EntropyData

2743 days ago by chlin

############################### # Data for "Entropy of a graph" by # Michael Dairyko, Leslie Hogben, Jephian C.-H. Lin, Joshua Lockhart, David Roberson, Simone Severini, Michael Young # program by Jephian C.H. Lin ############################### 
       
############################### # CONTENTS # Test of minimum entropy on connected graphs (Conjecture 1.2) # Test of min/max entropy on trees (Conjecture 1.3) # Graphs that fail the Renyi-Quantum Star Test (Table 1 in Section 2) # Test of minimum Renyi 2-entropy on connected graphs (Conjecture 3.3) # Examples for noncomparability (Section 4) # Necessary code ############################### 
       
############################### # To evaluate the cells below you must first evaluate the cells # listed as necessary code (at the end) ############################### 
       
############################### # Test of minimum entropy on connected graphs (Conjecture 1.2) ############################### 
       
for n in range(2,9): g0=graphs.CompleteBipartiteGraph(1,n-1); epy0=N(vnentropy(g0)); stg0=g0.canonical_label().graph6_string(); ming=stg0; minepy=epy0; for g in graphs.nauty_geng("%s 1:0 -c"%n): stg=g.canonical_label().graph6_string(); epy=N(vnentropy(g)); if epy<minepy: minepy=epy; ming=stg; print n, ["min",ming,minepy]; sshow(Graph(ming),ming); 
       
2 ['min', 'A_', 0.000000000000000]
3 ['min', 'BW', 0.811278124459133]
4 ['min', 'CF', 1.25162916738782]
5 ['min', 'D?{', 1.54879494069540]
6 ['min', 'E?Bw', 1.77095059445467]
7 ['min', 'F??Fw', 1.94733879618755]
8 ['min', 'G???F{', 2.09306920777189]
2 ['min', 'A_', 0.000000000000000]
3 ['min', 'BW', 0.811278124459133]
4 ['min', 'CF', 1.25162916738782]
5 ['min', 'D?{', 1.54879494069540]
6 ['min', 'E?Bw', 1.77095059445467]
7 ['min', 'F??Fw', 1.94733879618755]
8 ['min', 'G???F{', 2.09306920777189]






############################### # Test of min/max entropy on trees (Conjecture 1.3) ############################### 
       
for n in range(2,16): g0=graphs.CompleteBipartiteGraph(1,n-1); epy0=N(vnentropy(g0)); stg0=g0.canonical_label().graph6_string(); ming=stg0; maxg=stg0; minepy=epy0; maxepy=epy0; for g in graphs.nauty_geng("%s %s:%s -c"%(n,n-1,n-1)): stg=g.canonical_label().graph6_string(); epy=N(vnentropy(g)); if epy>maxepy: maxepy=epy; maxg=stg; if epy<minepy: minepy=epy; ming=stg; print n, ["min",ming,minepy],["max",maxg,maxepy]; multi_sshow([ming,maxg],"%s "%n+ming+" "+maxg); 
       
