JP2021174061A - Zsdd construction apparatus, zsdd construction method, and program - Google Patents

Abstract

To efficiently construct a ZSDD representing a family of edge collections that satisfies an order constraint on nodes of a graph.SOLUTION: A ZSDD construction apparatus according to one embodiment includes an input unit that inputs a graph G configured by a node collection V and an edge collection E, and an order constraint relating to the number of edges with the node as an end point, and a construction unit that constructs a ZSDD representing a family which is a subset of the collection E and is configured by a subset of the edges that satisfy an order constraint based on the graph G and a vtree corresponding the order constraint and the graph G.SELECTED DRAWING: Figure 7

Description

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

Given a set S, a set whose elements are subsets of S is called a family of sets. Various problems can be solved by reducing them to computational problems for the family of sets. For example, when a graph G = (V, E) is given in which the set of nodes is V and the set of edges consisting of pairs of nodes is E, the matching of the graph G is the other subset of E. A subset of edges that do not share a node with an edge. Therefore, the matching set can be represented by a family of sets with E as a support set. Of the matching, the problem of selecting the matching with the largest number of edges is called the maximum matching problem, and many applied problems are solved as a result of the maximum matching problem. The maximum matching problem can be solved as a problem of selecting the set having the largest number of elements from the set family if all the matching sets of the graph G are expressed as a set family.

By the way, in many cases, the calculation for the family of sets takes time proportional to the size of the family of sets. Therefore, when the size of the family of sets becomes enormous, it takes a lot of time to calculate it, and efficient processing becomes difficult. For example, since the number of matchings described above may be an exponential multiple of the total number of edges | E | in the worst case, it is inefficient to explicitly obtain a set of matchings and process it. ..

In view of these points, there is an approach to execute various operations on the family of sets at high speed by converting the family of sets into a data structure suitable for calculation and holding it. As one of such approaches, there is a method of expressing a family of sets using ZSDD (Zero-suppressed Sentential Decision Diagram) (Non-Patent Document 1). ZSDD is a data structure that expresses a family of sets as a directed acyclic graph (DAG) and can perform various operations on the family of sets 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 the family of sets. On the other hand, (1) a set family representing a set of matching graphs (that is, a set whose elements are a subset of edges that do not share a node with other edges. A method is known for efficiently constructing a ZSDD that represents one of a set family consisting of a subset of K elements among a subset of a set (family) and (2) a set (patent). Document 1).

JP-A-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 described above, ZSDD representing a family of sets other than the two types of family of sets (1) or (2) described above could not be constructed. On the other hand, there are many computational problems for the family of sets of edges that satisfy the degree constraint for the nodes of the graph. Therefore, there is a demand for a method for efficiently constructing a ZSDD that expresses such a family of sets.

One embodiment of the present invention has been made in view of the above points, and an object of the present invention is to efficiently construct a ZSDD representing a family of sets of edges satisfying a degree constraint for a node of a graph.

In order to achieve the above object, the ZSDD construction apparatus according to the embodiment includes a graph G composed of a set V of nodes and a set E of edges, and a degree constraint representing a constraint on the number of edges with the node as an end point. Based on the input unit for inputting, the graph G, the order constraint, and the vtree corresponding to the graph G, the set is a subset of the set E and is composed of a subset of edges satisfying the order constraint. It is characterized by having a construction unit for constructing a ZSDD representing a family of sets to be created.

It is possible to efficiently construct a ZSDD that represents a family of sets of edges that satisfy the order constraint for the nodes of the graph.

It is a figure which shows an example of vtree. It is a figure which shows an example of ZSDD. It is a figure which shows an example of the expression on the computer of ZSDD. It is a figure which shows an example of the functional structure of the ZSDD construction apparatus which concerns on this embodiment. It is a figure which shows an example of the hardware composition of the ZSDD construction apparatus which concerns on this embodiment. It is a flowchart which shows an example of the process executed by the ZSDD construction apparatus which concerns on this embodiment. It is a flowchart which shows an example of the ZSDD construction process which concerns on this Embodiment. It is a flowchart which shows an example of the process of construct (v, Z) which concerns on this embodiment. It is a flowchart which shows an example of the processing of terminal (v, m) which concerns on this embodiment. It is a flowchart which shows an example of the process of decomp (v, z) which concerns on this embodiment.

