CN110059405B - Construction method of high-quality Steiner minimum tree with differential evolution under X structure - Google Patents

Construction method of high-quality Steiner minimum tree with differential evolution under X structure Download PDF

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CN110059405B
CN110059405B CN201910306100.XA CN201910306100A CN110059405B CN 110059405 B CN110059405 B CN 110059405B CN 201910306100 A CN201910306100 A CN 201910306100A CN 110059405 B CN110059405 B CN 110059405B
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刘耿耿
吴海林
郭文忠
陈国龙
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Abstract

The invention relates to a high-quality Steiner minimum tree construction method with differential evolution under an X structure. The method can effectively optimize the wire length of the wiring tree and realize the high-quality Steiner minimum tree construction method with differential evolution under the X structure.

Description

High-quality Steiner minimum tree construction method with differential evolution under X structure
Technical Field
The invention relates to the technical field of computer aided design, in particular to a method for constructing a high-quality Steiner minimum tree with differential evolution under an X structure.
Background
The general wiring is an important link of the design of the very large scale integrated circuit, in the general wiring, the general wiring problem of a multi-terminal net is the problem of finding a wiring tree of a given pin set, and the Steiner minimum tree is used as an optimal model of the general wiring of the very large scale integrated circuit, and the problem to be solved by the current very large scale integrated circuit manufacturing process is how to construct the Steiner minimum tree.
At present, most algorithms for solving the wiring problem are related work based on a Manhattan structure as a model, but the manhattan structure is used for optimizing the wire length, and because the wiring trend is limited, only horizontal wiring and vertical wiring can be realized, the wiring area cannot be fully utilized, and the excessive redundancy of interconnection wire resources is caused. Therefore, when the optimization strategy based on the Manhattan structure is used for optimizing the length of the interconnection line, the optimization capability is limited.
Disclosure of Invention
In view of this, the present invention provides a high-quality Steiner minimum tree construction method with differential evolution under an X structure, which aims at optimizing the length of a routing line, and finally can obtain a high-quality Steiner minimum tree routing scheme for the X structure.
In order to achieve the purpose, the invention adopts the following technical scheme:
a high-quality Steiner minimum tree construction method with differential evolution under an X structure comprises the following steps:
step S1, initializing the population by adopting a Prim algorithm to obtain a population of an initial solution;
step S2, constructing an improved differential evolution algorithm and a traditional differential evolution algorithm;
step S3, judging whether the iteration times of the algorithm reach a threshold value, if not, performing variation operation on the population of the initial solution according to the improved differential evolution algorithm; if the threshold value is reached, performing variation operation on the population of the initial solution according to a traditional differential evolution algorithm;
s4, selecting the optimal individual in the population to enter the next generation of the population based on a greedy selection strategy;
and S5, looping the steps S3-S4 until the population iteration is finished to obtain a global optimal solution, namely the final wiring scheme.
Further, the variant operation formula of the improved differential evolution algorithm is as follows:
Vi(g)＝Xp1
wherein Xp1(g)，Xp2(g)，Xp3(g) Is three random individuals in the population of the g generation, and p1≠p2≠p3≠i，Vi(g) Variant individuals were generated for the ith individual in the population of the g generation.
The formula for the crossover operation is as follows:
Figure GDA0003552664540000021
wherein Vi(g)，Xi(g) Respectively the ith individual and the ith variant individual in the g generation population, XMi(g) Is the ith individual and the ith in the population of the g generationCross individuals resulted from individual variant individuals.
Wherein:
Figure GDA0003552664540000022
the difference set result of A and B is shown;
Figure GDA0003552664540000023
if B is an empty set, then two-point variation is adopted for A; if B is not empty, combining and searching a set strategy, adding the element in A as a to-be-selected edge into B, if B is an illegal tree after the edge which can be added into B in A is added, randomly connecting unconnected points into an edge and initializing a routing mode of the edge to be added into B until B is a legal tree;
Figure GDA0003552664540000031
indicating that a and B operate crossly with a probability cr.
Further, the variant operation formula of the conventional differential evolution algorithm is as follows:
Vi(g)＝Xp1(g)+F*(Xp2(g)-Xp3(g))
wherein Xp1(g)，Xp2(g)，Xp3(g) Is three random individuals in the population of the g generation, and p1≠p2≠p3≠i，Vi(g) Variant individuals were generated for the ith individual in the population of the g generation.
Wherein, F adopts a self-adaptive strategy, and the specific rule is as follows:
Figure GDA0003552664540000032
in the formula Fl＝0.1，Fu＝0.9，fb，fm，fwRespectively sorting results of three individual adaptive values selected in the variation process from large to small;
the cross-operation formula is as follows:
Figure GDA0003552664540000033
wherein Vi j(g) The jth bit of the ith individual is represented, cr adopts a self-adaptive strategy, and the specific rule is as follows:
Figure GDA0003552664540000034
wherein crl＝0.1，cru＝0.6，fi，fmin，fmax，
Figure GDA0003552664540000041
The adaptive values of the current individual, the minimum adaptive value individual in the population, the maximum adaptive value individual in the population and the average individual adaptive value in the population are respectively.
