DenseZDD: A Compact and Fast Index for Families of Sets ^{†}
Abstract
:1. Introduction
2. Preliminaries
2.1. Succinct Data Structures for Rank/Select
2.2. Succinct Data Structures for Trees
 $\mathit{depth}(P,i)$: the depth of the node at position i. (The depth of a root is 0.)
 $\mathit{preorder}(P,i)$: the preorder of the node at position i.
 $\mathit{level}\_\mathit{ancestor}(P,i,d)$: the position of the ancestor with depth d of the node at position i.
 $\mathit{parent}(P,i)$: the position of the parent of the node at position i (identical to $\mathit{level}\_\mathit{ancestor}$(P, i, $\mathit{depth}(P,i)1)$).
 $\mathit{degree}(P,i)$: the number of children of the node at position i.
 $\mathit{child}(P,i,d)$: the dth child of the node at position i.
2.3. ZeroSuppressed Binary Decision Diagrams
2.4. Problem of Existing ZDDs
3. Data Structure
3.1. DenseZDD
3.1.1. ZeroEdge Tree
3.1.2. Dummy Node Vector
3.1.3. OneChild Array
3.2. Convert Algorithm
Algorithm 1 Compute_Preorder: Algorithm that computes the preorder rank $\mathit{prank}(v)$ of each node v. Sets of nodes are implemented by arrays or lists in this code. 

Algorithm 2 Convert_ZDD_BitVectors ($v,paren,dummy,onechild$): Algorithm for obtaining the BP representation of the zeroedge tree, the dummy node vector, and the onechild array. 
Input: ZDD node v, list of parentheses $paren$, list of bits $dummy$, list of integers $onechild$

Algorithm 3 Construct_DenseZDD (W: a set of root nodes of ZDD): Algorithm for constructing the DenseZDD from a source ZDD. 
Output: DenseZDD $DZ$

4. ZDD Operations
4.1. $\mathit{index}(i)$
4.2. $\mathit{topset}(i,d)$
4.3. $\mathit{zero}(i)$
4.4. $\mathit{one}(i)$
4.5. $\mathit{count}(i)$
4.6. $\mathit{sample}(i)$
Algorithm 4 Count$(i)$: Algorithm that computes the cardinality of the family of sets represented by nodes reachable from a node i. The cardinalities are stored in an integer array C of length m, where m is the number of ZDD nodes. The initial values of all the elements in C are 0. 

Algorithm 5 Random_naive$(i,\mathit{empflag})$: Algorithm that returns a set uniformly and randomly chosen from the family of sets that is represented by a ZDD whose root is node i. Assume that Count has already been executed. The argument $\mathit{empflag}\in \{0,1\}$ means whether or not the current family of sets has the empty set. If $\mathit{empflag}=1$, this family has the empty set. 

Algorithm 6 Random_bin$(i,\mathit{empflag})$: Algorithm that returns a set uniformly and randomly chosen from the family of sets represented by the ZDD whose root is node i. This algorithm chooses the index by binary search on nodes linked by 0edges. 

5. Complexity Analysis
6. Hybrid Method
7. Other Decision Diagrams
7.1. Sets of Strings
7.2. Boolean Functions
8. Experimental Results
9. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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$\mathit{index}(v)$  Returns the index of node v. 
$\mathit{zero}(v)$  Returns the 0child of node v. 
$\mathit{one}(v)$  Returns the 1child of node v. 
$\mathit{getnode}(i,{v}_{0},{v}_{1})$  Generates (or makes a reference to) a node v 
with index i and two child nodes ${v}_{0}=\mathit{zero}(v)$ and ${v}_{1}=\mathit{one}(v)$.  
$\mathit{topset}(v,i)$  Returns a node with the index i reached by traversing only 0edges from v. 
If such a node does not exist, it returns the 0terminal node.  
$\mathit{member}(v,S)$  Returns $\mathit{true}$ if $S\in F(v)$, and returns $\mathit{false}$ otherwise. 
$\mathit{count}(v)$  Returns $F(v)$. 
$\mathit{sample}(v)$  Returns a set $S\in F(v)$ uniformly and randomly. 
$\mathit{offset}(v,i)$  Returns u such that $F(u)=\{\phantom{\rule{0.222222em}{0ex}}S\subseteq {U}_{n}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}S\in F,{e}_{i}\notin S\phantom{\rule{0.222222em}{0ex}}\}$. 
$\mathit{onset}(v,i)$  Returns u such that $F(u)=\{\phantom{\rule{0.222222em}{0ex}}S\backslash \left\{{e}_{i}\right\}\subseteq {U}_{n}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}S\in F,{e}_{i}\in S\phantom{\rule{0.222222em}{0ex}}\}$. 
${\mathit{apply}}_{\diamond}({v}_{1},{v}_{2})$  Returns v such that $F(v)=F({v}_{1})\diamond F({v}_{2})$, for $\diamond \in \{\cup ,\cap ,\backslash ,\oplus \}$. 
Data Set  n  $\left\mathit{F}\right$  $\parallel \mathit{F}\parallel $  #roots  #nodes 

