A method for crosstalk-aware track assignment based on lagrangian relaxation

Abstract

The present application relates to a kind of crosstalk-aware orbit allocation method based on Lagrangian relaxation. First, a kind of crosstalk-aware orbit allocation model is proposed, and mathematical formula is given to find the best orbit allocation scheme. Secondly, a pre-processing strategy based on look-up table and crosstalk violation variable reduction strategy are proposed to reduce the number of times of obstacle cost calculation and the number of crosstalk violation variables in the iterative optimization process. Then, the crosstalk constraint is transferred to the objective function, so that the original problem is converted into LRS. Finally, the classical subgradient method is used for iterative optimization to obtain a high-quality orbit allocation scheme.

Description

Technical Field

[0001] This invention relates to the field of computer-aided design technology for very large-scale integrated circuits, and in particular to a crosstalk sensing track allocation method based on Lagrange relaxation. Background Technology

[0002] With the continuous advancement of science and technology, the design and optimization of Very Large Scale Integration (VLSI) will fundamentally transform circuit design methods, change production methods, and boost industrial output, thereby improving people's living standards. As the core of electronic design technology, Electronic Design Automation (EDA) technology integrates computer technology, intelligent technology, and other technologies, greatly promoting the development of VLSI.

[0003] Physical design is a crucial and time-consuming step in the VLSI design flow. It typically involves partitioning, layout planning and placement, overall routing, and detailed routing. Routing is a critical step, directly impacting the chip's final performance. Routing consists of three steps: overall routing, track assignment, and detailed routing. In modern VLSI physical design, as feature sizes continue to shrink, the available routing area becomes increasingly limited, making routing optimization more challenging. Routing can be optimized at multiple stages of physical design, but the routing stage is the most effective. This is because at this stage, wire distribution can be planned and assigned according to design rules, while accurate pin and obstacle information is available. Due to the continuous shrinking of device size, crosstalk between adjacent interconnects has become a key challenge for VLSI timing convergence and functional correctness; therefore, crosstalk is one of the metrics for evaluating the routing feasibility of a routing scheme.

[0004] Crosstalk noise refers to the noise generated between adjacent signals on a chip due to capacitive coupling. This noise can cause high-low level switching in circuits, thus unexpectedly affecting other circuits, such as causing interference and bit errors. Therefore, in circuit design and layout, appropriate measures need to be taken to reduce the coupling between circuits to mitigate the impact of crosstalk noise. Figure 1 This is a schematic diagram of crosstalk noise. The signal that causes the interference is called the attacker, and the signal that is affected is called the victim. The coupling capacitance between the attacker and the victim is Cc, and the load capacitance is CL. When the attacker's output voltage V... a (t) When a signal switching occurs, this voltage change is coupled to the victim's output terminal through Cc, thereby causing the victim's output voltage V to change. s (t) changes.

[0005] The magnitude of crosstalk is usually quantified by the coupling capacitance, as shown in Equation (3).

[0006]

[0007] Where Cc(i,j) represents the coupling capacitance between wire i and wire j, f ij Len represents the switching factor. ij This represents the coupling length between conductor i and conductor j. Let represent the spacing between conductor i and conductor j, and α and β be technology-dependent constants. As shown in equation (1), this coupling capacitance is directly proportional to the coupling length of two adjacent wires and inversely proportional to the wire distance. Therefore, proper wiring design can effectively reduce coupling effects and ultimately reduce crosstalk noise in the circuit.