2 ['min', 'A_', 0.000000000000000] ['max', 'A_', 0.000000000000000]
3 ['min', 'BW', 0.811278124459133] ['max', 'BW', 0.811278124459133]
4 ['min', 'CF', 1.25162916738782] ['max', 'CL', 1.31887985851639]
5 ['min', 'D?{', 1.54879494069540] ['max', 'DBg', 1.69225792845736]
6 ['min', 'E?Bw', 1.77095059445467] ['max', 'E@YO', 1.98827090573713]
7 ['min', 'F??Fw', 1.94733879618755] ['max', 'F@HSO', 2.23367043592823]
8 ['min', 'G???F{', 2.09306920777189] ['max', 'G@GQSG',
2.44330385554387]
9 ['min', 'H????B~', 2.21691718668870] ['max', 'H@GOQIA',
2.62629723307920]
10 ['min', 'I??????~w', 2.32440939317156] ['max', 'I@GOOHA_O',
2.78866996973953]
11 ['min', 'J???????F~_', 2.41924070463685] ['max', 'J@GOOGAOS@?',
2.93460460391692]
12 ['min', 'K?????????^~', 2.50399752733485] ['max', 'K@GOOGA?Q@OA',
3.06712654828993]
13 ['min', 'L???????????~~', 2.58055765339473] ['max', 'L@GOOGA?O@GA_A',
3.18849548432230]
14 ['min', 'M?????????????~~_', 2.65032552934084] ['max',
'M@GOOGA?O@?AOA_@?', 3.30044394882930]
15 ['min', 'N???????????????^~w', 2.71437781726733] ['max',
'N@GOOGA?O@?A?AO@O?O', 3.40432961114319]
2 ['min', 'A_', 0.000000000000000] ['max', 'A_', 0.000000000000000]
3 ['min', 'BW', 0.811278124459133] ['max', 'BW', 0.811278124459133]
4 ['min', 'CF', 1.25162916738782] ['max', 'CL', 1.31887985851639]
5 ['min', 'D?{', 1.54879494069540] ['max', 'DBg', 1.69225792845736]
6 ['min', 'E?Bw', 1.77095059445467] ['max', 'E@YO', 1.98827090573713]
7 ['min', 'F??Fw', 1.94733879618755] ['max', 'F@HSO', 2.23367043592823]
8 ['min', 'G???F{', 2.09306920777189] ['max', 'G@GQSG', 2.44330385554387]
9 ['min', 'H????B~', 2.21691718668870] ['max', 'H@GOQIA', 2.62629723307920]
10 ['min', 'I??????~w', 2.32440939317156] ['max', 'I@GOOHA_O', 2.78866996973953]
11 ['min', 'J???????F~_', 2.41924070463685] ['max', 'J@GOOGAOS@?', 2.93460460391692]
12 ['min', 'K?????????^~', 2.50399752733485] ['max', 'K@GOOGA?Q@OA', 3.06712654828993]
13 ['min', 'L???????????~~', 2.58055765339473] ['max', 'L@GOOGA?O@GA_A', 3.18849548432230]
14 ['min', 'M?????????????~~_', 2.65032552934084] ['max', 'M@GOOGA?O@?AOA_@?', 3.30044394882930]
15 ['min', 'N???????????????^~w', 2.71437781726733] ['max', 'N@GOOGA?O@?A?AO@O?O', 3.40432961114319]













############################### # Graphs that fail the Renyi-quantum Star test ############################### 
       
### Graphs on 3 vertices that fail the Renyi-quantum Star test n=3; for stg in want[n]: g=Graph(stg); sshow(g,stg+" %s"%N(vnentropy(g))); 
       
### Graphs on 4 vertices that fail the Renyi-quantum Star test n=4; for stg in want[n]: g=Graph(stg); sshow(g,stg+" %s"%N(vnentropy(g))); 
       

### Graphs on 5 vertices that fail the Renyi-quantum Star test n=5; for stg in want[n]: g=Graph(stg); sshow(g,stg+" %s"%N(vnentropy(g))); 
       



### Graphs on 6 vertices that fail the Renyi-quantum Star test n=6; for stg in want[n]: g=Graph(stg); sshow(g,stg+" %s"%N(vnentropy(g))); 
       







### Graphs on 7 vertices that fail the Renyi-quantum Star test n=7; for stg in want[n]: g=Graph(stg); sshow(g,stg+" %s"%N(vnentropy(g))); 
       















### Graphs on 8 vertices that fail the Renyi-quantum Star test n=8; for stg in want[n]: g=Graph(stg); sshow(g,stg+" %s"%N(vnentropy(g))); 
       
















































############################### # Test of minimum Renyi alpha-entropy on connected graphs (Conjecture 3.3) ############################### 
       