rect1x10000  10,000  10,000  10,000  1  10,001 
rect5x2000  10,000  $3.2\times {10}^{16}$  $1.6\times {10}^{17}$  1  10,001 
rect100x100  10,000  $1.0\times {10}^{200}$  $1.0\times {10}^{202}$  1  10,001 
rect2000x5  10,000  $8.7\times {10}^{1397}$  $1.7\times {10}^{1401}$  1  10,001 
rect10000x1  10,000  1  10,000  1  10,001 
randomjoin256  32,740  $4.3\times {10}^{9}$  $1.7\times {10}^{10}$  1  25,743 
randomjoin2048  32,765  $1.7\times {10}^{13}$  $7.0\times {10}^{13}$  1  375,959 
randomjoin8192  32,768  $3.6\times {10}^{15}$  $1.4\times {10}^{16}$  1  $1.3\times {10}^{6}$ 
randomjoin16384  32,768  $2.8\times {10}^{16}$  $1.1\times {10}^{17}$  1  $1.9\times {10}^{6}$ 
bddqueen13  169  73,712  958,256  1  204,782 
bddqueen14  196  365,596  $5.1\times {10}^{6}$  1  911,421 
bddqueen15  225  $2.3\times {10}^{6}$  $3.4\times {10}^{7}$  1  $4.8\times {10}^{6}$ 
T40I10D100K:0.001  925  $7.0\times {10}^{7}$  $8.2\times {10}^{8}$  1  $1.1\times {10}^{6}$ 
T40I10D100K:0.0005  933  $2.0\times {10}^{8}$  $2.1\times {10}^{9}$  1  $6.5\times {10}^{6}$ 
T40I10D100K:0.0001  942  $1.2\times {10}^{10}$  $1.5\times {10}^{11}$  1  $1.9\times {10}^{8}$ 
accidents:0.1  76  $1.1\times {10}^{7}$  $1.1\times {10}^{8}$  1  36,324 
accidents:0.05  106  $8.3\times {10}^{7}$  $9.3\times {10}^{8}$  1  183,144 
accidents:0.01  167  $4.1\times {10}^{9}$  $5.3\times {10}^{10}$  1  $4.7\times {10}^{6}$ 
chess:0.1  62  $4.6\times {10}^{9}$  $6.4\times {10}^{10}$  1  $1.1\times {10}^{6}$ 
chess:0.05  67  $4.1\times {10}^{10}$  $6.2\times {10}^{11}$  1  $3.1\times {10}^{6}$ 
chess:0.01  72  $1.6\times {10}^{12}$  $2.9\times {10}^{13}$  1  $5.8\times {10}^{6}$ 
connect:0.05  87  $4.1\times {10}^{11}$  $6.9\times {10}^{12}$  1  331,829 
connect:0.01  110  $1.7\times {10}^{13}$  $3.2\times {10}^{14}$  1  $2.3\times {10}^{6}$ 
connect:0.005  116  $6.8\times {10}^{13}$  $1.3\times {10}^{15}$  1  $4.1\times {10}^{6}$ 
16adder_col  66  $9.7\times {10}^{6}$  $2.7\times {10}^{8}$  17  $1.5\times {10}^{6}$ 
C1908  66  $3.7\times {10}^{8}$  $1.1\times {10}^{10}$  25  133,379 
C3540  100  $5.0\times {10}^{6}$  $1.3\times {10}^{8}$  22  $1.5\times {10}^{6}$ 
C499  82  $4.9\times {10}^{11}$  $1.9\times {10}^{13}$  32  140,932 
C880  120  $2.4\times {10}^{6}$  $7.0\times {10}^{7}$  26  606,310 
comp  64  196,606  $6.0\times {10}^{6}$  3  589,783 
my_adder  66  655,287  $2.0\times {10}^{7}$  17  884,662 
Data Set  Size (byte)  Comp. Ratio  $\mathit{\delta}$  