[0008] The level of crosstalk noise largely depends on the interconnect layout design. Therefore, physical design techniques, especially routing techniques, are the most effective methods to reduce crosstalk noise. By optimizing the interconnect layout design, mutual interference between signal lines can be minimized, thereby improving chip performance and reliability. However, neither general routing nor detailed routing is the optimal stage for considering crosstalk noise. Figure 1 It is known that crosstalk noise is primarily a localized geometric effect, while overall routing provides a coarse routing scheme from a global perspective, thus failing to capture local conductor distribution. In the overall routing phase, the best approach is to prevent conductors from entering heavily congested areas, thereby reducing crosstalk noise. Therefore, overall routing is not the optimal stage for reducing crosstalk noise. On the other hand, in the detailed routing phase, conductor distribution is largely determined, and rewiring is limited to a small range, thus restricting crosstalk reduction. Addressing crosstalk at this stage implies that conductors might traverse areas with complex congestion, leading to difficult rewiring issues. Furthermore, the detailed routing phase requires handling complex design constraints, significantly increasing the workload of crosstalk reduction. Therefore, detailed routing is also not the optimal stage for reducing crosstalk. To address these issues, this paper proposes reducing crosstalk in the intermediate stage—the track allocation phase.

[0009] In a multi-layer cabling model, vertical and horizontal cabling layers are interleaved. Therefore, when considering crosstalk, we can safely ignore crosstalk noise between different layers. Due to their close geometric location, wires distributed on the same layer are the main source of crosstalk noise. Studies have shown that wire width has little effect on coupling effect; data shows that as the wire width increases by one to two times, the resulting change in coupling effect is approximately 0.4% to 7%. Therefore, the effect of wire width on crosstalk effect can be ignored. As shown in formula (1), the coupling effect is inversely proportional to the spacing between the two wires and directly proportional to the coupling length of the two wires. Therefore, it is reasonable to model two parallel wires assigned to adjacent tracks as crosstalk violations.

[0010] The track assignment problem has been proven to be NP-hard. Integer Linear Programming (ILP) is one of the mathematical programming methods for optimizing decision systems, showing promising applications in solving NP-hard problems and multivariate optimal decision problems. However, existing wired track assignment techniques are based on heuristics and are prone to getting trapped in local optima. Therefore, applying ILP techniques to obtain a high-quality track assignment scheme is crucial; however, ILP is a very time-consuming method. Summary of the Invention

[0011] In view of this, the purpose of this invention is to provide a crosstalk sensing orbit allocation method based on Lagrange relaxation, so as to obtain a high-quality orbit allocation scheme.

[0012] To achieve the above objectives, the present invention adopts the following technical solution: a crosstalk sensing orbit allocation method based on Lagrange relaxation, comprising three stages:

[0013] In the first stage, the track assignment environment is initialized, and tracks are created for each wiring strip based on the width of each layer of conductors and the minimum spacing requirements between conductors; conductors are extracted from the global and local nets using the conductor extraction method in NTA.

[0014] In the second phase, preliminary work for iterative optimization was completed, a crosstalk-aware ILP model was created for each wiring strip, the crosstalk violation variable was scaled down, and |I p ×T p | Lookup table of size;

[0015] In the third stage, the original problem is first transformed into a Lagrange slack subproblem (LRS), and a lower bound of the original problem is obtained by solving the LRS. Then, the solution of the LRS is modified into a feasible solution by a feasible solution construction method, and an upper bound of the original problem is obtained based on the feasible solution.

[0016] By iteratively updating the Lagrange multipliers to continuously narrow the values ​​of the upper and lower limits, the final orbital allocation scheme can be obtained.

[0017] In a preferred embodiment, the crosstalk-aware track assignment problem is modeled on a unit of wiring strip, with the initial mathematical formula as shown in Equation (1); the objective function is shown in Equation (1)(a)-(e), where (b) is the obstacle cost, (c) is the overlap cost, (d) is the line length cost, and (e) is the number of crosstalk violations; constraint (f) guarantees the following two conditions: (1) all wires within wiring strip p will be assigned to tracks; (2) each wire will be assigned to only one track; constraints (g) and (h) are used to calculate the number of overlaps on each segment; constraint (i) restricts the decision variable x. it It is a binary variable;

[0018] The track assignment model uses a segmentation strategy to calculate the overlap cost within a wiring strip. First, before track assignment, the wiring strip is divided into segments of unequal length based on the start and end coordinates of all conductors within it. Then, a binary constant c is calculated based on the conductor distribution. ik The value of ; then, in the solution process, the number of overlaps of all segments on each track is calculated using formula (g); when no conductor is assigned to a segment, the obtained onum is calculated using formula (g). tk The value is "-1"; according to onum tk In practice, the number of overlaps should be non-negative. Therefore, constraint (h) is introduced to ensure that the variable onum is non-negative. tk The legality of the data; finally, the product of the number of overlaps in each segment and the segment length is the overlap cost of the entire wiring strip.

