the following conditions[7]: ΓGi+ = sym(rev(ΓGi+)) ΓGi− = sym(rev(ΓGi−)), where rev(s) is a string obtained by reversing a string s and sym(s) is a string obtained by replacing a string s by their symmetric counterparts. 3) separate condition of sequence-pair: Separate condition is defined as follows[12]: A seq-pair satisfies separate condi- tion if any pair of common centroid groups is separated from each other in either Γ+ or Γ−. “Common centroid groups Gi and Gj are separated from each other in Γ+” means all the elements of Gi are before or after all the elements of Gj in Γ+. Note that every asymmetry cell, which does not belong to any common centroid group, is thought that it forms a common centroid group individually A similar condition “Group Clustered Placement condi- tion” is defined afterwords in [7]. Seq-pair (Γ+; Γ−) containing common centroid group Gi satisfies Group Clustered Place- ment condition if and only if there exist no cells u, v, x, and y in Gi such that Γ−1+ (u) < Γ−1+ (z) < Γ−1+ (v) and Γ−1− (x) < Γ −1 − (z) < Γ −1 − (y), where cell z does not belong to Gi If a seq-pair satisfies separate condition, it is easy to understand that the seq-pair satisfies Group Clustered Place- ment condition. On the other hand, some seq-pair satis- fies Group Clustered Placement condition, but does not sat- isfy separate condition. An example of seq-pair is S = (c1d1a1bsa2d2c2; a1d2c2c1d1bsa2) when G1 = {(a1, a2), bs} and G2 = {(c1, c2), (d1, d2)}. Seq-pair S satisfies Group Clus- tered Placement condition, but separate condition. A placement corresponds to S is shown in Fig.1(a). Note that seq-pair S is point-symmetric feasible. Such seq-pair, which satisfies Group Clustered Place- ment condition but does not satisfy separate condition, includes a pair of “crossing” group [14], and any placements satisfying the all constraints imposed by the seq-pair have the following property: Property of seq-pair with crossing groups: Suppose common centroid groups G1 and G2 are crossing, and the bounding rectangle of the minimum placements of all cells in G1 is r1 (see Fig.1(b)). and the bounding rectangle of the minimum placements of all cells in G2 is r2 (see Fig.1(c)). If all cells in G1 are placed in a bounding rectangle r1, cells in G2 cannot be placed in a bounding rectangle r2 (see Fig.1(a)), and vice versa. Then, the distance between cells and the common center point might become redundantly large, and immunity against the process gradient degrades. On the other hand, if a seq- pair satisfies separate condition, all cells in every common centroid group can be placed in the minimum bounding rectangle separately under the constraints imposed by the seq- pair. Therefore, we decide to use point-symmetric feasible seq- pair only which satisfies separate condition. III. PROPOSED DECODING METHOD In this paper, we propose a novel algorithm average- placement-method. The algorithm is obtained by replacing step 2 in [8], [15] by ours. Therefore, the main contribution of this paper is in step 2 of the algorithm. step 2 of the algorithm is very simple to place cells belonging to each common centroid group. The algorithm is based on average placement, which is known by a part of seq-pair users and was used as “average placement” in [13]. The biggest discovery is that the average placement of all cells in one common centroid group satisfies the common centroid constraint inherently. average placement is obtained by the very simple algorithm as follows: • Obtain bottom-left packing (each cell cannot move downward or leftward in the first quadrant without making any overlapping of cells) of all cells. • Obtain top-right packing of all cells. • Determine coordinates of all cells as the average value of the above two packings. However, there may exist redundant dead spaces in an average placement as shown in Fig.2(e). Therefore, an additional procedure follows to close the spaces. A. Algorithm average-placement-method Input: point-symmetric feasible sequence-pair S which satisfies separate condition; Output: Placement satisfying common centroid constraint; step 1: Based on the horizontal (left of) constraints imposed by the seq-pair S, a directed weighted graph Gh(V,E) (V : vertex set, E: edge set), called horizontal constraint graph is constructed as follows [9]. · V consists of source sv and vertices labeled with the corresponding names of all cells. · E consists of (sv, p) for every cell p, and (p, q) if and only if p is constrained to be left of q by the seq-pair. · Length of edge (p, q) is the width of the cell p. Similarly, the vertical constraint graph Gv is constructed using vertical (below) constraints and the height of each cell. step 2: foreach common centroid group Gi, do step 2-1, 2- 2, 2-3. step 2-1: Make an induced subgraph Gih of Gh induced by the vertices corresponding to cells in Gi. Make an induced subgraph Giv of Gv induced by the vertices corresponding to cells in Gi; Obtain bottom-left packing of all cells in Gi by the longest path algorithm. Obtain top-right packing of all cells in Gi; Determine relative coordinates of all cells as the mean value of the packings; 227 Let the relative coordinates of common center point be (XGi , YGi); step 2-2: Make an induced subgraph Gi+h of Gh induced by vertices which satisfies XGi < x(a) + a.width (a ∈ Gi). (1) Put a source vertex, and edges from the source to all vertices, whose weights are{ x(a)−XGi : (if x(a) < XGi) 0 : (if x(a) ≥ XGi) Then, determine x-coordinate of each cell as (the longest path length from the source to the corresponding vertex) plus XGi ; Determine the x-coordinate of the other cells as x(a2) = XGi − (a1.width+ x(a1)) since x-coordinate of at least one of cells in every pair has already determined. step 2-3: Determine y-coordinate of each cell similarly. step 3: Coordinates of all cells are determined by convex block packing algorithm, proposed in [12]. Detailed operations are as follows. step 3-1: Adjust weight of edges in Gh by using relative coordinates obtained in step 2 to obtain G′h. Weight of edge (a, b) is adjusted to a.width+ x(a) − x(b), where x(a) is relative x-coordinate of a in Gi if a ∈ Gi, and x(a) = 0 if a is an asymmetry cell, x(b) is similarly. Similarly adjust weight of edges in Gv to obtain G′v. step 3-2: Merge all the vertices corresponding to cells in each common centroid group on G′h and G ′ v into one “super- vertex” to obtain G′′h and G ′′ v , and calculate the longest path length from source to all vertices on G′′h and G ′′ v . And let these longest path lengths be x and y coordinates of corresponding cells or cells in the smallest coordinate in each common centroid group. For example, a point-symmetric feasible seq-pair S = (a1b1csb2a2d; b1a1csa2b2d), which satisfies separate condi- tion, and common centroid group G1 = {(a1, a2), (b1, b2), cs} are given. The size of a1 and a2 is 3 × 1, the size of b1 and b2 is 1 × 2, and the size of cs and d is 1 × 3. Constraint graphs Gh and Gv of the input seq-pair, ob- tained in step 1, are shown in Fig.2(a) and (b), respectively. Bottom-left packing, top-right packing, and average placement of cells in G1 in step 2-1 are shown in Fig.2(c), (d), and (e), respectively. G1+h and G 1+ v of G1, obtained in step 2-2 and step 2-3, are shown in Fig.2(f) and (g), respectively. Average- placement of cells in G1 is shown in Fig.2(h). At step 3, G′h, G ′ v, G ′′ h, and G ′′ v are shown in Fig.2(i), (j), (k), and (l), respectively. Resultant placement is shown in Fig.2(m). B. Theorem The proposed algorithm is based on the following theorem. Theorem 1: The proposed algorithm: average- placement-method can obtain a placement which satisfies the given common centroid constraints and the relative position constraints imposed by a seq-pair in O(n2) time. The placement has no overlapping of cells, and x-distance between cells in each common centroid group, which is equal to the width of the bounding