### Find connected graphs with min/max Renyi entropy for certain alpha print "n vertices, graph6_string of star, graph6_string of complete graph"; for n in range(2,9): print n,graphs.CompleteBipartiteGraph(1,n-1).canonical_label().graph6_string(),graphs.CompleteGraph(n).canonical_label().graph6_string(); print "-----" considered_alpha=[1.1,1.5,2,5,10]; for alpha in considered_alpha: print "alpha = %s"%alpha; for n in range(2,9): g0=graphs.PathGraph(n); epy0=N(Renyi_Entropy(g0,alpha,True)); stg0=g0.canonical_label().graph6_string(); ming=stg0; maxg=stg0; minepy=epy0; maxepy=epy0; for g in graphs.nauty_geng("%s -c"%n): stg=g.canonical_label().graph6_string(); epy=N(Renyi_Entropy(g,alpha,True)); if epy>maxepy: maxepy=epy; maxg=stg; if epy<minepy: minepy=epy; ming=stg; print n, [ming,minepy],[maxg,maxepy]; #sshow(Graph(ming),ming); #sshow(Graph(maxg),maxg); 
       
n vertices, graph6_string of star, graph6_string of complete graph
2 A_ A_
3 BW Bw
4 CF C~
5 D?{ D~{
6 E?Bw E~~w
7 F??Fw F~~~w
8 G???F{ G~~~~{
-----
alpha = 1.10000000000000
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 1.05667041327420] ['Bw', 1.07177346253629]
4 ['CF', 1.08834147425361] ['C~', 1.11612317403390]
5 ['D?{', 1.11000227345943] ['D~{', 1.14869835499703]
6 ['E?Bw', 1.12631457606697] ['E~~w', 1.17461894308802]
7 ['F??Fw', 1.13932242091158] ['F~~~w', 1.19623119885132]
8 ['G???F{', 1.15009765553062] ['G~~~~{', 1.21481404403907]
alpha = 1.50000000000000
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 1.29112382237127] ['Bw', 1.41421356237309]
4 ['CF', 1.46969384566991] ['C~', 1.73205080756888]
5 ['D?{', 1.59568932603071] ['D~{', 2.00000000000000]
6 ['E?Bw', 1.69133447569077] ['E~~w', 2.23606797749979]
7 ['F??Fw', 1.76737930768653] ['F~~~w', 2.44948974278318]
8 ['G???F{', 1.82982640098304] ['G~~~~{', 2.64575131106459]
alpha = 2
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 1.60000000000000] ['Bw', 2.00000000000000]
4 ['CF', 2.00000000000000] ['C~', 3.00000000000000]
5 ['D?{', 2.28571428571429] ['D~{', 4.00000000000000]
6 ['E?Bw', 2.50000000000000] ['E~~w', 5.00000000000000]
7 ['F??Fw', 2.66666666666667] ['F~~~w', 6.00000000000000]
8 ['G???F{', 2.80000000000000] ['G~~~~{', 7.00000000000000]
alpha = 5
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 4.19672131147541] ['Bw', 16.0000000000000]
4 ['CF', 7.57894736842106] ['C~', 81.0000000000001]
5 ['D?{', 10.4757033248082] ['D~{', 256.000000000000]
6 ['E?Bw', 12.8534704370180] ['E~~w', 625.000000000000]
7 ['F??Fw', 14.8008565310492] ['F~~~w', 1296.00000000000]
8 ['G???F{', 16.4100811618966] ['G~~~~{', 2401.00000000000]
alpha = 10
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 17.7574259102455] ['Bw', 512.000000000000]
4 ['CF', 57.6649290753764] ['C~', 19683.0000000000]
5 ['D?{', 109.951129000613] ['D~{', 262144.000000000]
6 ['E?Bw', 165.381705938758] ['E~~w', 1.95312500000000e6]
7 ['F??Fw', 219.195711295829] ['F~~~w', 1.00776960000000e7]
8 ['G???F{', 269.389388486430] ['G~~~~{', 4.03536070000001e7]
n vertices, graph6_string of star, graph6_string of complete graph
2 A_ A_
3 BW Bw
4 CF C~
5 D?{ D~{
6 E?Bw E~~w
7 F??Fw F~~~w
8 G???F{ G~~~~{
-----
alpha = 1.10000000000000
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 1.05667041327420] ['Bw', 1.07177346253629]
4 ['CF', 1.08834147425361] ['C~', 1.11612317403390]
5 ['D?{', 1.11000227345943] ['D~{', 1.14869835499703]
6 ['E?Bw', 1.12631457606697] ['E~~w', 1.17461894308802]
7 ['F??Fw', 1.13932242091158] ['F~~~w', 1.19623119885132]
8 ['G???F{', 1.15009765553062] ['G~~~~{', 1.21481404403907]
alpha = 1.50000000000000
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 1.29112382237127] ['Bw', 1.41421356237309]
4 ['CF', 1.46969384566991] ['C~', 1.73205080756888]
5 ['D?{', 1.59568932603071] ['D~{', 2.00000000000000]
6 ['E?Bw', 1.69133447569077] ['E~~w', 2.23606797749979]
7 ['F??Fw', 1.76737930768653] ['F~~~w', 2.44948974278318]
8 ['G???F{', 1.82982640098304] ['G~~~~{', 2.64575131106459]
alpha = 2
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 1.60000000000000] ['Bw', 2.00000000000000]
4 ['CF', 2.00000000000000] ['C~', 3.00000000000000]
5 ['D?{', 2.28571428571429] ['D~{', 4.00000000000000]
6 ['E?Bw', 2.50000000000000] ['E~~w', 5.00000000000000]
7 ['F??Fw', 2.66666666666667] ['F~~~w', 6.00000000000000]
8 ['G???F{', 2.80000000000000] ['G~~~~{', 7.00000000000000]
alpha = 5
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 4.19672131147541] ['Bw', 16.0000000000000]
4 ['CF', 7.57894736842106] ['C~', 81.0000000000001]
5 ['D?{', 10.4757033248082] ['D~{', 256.000000000000]
6 ['E?Bw', 12.8534704370180] ['E~~w', 625.000000000000]
7 ['F??Fw', 14.8008565310492] ['F~~~w', 1296.00000000000]
8 ['G???F{', 16.4100811618966] ['G~~~~{', 2401.00000000000]
alpha = 10
2 ['A_', 1.00000000000000] ['A_', 1.00000000000000]
3 ['BW', 17.7574259102455] ['Bw', 512.000000000000]
4 ['CF', 57.6649290753764] ['C~', 19683.0000000000]
5 ['D?{', 109.951129000613] ['D~{', 262144.000000000]
6 ['E?Bw', 165.381705938758] ['E~~w', 1.95312500000000e6]
7 ['F??Fw', 219.195711295829] ['F~~~w', 1.00776960000000e7]
8 ['G???F{', 269.389388486430] ['G~~~~{', 4.03536070000001e7]
############################### # Examples for noncomparability ############################### 
       