[0019]

[0020] Formula (1)(e) contains three quadratic terms, and the quadratic order of the decision variables will increase the computational complexity; replace each quadratic term with Boolean variables vio_up and vio_down; by introducing the following constraints, the quadratic form in Formula (1)(e) can be transformed into a linear term;

[0021]

[0022] By eliminating constraint (1)(i), ILP is relaxed to LP relaxation.

[0023] In a preferred embodiment, a lookup table-based preprocessing strategy is proposed to eliminate the computational number of redundant obstacle costs; after constructing the crosstalk-aware ILP model, a model of size |I is created for the current wiring strip. p ×T pA lookup table is used to store the overlap between all wires within the cabling strip and obstacles on all tracks within the cabling strip; during iterative optimization, the time to obtain the obstacle cost for each wire is reduced from the original... It is reduced to O(1).

[0024] In a preferred embodiment, during the conversion of the quadratic term to a linear term, two Boolean variables, vio_up and vio_down, are introduced. These linear variables consist of the following three dimensions: (1) all conductors to be assigned within the current wiring band; (2) all conductors that would cause crosstalk to the conductor i to be assigned; and (3) all available tracks within the current wiring band. Therefore, the magnitudes of vio_up and vio_down should be |I... p ×I p ×T p |;

[0025] For each wiring strip, the storage space size before and after applying the crosstalk violation variable reduction method is determined by |I p ×I p ×T p | Reduce to

[0026] In a preferred embodiment, the crosstalk constraint (2)(e) is relaxed and the term is transferred to the objective function, i.e., for each x it ×x jt+1 ,x it ×x jt-1 The term is multiplied by a non-negative LM and the constraint is moved to the objective function; the modified formula is called LRS, as shown in formula (3); through this relaxation method, crosstalk violation is handled simultaneously with overlap cost, obstacle cost, and line length cost;

[0027]

[0028] It is known that for any fixed LM, the optimal solution of LRS is less than or equal to the optimal solution of the primal problem; that is, the primal formula is the primal problem, and LM optimization is the dual problem.

[0029] The subgradient method is used to update the LMs to maximize the solution to the Lagrangian dual problem; specifically, the value of the LM in the current iteration depends on the LMλ in the previous iteration. ijt and step size θ ijt ;

[0030] λ ijt =λ′ ijt +θ ijt ×s n Formula (4)

[0031]

[0032] Where, λ ijt Let λ′ be the LM in the current iteration. ijt For LM in the previous iteration, θ ijt To update the step size; s n The result is derived from formula (2), as shown in formula (5);

[0033] To determine the step size, the classic subgradient calculation method is used as follows:

[0034]

[0035] Among them, Z UP Z is the upper bound of the original problem obtained using the feasible solution construction method. LB It is the lower bound of the original problem derived from solving the LRS problem. It is a scaling factor, S i It is a loosened constraint.

[0036] In a preferred embodiment, based on the solution obtained from the dual problem, it is observed that the solution is a 0-1 matrix, where each row of the matrix represents the allocation of a wire; let n0 represent the number of 0s in a row of the matrix, n1 represent the number of 1s in a row of the matrix, and T represent the number of tracks; on a row-by-row basis, the solution for each row is specifically divided into three types:

[0037] (1) When there is only one 1 in the row and the rest are 0, i.e. n0 = T-1, n1 = 1, the row satisfies the constraint and does not need to be solved again;

[0038] (2) When the number of 0s and 1s in the row is greater than 1, i.e., 1≤n0≤T-12≤n1≤T-1, update the elements with a value of 0 to the constant 0, and update the elements with a value of 1 to the variable x. it And solve the problem again for that row;

[0039] (3) When all elements in the row are 0 or 1, i.e. n0 = Tn1 = T, update all elements in the row as variables and solve the row again.

