JP7377493B2 - ZSDD construction device, ZSDD construction method and program - Google Patents

Description

The present invention relates to a ZSDD construction device, a ZSDD construction method, and a program.

When a set S is given, a set whose elements are subsets of S is called a set family. Various problems can be solved by reducing them to computational problems targeting set families. For example, given a graph G = (V, E) where V is a set of nodes and E is a set of edges consisting of pairs of nodes, matching of graph G means A subset consisting of edges that do not share nodes with edges. Therefore, the matching set can be expressed as a set family with E as a platform set. Among matching, the problem of selecting the matching with the largest number of edges is called the maximum matching problem, and many applied problems have been solved by reducing the problem to the maximum matching problem. The maximum matching problem can be solved as a problem of selecting the set with the maximum number of elements from the set family if all matching sets of the graph G are expressed as a set family.

By the way, calculations targeting a set family often require time proportional to the size of the set family. For this reason, when the size of a set family becomes enormous, it takes a lot of time to calculate it, making efficient processing difficult. For example, in the worst case, the number of matchings mentioned above may be an exponential multiple of the total number of edges |E|, so it is inefficient to explicitly find a set of matchings and perform processing on it. .

In view of these points, there is an approach to perform various operations on set families at high speed by converting set families into data structures suitable for calculation and storing them. One such approach is a method of expressing a set family using a ZSDD (Zero-suppressed Sentential Decision Diagram) (Non-Patent Document 1). ZSDD is a data structure that expresses a set family as a directed acyclic graph (DAG) and allows various operations on the set family to be executed in a time proportional to the number of edges of ZSDD.

When performing calculations using ZSDD, it is important to efficiently construct ZSDD that represents a set family. On the other hand, (1) a set family representing a set of matchings in a graph (i.e., a set whose elements are a subset of edges that do not share nodes with other edges, out of a subset of a set of edges; There is a known method for efficiently constructing a ZSDD that represents either a set family (family) or (2) a set family consisting of a subset of K elements among subsets of the platform set (patent). Reference 1).

Japanese Patent Application Publication No. 2018-041161

Masaaki Nishino, Norihito Yasuda, Shin-ichi Minato and Masaaki Nagata, "Zero-suppressed Sentential Decision Diagrams", In Proceedings of the 30th AAAI Conference on Artificial Intelligence, pp 1058-1066, 2016.

However, in Patent Document 1 mentioned above, it was not possible to construct a ZSDD expressing a set family other than the two types of set groups (1) or (2) above. On the other hand, there are many computational problems that target families of sets of edges that satisfy degree constraints on nodes in a graph. Therefore, there is a need for a method for efficiently constructing a ZSDD that represents such a set family.

One embodiment of the present invention has been made in view of the above points, and aims to efficiently construct a ZSDD that expresses a set family of sets of edges that satisfy degree constraints on nodes of a graph.

In order to achieve the above object, a ZSDD construction device according to an embodiment includes a graph G composed of a set V of nodes and a set E of edges, and a degree constraint expressing a constraint regarding the number of edges with the nodes as end points. an input unit for inputting , a subset of the set E, and a subset of edges that satisfy the degree constraint based on the graph G, the degree constraint, and the vtree corresponding to the graph G. and a construction unit that constructs a ZSDD representing a set family.

It is possible to efficiently construct a ZSDD that represents a set family of sets of edges that satisfy degree constraints on the nodes of a graph.

FIG. 2 is a diagram showing an example of a vtree. It is a figure showing an example of ZSDD. It is a figure which shows an example of the representation on the computer of ZSDD. FIG. 1 is a diagram showing an example of a functional configuration of a ZSDD construction device according to the present embodiment. It is a diagram showing an example of the hardware configuration of the ZSDD construction device according to the present embodiment. It is a flowchart which shows an example of the process performed by the ZSDD construction apparatus based on this embodiment. It is a flowchart which shows an example of ZSDD construction processing concerning this embodiment. 12 is a flowchart illustrating an example of construct(v, Z) processing according to the present embodiment. 12 is a flowchart illustrating an example of terminal (v, m) processing according to the present embodiment. 12 is a flowchart illustrating an example of decomp(v, z) processing according to the present embodiment.