Z  DZ  DZ${}_{\mathit{dc}}$  DZ  DZ${}_{\mathit{dc}}$  
rect1x10000  320,032  14,662  10,372  0.000  0.046  0.032 
rect5x2000  320,032  36,947  29,227  0.444  0.115  0.091 
rect100x100  320,032  38,014  29,648  0.498  0.119  0.093 
rect2000x5  320,032  38,078  32,100  0.500  0.119  0.100 
rect10000x1  320,032  38,078  34,048  0.500  0.119  0.106 
randomjoin256  823,760  792,703  279,719  0.978  0.962  0.340 
randomjoin2048  $1.2\times {10}^{7}$  $2.5\times {10}^{6}$  $1.6\times {10}^{6}$  0.821  0.210  0.135 
randomjoin8192  $4.0\times {10}^{7}$  $5.6\times {10}^{6}$  $4.7\times {10}^{6}$  0.424  0.139  0.115 
randomjoin16384  $6.0\times {10}^{7}$  $7.7\times {10}^{6}$  $6.8\times {10}^{6}$  0.145  0.128  0.113 
bddqueen13  $6.1\times {10}^{6}$  846,809  752,775  0.466  0.138  0.123 
bddqueen14  $2.7\times {10}^{7}$  $4.2\times {10}^{6}$  $3.7\times {10}^{6}$  0.510  0.153  0.136 
bddqueen15  $1.4\times {10}^{8}$  $2.5\times {10}^{7}$  $2.2\times {10}^{7}$  0.558  0.171  0.151 
T40I10D100K:0.001  $3.6\times {10}^{7}$  $8.0\times {10}^{6}$  $5.3\times {10}^{6}$  0.826  0.220  0.148 
T40I10D100K:0.0005  $2.1\times {10}^{8}$  $4.0\times {10}^{7}$  $3.0\times {10}^{7}$  0.748  0.191  0.141 
T40I10D100K:0.0001  $6.0\times {10}^{9}$  $1.2\times {10}^{9}$  $9.4\times {10}^{8}$  0.703  0.200  0.159 
accidents:0.1  $1.2\times {10}^{6}$  125,440  117,714  0.083  0.108  0.101 
accidents:0.05  $5.9\times {10}^{6}$  672,553  634,169  0.079  0.115  0.108 
accidents:0.01  $1.5\times {10}^{8}$  $2.0\times {10}^{7}$  $1.9\times {10}^{7}$  0.089  0.135  0.128 
chess:0.1  $3.7\times {10}^{7}$  $4.6\times {10}^{6}$  $4.4\times {10}^{6}$  0.098  0.127  0.120 
chess:0.05  $1.0\times {10}^{8}$  $1.3\times {10}^{7}$  $1.2\times {10}^{7}$  0.098  0.131  0.124 
chess:0.01  $1.8\times {10}^{8}$  $2.5\times {10}^{7}$  $2.3\times {10}^{7}$  0.118  0.135  0.127 
connect:0.05  $1.1\times {10}^{7}$  $1.3\times {10}^{6}$  $1.2\times {10}^{6}$  0.206  0.122  0.112 
connect:0.01  $7.3\times {10}^{7}$  $9.7\times {10}^{6}$  $9.0\times {10}^{6}$  0.204  0.133  0.124 
connect:0.005  $1.3\times {10}^{8}$  $1.8\times {10}^{7}$  $1.6\times {10}^{7}$  0.202  0.133  0.124 
16adder_col  $4.9\times {10}^{7}$  $6.2\times {10}^{6}$  $5.9\times {10}^{6}$  0.124  0.127  0.122 
C1908  $4.3\times {10}^{6}$  487,434  470,422  0.027  0.114  0.110 
C3540  $4.6\times {10}^{7}$  $5.9\times {10}^{6}$  $5.6\times {10}^{6}$  0.152  0.128  0.122 
C499  $4.5\times {10}^{6}$  513,322  499,158  0.009  0.114  0.111 
C880  $1.9\times {10}^{7}$  $2.5\times {10}^{6}$  $2.3\times {10}^{6}$  0.305  0.129  0.120 
comp  $1.9\times {10}^{7}$  $2.4\times {10}^{6}$  $2.2\times {10}^{6}$  0.234  0.127  0.119 
my_adder  $2.8\times {10}^{7}$  $3.8\times {10}^{6}$  $3.5\times {10}^{6}$  0.399  0.133  0.122 
Data Set  Conversion Time (s)  Getnode Time (s)  