### No (positive) correlation between matching number m(G) and entropy S(G) pairs=['E@FG', 'E@QW']; betas=[len(Graph(stg).matching()) for stg in pairs]; #matching numbers epys=[N(vnentropy(Graph(stg))) for stg in pairs]; for i in range(2): Graph(pairs[i]).show(figsize=[3,3],vertex_labels=False,vertex_size=50,title="m=%s, S=%s"%(betas[i],epys[i])); 
       

### No (negative) correlation between maximum degree Delta(G) and entropy S(G) pairs=['G???~C', 'G??GfK']; betas=[max(Graph(stg).degree_sequence()) for stg in pairs]; #maximum degrees epys=[N(vnentropy(Graph(stg))) for stg in pairs]; for i in range(2): Graph(pairs[i]).show(figsize=[3,3],vertex_labels=False,vertex_size=50,title="Delta=%s, S=%s"%(betas[i],epys[i])); 
       

### No (positive) correlation between diameter diam(G) and entropy S(G) pairs=['G?Ce?[', 'G?C_MS']; betas=[Graph(stg).diameter() for stg in pairs]; #diameters epys=[N(vnentropy(Graph(stg))) for stg in pairs]; for i in range(2): Graph(pairs[i]).show(figsize=[3,3],vertex_labels=False,vertex_size=50,title="diam=%s, S=%s"%(betas[i],epys[i])); 
       

