SQL/PGQ and graph theory

The story goes almost two decades back. I was studying computer engineering in College of Engineering Pune. Prof Vinayak Joshi had recently joined COEP and had research interest in discrete mathematic, especially Lattices. He needed some software for exploring some patterns in graphs. Matlab, though almost three decades old by then, was new to us and quite expensive. He approached my scholar friend Amit Bose and requested him to write a program for exploring the desired graph patterns. If memory serves, Amit wrote the program in C and quite possibly used TurboC as an IDE. That was a remarkable feat given that graphs could consume a huge memory, C is a very basic language and TurboC was clumsy and clunky back then. May be Amit knew how to use gcc and linux which allowed huge memory models. (I am sure today's software engineers haven't heard about memory models.) In the due course we studied discrete mathematics, graph theory and also lattices.

Few years later I joined post-graduate program in Computer Science and Engineering at IIT Bombay. The institute employed eminent faculty in the area of theoretical computer science, including graph theory. I worked as a teaching assistant to Prof. Ajit Diwan and Prof. Abhiram Ranade. I also took a course run by Prof. Sundar Vishwanathan. That developed my interest in the graph theory. But I felt any work in graph theory required a lot of patience (more details to follow) and it was too difficult and intangible. Like many others in my class, I dared not take a research project in the subject.

Graph theory still fascinates me. Graphs are still my first go-to-tool to model and solve any problem in profession or life. If graph theory fails I use other tools. Fast forward to the present, when I saw opportunity to work with graphs in RDBMS, I immediately grabbed it. Follow pgsql-hackers thread for more about that work. I don't know whether Prof. Ajit had graph databases in his mind when he said to our class, "Do you think that I can not offer projects in databases? I can offer a lot of them". But that's quite true. In the age of AI and analytics, graph databases are once again a hot topic.

In Prof. Sundar prescribed "Introduction to graph theory (second edition)" by Douglas West for his course. the exercises in that book were like puzzles for me. I liked to work those out myself. Most of the times I didn't. One such problem was related to "king". If memory serves me, it was 1.4.38 on page 66 of that book, which I still have with me. I spent hours trying to prove the theorem but did not succeed. I went to Prof. Sundar for help. He patiently listened to all the things I had tried to solve that problem. He said I was very close to the solution and any hint from him would be as good as the solution itself. He suggested that I sit in his room and try again. After an hour of struggle, I left his room without any success. The problem still haunts me.

I don't know whether Matlab is still expensive and whether Prof. Joshi is still using programs to explore his graphs. But if he is, SQL/PGQ might come handy. Having it in PostgreSQL means they can use it for free. All the database capabilities allow them to store and retrieve the graphs they have tried in their research. Let's take a simple example, of a king.

In a digraph, a king is a vertex from which every vertex is reachable by a path of length at most 2. In other words, if a vertex v is a king, it is connected to every other vertex by a path of length at most two. Let's see how to do that with SQL/PGQ. Assume a table "vertexes" which contains all the vertexes of the graph and a table "edges" which contains all the edges in the graph connecting those vertices.

create table vertexes (id int primary key,

                        name varchar(10));

create table edges (id int primary key,

                    src int references vertexes(id),

                    dest int references vertexes(id),

                    name varchar(10));

create property graph tournament

    vertex tables (vertexes default label)

    edge tables (edges source key (src) references vertexes(id)

                        destination key(dest) references vertexes(id)

                        default label)

Let's build the query to find a king step by step. First step would be to find all the nodes reachable from a given node by a path of length at most 2.

select src_name, dest_name 

    from graph_table (tournament 

                        match (src is vertexes)->{1,2}(dest is vertexes)

                        where (src.id <> dest.id)

                        columns (src.name as src_name, dest.name as dest_name))

    order by src_name;;

I have discussed most of the constructs in my previous posts on DBaaG and its components. {1, 2} is a new construct being used here which indicates that the path between src and dest is of length maximum 2. Thus it lists all src nodes and respective dest nodes which are connected to their respective src node by a path of length at most 2. Also notice that we are eliminating the src node being reported as dest node to simplify the next step in the query.