convert  const.  comp.  Z  DZ  DZ${}_{\mathit{dc}}$  
rect1x10000  0.007  0.009  0.008  0.001  0.001  0.005 
rect5x2000  0.006  0.015  0.011  0.000  0.001  0.006 
rect100x100  0.006  0.014  0.009  0.001  0.001  0.005 
rect2000x5  0.006  0.016  0.012  0.000  0.001  0.005 
rect10000x1  0.504  0.015  0.009  0.000  0.001  0.008 
randomjoin256  0.025  0.105  0.005  0.001  0.002  0.013 
randomjoin2048  0.254  0.263  0.001  0.036  0.037  0.189 
randomjoin8192  0.946  0.526  0.000  0.156  0.164  0.710 
randomjoin16384  1.463  0.692  0.010  0.235  0.278  1.123 
bddqueen13  0.175  0.087  0.003  0.009  0.017  0.159 
bddqueen14  0.926  0.415  0.019  0.059  0.074  0.692 
bddqueen15  6.217  2.438  0.142  0.426  0.402  3.498 
T40I10D100K:0.001  0.934  0.814  0.037  0.089  0.218  0.872 
T40I10D100K:0.0005  6.006  3.958  0.175  0.771  1.088  4.706 
T40I10D100K:0.0001  233.006  120.423  4.378  32.316  30.181  122.104 
accidents:0.1  0.026  0.040  0.023  0.002  0.005  0.033 
accidents:0.05  0.162  0.094  0.022  0.012  0.023  0.161 
accidents:0.01  5.901  1.949  0.075  0.785  0.657  4.568 
chess:0.1  1.149  0.455  0.016  0.142  0.145  1.130 
chess:0.05  3.319  1.263  0.085  0.471  0.414  2.895 
chess:0.01  5.829  2.408  0.098  0.847  0.729  4.662 
connect:0.05  0.289  0.136  0.002  0.023  0.037  0.227 
connect:0.01  2.287  0.945  0.033  0.297  0.268  1.625 
connect:0.005  4.377  1.716  0.080  0.579  0.491  2.996 
16adder_col  1.318  0.585  0.010  0.119  0.137  1.821 
C1908  0.085  0.070  0.016  0.006  0.011  0.147 
C3540  1.319  0.563  0.017  0.098  0.119  1.488 
C499  0.084  0.073  0.010  0.007  0.010  0.140 
C880  0.491  0.249  0.005  0.034  0.048  0.551 
comp  0.445  0.232  0.002  0.032  0.046  0.683 
my_adder  0.743  0.375  0.009  0.061  0.083  0.930 
Data Set  Traverse Time (s)  Search Time (s)  

Z  DZ  DZ${}_{\mathit{dc}}$  Z  DZ  DZ${}_{\mathit{dc}}$  
rect1x10000  0.000  0.002  0.002  4.563  0.012  0.014 
rect5x2000  0.000  0.002  0.002  2.082  0.014  0.015 
rect100x100  0.000  0.001  0.002  0.092  0.009  0.011 
rect2000x5  0.001  0.002  0.003  0.006  0.009  0.021 
rect10000x1  0.001  0.003  0.015  0.002  0.009  0.070 
randomjoin256  0.001  0.004  0.005  0.470  0.013  0.013 
randomjoin2048  0.021  0.057  0.065  3.772  0.014  0.015 
randomjoin8192  0.088  0.176  0.201  14.568  0.019  0.020 
randomjoin16384  0.144  0.269  0.306  25.244  0.016  0.016 
bddqueen13  0.013  0.054  0.237  0.014  0.005  0.007 
bddqueen14  0.068  0.259  0.998  0.015  0.005  0.006 
bddqueen15  0.420  1.421  4.778  0.016  0.005  0.006 
T40I10D100K:0.001  0.054  0.222  0.298  0.003  0.002  0.003 
T40I10D100K:0.0005  0.314  1.210  1.606  0.003  0.002  0.002 
T40I10D100K:0.0001  11.615  42.730  55.085  0.004  0.001  0.002 
accidents:0.1  0.002  0.007  0.028  0.003  0.000  0.000 
accidents:0.05  0.011  0.038  0.150  0.003  0.000  0.000 
accidents:0.01  0.369  1.165  4.507  0.003  0.000  0.000 
chess:0.1  0.075  0.251  1.000  0.003  0.000  0.000 
chess:0.05  0.218  0.707  2.640  0.003  0.000  0.000 
chess:0.01  0.394  1.276  3.911  0.003  0.000  0.000 
connect:0.05  0.022  0.069  0.169  0.003  0.000  0.000 
connect:0.01  0.169  0.492  1.219  0.003  0.000  0.000 
connect:0.005  0.316  0.906  2.250  0.003  0.000  0.000 
16adder_col  0.090  0.340  2.054  0.053  0.002  0.018 
C1908  0.007  0.030  0.174  0.169  0.221  1.851 
C3540  0.085  0.358  2.162  0.072  0.123  1.017 
C499  0.007  0.031  0.171  0.101  0.145  0.988 
C880  0.033  0.147  0.815  0.081  0.159  1.331 
comp  0.035  0.138  0.956  0.010  0.010  0.085 
my_adder  0.066  0.185  0.931  0.054  0.001  0.003 
Data Set  Count Time (sec)  Sample Time (sec)  

