X-architecture Steiner minimal tree algorithm based on multi-strategy optimization discrete differential evolution

Research status of RSMT and XSMT

Optimizing the wire length of SMT is a popular research direction, and there are many important research achievements. In Tang et al. (2020a), three kinds of sub problems and three kinds of general routing methods in Steiner tree construction were analyzed, and the research progress in two new technology modes was analyzed (Tang et al., 2020a). Chen et al. (2020a) introduced five commonly used swarm intelligence technologies and related models, as well as three classic routing problems: Steiner tree construction, global routing, and detailed routing. On this basis, the research status of Steiner minimum tree construction, wire length driven routing, obstacle avoidance routing, timing driven routing, and power driven routing were summarized (Chen et al., 2020a). In Liu et al. (2011), Rectilinear Steiner Minimal Tree (RSMT) based on Discrete Particle Swarm Optimization (DPSO) algorithm was proposed to effectively optimize the average wire length (Liu et al., 2011). Liu et al. (2014a) proposed a multi-layer obstacle avoidance RSMT construction method based on geometric reduction method (Liu et al., 2014a). Zhang, Ye & Guo (2016) proposed a heuristic for constructing a RSMT with slew constraints to maximize routing resources over obstacles (Zhang, Ye & Guo, 2016).

Teig (2002) adopted XSMT, which is superior to RSMT in terms of average wire length optimization (Teig, 2002). In Zhu et al. (2005), an XSMT construction method was proposed by side substitution and triangle contraction methods (Zhu et al., 2005). Liu et al. (2020c) constructed a multi-layer global router based on the X-architecture. Compared with other global routers, it had better performance in overflow and wire length (Liu et al., 2020c). Liu et al. (2015c) proposed a PSO-based multi-layer obstacle-avoiding XSMT, which used an effective penalty mechanism to help particles to avoid obstacles (Liu et al., 2015c). In Liu et al. (2020b), a novel DPSO and multi-stage transformation were used to construct XSMT and RSMT. The simulation results on industrial circuits showed that this method could obtain high-quality routing solutions (Liu et al., 2020b). Chen et al. (2020b) proposed an XSMT construction algorithm based on Social Learning Particle Swarm Optimization (SLPSO), which can effectively balance the exploration and exploitation capabilities (Chen et al., 2020b).

The present situation of DE and DDE algorithm

DE algorithm has high efficiency and powerful search ability in solving continuous optimization problems. In the past 20 years after its emergence, many scholars have proposed improved versions of DE algorithm. These improvements better balance the exploitation and exploration ability of DE, and show strong optimization ability on many problems.

An Self-adaptive DE (SaDE) algorithm was proposed in Qin, Huang & Suganthan (2008). In different stages of the evolution process, the value of control parameters is adjusted according to experience, which saves the trial and error cost of developers in the process of adjusting parameters (Qin, Huang & Suganthan, 2008). Rahnamayan, Tizhoosh & Salama (2008) proposed an algorithm for accelerating DE, using opposition-based DE and opposition-based learning methods to initialize population and realize generation jumping to accelerate convergence of DE. Subsequently, Wang, Wu & Rahnamayan (2011) proposed an improved version of accelerated DE, which could be used to solve high-dimensional problems. Wang, Cai & Zhang (2011) proposed Composite DE (CoDE). The algorithm proposed three generation strategies of trial vector and three control parameter settings, and randomly combined the generation strategies and control parameters. The experimental results showed that the algorithm had strong competitiveness (Wang, Cai & Zhang, 2011). Wang, Zeng & Chen (2015) combined adaptive DE algorithm with Back Propagation Neural Network (BPNN) to improve its prediction accuracy.

DDE algorithm is a derivative of DE, which can solve discrete problems. Many existing results have applied DDE algorithm to solve practical problems. In Pan, Tasgetiren & Liang (2008), DDE was used to solve the permutation flow shop scheduling problem with the total flow time criterion. For the total flow time criterion, its performance is better than the PSO algorithm proposed by predecessors (Pan, Tasgetiren & Liang, 2008). In Tasgetiren, Suganthan & Pan (2010), an ensemble of DDE (eDDE) algorithms with parallel populations was presented. eDDE uses different parameter sets and crossover operators for each parallel population, and each parallel parent population has to compete with the offspring populations produced by this population and all other parallel populations (Tasgetiren, Suganthan & Pan, 2010). Deng & Gu (2012) presented a Hybrid DDE (HDDE) algorithm for the no-idle permutation flow shop scheduling problem with makespan criterion. A new acceleration method based on network representation was proposed and applied to HDDE, and the local search of the inserted neighborhood in HDDE was effectively improved to balance global search and local development (Deng & Gu, 2012).