Wherein the fitness value calculation function is as follows:
Figure GDA0003552664540000042
wherein T isxSet of edge sets, l (e), representing a routing treei) Representing edge set element eiLength of (d).
Further, the greedy selection strategy is specifically as follows:
Figure GDA0003552664540000043
wherein: f (XM)i(g) Express intersecting individual XMi(g) Adapted value of f (M)i(g) Represents a current individual Xi(g) Adapted value of (A), Xi(g +1) represents an individual entering the next generation population.
Further, the threshold is set to 0.25 times of the maximum iteration number of the algorithm.
Compared with the prior art, the invention has the following beneficial effects:
the method is based on a minimum tree generation strategy, an edge point strategy, a greedy selection strategy and a discrete differential evolution algorithm, and can solve the problem that the optimization capacity of the wiring line length under the Manhattan structure is limited.
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FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is an example of the construction of the Steiner minimum tree with X structure according to an embodiment of the present invention;
FIG. 3 is a diagram illustrating a routing scheme for two pins under an edge-to-edge strategy according to an embodiment of the present invention;
FIG. 4 is a diagram illustrating the encoding of instances of particles in a population according to an embodiment of the present invention;
FIG. 5 is a diagram illustrating an improved mutation operation one in the operator design process, in accordance with an embodiment of the present invention;
FIG. 6 is a diagram illustrating a modified mutation operation II in the operator design process according to an embodiment of the present invention;
FIG. 7 is a diagram illustrating improved interleaving operations in the design of operators, in accordance with an embodiment of the present invention.
Detailed Description
The invention is further explained by the following embodiments in conjunction with the drawings.
Referring to fig. 1, the invention provides a method for constructing a high-quality Steiner minimum tree with differential evolution under an X structure, which comprises the following steps:
step S1, initializing the population by adopting a Prim algorithm to obtain a population of an initial solution; in this embodiment, a minimum spanning tree generation method is used to initialize a given pin point set, a manhattan distance between pin points is used as a weight, and a Prim algorithm is used to construct a minimum spanning tree for the pin point set
Step S2, constructing an improved differential evolution algorithm and a traditional differential evolution algorithm;
step S3, judging whether the iteration number of the algorithm reaches a threshold value threshold, namely the iteration number of the algorithm reaches 0.25 times of the maximum iteration number, and if not, performing variation operation on the population of the initial solution according to the improved differential evolution algorithm; if the threshold value is reached, performing variation operation on the population of the initial solution according to a traditional differential evolution algorithm;
s4, selecting the optimal individual in the population to enter the next generation of the population based on a greedy selection strategy;
and S5, looping the steps S3-S4 until the population iteration is finished to obtain a global optimal solution, namely the final wiring scheme.
In the improved differential evolution algorithm in the embodiment, the encoding strategy adopts edge point pair encoding particles, the edge point pair encoding is to encode the particles through n-1 edges of a spanning tree, n-1 Steiner point selection modes and one redundant bit for recording the length (adaptive value) of the spanning tree line, and each particle represents one solution for representing the wiring problem. FIG. 2 represents an example of the X-structured Steiner minimum tree building problem. Fig. 5 shows 4 selection modes of the Steiner point, which are respectively 0 selection, 1 selection, 2 selection and 3 selection from left to right, and are mainly defined by the routing modes of the edges. Fig. 4 shows the case of the particle encoding of the wiring example, which is known from the last bit of the encoding bit, and thus the line length of the particle is 102.071.
Secondly, because the solved problem is a discrete optimization problem, the mutation operator and the crossover operator of the traditional differential evolution algorithm are improved as follows:
the following set operations are defined:
Figure GDA0003552664540000061
shows the difference set result of A and B
Figure GDA0003552664540000062
Represents the elements in A
Figure GDA0003552664540000063
And adding the rule into B until B meets the condition of ending the operation.
Figure GDA0003552664540000064
Represents A and B or
Figure GDA0003552664540000065
Operate regularly in a cross-over manner
To this problem
Figure GDA0003552664540000066
The rule is that:
if B is empty set, then two points of variation are adopted for A
And secondly, if the B is not empty, combining and searching a set strategy, adding the element in the A as a to-be-selected edge into the B, if the B is still an illegal tree after the edge which can be added into the B in the A is added, randomly connecting unconnected points into an edge and initializing the routing mode of the edge to be added into the B until the B is a legal tree.
The modified mutation operation formula is as follows:
Figure GDA0003552664540000071
wherein Xp1(g)，Xp2(g)，Xp3(g) Is three random individuals in the population of the g generation, and p1≠p2≠p3≠i，Vi(g) Variant individuals were generated for the ith individual in the population of the g generation.
As shown in FIG. 5, in the mutation operation process of the algorithm, for the processing situation of the rule (r), three randomly selected individuals, p1, p2 and p3, are substituted into a formula,
Figure GDA0003552664540000072