[0040] In a preferred embodiment, the quality of a first allocation is evaluated using the weighted sum formula (7) in the construction of feasible solutions;

[0041] Eva()=α×blkcost+β×overlapcost+χ×wlcost+δ×crt_vio formula (7)

[0042] Here, α, β, χ and δ are user-defined parameters; because the wire passing through the obstacle is a key wiring problem, α is set to a very large number during the iterative optimization phase to avoid the wire from overlapping with the obstacle as much as possible; β, χ and δ are set to 1, 6 and 1 respectively.

[0043] Compared with the prior art, the present invention has the following beneficial effects: Through the crosstalk sensing track allocation scheme provided by the present invention, circuit designers can clearly observe which areas are severely congested and which areas are high-incidence areas of crosstalk noise in the entire wiring area. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of crosstalk noise.

[0045] Figure 2 This is a flowchart illustrating the overall design of the LR-TA according to a preferred embodiment of the present invention.

[0046] Figure 3 This is a schematic diagram illustrating the reduction of crosstalk violation variables in a preferred embodiment of the present invention;

[0047] Figure 4 This is a schematic diagram illustrating a feasible solution construction according to a preferred embodiment of the present invention. Detailed Implementation

[0048] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0049] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0050] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations according to this application; as used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise; furthermore, it should be understood that when the terms “comprising” and / or “including” are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0051] This invention proposes a crosstalk sensing orbit allocation method based on Lagrange relaxation, referencing... Figure 2-4 This method, known as LR-TA, consists of three phases: initialization, preparation, and iterative optimization. The input file is an overall routing scheme, and the output file is a crosstalk-aware track allocation scheme. The overall process is as follows: Figure 2 As shown.

[0052] Specifically, the design process includes the following:

[0053] 1. In the first stage, the present invention initializes the track assignment environment, creating tracks for each wiring strip based on the width of each layer of conductors and the minimum spacing requirements between conductors. In this stage, the present invention uses the conductor extraction method in NTA to extract conductors for each wiring strip from the global and local nets.

[0054] 2. In the second stage, this invention completed the preliminary work for iterative optimization, created a crosstalk-aware ILP model for each wiring strip, reduced the scale of the crosstalk violation variable, and established |I p ×T p | A lookup table of size. In the final stage, this invention first transforms the original problem into a Lagrange Relaxation Subproblem (LRS), and solves the LRS to obtain a lower bound for the original problem. Secondly, this invention modifies the solution of the LRS into a feasible solution using a feasible solution construction method, and derives an upper bound for the original problem based on the feasible solution.

[0055] 3. Finally, by iteratively updating the Lagrange multipliers to continuously narrow down the upper and lower limits, the final track allocation scheme is obtained. Through this crosstalk-aware track allocation scheme, circuit designers can clearly observe which areas in the entire wiring area are severely congested and which areas are high-incidence areas of crosstalk noise.

[0056] The specific details are explained in the following aspects:

[0057] (I) Crosstalk-aware ILP Model

[0058] This invention models the crosstalk-aware track assignment problem using a single wiring strip as a unit, with the initial mathematical formula shown in Equation (1). The objective function is shown in Equations (1)(a)-(e), where (b) is the obstacle cost, (c) is the overlap cost, (d) is the line length cost, and (e) is the number of crosstalk violations. Constraint (f) guarantees the following two conditions: (1) all wires within the wiring strip p will be assigned to tracks; (2) each wire will be assigned to only one track. Constraints (g) and (h) are used to calculate the number of overlaps on each segment. Constraint (i) restricts the decision variable x. it It is a binary variable.