XSMT problem

Unlike the traditional Manhattan structure, which only has horizontal and vertical connections, two connection methods of 45 and 135 are added to the XSMT problem (Liu, Chen & Guo, 2012; Liu et al., 2015a). This paper introduces the concept of Pseudo-Steiner (PS) point (Definition 1). The PS point exists in two interconnected pins. The fixation of PS point determines the connection method (Definition 2-5) of two pins.

An example of XSMT problem model is as follows. In a given set of pins {p₁, p₂,…, p_n}, p_i represents the i − th pin to be connected, and the corresponding coordinate is (x_i, y_i). Given 5 pins, the corresponding coordinates are shown in Table 1, and the corresponding pin layout is shown in Fig. 1.

• Definition 1 Pseudo-Steiner point. Except for pin points, other join points are called Pseudo-Steiner points, denoted as PS points.

• Definition 2 Selection 0. As shown in Fig. 2A, draw the vertical edge from A to point PS, and then draw the X-architecture edge from PS to B.

• Definition 3 Selection 1. As shown in Fig. 2B, draw the X-architecture edge from A to point PS, and then draw the vertical edge from PS to B.

• Definition 4 Selection 2. As shown in Fig. 2C, draw the vertical edge from A to PS, and then draw the horizontal edge from PS to B.

• Definition 5 Selection 3. As shown in Fig. 2D, draw the horizontal edge from A to PS, and then draw the vertical edge from PS to B.

Differential evolution algorithm

DE algorithm is a heuristic search algorithm based on modern intelligence theory. The particles of population cooperate and compete with each other to determine the search direction.

Encoding strategy

Property 1. The encoding strategy of edge-point pairs is suitable for DDE algorithm, and it can well record the structure of XSMT.

Suppose there are n pin points in the pin graph, and the corresponding Steiner tree has n − 1 edges and n − 1 PS points. Number each pin, determine an edge by recording two endpoints, and add a bit to record selection method of edge. Finally, a bit is added at the end to represent the fitness value of the particle, and the final encoding length is $3\times \left(n-1\right)+1$. The Steiner tree corresponding to pins in Table 1 is shown in Fig. 3, and the corresponding encoding is: 1 3 1 2 3 0 4 5 0 3 4 3 46.284.

Fitness function

Property 2. The wire length of XSMT is a key factor that affects global routing results, and the fitness value based on the wire length of XSMT can make the algorithm go in the direction of optimal wire length to the greatest extent.

In an edge set of a XSMT, all edges belong to one of the following four types: horizontal, vertical, 45° diagonal and 135° diagonal. Rotate a 45° diagonal counterclockwise 45° to form a vertical line and a 135° diagonal counterclockwise 45° to form a horizontal line, so that the four types of edges can be replaced by two types. Make the starting point number of all edges smaller than the ending point number, and then sort all edges according to the starting point number, and subtract the overlapping part of the edges. At this time, the total wire length of XSMT can be obtained.

The excellence of XSMT is determined by the total wire length. The smaller the wire length is, the higher the excellence of XSMT will be. Therefore, fitness value measured by XSMT-MoDDE is total wire length of particle. The fitness function of XSMT-MoDDE is shown in Eq. (5).

(5) $$fitness(T_x)={\sum }_{e_i\in T_x}length(e_i)$$

Initialization

Property 3. Prim algorithm can search an edge subset, which not only includes all the vertices in a connected graph, but also minimizes the sum of the weights of all the edges in subset. Selecting different starting points can get the same weight but different edge subsets. Prim algorithm is used to initialize population, so that particles in population have diversity and the solution space can be reduced at the same time.

Traditional DE algorithm directly uses Eq. (1) to initialize the population. However, for XSMT, if the random strategy is used to initialize each particle (i.e., randomly select a point as root, and use backtracking method to randomly select edges to build a legal tree), will lead to the problem that the solution space is too large to converge well. Therefore, this paper uses Prim algorithm to construct Minimum Spanning Tree (MST) to initialize population. The weight of each edge in MST is determined by Manhattan distance between each two pins. Each particle randomly selects a starting point s to generate a MST and randomly select a connection method for each edge of MST.

