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Figure from:Filling algorithms and analyses for layout density control

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Fig. 10. Finding the total area of a union of possibly intersecting rectangles using a sweep-line technique.

Figure 10 Finding the total area of a union of possibly intersecting rectangles using a sweep-line technique.

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Fig. 2. The layout is partitioned by r? (r = 4) distinct dissections, each dissection having window size w x w, into ((nr)/(w)) x ((nr)/(w)) tiles. Each w x w window (dark) consists of r? tiles. A pair of windows from different dissections may overlap. each other. In other words, density bounds are enforced only for windows from the fixed r-dissection defined below.
Fig. 2. The layout is partitioned by r? (r = 4) distinct dissections, each dissection having window size w x w, into ((nr)/(w)) x ((nr)/(w)) tiles. Each w x w window (dark) consists of r? tiles. A pair of windows from different dissections may overlap. each other. In other words, density bounds are enforced only for windows from the fixed r-dissection defined below.
Fig. 3. Worst case analysis of two design rules when density bounds are enforced (a) in all (w/r) x (w/r)-sized tiles of a fixed r-dissection, and (b) in all w x w-sized tiles with bottom-left comers at points (¢-(w/r),7-(w/r)), i,j =0,1,---,(n/w) (this corresponds to r? dissections into w x w-sized windows). For the first rule (a) the window W with dashed boundary contains (r — 1)? tiles with thick boundary (the first group) and the highlighted area (the second and third groups) can be completely or partially filled. For the second rule (b) the window W with dashed boundary can contain a square region R (the empty area in the center of W) that overlaps with any fixed r-dissection w X w window F (square with thick boundary) having largest intersection with W.
Fig. 3. Worst case analysis of two design rules when density bounds are enforced (a) in all (w/r) x (w/r)-sized tiles of a fixed r-dissection, and (b) in all w x w-sized tiles with bottom-left comers at points (¢-(w/r),7-(w/r)), i,j =0,1,---,(n/w) (this corresponds to r? dissections into w x w-sized windows). For the first rule (a) the window W with dashed boundary contains (r — 1)? tiles with thick boundary (the first group) and the highlighted area (the second and third groups) can be completely or partially filled. For the second rule (b) the window W with dashed boundary can contain a square region R (the empty area in the center of W) that overlaps with any fixed r-dissection w X w window F (square with thick boundary) having largest intersection with W.
Fig. 4. (a) A layout and (b) its corresponding Hanan grid.
Fig. 4. (a) A layout and (b) its corresponding Hanan grid.
Fig. 5. A maximum-density window may be slid (a) horizontally until it touches one of the rectangles; (b) the window may then be slid vertically until it touches one of the rectangles. After the sliding operations have been performed, (c) the window will abut one or more rectangles of the layout on two adjacent sides
Fig. 5. A maximum-density window may be slid (a) horizontally until it touches one of the rectangles; (b) the window may then be slid vertically until it touches one of the rectangles. After the sliding operations have been performed, (c) the window will abut one or more rectangles of the layout on two adjacent sides
Fig. 6. (a) ALG2 starts a window abutting a pivot rectangle and (b) slides the window to the right, stopping at each edge that intersects its perimet until the pivot abuts the opposite side of the window, (f) on the outside. (g)-(i) Other combinations of the pivot-window orientations are then explor This process is repeated for every rectangle, using each as a pivot in turn.
Fig. 6. (a) ALG2 starts a window abutting a pivot rectangle and (b) slides the window to the right, stopping at each edge that intersects its perimet until the pivot abuts the opposite side of the window, (f) on the outside. (g)-(i) Other combinations of the pivot-window orientations are then explor This process is repeated for every rectangle, using each as a pivot in turn.
Fig. 8. An arbitrary floating w x w-window W always contains a shrunk (r — 1) x (r — 1)-window of a fixed r-dissection, and is always covered by a bloated (r +1) x (r + 1)-window of the fixed r-dissection. In the figure, a standard r x r fixed-dissection window is shown with thick border. A floating window is shown in light gray. The white window is the bloated fixed-dissection window and the dark gray window is the shrunk fixed-dissection window.