[0059] In the crosstalk-aware track allocation model proposed in this invention, a segmentation strategy is used to calculate the overlap cost within a wiring strip. First, before track allocation, this invention divides the wiring strip into several segments of unequal length based on the start and end coordinates of all conductors within the strip. Then, a binary constant c is calculated based on the conductor distribution. ikThe value of is then used. Then, during the solution process, formula (g) is used to calculate the overlap of all segments on each track. When no conductor is assigned to a segment, the value of onum is calculated using formula (g). tk The value is "-1". According to onum tk In practice, the number of overlaps should be non-negative. Therefore, this invention introduces constraint (h) to ensure that the variable onum is non-negative. tk The legality of the data. Finally, the product of the number of overlaps in each segment and the segment length is the overlap cost of the entire wiring strip.

[0060]

[0061] Formula (1)(e) contains three quadratic terms, and the quadratic order of the decision variables would escalate the computational complexity. We can replace each quadratic term with Boolean variables vio_up and vio_down.

[0062] By introducing the following constraints, the quadratic form in equation (1)(e) can be transformed into a linear term.

[0063]

[0064] Due to computational complexity, the ILP formula suffers from severe runtime overhead, especially for practical wiring test cases. A simple acceleration technique is to relax the ILP to LP relaxation by eliminating constraint (1)(i). Clearly, the relaxed LP provides a lower bound for the original ILP problem. By observing the solution schemes provided by LP relaxation, we derive the decision variable x. it The value is 0.5, vio_up ijt+1 ,vio_down ijt-1 The value of is 0. In this case, all constraints are satisfied and the objective function is minimized. However, we would prefer to obtain x... it The value tends to be 0 or 1, thus providing effective guidance for track allocation. From this perspective, LP relaxation is unlikely to provide a reasonable track allocation scheme. Therefore, this invention proposes a Lagrange relaxation-based method to solve the crosstalk-aware track allocation problem, rather than the time-consuming ILP method or LP relaxation method.

[0065] (ii) Preprocessing strategy based on lookup table

[0066] As can be seen from the overall LR-TA process, in the relaxation iterative solution, the obstacle cost of the current conductor needs to be calculated in each iteration. In calculating the obstacle cost once, the algorithm needs to traverse all obstacle information within the wiring strip. For large-scale wiring strips, the calculation of obstacle cost consumes a significant amount of runtime. Therefore, this invention proposes a lookup table-based preprocessing strategy to eliminate redundant obstacle cost calculations. After constructing the crosstalk-aware ILP model, a model of size |I| is created for the current wiring strip. p ×T p A lookup table is used to store the overlap between all wires within the cabling strip and obstacles on all tracks within the cabling strip. During iterative optimization, the time required to obtain the obstacle cost for each wire is reduced from the original... It is reduced to O(1).

[0067] (III) Crosstalk Violation Variable Reduction Strategy

[0068] In the process of converting quadratic terms into linear terms, this invention introduces two Boolean variables, vio_up and vio_down. These linear variables consist of the following three dimensions: (1) all conductors to be assigned within the current wiring band; (2) all conductors that would cause crosstalk to the conductor i to be assigned; and (3) all available tracks within the current wiring band. Therefore, the magnitudes of vio_up and vio_down should be |I... p ×I p ×T p For large-scale wiring strips, crosstalk violation variables can consume considerable storage space. Therefore, a method for reducing the size of crosstalk violation variables is proposed to optimize storage space. In experiments, it was observed that a large number of wires in the second dimension are not accessed. Therefore, deleting unaccessed data when creating crosstalk violation variables can save space.

[0069] like Figure 3 As shown, the wiring strip contains 4 conductors (i1~i4) and 2 tracks (t1~t2). The given crosstalk relationships are as follows: (1) conductor i1 and conductor i2; (2) conductor i1 and conductor i3. Since this strategy only deletes part of the data in the second dimension, the space of the third dimension before and after the reduction will not change. Figure 3 The number of data points in the third dimension is not shown. Before implementing the method to reduce crosstalk violation variables, the creation status of crosstalk violation variables is as follows: Figure 3As shown in the rectangles, the white rectangle represents the number of violating variables before reduction, which is 16. Based on the conditions, only wires i1 and i2, and wires i1 and i3, have crosstalk violations. Therefore, I only need to record the crosstalk relationship between these two pairs of wires; other information can be omitted. After executing the crosstalk violation variable reduction method, the creation of crosstalk violation variables is as follows: Figure 3 The irregular rectangles in the diagram represent the number of crosstalk violation variables after reduction, specifically 5 (white rectangles). For each wiring strip, the storage space size before and after applying the crosstalk violation variable reduction method varies depending on |I p ×I p ×T p | Reduce to