Now we have to find node/s which is connected to all the nodes in such a way. To do that we simply count the distinct dest nodes an src node is reachable to. If this count is same as the number of vertexes in the graph but one, corresponding src node is the king.

select src_name, count(distinct dest_name) num_reachable_nodes 

    from graph_table (tournament

                match (src is vertexes)->{1,2}(dest is vertexes)

                where (src.id <> dest.id)

                columns (src.name as src_name, dest.name as dest_name))

    group by src_name

    having count(distinct dest_name) = (select count(*) - 1 from vertexes);

distinct in aggregate count makes sure to count each dest node only once. having clause filters every node which is not connected to all  the other nodes in the graph. If we populate vertex and edge tables as follows:

insert into vertexes values (1, 'a'), (2, 'b'), (3, 'c'), (4, 'd'), (5, 'e');

insert into edges values (1, 1, 2, 'a-b'), (2, 2, 3, 'b-c'), (3, 1, 3, 'a-c'), (4, 3, 4, 'c-d'), (5, 4, 5, 'd-e'), (6, 2, 1, 'b-a');

This will create a graph as shown in the figure below. Notice the cycle a->b->a. Node e is reachable from node a by a path of length 3. All other nodes are reachable from node a by a path of length at most 2.

Above query does not return any rows since there is no king in this graph.

Let's remove outlier node e and the edge connecting nodes d and e.

delete from edges where name = 'd-e';

delete from vertexes where name = 'e';

With that the above query returns two king nodes a and b.

src_name   num_reachable_nodes

---------- -------------------

a   | 3

b   | 3

PostgreSQL is loved by developers. Hope introduction of SQL/PGQ makes it popular among graph theory researchers as well.

Property graphs: elements, labels and properties

A property graph consists of three types of "things" in it: elements, labels and properties. 

Elements are nodes or edges in the graphs. They form the basic structure of a graph. An edge connects two nodes. Two nodes may be connected by multiple edges corresponding to different relationships between them.

Labels classify the elements. An element may belong to multiple classes and thus have multiple labels.

Properties are key-value pairs providing more information about an element. All the elements with the same label expose same set of keys or properties. A property of a given element may be exposed through multiple labels associated with that element.

Let's use the diagram here to understand the concepts better. There are three elements: N1, N2, the vertexes and an edge connecting them, labels L1 to L4, properties P1 to P7. The arrows connecting a label to an property indicates that that label exposes that property. E.g. label L3 exposes properties P2, P3, P5. Property P1 is exposed by both L1 and L2. An arrow between an element and a label indicates that that label is associated with that element. N1 has labels L1 and L2 whereas the edge has just one label L4. The properties that are associated with (and are exposed by) an element are decided by the labels associated with it. E.g. the properties P1, P2 and P4, which are union of properties associated with labels L1 and L2, are exposed by element N1. P1 has the same value v1 irrespective of which label is considered for this association. E.g height of a person will not change whether that person is classified as a teacher, businessman or a plumber. Similarly notice that the edge exposes properties P6 and P7 since it is labelled as L4.

SQL/PGQ's path pattern specification language allows to specify paths in terms of labels ultimately exposing the properties of individual paths that obey that patterns. E.g. (a IS L1 | L2)-[]->(b IS L3) COLUMNS (a.P3) will returns values of property P2 of all the nodes with labels L1 or L2. If you notice that N1 and N2 are the elements associated with either L1 or L2 or both. But N1 does not expose property P3. Hence we might expect that the above query would return an error. But instead the standard specified that it should report NULL, quite inline with the spirit of SQL NULL which means unknown.

The way I see it, a property can not exist without at least one label exposing it. A label can not exist without being associated with at least an element. But once defined, they have quite an independent existence.

PostgreSQL's memory allocations

There's a thread on hackers about recovering memory consumed by paths. A reference count is maintained in each path. Once paths are created for all the upper level relations that a given relation participates in, any unused paths, for which reference count is 0, are freed. This adds extra code and CPU cycles to traverse the paths, maintain reference counts and free the paths. Yet, the patch did not show any performance degradation. I was curious to know why. I ran a small experiment.