Hereinafter, an embodiment of the present invention will be described. In the present embodiment, the ZSDD construction device 10 capable of efficiently constructing a ZSDD representing a family of sets of edges satisfying the order constraint for the nodes of the graph will be described.

<Preparation>
First, symbols, concepts, terms, and the like used in the present embodiment will be described.

Let V be a set of nodes, E be a set of edges, and G = (V, E) be a graph. The edge e ∈ E is expressed as e = (u, v) by the pair of nodes u, v ∈ V. Further, the number of edges including v as an end point with respect to the vertex v ∈ V of the graph is called the order of v.

The family of sets is a set whose elements are subsets S ⊆ Z for a certain support set Z = {A, B, C, ...}. For example, if Z = {A, B, C, D}, then {{A, B}, {B}, {B, C}, {C, D}} is a family of sets with Z as a support. One). Further, φ is an empty set, 2 ^Z is a power set of Z (that is, a family of sets composed of all subsets of the platform set Z), and | Z | is the number of elements of Z. In the following, the elements of the support set (for example, A, B, C and D when Z = {A, B, C, D}) are referred to as items.

Define some operations on the family of sets. Family of f and g. At this time, a binary operation on the family of sets

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 described. ZSDD divides the family of sets by recursive (X, Y) -division, and the state of the division is expressed as DAG. Assuming that the sets X and Y satisfy the division of the support set Z, that is, X∪Y = Z and X∩Y = φ, the (X, Y) -division of the family of sets f (Z) with Z as the support 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, _pi (X) is a family of sets with X as a support set, and s _i (Y) is a family of sets with Y as a support set.

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

を満たすものとする。

Satisfy.

For example, when Z = {A, B, C, D}, X = {A, B}, Y = {C, D}, the family of sets {{A, B}, {B}, {B, C }, {C, D}} (X, Y) -division 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。

Will be. Hereinafter, (X, Y) -division is also referred to as {(p ₁ , s ₁ ), ..., (P _n , s _n )}. In addition, the prime _{p i,} a _{s i} is referred to as a sub.

(X, Y) -Division can be done recursively for the family of sets. _{That is, for example, p 1} = {{A, B}} in the above (X, Y) -division is further {({{B}}, {{A}}) by (B, A) -division. It can be divided into ({φ}, φ)}.

Then, vtree is introduced. vtree is a complete binary tree whose leaves are the nodes corresponding to each item included in the support set. An example of vtree is shown in FIG. FIG. 1 shows an example of 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. A node that is not a leaf node is also called an intermediate node.

The intermediate node v of the volt represents the division of a set of variables consisting of the leaf nodes of the subtree rooted at the intermediate node v. That is, the set of items corresponding to the subtree leaf node rooted at the intermediate node v, the set of items corresponding to the subtree leaf node rooted at the child node on the left side of v, and the child node on the right side of v as the root. Divide into a set of items corresponding to the Konoha node. The child node on the left side of the node v is also expressed ^{as v k} , and the child node on the right side ^{is also expressed 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 child node v 1} on the left side of the _{root node v 3} corresponds to {B} and {A}. } Corresponds to the division into. _{Similarly, the child node v 5} on the left side of the root node v ₃ corresponds to the division into {D} and {C}. Hereinafter, among the nodes of vtree, the node whose left child node is a leaf node is referred to as a determined vtree node. A node that is not a determined vtee node is called a split vtree node. In the example shown in FIG. 1, the node _{v 1} and _{v 5} are determined vtree node.

ZSDD can be expressed as a DAG by recursively dividing the family of sets based on a given vtree. Assuming that α is ZSDD and the family of sets represented by α is <α>, 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, the ZSDD corresponding to a certain (X, Y) -division is called a decomposed ZSDD.