Fig. 8. An arbitrary floating w x w-window W always contains a shrunk (r — 1) x (r — 1)-window of a fixed r-dissection, and is always covered by a bloated (r +1) x (r + 1)-window of the fixed r-dissection. In the figure, a standard r x r fixed-dissection window is shown with thick border. A floating window is shown in light gray. The white window is the bloated fixed-dissection window and the dark gray window is the shrunk fixed-dissection window.
Fig. 9. Multilevel density analysis algorithm. A. Minimizing Density Variation in the Fixed-Dissection Regime BloatMax (maximum bloated window filled area found so far). The algorithm terminates when the relative gap between Max and BloatMax is at most 2-«¢, and then outputs the middle of the range (Max, BloatMax).
Fig. 9. Multilevel density analysis algorithm. A. Minimizing Density Variation in the Fixed-Dissection Regime BloatMax (maximum bloated window filled area found so far). The algorithm terminates when the relative gap between Max and BloatMax is at most 2-«¢, and then outputs the middle of the range (Max, BloatMax).
The constraints (2) imply that features can only be added, and cannot be deleted from any tile. The slack constraints (3) are computed for each tile. If a tile 7}; is originally overfilled, then we set slack(T;;) = 0. From the linear programming solution, the values of p;; indicate the fill amount to be inserted in each tile 7;;. The constraint (4) says that no window can have density more than U after filling unless it was overfilled initially, i.c., such a window cannot increase its density. The number of variables and the number of constraints in the linear program are both O((nr/w)?). In practice, even for a large die and a user requirement of high accuracy, we might have n = 15000, w = 3000, , = 10, which yields a linear program of tractable size. Equation (5) implies that the auxiliary variable M/ is the lower bound on all window densities. The linear programming seeks to maximize M, thus achieving the min-variation objective.
The constraints (2) imply that features can only be added, and cannot be deleted from any tile. The slack constraints (3) are computed for each tile. If a tile 7}; is originally overfilled, then we set slack(T;;) = 0. From the linear programming solution, the values of p;; indicate the fill amount to be inserted in each tile 7;;. The constraint (4) says that no window can have density more than U after filling unless it was overfilled initially, i.c., such a window cannot increase its density. The number of variables and the number of constraints in the linear program are both O((nr/w)?). In practice, even for a large die and a user requirement of high accuracy, we might have n = 15000, w = 3000, , = 10, which yields a linear program of tractable size. Equation (5) implies that the auxiliary variable M/ is the lower bound on all window densities. The linear programming seeks to maximize M, thus achieving the min-variation objective.
Fig. 11. “Basket-weaving” of the fill pattern so that long conductors on adjacent layers will have identical coupling to the fill. With the pattern in (a), each vertical or horizontal crossover line will have the same overlap capacitance to fill. On the other hand, with the fill pattern in (b) two cross overs can have different coupling to fill.
Fig. 11. “Basket-weaving” of the fill pattern so that long conductors on adjacent layers will have identical coupling to the fill. With the pattern in (a), each vertical or horizontal crossover line will have the same overlap capacitance to fill. On the other hand, with the fill pattern in (b) two cross overs can have different coupling to fill.
Fig. 12. (a) Given a layout, we create a grounded fill pattern by (b) first creating horizontal stripes, and then (c) spanning these stripes using a small number of vertical lines.
Fig. 12. (a) Given a layout, we create a grounded fill pattern by (b) first creating horizontal stripes, and then (c) spanning these stripes using a small number of vertical lines.
Fig. 13. Two patterns with maximum perimeter. (a) The pattern Prin with minimum possible area and (b) the pattern Pmax with maximum area.
Fig. 13. Two patterns with maximum perimeter. (a) The pattern Prin with minimum possible area and (b) the pattern Pmax with maximum area.