An embodiment of the present invention will be described below. In this embodiment, a ZSDD construction device 10 that can efficiently construct a ZSDD that expresses a set family of sets of edges that satisfy degree constraints on nodes of a graph will be described.

<Preparation>
First, symbols, concepts, terms, etc. used in this embodiment will be explained.

Let G=(V, E) be a graph, where V is a set of nodes and E is a set of edges. An edge e∈E is represented by a pair of nodes u, v∈V as e=(u,v). Further, for a vertex v∈V of a graph, the number of edges that include v as an end point is called the degree of v.

A set family is a set whose elements are a subset S⊆Z of a certain platform set Z={A, B, C, . . . }. For example, if Z = {A, B, C, D}, then {{A, B}, {B}, {B, C}, {C, D}} is the set family (of 1). Further, φ is an empty set, ^2Z is a power set of Z (that is, a set family consisting of all subsets of the platform set Z), and |Z| represents the number of elements of Z. Note that, hereinafter, the elements of the platform set (for example, A, B, C, and D when Z={A, B, C, D}) are referred to as items.

Define some operations on set families. Let f and g be a set family. In this case, the binary operation on the set family

をそれぞれ以下で定義する。

are defined below.

次に、ＺＳＤＤについて説明する。ＺＳＤＤは集合族を再帰的な（Ｘ，Ｙ）－分割によって分割し、その分割の様子をＤＡＧとして表現したものになる。集合Ｘ，Ｙが台集合Ｚの分割、すなわち、Ｘ∪Ｙ＝ＺかつＸ∩Ｙ＝φを満たすとすると、Ｚを台集合とする集合族ｆ（Ｚ）の（Ｘ，Ｙ）－分割は、

Next, ZSDD will be explained. ZSDD divides a set group by recursive (X, Y)-division, and the division is expressed as a DAG. If the sets X and Y are a partition of the platform set Z, that is, X∪Y=Z and X∩Y=φ, then the (X, Y)-division of the set family f(Z) with Z as the platform set is ,

と表される。ここで、各ｉ＝１，・・・，ｎに対して、ｐ_ｉ（Ｘ）はＸを台集合とする集合族、ｓ_ｉ（Ｙ）はＹを台集合とする集合族である。

It is expressed as Here, for each i=1, . . . , n, p _i (X) is a set family whose platform set is X, and s _i (Y) is a set family whose platform set is Y.

At this time, in (X, Y)-division, for all i, j (however, 1≦i<j≦n),

を満たすものとする。

The following shall be satisfied.

For example, if Z = {A, B, C, D}, X = {A, B}, Y = {C, D}, then the set family {{A, B}, {B}, {B, C The (X, Y)-division of }, {C, D}} is

となる。なお、以降では、（Ｘ，Ｙ）－分割を｛（ｐ_１，ｓ_１），・・・，（ｐ_ｎ，ｓ_ｎ）｝とも表す。また、ｐ_ｉをプライム、ｓ_ｉをサブと呼ぶ。

becomes. Note that hereinafter, (X, Y)-division is also expressed as {(p ₁ , s ₁ ), . . . , (p _n , s _n )}. Further, p _i is called a prime, and s _i is called a sub.

The (X,Y)-partition can be performed recursively on the set family. That is, for example, p ₁ = {{A, B}} in the above (X, Y)-division becomes {({{B}}, {{A}}), ({φ}, φ)}.

Next, install vtree. A vtree is a complete binary tree whose leaves are nodes corresponding to each item included in the table set. An example of a vtree is shown in FIG. FIG. 1 shows an example of a vtree for four items A, B, C, and D. In the vtree shown in FIG. 1, node v ₃ is a root node, nodes v ₀ , v ₂ , v ₄ and v ₆ are leaf nodes, and nodes v ₁ and v ₅ are nodes that are neither roots nor leaves. Note that nodes that are not leaf nodes are also referred to as intermediate nodes.