As an example, ZSDD representing the family of sets {{A, B}, {B}, {B, C}, {C, D}} is shown in FIG. The nodes represented by circles in FIG. 2 represent the decomposed ZSDD represented by the DAG rooted at that node. Such a node is called a decision ZSDD node. The number in the circle representing the decision ZSDD node represents the node corresponding to this decision ZSDD node among the nodes of vtree. For example, in ZSDD shown in FIG. 2, determined ZSDD node number "3" is written in the circle represents that correspond to the node _{v 3} of vtree shown in FIG. Similarly, decision ZSDD node number "1" is written in the circle, indicates that correspond to the node v ₁ of vtree shown in FIG. 1, the number "5" is written in the circle determining ZSDD node represents that correspond to the node _{v 5} of vtree shown in FIG. In addition, ZSDD may have a plurality of determination ZSDD nodes in which the same number is described in a circle.

Further, a pair of nodes having a determined ZSDD node as a parent node is called an element node pair, and is represented by a square pair in FIG. This element node pair represents a prime and sub pair ( _pi , s _i ) in the (X, Y) -division {(p ₁ , s ₁ ), ..., (P _n , s _n)}. ing. Of the element node pairs, the square on the left represents the prime and the square on the right represents the sub. Each of the prime and sub represents "termination ZSDD" or "pointer to another decomposition ZSDD", and when representing termination ZSDD, a symbol representing the corresponding termination ZSDD is described in the square, and to another decomposition ZSDD. When representing a pointer to, an arrow to the corresponding decision ZSDD node in another split ZSDD is provided. This represents a (X, Y) -division of the decision ZSDD node and its 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 and sub pair. 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 is a decision ZSDD node that corresponds to (that is, if there is a decision ZSDD node that has a group of element nodes representing any of the above three as child nodes), the first two are in α and the remaining 1 Replace one with ⊥.

Here, ZSDD can be realized by using an array on a computer. For example, FIG. 3 shows a case where the ZSDD shown in FIG. 2 is realized on a computer. In FIG. 3, _{the decision ZSDD node corresponding to the node v 3} of the vtree is realized by 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. Is realized by an array of 1 element each. At this time, each element of the array stores an element node pair that is a child node of the determination ZSDD node corresponding to the array. In other words, one element stores the prime and sub pairs represented by one element node pair. At this time, if the prime or sub represents the terminal ZSDD, the symbol (X, ⊥, ε, etc.) representing the terminal ZSDD is stored, otherwise (that is, if it represents a pointer to another decomposed ZSDD), the other Decision Holds the computer address to the ZSDD node. In the example shown in FIG. 3, it is represented by a pair of an ID indicating a node of vtree and an index of a determination ZSDD node corresponding to this ID. For example, in FIG. 3, "1: 1", among the decision ZSDD node corresponding to node _{v 1} of Vtree, representing the address of the array for implementing the decision ZSDD node of the index "1". Similarly, for example, "5: 1", among the decision ZSDD node corresponding to a node _{v 5} of Vtree, representing the address of the array for implementing the decision ZSDD node of the index "1". The reason why the index of the decision ZSDD node is necessary is that, as described above, a plurality of decision ZSDD nodes may correspond to one node of vtree.

<Family of sets of edges that satisfy order constraints>
Given a set of edges S ⊆ E, we define the number of edges e ∈ S including the node v ∈ V as an endpoint as deg (S, v). In this embodiment, a degree constraint is expressed by assigning an integer of 0 or more to each vertex v ∈ V. Assuming that the integer assigned to the vertex v is δ ^* (v), the set S⊆E of edges that satisfies the degree constraint is deg (S, v) = δ ^* (v) for all v ∈ V. It is S that satisfies. Hereinafter, the ZSDD construction device 10 for constructing the ZSDD representing the family of sets representing such a set of S will be described.

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

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

The input unit 101 accepts the graph G = (V, E) and the order constraint δ ^* (v) as inputs. The input unit 101 may input the graph G and the order constraint δ ^* (v) from an arbitrary input source set in advance. Examples of such an input source include a memory device 206 described later, a recording medium 203a described later, another device or system connected via a communication network, and the like.