is empty, then
Figure GDA0003552664540000073
The operation adopts a rule of firstly, so-called two-point mutation, namely, a side m1 to be subjected to mutation operation is randomly selected for a p1 individual to be deleted, the p1 individual is divided into two subtrees after deletion, two pin points are randomly selected from the two subtrees respectively by combining and searching a set strategy, and the two pin points are connected and used as an individual generated by mutation after a routing mode of the side is randomly initialized.
As shown in FIG. 6, in the variant operation of the algorithm, for the processing situation of rule 2, three randomly selected individuals p1, p2 and p3 are substituted into a formula,
Figure GDA0003552664540000074
is not empty, then
Figure GDA0003552664540000075
The operation adopts the rule II that the edges in the p1 individuals are continuously added into
Figure GDA0003552664540000076
In the result of (2), combining and searching the strategy until
Figure GDA0003552664540000077
Forming a complete tree finish
According to the operations defined above, the formula of the improved interleaving operation is as follows:
Figure GDA0003552664540000078
wherein Vi(g)，Xi(g) Respectively the ith individual and the ith variant individual in the g generation population, XMi(g) Generating crossed individuals for the ith individual and the ith variant individual in the population of the g generation.
As shown in fig. 7, the cross operation process of the algorithm is given, and the variant individuals and the current individuals generated by the variant operation are cross-operated, where
Figure GDA0003552664540000081
The rules are as follows: and performing cross operation on the variant individuals generated by the variation and the current individuals and performing cross according to the probability cr. After the mutation operation, the generated mutation individuals m and the current individuals i are subjected to cross operation, the common edges of the two individuals are used as an initial edge set in the cross mode, the rest edges are used as an edge set to be selected, and legal edges are continuously selected from the edge set to be selected and added into the initial edge set until the legal edges are added to the initial edge set by combining and searching the set strategy until the legal edges are selected from the edge set to be selectedA complete tree is constructed, and the result is taken as a crossed individual generated after crossing.
In this embodiment, when the iteration number of the algorithm exceeds the threshold, that is, the iteration number of the algorithm reaches 0.25 times of the maximum iteration number, only the edge routing manner is operated, so that the population result obtained in the previous period is directly subjected to iteration operation by using a mutation operator and a crossover operator of the conventional differential evolution algorithm, and the mutation operation formula of the conventional differential evolution algorithm is as follows:
Vi(g)＝Xp1(g)+F*(Xp2(g)-Xp3(g))
wherein Xp1(g)，Xp2(g)，Xp3(g) Is three random individuals in the population of the g generation, and p1≠p2≠p3≠i，Vi(g) Variant individuals were generated for the ith individual in the population of the g generation.
Wherein, F adopts a self-adaptive strategy, and the specific rule is as follows:
Figure GDA0003552664540000082
in the formula Fl＝0.1，Fu＝0.9，fb，fm，fwRespectively sorting results of three individual adaptive values selected in the variation process from large to small;
the cross-operation formula is as follows:
Figure GDA0003552664540000091
wherein
Figure GDA0003552664540000092
The jth bit of the ith individual is represented, cr adopts a self-adaptive strategy, and the specific rule is as follows:
Figure GDA0003552664540000093
wherein crl＝0.1，cru＝0.6，fi，fmin，fmax，
Figure GDA0003552664540000094
The adaptive values of the current individual, the minimum adaptive value individual in the population, the maximum adaptive value individual in the population and the average individual adaptive value in the population are respectively.
Wherein the fitness value calculation function is as follows:
Figure GDA0003552664540000095
wherein T isxSet of edge sets, l (e), representing a routing treei) Representing edge set element eiLength of (d).
In this embodiment, the greedy selection policy specifically includes:
Figure GDA0003552664540000096
wherein: f (XM)i(g) Express intersecting individual XMi(g) Adapted value of f (X)i(g) Represents the current individual X)i(g) Adapted value of (A), Xi(g +1) represents an individual entering the next generation population.
To verify the effectiveness of the present invention, the method result of the minimum spanning tree initialization is compared with the method result of the random initialization, and as shown in table 1, the present invention achieves an average optimization rate of 35.72% in solving the problem.
TABLE 1 optimization rates due to minimum spanning Tree strategy
Figure GDA0003552664540000097
Figure GDA0003552664540000101
To verify the effectiveness of the present invention, the present invention was compared to two Steiner minimum tree construction methods in ten sets of test circuits, as shown in Table 2. Considering the optimization goal of the total wiring length, the optimization rate of the method is averagely 9.74% and 0.51% compared with the RSMT (rectangular structure Steiner minimum tree) and the OSMT (X structure Steiner minimum tree) in the other two existing methods. The optimization effect is most obvious when the method is compared with the RSMT, and the line length optimization rate of 9.74% can be achieved. It can be seen that the method has good optimization capability in solving the problem of the Steiner minimum tree with the X structure.
Table 2 comparison of our algorithm with the other two algorithms in terms of routing bus length
Figure GDA0003552664540000102
The above description is only a preferred embodiment of the present invention, and all equivalent changes and modifications made in accordance with the claims of the present invention should be covered by the present invention.