rectangle of the placed cells in the common centroid group, is minimum, and y-distance between cells in each common centroid group, which is equal 1 s 0 0 3 1 1b1 a1 a2 b2 cs d 1 3 (a) Gh obtained in step 1 0 1 b1 a1 a2 b2 cs s 0 0 2 d 0 (b) Gv obtained in step 1 cs b1 a1 b2 a2 (c) bottom-left packing obtained in step 2-1 cs b1 a1 b2 a2 (d) top-right packing obtained in step 2-1 cs b1 a1 b2 a2 (e) average placement obtained in step 2-1 -0.5 1 1 a2 b2 css 0 0 (f) G1+h obtained in step 2-2 b1 a1 b2cs s -1.5 2 -1.5 0 -0.5 (g) G1+v obtained in step 2-3 cs b1 a1 b2 a2 (h) average-placement obtained in step 2 s b1 a1 a2 b2 cs d 5 7 0 -2 0 0 0 0 (i) G′h obtained in step 3-1 0 b1 a1 a2 b2 cs s 0 0 d 0 0 0 (j) G′v obtained in step 3-1 α ds 0 7 (k) G′′H obtained in step 3 α s d 00 (l) G′′V obtained in step 3 cs b1 b2 d a1 a2 (m) resultant placement of average-placement-method Fig. 2. Example of Algorithm average-placement-method: G1 = {(a1, a2), (b1, b2), cs}, seq-pair = (a1b1csb2a2d; b1a1csa2b2d) to the height of the bounding rectangle of the placed cells in the common centroid group, is also minimum. (Proof) It is easy to understand all the steps can be carried out if the input seq-pair is a point-symmetric feasible sequence- pair satisfying separate condition. There are no overlapping of cells in the average placement obtained in step 2-1 from the following Lemma 1. In step 2-2, cells overlapped with x-axis and y-axis are fixed, and cells in each quadrant approach to common center point in the quadrant. Since the movement of the approach is determined by the longest path algorithm on constraint graphs, any cells do not overlap each other in the resultant placement. The time complexity of step 1 and step 2 is obviously O(n2), and that of step 3 is O(n2), which is shown in [12]. So the total time complexity of this proposed decoding algorithm is O(n2). 228 TABLE I. EXPERIMENTAL COMPARISON OF PROPOSED DECODING ALGORITHM (ATHLON 2.7GHZ) WITH METHODS PROPOSED IN [7] (XEON 2.20GHZ) AND [8] (ATHLON 2.7GHZ). THE UPPER VALUE IS CPU TIME, AND THE LOWER VALUE IS A AREA RATIO. benchmark #cell #group Xiao[7] Ue[8] Proposed biasynth 2p4g 65 8+12+5 18.2 [s]1.0407 141.6 [s] 1.0888 147.3 [s] 1.0857 lnamixbias 2p4g 110 16+6+6+12+4 58.7 [s] 1.0450 222.4 [s] 1.1239 220.9 [s] 1.1155 The following lemma is known and used by a part of seq- pair users. Lemma 1: Any two cells do not overlap each other in average placement obtained by any seq-pair. (Proof) Focus a relation of any two cells. Suppose Γ−1+ (a) < Γ −1 + (b) and Γ −1 − (a) < Γ −1 − (b), namely, cell a is left of cell b. Then, from the constraint imposed by the seq-pair, the right side of cell a is not greater than the left side of cell b on the bottom-left packing, namely xbl(a) + a.width ≤ xbl(b) (2) where xbl(a) is the x-coordinate of the left side of cell a on the bottom-left packing. Next, from the constraint imposed by the seq-pair, the right side of cell a is not greater than the left side of cell b on the top-right packing, namely xtr(a) + a.width ≤ xtr(b) (3) where xtr(a) is the x-coordinate of the left side of cell a on the top-right packing. The next inequality (4) is obtained by adding inequality (2) and (3), before dividing both of them by two. xbl(a) + xtr(a) 2 + a.width ≤ x bl(b) + xtr(b) 2 (4) Hence, cell a and b do not overlap each other in any average placement. Similar proof can be possible if cell a is above cell b. IV. EXPERIMENTAL COMPARISONS In order to confirm effectiveness of the proposed decoding algorithm, we implemented our proposed decoding algorithm with Simulated Annealing in C language, and carried out experiments by Athlon 2.7GHz for two data used in [4] and [7], which are “biasynth 2p4g” and “lnamixbias 2p4g”. “biasynth 2p4g” consists of 65 cells with three common centroid groups. “lnamixbias 2p4g” consists of 110 cells with five common centroid groups. Table I shows experimental comparisons of results of the proposed method with those shown