An intermediate node v of a vtree represents a division of a set of variables made up of leaf nodes of a subtree whose root is the intermediate node v. In other words, the set of items corresponding to the leaf nodes of the subtree whose root is the intermediate node v, the set of items corresponding to the leaf nodes of the subtree whose root is the left child node of v, and the set of items corresponding to the leaf nodes of the subtree whose root is the right child node of v. and a set of items corresponding to the leaf nodes of the subtree. Note that the left child node of node v is also expressed as v ^k and the right child node as v ^r .

For example, in the vtree shown in FIG. 1, the root node v ₃ corresponds to the division into {A, B} and {C, D}, and the left child node v ₁ of the root node v ₃ corresponds to the division into {B} and {A }. Similarly, the left child node v ₅ of the root node v ₃ corresponds to the division into {D} and {C}. Hereinafter, among the nodes of a vtree, a node whose left child node is a leaf node will be referred to as a determined vtree node. Further, a node that is not a determined vtee node is called a split vtree node. In the example shown in FIG. 1, nodes v ₁ and v ₅ are the decision vtree nodes.

ZSDD can be expressed as a DAG by recursively partitioning a set family based on a given vtree. Let α be ZSDD, and let the set family represented by α be <α>, then ZSDD is recursively defined as follows.

ここで、⊥及びεを定数ＺＳＤＤと呼び、Ｘ及び±ＸをリテラルＺＳＤＤと呼ぶ。また、定数ＺＳＤＤとリテラルＺＳＤＤとをあわせて終端ＺＳＤＤと呼ぶ。一方、ある（Ｘ，Ｙ）－分割に対応するＺＳＤＤは分解ＺＳＤＤと呼ぶ。

Here, ⊥ and ε are called constant ZSDD, and X and ±X are called literal ZSDD. Further, the constant ZSDD and the literal ZSDD are collectively referred to as a terminal ZSDD. On the other hand, a ZSDD corresponding to a certain (X, Y)-division is called a decomposition ZSDD.

As an example, FIG. 2 shows a ZSDD representing the set family {{A, B}, {B}, {B, C}, {C, D}}. A node represented by a circle in FIG. 2 represents a decomposition ZSDD expressed by a DAG whose root is that node. Such a node is called a decision ZSDD node. The number in the circle representing the determined ZSDD node represents the node corresponding to this determined ZSDD node among the nodes of the vtree. For example, in the ZSDD shown in FIG. 2, the determined ZSDD node with the number "3" written in a circle indicates that it corresponds to node _v3 of the vtree shown in FIG. Similarly, the decision ZSDD node with the number "1" written in the circle corresponds to node v1 of the vtree shown in Figure 1, and the number "5" written in the circle means that it corresponds to node _v1 of the vtree shown in Figure 1. The determined ZSDD node corresponds to node _v5 of the vtree shown in FIG. 1. Note that in the ZSDD, there may be a plurality of determined ZSDD nodes in which the same number is written in a circle.

Further, a pair of nodes having the determined ZSDD node as a parent node is called an element node pair, and is represented by a pair of squares in FIG. 2. This element node pair represents a certain prime-sub pair (p i , s _i ) in the (X, Y)-partition {(p ₁ , s ₁ ), ..., ( _{p n} , s _n )}. ing. Among the element node pairs, the square on the left side represents the prime, and the square on the right side represents the sub. Each of the prime and sub represents a "terminal ZSDD" or a "pointer to another decomposition ZSDD", and when representing a terminal ZSDD, a symbol representing the corresponding terminal ZSDD is written in a square, and a pointer to another decomposition ZSDD is indicated. When representing a pointer, an arrow pointing to the corresponding determined ZSDD node in another divided ZSDD is indicated. As a result, an (X, Y)-division is expressed, which is a determined ZSDD node and a set of element node pairs that are its child nodes. That is, each element node pair that is a child node of the decision ZSDD node corresponds to a prime-sub pair. Note that in ZSDD, pairs whose sub is ⊥ are omitted. Therefore, during ZSDD,

に対応するような決定ＺＳＤＤノードが存在する場合（つまり、上記の３つのいずれかを表す要素ノード群を子ノードとして持つ決定ＺＳＤＤノードが存在する場合）、最初の２つはαに、残り１つは⊥に置き換える。

If there exists a decision ZSDD node that corresponds to Replace one with ⊥.

