Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/graphical.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Test sequences for graphiness. |
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2 """ |
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3 import heapq |
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4 import networkx as nx |
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5 |
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6 __all__ = [ |
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7 "is_graphical", |
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8 "is_multigraphical", |
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9 "is_pseudographical", |
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10 "is_digraphical", |
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11 "is_valid_degree_sequence_erdos_gallai", |
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12 "is_valid_degree_sequence_havel_hakimi", |
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13 ] |
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14 |
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15 |
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16 def is_graphical(sequence, method="eg"): |
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17 """Returns True if sequence is a valid degree sequence. |
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18 |
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19 A degree sequence is valid if some graph can realize it. |
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20 |
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21 Parameters |
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22 ---------- |
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23 sequence : list or iterable container |
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24 A sequence of integer node degrees |
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25 |
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26 method : "eg" | "hh" (default: 'eg') |
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27 The method used to validate the degree sequence. |
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28 "eg" corresponds to the Erdős-Gallai algorithm, and |
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29 "hh" to the Havel-Hakimi algorithm. |
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30 |
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31 Returns |
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32 ------- |
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33 valid : bool |
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34 True if the sequence is a valid degree sequence and False if not. |
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35 |
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36 Examples |
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37 -------- |
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38 >>> G = nx.path_graph(4) |
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39 >>> sequence = (d for n, d in G.degree()) |
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40 >>> nx.is_graphical(sequence) |
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41 True |
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42 |
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43 References |
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44 ---------- |
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45 Erdős-Gallai |
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46 [EG1960]_, [choudum1986]_ |
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47 |
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48 Havel-Hakimi |
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49 [havel1955]_, [hakimi1962]_, [CL1996]_ |
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50 """ |
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51 if method == "eg": |
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52 valid = is_valid_degree_sequence_erdos_gallai(list(sequence)) |
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53 elif method == "hh": |
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54 valid = is_valid_degree_sequence_havel_hakimi(list(sequence)) |
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55 else: |
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56 msg = "`method` must be 'eg' or 'hh'" |
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57 raise nx.NetworkXException(msg) |
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58 return valid |
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59 |
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60 |
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61 def _basic_graphical_tests(deg_sequence): |
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62 # Sort and perform some simple tests on the sequence |
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63 deg_sequence = nx.utils.make_list_of_ints(deg_sequence) |
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64 p = len(deg_sequence) |
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65 num_degs = [0] * p |
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66 dmax, dmin, dsum, n = 0, p, 0, 0 |
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67 for d in deg_sequence: |
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68 # Reject if degree is negative or larger than the sequence length |
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69 if d < 0 or d >= p: |
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70 raise nx.NetworkXUnfeasible |
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71 # Process only the non-zero integers |
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72 elif d > 0: |
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73 dmax, dmin, dsum, n = max(dmax, d), min(dmin, d), dsum + d, n + 1 |
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74 num_degs[d] += 1 |
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75 # Reject sequence if it has odd sum or is oversaturated |
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76 if dsum % 2 or dsum > n * (n - 1): |
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77 raise nx.NetworkXUnfeasible |
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78 return dmax, dmin, dsum, n, num_degs |
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79 |
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80 |
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81 def is_valid_degree_sequence_havel_hakimi(deg_sequence): |
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82 r"""Returns True if deg_sequence can be realized by a simple graph. |
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83 |
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84 The validation proceeds using the Havel-Hakimi theorem. |
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85 Worst-case run time is $O(s)$ where $s$ is the sum of the sequence. |
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86 |
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87 Parameters |
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88 ---------- |
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89 deg_sequence : list |
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90 A list of integers where each element specifies the degree of a node |
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91 in a graph. |
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92 |
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93 Returns |
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94 ------- |
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95 valid : bool |
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96 True if deg_sequence is graphical and False if not. |
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97 |
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98 Notes |
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99 ----- |
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100 The ZZ condition says that for the sequence d if |
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101 |
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102 .. math:: |