### No (negative) correlation between degree difference Delta(G)-delta(G) and entropy S(G) pairs=['G???~C', 'G??GfK']; [4, 5] betas=[max(Graph(stg).degree_sequence())-min(Graph(stg).degree_sequence()) for stg in pairs]; #D-d's epys=[N(vnentropy(Graph(stg))) for stg in pairs]; for i in range(2): Graph(pairs[i]).show(figsize=[3,3],vertex_labels=False,vertex_size=50,title="D-d=%s, S=%s"%(betas[i],epys[i])); 
       

################################ Necessary Code #################################################### 
       
### Code for better graph illustration def sshow(g,stg=None): if stg==None: stg=g.graph6_string(); g.show(figsize=[2,2],vertex_labels=False,vertex_size=50,title="%s: %s"%(g.order(),stg)); def multi_sshow(all_graphs,text=None,final_figsize=None): """ Input: all_graphs: a list of graph strings text: a string shown on the output picture, default is to show the strings of all_graphs final_figsize: the figsize of the output picture, default is [12,2] Output: a picture of all graphs in all_graphs in a row, plus text as the title. """ pic=Graph(0).plot(); shift=0; gap=1; for stg in all_graphs: g=Graph(stg); draft=g.plot(save_pos=True); xy_range=draft.get_axes_range(); local_shift=-xy_range["xmin"]; x_range=xy_range["xmax"]-xy_range["xmin"]; gpos=g.get_pos(); for v in g.vertices(): gpos[v][0]+=shift+local_shift; gpic=g.plot(figsize=[2,2],vertex_labels=False,vertex_size=50,pos=gpos); pic+=gpic; shift+=x_range+gap; pic+=line([(shift-0.5*gap,-0.5),(shift-0.5*gap,0.5)],linestyle="--"); pic.axes(False); if text==None: text=""; for stg in all_graphs: text+="%s "%stg; if final_figsize==None: final_figsize=[12,2]; pic.show(figsize=final_figsize, title=text); 
       
### Code for computing related parameters # compute von Neuman entropy S(g) def vnentropy(g,all_info=False): L=g.laplacian_matrix(); tr=L.trace(); nL=(1/tr)*L; eigs=nL.eigenvalues(); if all_info==True: print "tr=%s"%tr, eigs; entropy=0; for eig in eigs: if eig!=0: entropy-=eig*log(eig)/log(2); return entropy; # compute tr_2(g) (mode "A") or compare tr_2(G) to tr_2(K_{1,n-1}) (mod "B") def quadratic_bound(g,mode="A"): n=g.order(); if mode!="B": deg_seq=g.degree_sequence(); dG=0; dG2=0; for i in range(n): dG+=deg_seq[i]; dG2+=deg_seq[i]^2; A=(dG^2)/(dG2+dG); if mode=="A": return A; else: B=(2*n-2)/(n^(n/(2*n-2))); if mode=="B": return B; if mode=="AB": return [A,B]; if mode=="check": return A>B; # compute spec(rho(g)) def LondonSpectrum(g): L=g.laplacian_matrix(); tr=L.trace(); nL=(1/tr)*L; eigs=nL.eigenvalues(); return eigs; # compute Renyi alpha-entropy H_alpha(g) (or if no_log=TRUE compute 1/sum(lambda^alpha)) def Renyi_Entropy(g,alpha, no_log=False): """ Return the Renyi entropy of g for the input alpha, presumably alpha>1. """ eigs=LondonSpectrum(g); alpha_sum=sum(exp(alpha*log(eig)) for eig in eigs); if no_log: return 1/alpha_sum; else: return 1/(1-alpha)*log(alpha_sum)/log(2); 
       
### Code that obtain the data for graphs that fail the Renyi-quantum Star Test ### The output is recorded in the next cell, so don't need to run this cell. want={}; for n in range(2,11): want[n]=[]; counter=0; B=quadratic_bound(Graph(n),"B"); for g in graphs.nauty_geng("%s -c"%n): counter+=1; A=quadratic_bound(g,"A"); if A<B: want[n].append(g.canonical_label().graph6_string()); print n, [len(want[n]),counter],N(len(want[n])/counter,digits=2); 
       