in [7] and [8]. “Area ratio” is defined by the following equation: Area ratio = (area of bounding box)(total area of all cells) . We cannot simply compare the results of [7], which are the best one, with the others because [7] considered one group in “biasynth 2p4g” and two groups in “lnamixbias 2p4g” are 1- D symmetry groups but the other two methods considered all groups in both data are common centroid groups. The area of placement obtained by the proposed decoding algorithm is less than that by the method proposed in [8] though the CPU time is almost similar. Resultant placement of “biasynth 2p4g” and “lnamixbias 2p4g”, obtained by our proposed decoding algorithm, are shown in Fig.3 and Fig.4 respectively. These placements show that every common centroid group is packed closely and satisfies the close proximity. m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15 m16 m17 m18 m19 m20 m21 m22 m23 m24m25m26 m27 m28 m29 m30 m31 m32 m33 m34 m35 m36 m37 m38 m39 m40 m41 m42 m43 m44 m45 m46 m47 m48 m49 m50 m51 m52 m53 m54m55 m56 m57 m58 m59 m60 m61 m62 m63 m64 m65 Fig. 3. Resultant placement of “biasynth 2p4g” obtained by our proposed method (Time: 147.3[sec], Area ratio: 108.57[%]) m1m2 m3m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15 m16 m17 m18 m19 m20 m21m22 m23 m24 m25 m26 m27 m28 m29 m30 m31 m32 m33 m34 m35 m36 m37 m38 m39 m40 m41 m42 m43 m44 m45 m46 m47 m48 m49 m50 m51 m52 m53 m54 m55 m56 m57 m58 m59 m60 m61 m62 m63 m64 m65 m66 m67 m68 m69 m70m71 m72m73 m74 m75 m76 m77 m78 m79 m80 m81 m82 m83 m84 m85 m86 m87 m88 m89 m90 m91 m92 m93 m94 m95 m96 m97 m98 m99 m100 m101 m102 m103 m104 m105 m106 m107 m108 m109 m110 Fig. 4. Resultant placement of “lnamixbias 2p4g” obtained by our proposed method (Time: 220.9[sec], Area ratio: 111.55[%]) V. CONCLUSION In this paper, we proposed a novel decoding algorithm average-placement-method for point-symmetric feasible sequence-pair which satisfies separate condition. The decoding algorithm can obtain placement satisfying common centroid constraints and relative position constraints imposed by the seq-pair in O(n2) time. The placement has no overlapping of cells, and x-distance between cells in each common centroid group is minimum, and y-distance between cells in each com- mon centroid group is also minimum. The algorithm is based on the fact that any average placement obtained from point- symmetric feasible sequence-pair satisfies common centroid constraint. Experimental results show that every common cen- troid group is packed closely and satisfies the close proximity. REFERENCES [1] M. J. McNutt, S. LeMarquis, and J. L. Dunkley, “Systematic capacitance matching errors and corrective layout procedures ”, Proc. IEEE JSSC, pp. 611–616, 1994. [2] J. B. Shyu, G. C. Temes and F. Krummenacher. “Random Error Effects in Matched MOS Capacitors and Current Sources”, Proc. IEEE J. of Solid-State Circuits, pp.948–955, 1984. [3] M. Kaneko, M. Masuda, and T. 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Fujiyoshi: “A Method of Analog IC Placement with Common Centroid Constraints”, IEICE Trans, Fundamentals, vol.E97- A, no.1, pp.339–346, 2014. 229 /ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing false /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams > setpagedevice HistoryItem_V1 TrimAndShift Range: all pages Trim: fix size 8.500 x 11.000 inches / 215.9 x 279.4 mm Shift: move up by 14.40 points Normalise (advanced option): 'original' 32 D:20140625080313 792.0000 US Letter Blank 612.0000 Tall 1 0 No 675 322 Fixed Up 14.4000 0.0000 Both AllDoc PDDoc Uniform 0.0000 Top QITE_QuiteImposingPlus2 Quite Imposing Plus 2.9 Quite Imposing Plus 2 1 4 3 4 1 HistoryItem_V1 TrimAndShift Range: all pages Trim: none Shift: move left by 3.60 points Normalise (advanced option): 'original' 32 1 0 No 675 322 Fixed Left 3.6000 0.0000 Both AllDoc PDDoc None 0.0000 Top QITE_QuiteImposingPlus2 Quite Imposing Plus 2.9 Quite Imposing Plus 2 1 4 3 4 1 HistoryList_V1 qi2base