The relevant pseudo code is shown in Algorithm 1, where T is edge set of MST, s is starting point, U is point set of MST, V is pin set, P is population, and N is population size. From Lines 1–18 is the function to generate MST. Lines 2–3 randomly select a starting point s and add it to the set U. Line 4 initializes the edge set T. Line 6 selects a visited point i from the set U, and Line 7 sets the minimum cost to infinity. Lines 8–13 select a unvisited point j from the adjacent points of point i, the edge ij with the least cost will be selected and added to set T, and the point j is marked as visited and added to set U. The MST algorithm ends when the set U is the same as the set V, and Line 17 returns a randomly generated MST. Lines 21–24 construct the population, and the initial particle is an MST generated by function PRIMALGORITHM.

Elite selection and cloning strategy

Property 4. This strategy proposes two particle mutation strategies based on set, which can mutate elite particles in a very short time. The elite particles are cloned and mutated, and the optimal particle is selected based on greedy strategy to construct a elite buffer with high quality in a short time.

Algorithm flow

(1) Selection: Sort population according to fitness value, and select the first n particles to form an elite population, n = k × N. k is elite ratio, and the best result can be obtained when k is selected as 0.2 after experimental verification.

(2) Cloning: Clone the particles of the elite population to form a cloned population C. The number of cloned particles is calculated according to Eq. (6).

(6) $$N_i=round\left({\displaystyle \frac{N}{i}}\right)$$where i is rank of the particle in original population, and round() is rounding down function.

(3) Mutation: The mutation strategy adopts connection method mutation or topology mutation, and two strategies are shown in Fig. 4. Figure 4A shows the mutation process of connection method. The connection method of Line AB is changed from selection 3 to selection 0. Figure 4B shows the mutation process of topology. Line AB is selected to be disconnected and then connected to Line BC. Each cloned particle is assigned to a mutation strategy to form a mutated particle.

For particles that adopt connection method, randomly select a edges, and the value of a is determined according to the number of edges, as shown in Eq. (7), where n is the number of pins. Then change the connection method of the selected edge.

(7) $$a=max\left\{1,round\left({\displaystyle \frac{n-1}{10}}\right)\right\}$$

For particles that adopt topology mutation, one edge is randomly disconnected in XSMT to form two sub-XSMTs, and then respectively select a point from the two sub-XSMTs to connect. This process adopts the idea of Disjoint Set Union (DSU) to ensure that a legal tree is obtained after mutation.

(4) Extinction: Select the trial elite particle m_best with the best fitness value in the mutated population. If $f\left(m_{best}\right)$ is better than $f\left(g_{best}\right)$, then m_best will be added to the elite buffer, and all other particles will die, otherwise, all particles in the mutation population will die. If the elite buffer is full, the particle with the worst fitness value will be popped and new particle will be pushed.

The pseudo code of the elite selection and cloning strategy is shown in Algorithm 2, where S represents elite population, M represents mutated population, the inputs are Population P and its size, and the output E represents the elite buffer. Lines 1–9 are selection function, Line 2 calculates the number n of elite particles, Line 3 initializes the Set S, Line 4 establishes a minimum heap according to the fitness value of the population particles, and Lines 5–6 take n elite particles from the top of the minimum heap in turn. Lines 11–28 are the processes of cloning, mutation and extinction. Line 12 initializes Set E, Line 14 initializes Set M, Line 15–20 are cloning and mutation process, Line 15 clones elite particles, Line 16 selects a mutation strategy randomly, and Line 20 adds mutated elite particles to Set M. Lines 22–23 construct two minimum heaps through Set P and Set M. Line 24 compares the tops of the two minimum heaps to determine whether the trial elite particles are saved or died.

Algorithm 2:

Elite selection and cloning strategy.