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

The ZSDD construction unit 103 constructs a ZSDD that represents a family of sets of edges that satisfy the order constraint.

The output unit 104 outputs the ZSDD constructed by the ZSDD construction unit 103. The output unit 104 may output ZSDD to an arbitrary output destination set in advance. Examples of such an output destination include a memory device 206 described later, a recording medium 203a described later, another device or system connected via a communication network, and the like.

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

As shown in FIG. 5, the ZSDD construction device 10 according to the present embodiment is realized by a general computer or a computer system, and includes an input device 201, a display device 202, an external I / F 203, and a communication I / F 204. It has a processor 205 and a memory device 206. Each of these hardware is communicably connected via 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 or the like. The ZSDD construction device 10 does not have to have at least one of the input device 201 and the display device 202.

The 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 or write the recording medium 203a via the external I / F 203. The recording medium 203a may store one or more programs that realize each functional unit (input unit 101, vtree acquisition unit 102, ZSDD construction unit 103, and output unit 104) of the ZSDD construction device 10.

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.

The communication I / F 204 is an interface for connecting the ZSDD construction device 10 to the communication network. One or more programs that realize each functional unit of 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 units such as a CPU (Central Processing Unit) and a GPU (Graphics Processing Unit). Each functional unit included in the ZSDD construction device 10 is realized, for example, by a process in which one or more programs stored in the memory device 206 are executed by the processor 205.

The memory device 206 is, for example, various storage devices such as an 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 can realize various processes described later by having the hardware configuration shown in FIG. The hardware configuration shown in FIG. 5 is an example, and the ZSDD construction device 10 may have another hardware configuration. For example, the ZSDD construction device 10 may have a plurality of processors 205 or a plurality of memory devices 206.

<Details of processing>
Next, the process of constructing the ZSDD representing the family of sets of edges satisfying the order constraint by 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 processing executed by the ZSDD construction device 10 according to the present embodiment.

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

Next, the vtree acquisition unit 102 acquires the vtree of the graph G input in step S101 above (step S102). As described above, the vtree acquisition unit 102 may input the vtree from another device or system connected via the communication network, or may input the vtree from the recording medium 203a or the like. You may calculate vtee from G. When calculating vtree from graph G, for example, vtree may be calculated (constructed) from graph G by the method described in Patent Document 1 above.

Next, the ZSDD construction unit 103 uses the graph G and the order constraint δ ^* (v) input in step S101 above and the vtree acquired in step S102 above to set the edges satisfying the order constraint. ZSDD Z representing the family of sets of is constructed (step S103). The details of this step (ZSDD construction process) will be described later.

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

≪ZSDD construction process≫
Next, the details of the ZSDD construction process in step S103 will be described with reference to FIG. 7. FIG. 7 is a flowchart showing an example of the ZSDD construction process according to the present embodiment. The execution of the ZSDD construction process is started by calling a predetermined function having the root node v of the graph G and the voltre and the order constraint δ ^{* (v) as arguments in step S102.}

The ZSDD construction unit 103 accepts the root node v of the graph G and the voltre and the order constraint δ ^* (v) as inputs (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 the determination ZZDD node corresponding to the node v of the volt and its label. The label is identification information for checking whether or not an equivalent decision ZSDD node exists. The label differs depending on the type of ZSDD to be constructed, but in the case of ZSDD representing a family of sets of edges that satisfy the order constraint, an array containing an integer value of | V | dimension is used as the label (that is, the label is). , | V | Represented by a dimensional vector.) This array is represented by δ (), and the integer value corresponding to the node u of vtree (that is, the element corresponding to the node u among the elements of the array δ ()) is represented by δ (u). Further, rootState () is a function that returns a pair of the determination ZSDD node and the label δ ^{* after setting δ *} as the label of the determination ZSDD node that is the root of ZSDD. Note that δ ^* is an | V | dimensional array (vector) having all u ∈ V degree constraints δ ^* (u) as elements, or a function equivalent to the array. Further, the determination ZSDD node that is the root of ZSDD may be determined by, for example, the same process as getRoot () described in Patent Document 1 described above. The decision ZSDD that is the root of ZSDD is the node that corresponds to the root node of vtree.