2 [0, 1] 0.00
3 [1, 2] 0.50
4 [2, 6] 0.33
5 [4, 21] 0.19
6 [8, 112] 0.071
7 [16, 853] 0.019
8 [49, 11117] 0.0044
9 [106, 261080] 0.00041
10 [307, 11716571] 0.000026
2 [0, 1] 0.00
3 [1, 2] 0.50
4 [2, 6] 0.33
5 [4, 21] 0.19
6 [8, 112] 0.071
7 [16, 853] 0.019
8 [49, 11117] 0.0044
9 [106, 261080] 0.00041
10 [307, 11716571] 0.000026
### Data for graphs that fail the Renyi-quantum Star Test want={2: [], 3: ['BW'], 4: ['CF', 'CL'], 5: ['D?{', 'D@s', 'D@{', 'DBg'], 6: ['E?Bw', 'E?Fg', 'E?Fw', 'E@FG', 'E?NG', 'E?NW', 'E?Nw', 'E@QW'], 7: ['F??Fw', 'F??Ng', 'F??Nw', 'F?CNG', 'F??^G', 'F??^o', 'F??^W', 'F??^w', 'F??}O', 'F??}W', 'F??}w', 'F?CeW', 'F@?Mw', 'F?Cfw', 'F?CmW', 'F?DcW'], 8: ['G???F{', 'G???Ns', 'G???N{', 'G??GNc', 'G???^c', 'G???^w', 'G???^k', 'G???^{', 'G???~G', 'G???~C', 'G???~W', 'G???~K', 'G???~w', 'G???~[', 'G???~{', 'G??@}[', 'G??@}{', 'G??@~{', 'G??GfK', 'G?C?N[', 'G??Gf{', 'G??GnC', 'G?C?^G', 'G??GnS', 'G?C?^K', 'G??GnK', 'G??Gns', 'G??O^[', 'G??Gn{', 'G?C_MS', 'G??HeK', 'G?C@M[', 'G??He[', 'G?C`E{', 'G@??]{', 'G??He{', 'G@??^{', 'G??G^c', 'G??G^k', 'G??G^{', 'G??HmK', 'G??P][', 'G??Hm[', 'G??G~K', 'G??G~[', 'G??u?[', 'G?CaC[', 'G?CcB{', 'G?CaF{'], 9: ['H????B~', 'H????F|', 'H????F~', 'H???GFx', 'H????Nx', 'H????N}', 'H????Nz', 'H????N~', 'H????^q', 'H????^p', 'H????^u', 'H????^r', 'H????^}', 'H????^v', 'H????^~', 'H????~a', 'H????~e', 'H????~b', 'H????~m', 'H????~f', 'H????~}', 'H????~n', 'H????~~', 'H???@~N', 'H???@~^', 'H???@~~', 'H???B}~', 'H???GRr', 'H??G?Fv', 'H???GR~', 'H???GVp', 'H??G?Nq', 'H???GVt', 'H??G?Nr', 'H???GV{', 'H???GVr', 'H???GV|', 'H???ONv', 'H???GV~', 'H??G_Fd', 'H???Grb', 'H?C??Nm', 'H?C?GRf', 'H??G?ff', 'H???Grf', 'H??G_bn', 'H?C??Nn', 'H???Grn', 'H?C??N~', 'H???GNx', 'H???GNz', 'H???GN~', 'H???oN`', 'H???SLu', 'H???oNd', 'H???Gvb', 'H??G?nb', 'H???Gvd', 'H???Onf', 'H?C??^f', 'H???Gvf', 'H???Gvl', 'H???_^n', 'H???Gvn', 'H?C??^v', 'H???Gv~', 'H???HrB', 'H??G@fF', 'H???HrF', 'H?C?@NN', 'H???HrN', 'H??C?x~', 'H?C?@N^', 'H@???