PARAMETERS OF THREE INDUSTRY TEST CASES
PARAMETERS OF THREE INDUSTRY TEST CASES
Fig. 14. The x-axis represents the area and the y-axis represents the perimeter of the filling pattern. The highlighted region with vertices Pp, Pymin. Pmax, and P; represents the combinations of area and perimeter for which there exist filling patterns. The pattern with the area S minimum perimete: is the largest square which can be embedded in R. Before the pattern area reaches S, the minimum perimeter grows quadratically; when the are< exceeds S, the minimum perimeter grows linearly.
Fig. 14. The x-axis represents the area and the y-axis represents the perimeter of the filling pattern. The highlighted region with vertices Pp, Pymin. Pmax, and P; represents the combinations of area and perimeter for which there exist filling patterns. The pattern with the area S minimum perimete: is the largest square which can be embedded in R. Before the pattern area reaches S, the minimum perimeter grows quadratically; when the are< exceeds S, the minimum perimeter grows linearly.
Ultra-1 with 256-MB RAM. In practice, we find that the multilevel analysis is preferable to the exact method of ALG3, since the latter has runtimes on the order of tens of CPU minutes for the same test cases (see [13] for ALG3 runtimes on slightly variant layouts of the same standard-cell designs). Table III shows the analogous results for the fixed-dissection approach, which we understand to be used in industry. The two tables show that large values of r, and correspondingly large runtimes, will be required for the fixed-dissection approach to find the window densities that the multilevel analysis can find in only a few seconds.
Ultra-1 with 256-MB RAM. In practice, we find that the multilevel analysis is preferable to the exact method of ALG3, since the latter has runtimes on the order of tens of CPU minutes for the same test cases (see [13] for ALG3 runtimes on slightly variant layouts of the same standard-cell designs). Table III shows the analogous results for the fixed-dissection approach, which we understand to be used in industry. The two tables show that large values of r, and correspondingly large runtimes, will be required for the fixed-dissection approach to find the window densities that the multilevel analysis can find in only a few seconds.
EXPERIMENTAL RESULTS SHOWING CPU TIMEs FoR THREE VARIANT LP APPROACHES TO COMPUTING OPTIMAL FILL AMOUNTS: CPU TIMEs For FILL GENERATION ARE ALSO SHOWN are reasonably close to the best possible values.!° Larger r values may be used to reduce the potential variation from uniform density. For the multilevel density analysis, LP and fill generation, we observe somewhat lower values of (/, and one large runtime for the L2 benchmark with r = 8. The floating deviation LP, which allows the user to bound the density variation between minimum- and maximum-density floating windows, shows steady improvement in solution quality with increasing 7, just as we expect. In practice, we believe that the floating deviation LP is attractive for its control over arbitrary windows; the multilevel LP is also attractive for its data flow directly from multilevel density analysis.
EXPERIMENTAL RESULTS SHOWING CPU TIMEs FoR THREE VARIANT LP APPROACHES TO COMPUTING OPTIMAL FILL AMOUNTS: CPU TIMEs For FILL GENERATION ARE ALSO SHOWN are reasonably close to the best possible values.!° Larger r values may be used to reduce the potential variation from uniform density. For the multilevel density analysis, LP and fill generation, we observe somewhat lower values of (/, and one large runtime for the L2 benchmark with r = 8. The floating deviation LP, which allows the user to bound the density variation between minimum- and maximum-density floating windows, shows steady improvement in solution quality with increasing 7, just as we expect. In practice, we believe that the floating deviation LP is attractive for its control over arbitrary windows; the multilevel LP is also attractive for its data flow directly from multilevel density analysis.
Fig. 15. The two fill patterns considered in Raphael simulations: (a) 1 x 1 squares separated by 1 unit and (b) 10x 1 rectangles separated by 1 unit horizontally and 4 units vertically. The fill pattern (b) was used for the simulations reported in Table V.
Fig. 15. The two fill patterns considered in Raphael simulations: (a) 1 x 1 squares separated by 1 unit and (b) 10x 1 rectangles separated by 1 unit horizontally and 4 units vertically. The fill pattern (b) was used for the simulations reported in Table V.