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103 |d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)} |
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104 |
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105 then d is graphical. This was shown in Theorem 6 in [1]_. |
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106 |
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107 References |
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108 ---------- |
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109 .. [1] I.E. Zverovich and V.E. Zverovich. "Contributions to the theory |
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110 of graphic sequences", Discrete Mathematics, 105, pp. 292-303 (1992). |
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111 |
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112 [havel1955]_, [hakimi1962]_, [CL1996]_ |
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113 |
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114 """ |
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115 try: |
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116 dmax, dmin, dsum, n, num_degs = _basic_graphical_tests(deg_sequence) |
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117 except nx.NetworkXUnfeasible: |
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118 return False |
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119 # Accept if sequence has no non-zero degrees or passes the ZZ condition |
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120 if n == 0 or 4 * dmin * n >= (dmax + dmin + 1) * (dmax + dmin + 1): |
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121 return True |
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122 |
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123 modstubs = [0] * (dmax + 1) |
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124 # Successively reduce degree sequence by removing the maximum degree |
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125 while n > 0: |
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126 # Retrieve the maximum degree in the sequence |
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127 while num_degs[dmax] == 0: |
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128 dmax -= 1 |
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129 # If there are not enough stubs to connect to, then the sequence is |
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130 # not graphical |
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131 if dmax > n - 1: |
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132 return False |
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133 |
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134 # Remove largest stub in list |
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135 num_degs[dmax], n = num_degs[dmax] - 1, n - 1 |
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136 # Reduce the next dmax largest stubs |
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137 mslen = 0 |
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138 k = dmax |
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139 for i in range(dmax): |
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140 while num_degs[k] == 0: |
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141 k -= 1 |
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142 num_degs[k], n = num_degs[k] - 1, n - 1 |
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143 if k > 1: |
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144 modstubs[mslen] = k - 1 |
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145 mslen += 1 |
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146 # Add back to the list any non-zero stubs that were removed |
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147 for i in range(mslen): |
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148 stub = modstubs[i] |
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149 num_degs[stub], n = num_degs[stub] + 1, n + 1 |
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150 return True |
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151 |
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152 |
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153 def is_valid_degree_sequence_erdos_gallai(deg_sequence): |
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154 r"""Returns True if deg_sequence can be realized by a simple graph. |
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155 |
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156 The validation is done using the Erdős-Gallai theorem [EG1960]_. |
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157 |
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158 Parameters |
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159 ---------- |
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160 deg_sequence : list |
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161 A list of integers |
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162 |
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163 Returns |
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164 ------- |
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165 valid : bool |
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166 True if deg_sequence is graphical and False if not. |
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167 |
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168 Notes |
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169 ----- |
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170 |
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171 This implementation uses an equivalent form of the Erdős-Gallai criterion. |
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172 Worst-case run time is $O(n)$ where $n$ is the length of the sequence. |
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173 |
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174 Specifically, a sequence d is graphical if and only if the |
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175 sum of the sequence is even and for all strong indices k in the sequence, |
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176 |
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177 .. math:: |
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178 |
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179 \sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{j=k+1}^{n} \min(d_i,k) |
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180 = k(n-1) - ( k \sum_{j=0}^{k-1} n_j - \sum_{j=0}^{k-1} j n_j ) |
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181 |
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182 A strong index k is any index where d_k >= k and the value n_j is the |
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183 number of occurrences of j in d. The maximal strong index is called the |
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184 Durfee index. |
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185 |
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186 This particular rearrangement comes from the proof of Theorem 3 in [2]_. |
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187 |
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188 The ZZ condition says that for the sequence d if |
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189 |
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190 .. math:: |
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191 |d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)} |
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192 |
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193 then d is graphical. This was shown in Theorem 6 in [2]_. |
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194 |
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195 References |
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196 ---------- |
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197 .. [1] A. Tripathi and S. Vijay. "A note on a theorem of Erdős & Gallai", |
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198 Discrete Mathematics, 265, pp. 417-420 (2003). |