^~', 'H???WNp', 'H???G^p', 'H??GGNr', 'H???G^r', 'H???G^x', 'H???G^}', 'H???G^z', 'H???G^v', 'H???G^~', 'H???PnF', 'H???HvF', 'H???G~f', 'H???G~n', 'H???WZv', 'H???WZ~', 'H??G_Nb', 'H???Wjb', 'H??G_Nf', 'H??OGVn', 'H???Wj~', 'H???XbB', 'H???sHf', 'H???Xb~', 'H???XfB', 'H??G`BF', 'H?C_?FN', 'H??Gc@~', 'H??G`B~', 'H??G`FF', 'H@??GV~', 'H?C?GNn', 'H?C?GN~'], 10: ['I??????~w', 'I?????@~g', 'I?????@~w', 'I????C@~G', 'I?????B~G', 'I?????B~o', 'I?????B~W', 'I?????B~w', 'I?????F}O', 'I?????F}G', 'I?????F}o', 'I?????F}W', 'I?????F~o', 'I?????F}w', 'I?????F~w', 'I?????N{O', 'I?????N{G', 'I?????N{o', 'I?????N{W', 'I?????N|o', 'I?????N{w', 'I?????N~o', 'I?????N|w', 'I?????N~w', 'I?????^wo', 'I?????^xo', 'I?????^ww', 'I?????^zo', 'I?????^xw', 'I?????^~o', 'I?????^zw', 'I?????^~w', 'I?????~ro', 'I?????~rw', 'I?????~vw', 'I?????~~w', 'I????@~nw', 'I????@~~w', 'I????CC}W', 'I???G?@}w', 'I????CC~w', 'I????CD}G', 'I???G?B}O', 'I????CD}g', 'I???G?B}W', 'I????CD~_', 'I????CD}W', 'I????CD~g', 'I????GB}w', 'I????CD~w', 'I???GO@{g', 'I????CK{W', 'I??G??B|o', 'I??G?CC{w', 'I???G?H{w', 'I????CK{w', 'I???GOG|w', 'I??G??B|w', 'I????CK|w', 'I????CK~w', 'I????CB~G', 'I????CB~W', 'I????CB~w', 'I????WB{G', 'I???G?J{O', 'I????CL{_', 'I????CL{G', 'I????IB]o', 'I??G??F{o', 'I????WB{g', 'I????CL|_', 'I????CL{W', 'I???G?J{W', 'I????CL{g', 'I????OF|o', 'I????WB|g', 'I????CL~_', 'I????GJ{w', 'I??G??F{w', 'I????CL{w', 'I????CL|g', 'I????CL~o', 'I????OF|w', 'I????CL|w', 'I????CL~g', 'I????CL~w', 'I???G?Xwo', 'I????C[wW', 'I??G??Rxo', 'I????C[xo', 'I???G?Xww', 'I????C[ww', 'I?C???Fzo', 'I????C[zo', 'I??G??Rxw', 'I????C[xw', 'I?C???F~o', 'I?C???Fzw', 'I????C[zw', 'I?C???F~w', 'I????KB}G', 'I????CF~?', 'I????CF}G', 'I????CF~O', 'I???GCB}W', 'I????CF}W', 'I????CF~G', 'I????CF~o', 'I????CF~W', 'I????CF}w', 'I????CF~w', 'I????C\\wO', 'I????GZwo', 'I????C\\wo', 'I????C\\wW', 'I????F`Fg', 'I????GZww', 'I??G??Vww', 'I????C\\ww', 'I????C\\xg', 'I????OVxw', 'I?C???Nxw', 'I????C\\xw', 'I????C\\zg', 'I????_Nzw', 'I????C\\zw', 'I?C???N|w', 'I????C\\~w', 'I???G?xow', 'I????C{ow', 'I??G??rpw', 'I????C{pw', 'I?C???frw', 'I????C{rw', 'I???C?]^w', 'I?C???fvw', 'I@????N~w', 'I????CN{O', 'I????CN{o', 'I????CN{W', 'I????CN|G', 'I????CN|o', 'I????CN|W', 'I????CN~G', 'I????CN{w', 'I????CN~o', 'I????CN~W', 'I????CN|w', 'I????CN~w', 'I????Ovpw', 'I????C|pw', 'I????FLNg', 'I????