Claims (4)

1. A high-quality Steiner minimum tree construction method with differential evolution under an X structure is characterized by comprising the following steps:
step S1: initializing the population by adopting a Prim algorithm to obtain a population of an initial solution;
step S2: constructing an improved differential evolution algorithm and a traditional differential evolution algorithm;
step S3: judging whether the iteration times of the algorithm reach a threshold value or not, and if not, performing variation operation on the population of the initial solution according to an improved differential evolution algorithm; if the threshold value is reached, performing variation operation on the population of the initial solution according to a traditional differential evolution algorithm;
step S4: based on a greedy selection strategy, selecting the optimal individual in the population to enter the next generation of the population;
step S5: looping the steps S3-S4 until the population iteration is finished to obtain a global optimal solution, namely the final wiring scheme;
the variant operation formula of the improved differential evolution algorithm is as follows:
Figure FDA0003552664530000011
wherein Xp1(g)，Xp2(g)，Xp3(g) Is three random individuals in the g generation population, and p1≠p2≠p3≠i，Vi(g) Variant individuals generated for the ith individual in the population of the g generation;
the formula for the crossover operation is as follows:
Figure FDA0003552664530000012
wherein Vi(g) Denotes the i-th variant individual in the g-th generation population, Xi(g) Is the ith individual in the g generation population, XMi(g) Generating crossed individuals for the ith individual and the ith variant individual in the population of the g generation;
wherein:
Figure FDA0003552664530000026
the difference set result of A and B is shown;
Figure FDA0003552664530000021
if B is an empty set, then two-point variation is adopted for A; if B is not empty, combining and searching a set strategy, adding the element in A as a to-be-selected edge into B, if B is an illegal tree after the edge which can be added into B in A is added, randomly connecting unconnected points into an edge and initializing a routing mode of the edge to be added into B until B is a legal tree;
Figure FDA0003552664530000022
indicating that a and B operate with a cross at a probability cr.
2. The method for constructing a high-quality Steiner minimum tree with differential evolution under an X structure according to claim 1, wherein: the variant operation formula of the conventional differential evolution algorithm is as follows:
Vi(g)＝Xp1(g)+F*(Xp2(g)-Xp3(g))
wherein Xp1(g)，Xp2(g)，Xp3(g) Is three random individuals in the population of the g generation, and p1≠p2≠p3≠i，Vi(g) Variant individuals generated for the ith individual in the population of the g generation;
wherein, F adopts a self-adaptive strategy, and the specific rule is as follows:
Figure FDA0003552664530000023
in the formula Fl＝0.1，Fu＝0.9，fb，fm，fwRespectively sorting results of three individual adaptive values selected in the variation process from large to small;
the cross-operation formula is as follows:
Figure FDA0003552664530000024
wherein
Figure FDA0003552664530000025
The jth bit of the ith individual is represented, cr adopts a self-adaptive strategy, and the specific rule is as follows:
Figure FDA0003552664530000031
wherein crl＝0.1，cru＝0.6，fi，fmin，fmax，
Figure FDA0003552664530000033
The adaptive values of the current individual, the minimum adaptive value individual in the population, the maximum adaptive value individual in the population and the average individual adaptive value in the population are respectively.
3. The method for constructing a high-quality Steiner minimum tree with differential evolution under an X structure according to claim 1, wherein: the greedy selection strategy specifically comprises:
Figure FDA0003552664530000032
wherein: f (XM)i(g) Express intersecting individual XMi(g) Adapted value of f (X)i(g) Represents a current individual Xi(g) Adapted value of (A), Xi(g +1) represents an individual entering the next generation population.
4. The method for constructing a high-quality Steiner minimum tree with differential evolution under an X structure according to claim 1, wherein: the threshold is set to 0.25 times the number of iterations of the algorithm up to the maximum number of iterations.
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