Next, the ZSDD construction unit 103 calls construct (v, Z) (step S203). construct (v, Z) is a function that recursively constructs a ZSDD having a decision ZSDD node as a root node corresponding to a node v or less of the node v of vtree. 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 ZZDD determination nodes that make up Z. For details of reduce (Z), refer to, for example, Patent Document 1 above.

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

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

The ZSDD construction unit 103 repeatedly executes the processes of steps S311 to S314 for each of the determined ZSDD nodes 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 to S322 for each of ^{the label sets (m k} , m ^r ) obtained as the execution result of decomp (v, z). .. Here, decomp (v, z) is a function that returns a list ^{of label sets (m k} , m ^r ) of each of the prime and sub labels that are child nodes based on the label m (z) of the determined ZSDD node z. be. The details of decomp (v, z) will be described later.

In step S321, ZSDD construction unit 103 uses each of ^{m k} and ^{m r,} performs the process of step S331~ step S333. That, ZSDD construction unit 103, and executes the processing of step S331~ step S333 by using the ^{m k,} executes the processing of step S331~ step S333 using the ^{m r.} In the following, one of k or r expressed in 〇, m ^〇 denote the one of the m ^k or m ^r.

In step S331, ZSDD construction unit 103 of the node v of Vtree, select the node v ^〇 corresponding to m ^〇, is v ^〇 selected determines whether a leaf node of Vtree. node v ^〇 corresponding to m ^〇, if m ^〇 is ^{m k} is ^{v k,} if m ^〇 is ^{m r} a ^{v r.}

When it is determined in step S331 above that v ^〇 is a leaf node of vtree (YES in step S331), the ZSDD construction unit 103 is the node (termination) obtained as the execution result of ^{terminal (v 〇} , m ^〇). ZSDD) is ^{substituted for z 〇} (step S332). z ^〇, if m ^〇 is a ^{m k} is ^{z k,} if m ^〇 is a ^{m r} a ^{z r.} Further, terminal (v ^〇 , m ^〇 ) is a function for obtaining the terminal ZSDD corresponding to the ^{label m 〇} when the node v ^〇 is a leaf node of voltre. The details of terminal (v ^〇 , m ^〇 ) will be described later.

On the other hand, when ^{it is determined in step S331 above that v 〇} is not a leaf node of volt (NO in step S331), the ZSDD construction unit 103 is 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), the decision case where those with a label m ^● Among decision ZSDD node corresponding to the node v ^〇 already determined whether present in Z [v ^〇], there It is a function that outputs a ZSDD node, creates a ^{new decision ZSDD node with a label m 〇} if it does not exist ^{, stores it in Z [v 〇} ], and then outputs the decision ZSDD node.

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

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

In step S314, the ZSDD construction unit 103 executes the processes of steps S341 to S342 by using each of ^{v k} and v ^r. That is, the ZSDD construction unit 103 executes the processes of steps S341 to S342 using ^{v k} , and executes the processes of steps S341 to S342 using ^{v r.} In the following, as in the above, v ^〇 represents either v ^k or v ^r.

In step S341, the ZSDD construction unit 103 ^{determines whether or not v 〇} is an intermediate node of the volt (that is, a node other than the leaf node). Then, when it is determined in step S341 that v ^〇 is an intermediate node of volt (YES in step S341), the ZSDD construction unit 103 recursively ^{calls construct (v 〇} , Z) (step S342). As a result, ZSDD is recursively constructed.

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

The ZSDD construction unit 103 sets m for 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 vtree (that is, in this case, the array and the function are equivalent).

Next, the ZSDD construction unit 103 _{determines whether or not δ (u 1} ) = δ (u ₂ ) = 0 (step S402). Here, u ₁ and u ₂ are nodes at both ends of the edge e corresponding to the node v of the voltre.