Here, ZSDD can be realized on a computer using an array. For example, FIG. 3 shows a case in which the ZSDD shown in FIG. 2 is implemented on a computer. In FIG. 3, the decision ZSDD node corresponding to the node v ₃ of the vtree is realized as an array of 3 elements, and the decision ZSDD node corresponding to the node v ₁ of the vtree and the decision ZSDD node corresponding to the node v ₅ of the vtree are realized as an array with 3 elements. are each realized as an array with one element. At this time, each element of the array stores a pair of element nodes that are child nodes of the determined ZSDD node corresponding to the array. In other words, one element stores a prime-sub pair represented by one element node pair. At this time, if the prime or sub represents the terminal ZSDD, a symbol (X, ⊥, ε, etc.) representing the terminal ZSDD is stored, and if not (that is, if it represents a pointer to another decomposition ZSDD), another symbol is stored. Determination of ZSDD holds the address on the computer to the node. In the example shown in FIG. 3, it is represented by a pair of an ID indicating a vtree node and an index of the determined ZSDD node corresponding to this ID. For example, "1:1" in FIG. 3 represents the address of an array that implements the determined ZSDD node with index "1" among the determined ZSDD nodes corresponding to node v ₁ of the vtree. Similarly, for example, "5:1" represents the address of an array that implements the determined ZSDD node with index "1" among the determined ZSDD nodes corresponding to node _v5 of the vtree. Note that the reason why the index of the determined ZSDD node is necessary is that, as described above, a plurality of determined ZSDD nodes may correspond to one node of a vtree.

<Set family of sets of edges that satisfy degree constraints>
When a set of edges S⊆E is given, the number of edges e∈S that include node v∈V as an end point is defined as deg(S, v). In this embodiment, the degree constraint is expressed by assigning an integer greater than or equal to 0 to each vertex v∈V. If the integer assigned to vertex v is δ ^* (v), then the set of edges S⊆E that satisfies the degree constraint is deg(S, v) = δ ^* (v) for all v∈V It is an S that satisfies the following. Hereinafter, a ZSDD construction device 10 that constructs a ZSDD expressing a set family representing such a set of S will be described.

<Functional configuration>
The functional configuration of the ZSDD construction device 10 according to this embodiment will be described with reference to FIG. 4. FIG. 4 is a diagram showing an example of the functional configuration of the ZSDD construction device 10 according to this embodiment.

As shown in FIG. 4, the ZSDD construction device 10 according to this embodiment includes an input section 101, a vtree acquisition section 102, a ZSDD construction section 103, and an output section 104.

The input unit 101 receives the graph G=(V,E) and the degree constraint δ ^* (v) as input. Note that the input unit 101 may input the graph G and the degree constraint δ ^* (v) from any preset input source. Examples of such input sources include a memory device 206, which will be described later, a recording medium 203a, which will be described later, and other devices or systems connected via a communication network.

The vtree acquisition unit 102 acquires a vtree to be used when constructing the ZSDD. Note that the vtree acquisition unit 102 may input a vtree from another device or system connected via a communication network, may input a vtree from a recording medium 203a described later, or may input a vtree from a graph G. vtee may be calculated.

The ZSDD construction unit 103 constructs a ZSDD representing a set family of sets of edges that satisfy the degree constraint.

The output unit 104 outputs the ZSDD constructed by the ZSDD construction unit 103. Note that the output unit 104 may output the ZSDD to any preset output destination. Examples of such output destinations include a memory device 206 (described later), a recording medium 203a (described later), and other devices or systems connected via a communication network.

<Hardware configuration>
Next, the hardware configuration of the ZSDD construction device 10 according to this embodiment will be described with reference to FIG. 5. FIG. 5 is a diagram showing an example of the hardware configuration of the ZSDD construction device 10 according to this embodiment.

As shown in FIG. 5, the ZSDD construction device 10 according to this embodiment is realized by a general computer or computer system, and includes an input device 201, a display device 202, an external I/F 203, a communication I/F 204, It has a processor 205 and a memory device 206. Each of these pieces of hardware is communicably connected via a bus 207.