[0070] (iv) Optimization strategy based on Lagrange relaxation

[0071] The Lagrange relaxation method is a technique for solving optimization problems with difficult constraints. It reduces the complexity of solving the problem by moving some or all of the difficult constraints of the original problem to the objective function. In the new objective function, each new objective function term is multiplied by an LM. This invention relaxes the crosstalk constraint (2)(e) and transfers this term to the objective function, that is, each x it ×x jt+1 ,x it ×x jt-1 The term is multiplied by a non-negative LM and the constraint is shifted to the objective function. The modified formula is called LRS, as shown in formula (3). Through this relaxation method, crosstalk violations are handled simultaneously with overlap costs, barrier costs, and line length costs.

[0072]

[0073] It is known that for any fixed LM, the optimal solution of LRS is less than or equal to the optimal solution of the primal problem. That is, the primal problem is the primal problem, and LM optimization is the dual problem. Therefore, the Lagrangian dual problem is to minimize the LRS problem by updating the LMs accordingly.

[0074] This invention uses a subgradient method to update LMs to maximize the solution to the Lagrangian dual problem. Specifically, the value of LM in the current iteration depends on LMλ in the previous iteration. ijt and step size θ ijt .

[0075] λ ijt =λ′ ijt +θ ijt ×s n Formula (4)

[0076]

[0077] Where, λ ijt Let λ′ be the LM in the current iteration. ijt For LM in the previous iteration, θ ijt To update the step size. s n The result is derived from formula (2), as shown in formula (5).

[0078] To determine the step size, this invention employs the classic subgradient calculation method as follows:

[0079]

[0080] Among them, Z UP Z is the upper bound of the original problem obtained using the feasible solution construction method. LB It is the lower bound of the original problem derived from solving the LRS problem. It is a scaling factor, S i It is a loosened constraint.

[0081] (v) Construction of feasible solutions

[0082] In the previous section, this invention incorporated the complex constraints that caused the problem into the objective function, while maintaining the linearity of the objective function. The aim was to enable the transformed problem to be solved in polynomial time or, due to the reduced constraints, to be solved quickly, thus aiding in solving the original problem. Comparing the original and dual problems, we can see that the feasible region for solving the dual problem is expanded. Therefore, the solution obtained from solving the dual problem may not be a feasible solution to the original problem. In this case, based on the characteristics of the orbital assignment problem, a heuristic method is needed to correct this infeasible solution into a feasible solution to the original problem.

[0083] The solution obtained from the dual problem can be observed to be a 0-1 matrix, where each row represents the allocation of a wire. Let n0 represent the number of 0s in a row of the matrix, n1 represent the number of 1s in a row, and T represent the number of tracks. The solutions for each row can be categorized into three types, working row by row:

[0084] (1) The row contains only one 1 and all the others are 0, i.e., n0 = T-1, n1 = 1;

[0085] (2) The number of 0s and the number of 1s in the row are greater than 1, i.e., 1≤n0≤T-12≤n1≤T-1.

[0086] (3) All elements in the row are either 0 or 1, i.e., n0 = Tn1 = T;

[0087] For the three different situations mentioned above, the present invention designs corresponding correction methods to transform the current solution into a feasible solution:

[0088] (1) When n0 = T-1 and n1 = 1, the row satisfies the constraints and does not need to be solved again;

[0089] (2) When 1≤n0≤T-12≤n1≤T-1, update the elements with a value of 0 to the constant 0, and update the elements with a value of 1 to the variable x. it Then solve the problem again for that row.