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

On the other hand, when it is determined in step S402 above that δ (u ₁ ) = δ (u ₂ ) = 0 (NO in step S302), the ZSDD construction unit 103 has δ (u ₁ ) = δ (u _2). ) = 1 (step S404).

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

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

≪decomp (v, z) ≫
Next, the details of the processing when decomp (v, z) is called will be described with reference to FIG. FIG. 10 is a flowchart showing an example of the decomp (v, z) process according to the present 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). As described above, m (z) is the label of the determined ZSDD node z.

Next, ZSDD construction unit 103, enumBoundedArray (δ, v) for each of the [delta] ^k obtained as the execution result of repeats steps S511~ step S512 (step S503). The enumBoundedArray (δ, v) enumerates all the ^{labels δ k that can be taken by the node corresponding to the node v k of} vtree (that is, the child node on the left side of the node v) among the child nodes of the determined ZSDD node having the label δ. It is a function to do. Specifically, the u∈V ^{(v k)} ∩ For u as a ^{(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 as, 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 as, δ ^k (u) = 0, ^{and the labels δ k having δ k} (u) as an element are listed in order. Here, V (v) represents a set of all nodes that are the end points of the edges corresponding to the leaf nodes of the partial vtree (that is, the subtree of vtree) having 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 that is 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 a set of labels (δ ^k , δ ^r ) to label Pairs.

Then, in step S504, the ZSDD construction unit 103 outputs labelPairs to the caller. As a result, a list of label sets is output to the caller. <Summary>
As described above, the ZSDD construction device 10 according to the present embodiment can efficiently construct a ZSDD that represents a family of sets of edges that satisfy the order constraint. Therefore, by using the ZSDD construction device 10 according to the present embodiment, the ZSDD that can be used as an index for the family of sets operation can be efficiently obtained, and the bottleneck related to the ZSDD construction is eliminated. Further, since the ZSDD construction device 10 according to the present embodiment can construct a ZSDD representing a family of sets of edges satisfying the order constraint, it can contribute to the expansion of the applicable range of the processing using the ZSDD. can.

The present invention is not limited to the above-described embodiment disclosed in detail, and various modifications and modifications, combinations with known techniques, and the like are possible without departing from the scope of claims.

10 ZSDD construction unit 101 Input unit 102 vtree acquisition unit 103 ZSDD construction unit 104 Output unit 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 on the number of edges with the node as an end point.
Based on the graph G, the order constraint, and the vtree corresponding to the graph G, ZSDD represents a family of sets that is a subset of the set E and is composed of a subset of edges that satisfy the order constraint. With the construction department to build
ZSDD construction device characterized by having. The construction unit
As the root node of the ZSDD, a node corresponding to the root node of the vtree is set, and for the root node of the ZSDD, a vector having the degree constraint of each node of the graph as an element is set as a label.
After setting the root node of the ZSDD as a decision node, the ZSDD with respect to the decision node until the child node of the vtree node corresponding to the decision node among the vtrees representing the node of the vtree becomes a leaf node. The ZSDD construction device according to claim 1, wherein the ZSDD is constructed by repeatedly setting the child node of the ZSDD and setting the child node of the ZSDD as a determination node. The child node of the ZSDD includes a prime representing a child node on the left side of the decision node and a sub representing a child node on the right side of the decision node.
The construction unit
2. The second aspect of the present invention is to generate a child node of the ZSDD set for the determination node by using a list of a set of a label that can be taken by the prime and a label that can be taken by the sub. The ZSDD construction device described. The construction unit
When the child node of the vtree node corresponding to the decision node becomes a leaf node, the terminal note of the ZSDD is set as the child node of the ZSDD by 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, the decision node corresponding to the child node of the vtree node is used by using the child node of the vtree node and the label that the prime or sub can take. The ZSDD construction device according to claim 3, wherein a determination 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 on 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 family of sets that is a subset of the set E and is composed of a subset of edges that satisfy the order constraint. And the construction procedure to build
ZSDD construction method, characterized in that a computer executes. 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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