The input device 201 is, for example, a keyboard, a mouse, a touch panel, or the like. The display device 202 is, for example, a display. Note that the ZSDD construction device 10 does not need to have at least one of the input device 201 and the display device 202.

External I/F 203 is an interface with an external device. The external device includes a recording medium 203a and the like. The ZSDD construction device 10 can read from and write to the recording medium 203a via the external I/F 203. The recording medium 203a may store one or more programs that implement each functional unit (input unit 101, vtree acquisition unit 102, ZSDD construction unit 103, and output unit 104) included in the ZSDD construction device 10.

Note that the recording medium 203a includes, for example, a CD (Compact Disc), a DVD (Digital Versatile Disk), an SD memory card (Secure Digital memory card), a USB (Universal Serial Bus) memory card, and the like.

Communication I/F 204 is an interface for connecting ZSDD construction device 10 to a communication network. Note that one or more programs that implement each functional unit included in the ZSDD construction device 10 may be acquired (downloaded) from a predetermined server device or the like via the communication I/F 204.

The processor 205 is, for example, various arithmetic devices such as a CPU (Central Processing Unit) or a GPU (Graphics Processing Unit). Each functional unit included in the ZSDD construction device 10 is realized, for example, by processing that is executed by the processor 205 by one or more programs stored in the memory device 206.

The memory device 206 is, for example, various storage devices such as a HDD (Hard Disk Drive), an SSD (Solid State Drive), a RAM (Random Access Memory), a ROM (Read Only Memory), and a flash memory.

The ZSDD construction device 10 according to the present embodiment has the hardware configuration shown in FIG. 5, so that it can implement various processes described below. Note that the hardware configuration shown in FIG. 5 is an example, and the ZSDD construction device 10 may have other hardware configurations. For example, the ZSDD construction device 10 may have multiple processors 205 or multiple memory devices 206.

<Processing details>
Next, a process of constructing a ZSDD representing a set family of a set of edges satisfying a degree constraint using the ZSDD construction apparatus 10 according to the present embodiment will be described with reference to FIG. FIG. 6 is a flowchart showing an example of a process executed by the ZSDD construction device 10 according to the present embodiment.

The input unit 101 receives the graph G=(V,E) and the degree constraint δ ^* (v) as input (step S101).

Next, the vtree acquisition unit 102 acquires the vtree of the graph G input in step S101 above (step S102). Note that, as described above, the vtree acquisition unit 102 may input a vtree from another device or system connected via a communication network, may input a vtree from the recording medium 203a, etc. Vtee may be calculated from G. When calculating a vtree from the graph G, the vtree may be calculated (constructed) from the graph G using the method described in Patent Document 1 mentioned above, for example.

Next, the ZSDD construction unit 103 uses the graph G and degree constraint δ ^* (v) input in step S101 above, and the vtree acquired in step S102 above to create a set of edges that satisfy the degree constraint. A ZSDD Z representing the set family of is constructed (step S103). Note that details of this step (ZSDD construction processing) will be described later.

Then, the output unit 104 outputs the ZSDD Z constructed in step S103 above (step S104).

≪ZSDD construction process≫
Next, details of the ZSDD construction process in step S103 above will be explained with reference to FIG. FIG. 7 is a flowchart illustrating an example of ZSDD construction processing according to this embodiment. Note that the ZSDD construction process is started in step S102 above by calling a predetermined function that uses the graph G, the root node v of the vtree, and the degree constraint δ ^* (v) as arguments.

The ZSDD construction unit 103 receives the graph G, the root node v of the vtree, and the degree constraint δ ^* (v) as input (step S201).

