Algorithm 1.

Adjusted Single Queue Equi-Energy algorithm (ASQEE)

Assign $X_1^{(W)}$ an initial ordering, set ${\widehat{D}}_l=\varnothing ,l=1,2,\cdots ,W.$

For n = 1,2,···

For l = W,W − 1,··· ,1

1: if n > (W − l)(B + N) (B is the burn in period) then

2:     do

3:     if l = W or if ${\widehat{D}}_{I\left(X_{n-1}^{(i)}\right)}\ne$ ∅ then

4:         Perform local M-H move to update $X_{n-1}^{(l)}$ by $X_n^{(l)}$ with target distribution πl

5:     else if l < W AND if ${\widehat{D}}_{I\left(X_{n-1}^{(l)}\right)}\ne$ ∅ then

6:         Generate μ ∼ U(0,1)

7:         if μ > pee then

8:             Perform local M-H move to update $X_{n-1}^{(l)}$ by $X_n^{(l)}$ with target distribution πl

9:         else if $\mu ⩽p_{ee}$ then

10:             Uniformly pick a state y from ${\widehat{D}}_{I\left(X_{n-1}^{(l)}\right)}({\widehat{D}}_{I\left(X_{n-1}^{(l)}\right)}$ is the union of all previ-ous energy rings with similar energy level) and $X_n^{(l)}\leftarrow y$ with probability $min\left(1,\frac{{\pi }_l(y)Q\left(y;X_{n-1}^{(l)}\right)}{{\pi }_l\left(X_{n-1}^{(l)}\right)Q\left(X_{n-1}^{(l)};y\right)}\right)(Q(.,.)$ is the transition kernel function; $X_n^{(l)}\leftarrow X_{n-1}^{(l)}$ with the remaining probability

11:         end if

12:     end if

13:     if n > (W − l)(B + N) + B then

14:         ${\widehat{D}}_{I\left(X_n^{(l)}\right)}\leftarrow {\widehat{D}}_{I\left(X_n^{(l)}\right)}+\left\{X_n^{(l)}\right\}$

15:     end if

16: end if