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199 .. [2] I.E. Zverovich and V.E. Zverovich. "Contributions to the theory |
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200 of graphic sequences", Discrete Mathematics, 105, pp. 292-303 (1992). |
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201 |
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202 [EG1960]_, [choudum1986]_ |
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203 """ |
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204 try: |
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205 dmax, dmin, dsum, n, num_degs = _basic_graphical_tests(deg_sequence) |
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206 except nx.NetworkXUnfeasible: |
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207 return False |
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208 # Accept if sequence has no non-zero degrees or passes the ZZ condition |
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209 if n == 0 or 4 * dmin * n >= (dmax + dmin + 1) * (dmax + dmin + 1): |
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210 return True |
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211 |
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212 # Perform the EG checks using the reformulation of Zverovich and Zverovich |
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213 k, sum_deg, sum_nj, sum_jnj = 0, 0, 0, 0 |
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214 for dk in range(dmax, dmin - 1, -1): |
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215 if dk < k + 1: # Check if already past Durfee index |
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216 return True |
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217 if num_degs[dk] > 0: |
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218 run_size = num_degs[dk] # Process a run of identical-valued degrees |
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219 if dk < k + run_size: # Check if end of run is past Durfee index |
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220 run_size = dk - k # Adjust back to Durfee index |
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221 sum_deg += run_size * dk |
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222 for v in range(run_size): |
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223 sum_nj += num_degs[k + v] |
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224 sum_jnj += (k + v) * num_degs[k + v] |
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225 k += run_size |
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226 if sum_deg > k * (n - 1) - k * sum_nj + sum_jnj: |
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227 return False |
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228 return True |
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229 |
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230 |
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231 def is_multigraphical(sequence): |
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232 """Returns True if some multigraph can realize the sequence. |
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233 |
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234 Parameters |
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235 ---------- |
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236 sequence : list |
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237 A list of integers |
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238 |
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239 Returns |
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240 ------- |
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241 valid : bool |
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242 True if deg_sequence is a multigraphic degree sequence and False if not. |
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243 |
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244 Notes |
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245 ----- |
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246 The worst-case run time is $O(n)$ where $n$ is the length of the sequence. |
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247 |
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248 References |
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249 ---------- |
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250 .. [1] S. L. Hakimi. "On the realizability of a set of integers as |
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251 degrees of the vertices of a linear graph", J. SIAM, 10, pp. 496-506 |
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252 (1962). |
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253 """ |
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254 try: |
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255 deg_sequence = nx.utils.make_list_of_ints(sequence) |
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256 except nx.NetworkXError: |
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257 return False |
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258 dsum, dmax = 0, 0 |
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259 for d in deg_sequence: |
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260 if d < 0: |
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261 return False |
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262 dsum, dmax = dsum + d, max(dmax, d) |
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263 if dsum % 2 or dsum < 2 * dmax: |
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264 return False |
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265 return True |
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266 |
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267 |
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268 def is_pseudographical(sequence): |
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269 """Returns True if some pseudograph can realize the sequence. |
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270 |
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271 Every nonnegative integer sequence with an even sum is pseudographical |
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272 (see [1]_). |
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273 |
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274 Parameters |
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275 ---------- |
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276 sequence : list or iterable container |
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277 A sequence of integer node degrees |
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278 |
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279 Returns |
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280 ------- |
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281 valid : bool |
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282 True if the sequence is a pseudographic degree sequence and False if not. |
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283 |
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284 Notes |
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285 ----- |
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286 The worst-case run time is $O(n)$ where n is the length of the sequence. |
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287 |
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288 References |
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289 ---------- |
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290 .. [1] F. Boesch and F. Harary. "Line removal algorithms for graphs |
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291 and their degree lists", IEEE Trans. Circuits and Systems, CAS-23(12), |
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292 pp. 778-782 (1976). |
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293 """ |
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294 try: |
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295 deg_sequence = nx.utils.make_list_of_ints(sequence) |
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296 except nx.NetworkXError: |
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297 return False |