Next, the ZSDD construction unit 103 stores the output value of rootState() in the array Z[v] (step S202). Here, Z[v] is an array that stores a set of a determined ZSDD node corresponding to node v of the vtree and its label. The label is identification information for checking whether an equivalent decision ZSDD node exists. Labels differ depending on the type of ZSDD being constructed, but in the case of a ZSDD that represents a set family of edges that satisfy the degree constraint, an array that stores integer values of the |V| dimension is used as the label (in other words, the label is , |V|-dimensional vector). This array is expressed as δ(), and the integer value corresponding to node u of the vtree (that is, the element corresponding to node u among the elements of array δ()) is expressed as δ(u). RootState() is a function that sets δ ^* as the label of the determined ZSDD node that is the root of the ZSDD, and then returns a pair of the determined ZSDD node and label δ ^* . Note that δ ^* is a |V|-dimensional array (vector) whose elements are the order constraints δ ^* (u) for all u∈V, or a function equivalent to the array. Further, the determined ZSDD node that becomes the root of the ZSDD may be determined by, for example, the same process as getRoot() described in Patent Document 1 mentioned above. The determined ZSDD node that becomes the root of the ZSDD is the node that corresponds to the root node of the vtree.

Next, the ZSDD construction unit 103 calls construct(v, Z) (step S203). construct(v, Z) is a function that recursively constructs a ZSDD whose root node is the determined ZSDD node corresponding to nodes below node v of vtree. Note that the details of construct(v, Z) will be described later.

Next, the ZSDD construction unit 103 updates Z to reduce (Z) (step S204). reduce(Z) is a function that reduces the size of ZSDD by deleting redundant ones among the ZSDD decision nodes that constitute Z. For details of reduce (Z), please refer to the above-mentioned Patent Document 1, for example.

Then, the ZSDD construction unit 103 outputs ZSDD Z to the caller (step S205).

≪construct(v, Z)≫
Next, details of the process when construct(v, Z) is called will be explained with reference to FIG. FIG. 8 is a flowchart illustrating an example of construct(v, Z) processing according to this embodiment.

The ZSDD construction unit 103 repeatedly executes the processes of steps S311 to S314 for each determined ZSDD node z stored in Z[v] (step S301). In step S311, the ZSDD construction unit 103 initializes the array elems to φ. In step S312, the ZSDD construction unit 103 repeatedly executes the processes of steps S321 and S322 for each label set (m ^k , m ^r ) obtained as a result of executing decomp(v, z). . Here, decomp(v, z) is a function that returns a list of label pairs (m ^k , m ^r ) for each prime and sub node, based on the label m(z) of the determined ZSDD node z. be. Note that details of decomp(v, z) will be described later.

In step S321, the ZSDD construction unit 103 executes the processes of steps S331 to S333 using each of m ^k and m ^r . That is, the ZSDD construction unit 103 uses m ^k to execute the processes of steps S331 to S333, and uses m ^r to execute the processes of steps S331 to S333. Hereinafter, either k or r will be represented by 0, and m ⁰ will represent either m ^k or m ^r .

In step S331, the ZSDD construction unit 103 selects the node v ⁰ corresponding to m ⁰ from among the nodes v of the vtree, and determines whether the selected v ⁰ is a leaf node of the vtree. The node v ^〇 corresponding to m ^〇 is v ^k when m ^〇 is m ^k , and is v ^r when m ^〇 is m ^r .

If it is determined in the above step S331 that v ^〇 is a leaf node of the vtree (YES in step S331) ^, the ZSDD construction unit ¹⁰³ creates a node (terminal ZSDD) is substituted into ^z〇 (step S332). z ^〇 is z ^k when m ^〇 is m ^k , and z ^r when m ^〇 is m ^r . Further, terminal(v ^〇 , m ^〇 ) is a function that calculates the terminal ZSDD corresponding to the label m ^〇 when the node v ^〇 is a leaf node of the vtree. Note that details of terminal(v ^〇 , m ^〇 ) will be described later.

On the other hand, if it is determined in the above step S331 that v ^〇 is not a leaf node of the vtree (NO in step S331), the ZSDD construction unit 103 uses the node obtained as the execution result of unique(v ^〇 , m ^〇 ) is assigned to ^z〇 (step S333). unique(v ^〇 , m ^〇 ) determines whether a decision ZSDD node with label m ^〇 corresponding to node v ^〇 already exists in Z[v ^〇 ], and if so, the decision This is a function that outputs a ZSDD node, and if it does not exist, creates a new decision ZSDD node with a label m ^〇 , stores it in Z[v ^〇 ], and then outputs the decision ZSDD node.