Require: P, N

Ensure: E

1: function SELECTION(P)

2:   n ← k × N

3:   S ← 0

4:   H ← heap(P)

5:   for i ←1 to n do

6:    S ∪ H.top()

7:   end for

8:   return S, n

9:  end function

10:

11: function CLONEMUTATIONANDEXTINCTION(S, n)

12:   E ← 0

13:   for i ← 1 to n do

14:     M ← 0

15:     for j ← 1 to n/i do

16:      method ←random(0,1)

17:      if method == 0 then m ← connection_method_mutation()

18:      else m ← topology_mutation()

19:      end if

20:      M ∪ m

21:     end for

22:     H1 ← heap(M)

23:     H2 ← heap(P)

24:     if H1.top() < H2.top() then E ∪ H1.top()

25:     end if

26:   end for

27:   return E

28: end function

DOI: 10.7717/peerj-cs.473/table-11

Novel multiple mutation strategy

Property 5. The three novel mutation strategies proposed in this paper introduce the idea of set operations. Under the premise of reasonable computing time, through adjusting edge set of current particle and edge set of other particle, some substructures in XSMT are changed to search for a better combination of substructures.

In DE algorithm, there are six commonly used mutation strategies (Epitropakis et al., 2011), and each strategy uses different basis vectors and differential vectors. The mutation formulas are shown below.

(8) $$V_i^g={X_{r1}}^g+F({X_{r2}}^g-{X_{r3}}^g)$$

(9) $$V_i^g={X_{r1}}^g+F_1({X_{r2}}^g-{X_{r3}}^g)+F_2({X_{r4}}^g-{X_{r5}}^g)$$

(10) $$V_i^g={X_{best}}^g+F({X_{r1}}^g-{X_{r2}}^g)$$

(11) $$V_i^g={X_{best}}^g+F_1({X_{r1}}^g-{X_{r2}}^g)+F_2({X_{r3}}^g-{X_{r4}}^g)$$

(12) $$V_i^g=X_i^g+F({X_{best}}^g-X_i^g)$$

(13) $$V_i^g={X_{r0}}^g+F_1({X_{best}}^g-{X_{r0}}^g)+F_2({X_{r1}}^g-{X_{r2}}^g)$$where $X_r^g$ represents a random particle in population, $X_{best}^g$ represents the global optimal solution, and F represents learning factor.

Two operating rules

In XSMT-MoDDE algorithm, a particle represents a XSMT. Addition and subtraction operations in the above mutation formulas cannot be directly used in discrete problems. This paper defines two new calculation methods (Definition 6–7).

A is the edge set of particle X₁, B is the edge set of particle X₂, and the full set is A ∪ B. There are two definitions as follows:

Definition 6 A $\odot$ B. $\odot$ is expressed as finding the symmetric difference of A and B, which is (A ∪ B) − (A ∩ B), as shown in Fig. 5A.

Definition 7 A $\oplus$ B. First calculate Set C, C = A − B, and then add the edges of Set B to Set C until Set C can form a legal tree, as shown in Fig. 5B.

Adaptive learning factor

Property 6. Learning factor is a key parameter to determine the performance of DDE algorithm, which has a decisive influence on the exploitation and exploration ability of algorithm. This paper proposes an adaptive learning factor based on set operation for the first time to effectively balance the search ability of XSMT-MoDDE algorithm.

Refining strategy

Property 7. Refining strategy minimizes wire length of XSMT under the determined topology within a reasonable time.

There may still be space for optimization for the optimal particles at the end of iteration. In order to search for a better result, a refining strategy is proposed. The steps of algorithm are as follows:

(1) Calculate degree of each Point p_i in the optimal particle. The degree is defined as the number of edges connected to point, denoted as d_i;

(2) There are 4 kinds of edges in X-architecture. If the degree of Point p_i is d_i, there are 4^d_i types of substructures corresponding to the point. The set of all substructures corresponding to Point p_i is S, and edge Set E is obtained when the substructures corresponding to Points p₁ − p_{i − 1} have been determined. Calculate common wire length l between Substructure s_i in Set S and Set E, select Substructure s_i corresponding to the largest l, and add the edges of s_i to the Set E. The algorithm ends until all points have been visited.

The pseudo code of the refining strategy algorithm is shown in Algorithm 4, where X represents the target particle obtained by the XSMT-MoDDE algorithm, n represents the point number of XSMT, and R represents the refined particle. Line 2 initializes Set R. Lines 3-20 search for the optimal substructure corresponding to each point. Line 4 calculates the degree of Point p_i, Line 5 initializes maximum common wire length, and Line 6 initializes the optimal substructure set. Lines 7–14 calculate common wire length and update the largest common wire length. Lines 15–19 store the edges in the optimal substructure into Set R.