_nrw', 'I????C|rw', 'I?C??@fbw', 'I????FkFw', 'I????C^ww', 'I????C^zW', 'I????C^xw', 'I????C^zw', 'I????C^~w', 'I????KE}G', 'I????KE}W', 'I???GCD}W', 'I????KE}w', 'I????KE~w', 'I???GOB{G', 'I???GCH{G', 'I??G?CD{g', 'I??GOCC{W', 'I???GOB{W', 'I???OCD|G', 'I????KI|G', 'I????KI{W', 'I???GOB~G', 'I??G?GB{w', 'I???GOD{w', 'I????KI~G', 'I???GOB{w', 'I????SE|W', 'I???GOD}w', 'I???OCD|w', 'I????KI~W', 'I????KI~w', 'I???GCWwW', 'I????KWwW', 'I??GOA@\\o', 'I???OCSxW', 'I????KWxg', 'I????YA[w', 'I??GO_Oxw', 'I???_CKzW', 'I????KWzg', 'I????oExw', 'I????WQxw', 'I?CO??Bzw', 'I????cKzw', 'I????KW~w', 'I???GGJ{w', 'I????KM{w', 'I???GGJ}w', 'I????KM|w', 'I????SM|w', 'I????KM~w', 'I???GCXwW', 'I????KYwW', 'I???GCXww', 'I????KYxw', 'I???OCTxw', 'I????SUzw', 'I???_CLzw', 'I????cM~w', 'I????Kwow', 'I????Wqow', 'I????KF}g', 'I????KF}W', 'I????KF~g', 'I????KF}w', 'I????KF~w', 'I????KJ{G', 'I???GCJ{W', 'I????KJ{W', 'I???OCF|W', 'I????KJ|g', 'I????KJ{w', 'I????SF|W', 'I????SF~g', 'I????SF|w', 'I????KJ~W', 'I????KJ~w', 'I????KwoW', 'I????WRwg', 'I????KXwg', 'I????KXwW', 'I????YB[w', 'I????oFxw', 'I????WRxw', 'I????cLzw', 'I????KX~w', 'I????KZww', 'I????Kxow', 'I????Wrow', 'I???GOOww', 'I??G_?@xw', 'I???GQ?^w', 'I???GOO~w', 'I???OGPwg', 'I??G?CPxg', 'I??G_?BxW', 'I??GGQ?[w', 'I??G?CSww', 'I???GQG\\o', 'I???GOPww', 'I?C??CDzg', 'I??G?CPzg', 'I?C??CKxw', 'I?C?GOOxw', 'I???O_Dxw', 'I???GOPxw', 'I?C?G?B~W', 'I?C??CDzw', 'I???G_Hzw', 'I?C??CD~w', 'I???GOoow', 'I??GQA?Lw', 'I??GAE?Ng', 'I??G_?`pw', 'I??G?Copw', 'I??G_a?Zw', 'I??G_A@^o', 'I?C_??Brw', 'I?C??Ccrw', 'I???GOorw', 'I??G`?_vw', 'I?C?C?E^w', 'I??G_?`vw', 'I?C_??B~w', 'I??G?CB|G', 'I??G?CB|W', 'I??G?CB~G', 'I???GOH|w', 'I??G?CB|w', 'I??G?CB~w', 'I???GORwW', 'I???GORww', 'I???O_Fxw', 'I???OGRxw', 'I???_OFzw', 'I???GORzw', 'I???GOR~w', 'I???GOpow', 'I???OGF{w', 'I??G?CF{w', 'I??G?CF|w', 'I???OGF}w', 'I??G?CF}w', 'I???GOF~w', 'I??G?CRxw', 'I?C??CFzw', 'I??G?CRzw', 'I??G?CR~w', 'I???GSK~w', 'I???GKF~w', 'I??HAA?Fw', 'I??GaA?Nw', 'I??G`A?^w', 'I??G`@?~w']};