In step S322, the ZSDD construction unit 103 adds the pair of z ^k and z ^r obtained by repeating step S321 above to elems.

In step S313, the ZSDD construction unit 103 sets the elems obtained by repeating step S312 above as child nodes of the determined ZSDD node z.

In step S314, the ZSDD construction unit 103 executes the processes of steps S341 to S342 using each of v ^k and v ^r . That is, the ZSDD construction unit 103 uses v ^k to execute steps S341 to S342, and uses v ^r to execute steps S341 to S342. Note that hereinafter, similarly to the above, v ⁰ represents either v ^k or v ^r .

In step S341, the ZSDD construction unit 103 determines whether v ^〇 is an intermediate node of the vtree (that is, a node other than a leaf node). If it is determined in step S341 that v ⁰ is an intermediate node of the vtree (YES in step S341), the ZSDD construction unit 103 recursively calls construct(v ⁰ , Z) (step S342). As a result, ZSDD is recursively constructed.

≪terminal(v, m)≫
Next, details of the process when terminal(v, m) is called will be explained with reference to FIG. FIG. 9 is a flowchart illustrating an example of terminal (v, m) processing according to this embodiment.

The ZSDD construction unit 103 sets m to the array δ() (step S401). That is, the ZSDD construction unit 103 sets each element of the label m for each element of the array δ(). Although δ() is an array, it can be regarded as a function that returns an integer value for each node of the vtree (that is, in this case, the array and the function are equivalent).

Next, the ZSDD construction unit 103 determines whether δ(u ₁ )=δ(u ₂ )=0 (step S402). Here, u ₁ and u ₂ are nodes at both end points of edge e corresponding to node v of vtree.

If it is determined in step S402 that δ(u ₁ )=δ(u ₂ )=0 (YES in step S402), the ZSDD construction unit 103 outputs ε to the caller as the terminal ZSDD (step S403).

On the other hand, if it is determined in step S402 that δ(u ₁ )=δ(u ₂ )=0 (NO in step S302), the ZSDD construction unit 103 calculates δ(u ₁ )=δ(u ₂ )=1 (step S404).

If it is determined in step S404 that δ(u ₁ )=δ(u ₂ )=1, the ZSDD construction unit 103 outputs L(v) to the caller as the terminal ZSDD (step S405). L(v) represents edge e corresponding to vtree node v.

On the other hand, if it is not determined in step S404 that δ(u ₁ )=δ(u ₂ )=1, the ZSDD construction unit 103 outputs ⊥ to the caller as the terminal ZSDD (step S406). .

≪decomp(v, z)≫
Next, details of the process when decomp(v, z) is called will be explained with reference to FIG. FIG. 10 is a flowchart illustrating an example of decomp(v, z) processing according to this embodiment.

The ZSDD construction unit 103 initializes the array labelPairs to φ (step S501). Next, the ZSDD construction unit 103 sets m(z) for the array δ() (step S502). Note that, as described above, m(z) is the label of the determined ZSDD node z.

Next, the ZSDD construction unit 103 repeatedly executes the processes of steps S511 to S512 for each of δ ^k obtained as a result of executing enumBoundedArray(δ, v) (step S503). enumBoundedArray(δ, v) enumerates all labels δ ^k that can be taken by the node corresponding to node v ^k of vtree (that is, the left child node of node v) among the child nodes of the decision ZSDD node with label δ. This is a function that does Specifically, for u such that u∈V(v ^k )∩(v ^r ), 0≦δ ^k (u)≦δ(u),

であるようなｕに対してはδ^ｋ（ｕ）＝δ（ｕ）とし、

For u such that δ ^k (u)=δ(u),

であるようなｕに対してはδ^ｋ（ｕ）＝０として、δ^ｋ（ｕ）を要素とするラベルδ^ｋを順に列挙する。ここで、Ｖ（ｖ）は、ｖを根ノードとする部分ｖｔｒｅｅ（つまり、ｖｔｒｅｅの部分木）の葉ノードに対応するエッジの端点となる全てのノードの集合を表す。

For u such that δ ^k (u)=0, labels δ ^k having δ ^k (u) as an element are sequentially listed. Here, V(v) represents a set of all nodes that are end points of edges corresponding to leaf nodes of a partial vtree (that is, a partial tree of the vtree) with v as the root node.