The algorithm flow of XSMT-MoDDE

The algorithm flow chart of XSMT-MoDDE is shown in Fig. 6, and the detailed flow is as follows:

1. Initialize threshold, population size, adaptive learning factor F, and adaptive crossover probability cr.

2. Use Prim algorithm to construct initial particles and generate initial population.

3. Check the current stage: early stage or late stage of iteration.

4. Select a mutation strategy from the corresponding mutation strategy pool according to the current stage. Obtain the mutated particles according to the mutation strategy.

5. Obtain the trial particles according to the crossover operator.

6. Obtain the next generation of particles according to the selection operator.

7. Adopt elite selection and cloning strategy, and update the elite buffer after four steps of selection, clone, mutation, and extinction.

8. Update adaptive learning factor and adaptive crossover probability by Eqs. (21) and (22).

9. Check the number of iterations, and end the iteration if the termination condition is met, otherwise, return to Step (3).

10. At the end of XSMT-MoDDE algorithm, refining strategy is adopted to obtain the target solution.

Complexity analysis of XSMT-MoDDE algorithm

Property 8. When the population size is m and the number of pins is n, the time complexity of one iteration is O(mnlogn).

Verify the effectiveness of multi-strategy optimization

Experiment 1: In order to verify the effectiveness of the multi-strategy optimization DDE algorithm in constructing XSMT, this experiment will compare the results of XSMT-MoDDE algorithm and XSMT-DDE algorithm. Experimental results are shown in Tables 2 and 3. Table 2 is the optimization results of wire length, and Table 3 is the optimization results of standard deviation. The results show that multi-strategy optimization can achieve an average wire length optimization rate of 2.35% and a standard deviation optimization rate of 95.69%. This experiment proves that multi-strategy optimization has a powerful effect on wire length reduction, and at the same time greatly increases the stability of DDE.

Verify the effectiveness of refining strategy

Experiment 2: In order to verify the effectiveness of the refining strategy, this experiment will compare the results of refined XSMT-MoDDE algorithm and XSMT-MoDDE algorithm. The experiment result is shown in Tables 4 and 5. Table 4 is the optimization results of wire length, and Table 5 is the optimization results of standard deviation. The results show that refining strategy can achieve an average wire length optimization rate of 0.50% and a standard deviation optimization rate of 37.30%. From the experimental results and the above complexity analysis, it can be seen that after XSMT-MoDDE algorithm is over, refining strategy only takes a short time to obtain a lot of optimization of wire length and standard deviation. Regardless of whether refining strategy is added or not, both can always obtain accurate solutions in circuits with less than 10 pins. Refining strategy has more significant optimization effects in larger circuits.

Algorithm comparison experiment

Experiment 3: To compare the performance of XSMT-MoDDE algorithm with other heuristic algorithms, we compare the results of XSMT constructed by MoDDE algorithm, DDE algorithm, Artificial Bee Colony (ABC) algorithm, and Genetic Algorithm (GA). The experimental results are shown in Tables 6, 7, and 8. XSMT-MoDDE compares with XSMT-DDE, XSMT-ABC, and XSMT-GA, the average wire length is reduced by 2.40%, 1.74%, and 1.77%, the optimal wire length is reduced by 1.26%, 1.55%, and 1.77%, and the standard deviation is reduced by 95.65%, 33.52%, and 28.61%. Experimental results show that XSMT-MoDDE is better than XSMT-DE, XSMT-ABC, and XSMT-GA in both the wire length and standard deviation indicators. Compared with other algorithms, this algorithm still has excellent stability on the basis of having better wire length results.

Experiment 4: In the stage of global routing, there are tens of thousands of nets on the circuit board, and pins inside net need to be interconnected. This paper uses XSMT-MoDDE algorithm to optimize wire length of global routing. This experiment adopts the benchmark provided by IBM, and XSMT-MoDDE algorithm, SAT algorithm, and KNN algorithm are used to construct XSMT. The experimental results are shown in Table 9. Compared with SAT and KNN, XSMT-MoDDE optimizes wire length by 10.05% and 8.86% respectively. Experimental results show that XSMT-MoDDE can greatly shorten the wire length in the construction of multi-nets XSMT problem, and provide effective guidance for global routing.

Finally, for a better understanding the results of XSMT-MoDDE algorithm, we use Matlab to simulate the final XSMT diagrams. We choose Circuit 11 and Circuit 12 in Table 7 as representatives, as shown in Figs. 7A and 7B.