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298 return sum(deg_sequence) % 2 == 0 and min(deg_sequence) >= 0 |
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299 |
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300 |
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301 def is_digraphical(in_sequence, out_sequence): |
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302 r"""Returns True if some directed graph can realize the in- and out-degree |
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303 sequences. |
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304 |
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305 Parameters |
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306 ---------- |
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307 in_sequence : list or iterable container |
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308 A sequence of integer node in-degrees |
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309 |
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310 out_sequence : list or iterable container |
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311 A sequence of integer node out-degrees |
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312 |
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313 Returns |
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314 ------- |
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315 valid : bool |
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316 True if in and out-sequences are digraphic False if not. |
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317 |
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318 Notes |
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319 ----- |
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320 This algorithm is from Kleitman and Wang [1]_. |
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321 The worst case runtime is $O(s \times \log n)$ where $s$ and $n$ are the |
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322 sum and length of the sequences respectively. |
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323 |
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324 References |
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325 ---------- |
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326 .. [1] D.J. Kleitman and D.L. Wang |
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327 Algorithms for Constructing Graphs and Digraphs with Given Valences |
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328 and Factors, Discrete Mathematics, 6(1), pp. 79-88 (1973) |
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329 """ |
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330 try: |
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331 in_deg_sequence = nx.utils.make_list_of_ints(in_sequence) |
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332 out_deg_sequence = nx.utils.make_list_of_ints(out_sequence) |
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333 except nx.NetworkXError: |
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334 return False |
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335 # Process the sequences and form two heaps to store degree pairs with |
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336 # either zero or non-zero out degrees |
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337 sumin, sumout, nin, nout = 0, 0, len(in_deg_sequence), len(out_deg_sequence) |
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338 maxn = max(nin, nout) |
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339 maxin = 0 |
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340 if maxn == 0: |
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341 return True |
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342 stubheap, zeroheap = [], [] |
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343 for n in range(maxn): |
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344 in_deg, out_deg = 0, 0 |
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345 if n < nout: |
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346 out_deg = out_deg_sequence[n] |
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347 if n < nin: |
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348 in_deg = in_deg_sequence[n] |
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349 if in_deg < 0 or out_deg < 0: |
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350 return False |
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351 sumin, sumout, maxin = sumin + in_deg, sumout + out_deg, max(maxin, in_deg) |
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352 if in_deg > 0: |
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353 stubheap.append((-1 * out_deg, -1 * in_deg)) |
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354 elif out_deg > 0: |
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355 zeroheap.append(-1 * out_deg) |
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356 if sumin != sumout: |
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357 return False |
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358 heapq.heapify(stubheap) |
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359 heapq.heapify(zeroheap) |
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360 |
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361 modstubs = [(0, 0)] * (maxin + 1) |
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362 # Successively reduce degree sequence by removing the maximum out degree |
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363 while stubheap: |
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364 # Take the first value in the sequence with non-zero in degree |
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365 (freeout, freein) = heapq.heappop(stubheap) |
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366 freein *= -1 |
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367 if freein > len(stubheap) + len(zeroheap): |
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368 return False |
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369 |
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370 # Attach out stubs to the nodes with the most in stubs |
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371 mslen = 0 |
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372 for i in range(freein): |
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373 if zeroheap and (not stubheap or stubheap[0][0] > zeroheap[0]): |
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374 stubout = heapq.heappop(zeroheap) |
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375 stubin = 0 |
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376 else: |
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377 (stubout, stubin) = heapq.heappop(stubheap) |
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378 if stubout == 0: |
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379 return False |
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380 # Check if target is now totally connected |
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381 if stubout + 1 < 0 or stubin < 0: |
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382 modstubs[mslen] = (stubout + 1, stubin) |
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383 mslen += 1 |
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384 |
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385 # Add back the nodes to the heap that still have available stubs |
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386 for i in range(mslen): |
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387 stub = modstubs[i] |
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388 if stub[1] < 0: |
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389 heapq.heappush(stubheap, stub) |
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390 else: |
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391 heapq.heappush(zeroheap, stub[0]) |
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392 if freeout < 0: |
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393 heapq.heappush(zeroheap, freeout) |
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394 return True |