[0090] (3) When n0 = T and n1 = T, update all elements of the row as variables and re-solve the row;

[0091] like Figure 4 As shown, the wiring strip contains 6 wires (i1~i6) and 3 tracks (t1~t3). The left matrix is ​​the solution obtained when solving the dual problem—matrix 1, and the right matrix is ​​the correction process of applying the feasible solution construction method—matrix 2. Both matrices are 6×3 in size. The first row of matrix 1 belongs to type (1), the second row of matrix 1 belongs to type (2), and the third to sixth rows of matrix 1 belong to type (3). For the three different types, the present invention adopts a correction method one by one. The first row of matrix 1 satisfies the constraints, so according to the correction method (1), this row does not need to be solved again, corresponding to the first row of matrix 2. The second row of matrix 1 belongs to type (2). According to the correction method (2), the present invention sets the two elements 1 of this row as variables x respectively. 21 x 22 Set element 0 to a constant 0 and re-evaluate the row, which corresponds to the second row of matrix 2. The third to sixth rows of matrix 1 belong to type (3). According to the correction method (3), this invention sets all elements of each row to variables and re-evaluates the row, which corresponds to the third to sixth rows of matrix 2.

[0092] Experiments have shown that simultaneously reducing line length cost, obstacle cost, overlap cost, and crosstalk violation count is a challenging task; when one cost is improved, the other three costs worsen. Therefore, balancing these four costs during optimization is crucial. To achieve this goal, this invention uses a weighted sum formula (7) to evaluate the quality of a primary allocation in the construction of feasible solutions.

[0093] Eva()=α×blkcost+β×overlapcost+χ×wlcost+δ×crt_vio Formula (7)

[0094] Here, α, β, χ, and δ are user-defined parameters. Because wires passing through obstacles is a critical wiring viability issue, α is set to a very large number during the iterative optimization phase to minimize the overlap between wires and obstacles. β, χ, and δ are set to 1, 6, and 1, respectively.

Claims

1. A crosstalk sensing track allocation method based on Lagrange relaxation, characterized in that, It includes three stages: In the first stage, the track assignment environment is initialized, and tracks are created for each wiring strip based on the width of each layer of conductors and the minimum spacing requirements between conductors; conductors are extracted from the global and local nets using the conductor extraction method in NTA. In the second phase, preliminary work for iterative optimization was completed, including creating a crosstalk-aware ILP model for each wiring strip, scaling down the crosstalk violation variables, and establishing... A lookup table for sizes; In the third stage, the original problem is first transformed into the Lagrange slack subproblem LRS, and solving LRS yields a lower bound for the original problem. Secondly, the solution of LRS is modified into a feasible solution through the feasible solution construction method, and an upper limit of the original problem is obtained based on the feasible solution; By iteratively updating the Lagrange multipliers to continuously narrow the values ​​of the upper and lower limits, the final orbital allocation scheme can be obtained. The track allocation problem with crosstalk sensing is modeled using a single wiring strip as a unit, and the initial mathematical formula is shown in formula (1). The objective function is shown in equations (1)(a)-(e), where (b) is the obstacle cost, (c) is the overlap cost, (d) is the line length cost, and (e) is the number of crosstalk violations; constraint (f) guarantees the following two conditions: (1) all wires within the wiring strip p will be assigned to tracks; (2) each wire will be assigned to only one track; constraints (g) and (h) are used to calculate the number of overlaps on each segment; constraint (i) restricts the decision variables. It is a binary variable; The track assignment model uses a segmentation strategy to calculate the overlap cost within a wiring strip. First, before track assignment, the wiring strip is divided into several segments of unequal length based on the start and end coordinates of all conductors within it. Then, a binary constant is calculated based on the conductor distribution. The value is then used; subsequently, during the solution process, formula (g) is used to calculate the overlap of all segments on each track; when no conductor is assigned to a segment, the result is calculated using formula (g). The value is "-1"; according to In practice, the number of overlaps should be non-negative. Therefore, constraint (h) is introduced to ensure that the variable... The legality of the data; finally, the product of the number of overlaps in each segment and the segment length is the overlap cost of the entire wiring strip. Formula (1) in, , , and These are user-defined parameters; Formula (1)(e) contains three quadratic terms, and the quadratic order of the decision variables leads to an increase in computational complexity; using Boolean variables... To replace each quadratic term; By introducing the following constraints, the quadratic form in equation (1)(e) can be transformed into a linear term; Official (2) By eliminating constraint formula (1)(i), ILP is relaxed to LP relaxation.