D  DZ  DZ${}_{\mathit{dc}}$  Z  DZ (naive)  DZ (bin)  DZ${}_{\mathit{dc}}$ (naive)  DZ${}_{\mathit{dc}}$ (bin)  
rect1x10000  0.002  0.002  0.003  5.375  4.813  0.014  5.527  0.014 
rect5x2000  0.001  0.002  0.003  9.125  5.126  0.063  4.825  0.062 
rect100x100  0.002  0.003  0.003  10.150  5.155  0.816  5.176  0.812 
rect2000x5  0.004  0.005  0.007  8.250  5.765  7.142  5.773  7.171 
rect10000x1  0.001  0.004  0.016  0.001  0.011  0.012  0.074  0.075 
randomjoin256  0.003  0.007  0.007  1.035  0.535  0.036  0.564  0.035 
randomjoin2048  0.067  0.091  0.102  7.650  4.254  0.048  4.056  0.048 
randomjoin8192  0.256  0.305  0.336  22.989  16.830  0.056  15.663  0.056 
randomjoin16384  0.393  0.455  0.508  31.561  24.036  0.058  23.357  0.059 
bddqueen13  0.029  0.077  0.265  0.037  0.026  0.026  0.026  0.026 
bddqueen14  0.149  0.380  1.146  0.042  0.029  0.031  0.030  0.031 
bddqueen15  0.876  2.226  5.664  0.047  0.033  0.035  0.034  0.035 
T40I10D100K:0.001  0.187  0.339  0.418  0.947  0.338  0.047  0.323  0.045 
T40I10D100K:0.0005  1.153  1.925  2.338  0.954  0.479  0.049  0.495  0.047 
T40I10D100K:0.0001  36.329  67.113  79.435  0.978  0.576  0.061  0.572  0.066 
accidents:0.1  0.006  0.011  0.033  0.077  0.077  0.036  0.075  0.033 
accidents:0.05  0.031  0.059  0.175  0.108  0.106  0.040  0.105  0.040 
accidents:0.01  0.957  2.066  5.474  0.169  0.165  0.050  0.164  0.048 
chess:0.1  0.208  0.413  1.181  0.062  0.061  0.050  0.061  0.050 
chess:0.05  0.591  1.188  3.158  0.067  0.065  0.056  0.066  0.057 
chess:0.01  1.061  2.115  4.817  0.073  0.067  0.073  0.068  0.071 
connect:0.05  0.059  0.108  0.212  0.088  0.085  0.066  0.084  0.064 
connect:0.01  0.435  0.828  1.575  0.111  0.105  0.076  0.105  0.075 
connect:0.005  0.804  1.557  2.925  0.118  0.109  0.078  0.111  0.077 
16adder_col  0.224  0.505  2.260  0.167  0.246  0.474  0.246  0.483 
C1908  0.016  0.045  0.191  0.259  0.658  1.386  0.662  1.398 
C3540  0.199  0.530  2.386  0.150  0.349  0.715  0.352  0.727 
C499  0.017  0.045  0.189  0.455  1.241  2.500  1.250  2.529 
C880  0.079  0.214  0.904  0.084  0.233  0.480  0.238  0.485 
comp  0.081  0.199  1.038  0.035  0.055  0.109  0.055  0.109 
my_adder  0.135  0.278  1.043  0.081  0.232  0.416  0.234  0.424 
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Denzumi, S.; Kawahara, J.; Tsuda, K.; Arimura, H.; Minato, S.i.; Sadakane, K. DenseZDD: A Compact and Fast Index for Families of Sets ^{†}. Algorithms 2018, 11, 128. https://doi.org/10.3390/a11080128
Denzumi S, Kawahara J, Tsuda K, Arimura H, Minato Si, Sadakane K. DenseZDD: A Compact and Fast Index for Families of Sets ^{†}. Algorithms. 2018; 11(8):128. https://doi.org/10.3390/a11080128
Chicago/Turabian StyleDenzumi, Shuhei, Jun Kawahara, Koji Tsuda, Hiroki Arimura, Shinichi Minato, and Kunihiko Sadakane. 2018. "DenseZDD: A Compact and Fast Index for Families of Sets ^{†}" Algorithms 11, no. 8: 128. https://doi.org/10.3390/a11080128