In step S511, the ZSDD construction unit 103 sets the execution result of calcSubArray(δ, δ ^k , v) to δ ^r . calcSubArray(δ, δ ^k , v) is a function that calculates δ ^r (u)=δ(u)−δ ^k (u) for each u∈V. Therefore, δ ^r is a vector to be used as an element of each δ ^r (u) obtained as a result of executing calcSubArray(δ, δ ^k , v).

In step S512, the ZSDD construction unit 103 adds the label pair (δ ^k , δ ^r ) to labelPairs.

Then, in step S504, the ZSDD construction unit 103 outputs labelPairs to the caller. This will output a list of label pairs to the caller <Summary>
As described above, the ZSDD construction device 10 according to the present embodiment can efficiently construct a ZSDD that expresses a set family of sets of edges that satisfy a degree constraint. Therefore, by using the ZSDD construction device 10 according to the present embodiment, it is possible to efficiently obtain a ZSDD that can be used as an index for set family operations, thereby eliminating bottlenecks related to ZSDD construction. Furthermore, the ZSDD construction device 10 according to the present embodiment can construct a ZSDD that expresses a set family of edges that satisfy a degree constraint, and therefore can contribute to expanding the scope of application of processing using ZSDD. can.

The present invention is not limited to the above-described specifically disclosed embodiments, and various modifications and changes, combinations with known techniques, etc. are possible without departing from the scope of the claims.

10 ZSDD construction device 101 Input section 102 vtree acquisition section 103 ZSDD construction section 104 Output section 201 Input device 202 Display device 203 External I/F
203a Recording medium 204 Communication I/F
205 processor 206 memory device 207 bus

Claims (6)

an input unit for inputting a graph G composed of a set V of nodes and a set E of edges, and a degree constraint representing a constraint regarding the number of edges with the node as an end point;
Based on the graph G, the degree constraint, and the vtree corresponding to the graph G, ZSDD represents a set family that is a subset of the set E and is composed of a subset of edges that satisfy the degree constraint. a construction section that constructs the
A ZSDD construction device comprising: The construction section is
A node corresponding to the root node of the vtree is set as the root node of the ZSDD, and a vector whose elements are degree constraints of each node of the graph G is set as a label for the root node of the ZSDD,
After setting the root node of the ZSDD as a decision node, the ZSDD is applied to the decision node until a child node of the vtree node corresponding to the decision node among the vtrees representing the nodes of the vtree becomes a leaf node. 2. The ZSDD construction device according to claim 1, wherein the ZSDD is constructed by repeatedly setting a child node of the ZSDD and setting the child node of the ZSDD as a decision node. The child nodes of the ZSDD include a prime representing a left child node of the decision node and a sub representing a right child node of the decision node,
The construction section is
3. A child node of the ZSDD to be set for the decision node is generated using a list of pairs of labels that can be taken by the prime and labels that can be taken by the sub. The ZSDD construction device described. The construction section is
When the child node of the vtree node corresponding to the decision node becomes a leaf node, the terminal note of the ZSDD is changed to the child node of the ZSDD using the leaf node and the label that the prime or the sub can take. Set as,
If the child node of the vtree node corresponding to the decision node is not a leaf node, use the child node of the vtree node and the label that the prime or the sub can take to create the decision node corresponding to the child node of the vtree node. 4. The ZSDD construction device according to claim 3, wherein a decision node having a label that can be taken by the prime or the sub is set as a child node of the ZSDD. an input procedure for inputting a graph G composed of a set V of nodes and a set E of edges, and a degree constraint representing a constraint regarding the number of edges with the node as an end point;
Based on the graph G, the degree constraint, and the vtree corresponding to the graph G, ZSDD represents a set family that is a subset of the set E and is composed of a subset of edges that satisfy the degree constraint. The construction steps to build the
A ZSDD construction method characterized in that a computer executes the following. A program for causing a computer to function as the ZSDD construction device according to any one of claims 1 to 4.

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