2. The crosstalk sensing track allocation method based on Lagrange relaxation according to claim 1, characterized in that, A lookup table-based preprocessing strategy is proposed to eliminate redundant obstacle cost computations. After constructing the crosstalk-aware ILP model, create a new model of size [size missing] for the current wiring strip. A lookup table is used to store the overlap between all wires in the cabling strip and obstacles on all tracks in the cabling strip; During the iterative optimization process, the time required to obtain the obstacle cost for each wire was reduced from the original... Reduced to .

3. The crosstalk sensing track allocation method based on Lagrange relaxation according to claim 1, characterized in that, In the process of transforming a quadratic term into a linear term, two Boolean variables are introduced. The linear variable consists of the following three dimensions: (1) all conductors to be assigned within the current wiring band; (2) all conductors that will have crosstalk conflicts with the conductor i to be assigned; (3) all available tracks within the current wiring band; from this, it can be deduced that... The size should be ; For each wiring strip, the storage space size after applying the crosstalk violation variable reduction method is... Reduce to .

4. The crosstalk sensing track allocation method based on Lagrange relaxation according to claim 1, characterized in that, Relaxing crosstalk constraint formula (2) and transferring this term to the objective function, that is, each The term is multiplied by a non-negative LM and the constraint is moved to the objective function; the modified formula is called LRS, as shown in formula (3); through this relaxation method, crosstalk violation is handled simultaneously with overlap cost, obstacle cost, and line length cost; Official (3) It is known that for any fixed LM, the optimal solution of LRS is less than or equal to the optimal solution of the primal problem; that is, the primal formula is the primal problem, and LM optimization is the dual problem. The subgradient method is used to update the LMs to maximize the solution to the Lagrange dual problem; specifically, the value of the LM in the current iteration depends on the LM in the previous iteration. and step length ; Official (4) Official (5) in, For the LM in the current iteration, For the LM in the previous iteration, To update the step size; s n The result is derived from formula (2), as shown in formula (5); To determine the step size, the classic subgradient calculation method is used as follows: Official (6) Among them, Z UP Z is the upper bound of the original problem obtained using the feasible solution construction method. LB It is the lower bound of the original problem derived from solving the LRS problem. It is a scaling factor, S i It is a loosened constraint.

5. The crosstalk sensing track allocation method based on Lagrange relaxation according to claim 1, characterized in that, Based on the solution obtained from the dual problem, we observe that the solution is a 0-1 matrix, where each row represents the allocation of a wire. Let n0 represent the number of 0s in a row of the matrix, n1 represent the number of 1s in a row of the matrix, and T represent the number of tracks. We then categorize the solutions for each row into three types, working row by row: (1) When the row contains only one 1 and all the others are 0, that is... If the row satisfies the constraints, it does not need to be solved again; (2) When the number of 0s and 1s in the row is greater than 1, i.e. 1≤n0≤T-1 2≤n1≤T-1, update the elements with a value of 0 to the constant 0, and update the elements with a value of 1 to the variable. And solve the problem again for that row; (3) When all elements in the row are 0 or 1, i.e. n0 = T n1 = T, update all elements in the row as variables and solve the row again.

6. The crosstalk sensing track allocation method based on Lagrange relaxation according to claim 5, characterized in that, The quality of a first allocation is evaluated using the weighted sum formula (7) in the construction of feasible solutions; Official (7) , and Set them to 1, 6, and 1 respectively.

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