Experiment

I wrote an extension palloc_test which adds two SQL-callable functions palloc_pfree() and mem_context_free() written in C. Function definitions can be found here. The first function palloc's some memory and then pfree's it immediately. Other function just palloc's but never pfrees, assuming that the memory will be freed when the per-tuple memory context is freed. Both functions take the number of iterations and size of memory allocated in each iteration respectively as inputs. These functions return amount of time taken to execute the loop allocating memory. It appears that the first function spends CPU cycles to free memory and the second one doesn't. So the first one should be slower than the second one.

Results

The table below shows the amount of time reported by the respective functions to execute the loop as many times as the value in the first column, each iteration allocating 100 bytes. The figure shows the same as a plot. The time taken to finish the loop increases linearly for both the function indicating that the palloc logic is O(n) in terms of number of allocations. But the lines cross each other around 300K allocations.

 

count	palloc_pfree	memory context reset
100	0.0029	0.007124
100100	2.5646	5.079862
200100	5.1682	10.375552
300100	7.6373	15.704286
400100	10.1827	19.038238
500100	12.7013	23.847599
600100	15.2838	28.708501
700100	17.8255	36.982928
800100	20.3718	41.863638
900100	23.0706	44.332727
1000100	51.3311	52.546201
2000100	56.7407	104.747792
3000100	76.3961	154.225157
4000100	102.3415	206.510045
5000100	126.1954	256.367685
6000100	155.8812	314.178951
7000100	179.9267	367.597501
8000100	206.2112	420.003351
9000100	234.7584	474.137076

Inference and conclusion

This agrees with the observations I posted on the thread. Instead of letting all the useless path to be freed when query finishes, freeing them periodically during planning is time efficient as well as memory efficient. It compensates for the extra CPU cycles spent to maintain reference counts, traverse and free paths.

The actual memory allocation and freeing pattern as implemented in that patch is different from that in the experiment, so it might be worth repeating those experiments by simulating similar pattern.

I used chunk size of 100 since I thought it's closer to the order of average path size. But it might be worth repeating the experiment with larger chunk sizes to generalize the result.

DBaaG with SQL/PGQ

For those who have studied ERD-lore, it's not new that a relational database is very much like a graph. But it has taken SQL, more than 30 years since it became a standard and almost half a century since its inception to incorporate construct that will allow a DataBase to be treated as a Graph, DBaaG. This is surprising given that SQL was developed as language for relational databases which are modeled using ER diagrams. Better late than never. SQL/PGQ has arrived as 16th part of SQL:2023.

Entity Relationship Diagram, ERD in short, is a tool to model and visualize a database as entity types (which classify the things of interest) and relationships that can exist between them. Entity types and the relationships both map to relations in a Relational DataBase Management System (RDBMS in short). The rows in the relations represent entities (instances of entity types) and relationship between entities respectively. Fig. 1 below shows an ERD for a hypothetical shop.

This diagram very much looks like a graph with entity types represented as nodes and relationships represented by edges. That's exactly what SQL/PGQ is about. It adds language constructs to SQL to present underlying database as a "Property Graph". For example, property graph definition corresponding to the above ERD would look like

CREATE PROPERTY GRAPH shop

VERTEX TABLES (

    CreditCard label Payment,

    BankAccount label Payment,

    Person label Customer,

    Company label Customer,

    Trust label Customer,

    Wishlist label ProdLink,

    Order label ProdLink,

    Product)

EDGE TABLES (

    CCOwns label Owns

    BAHolds lable Owns,

    CustOrders label CustLink,

    CustWishlist label CustLink,

    CompanyOrders label CustLink,

    CompanyWishlist label CustLink,

    TrustOrders label CustLink,

    TrustWishlist label CustLink,

    OrderCCPayment label OrderPayment,

    OrderBAPayment label OrderPayment,

    OrderItems label ItemLink,

    WishlistItems label ItemLink);

 

Clever readers may have noticed that some of the entity types have some commonality. CreditCard and BankAccount are both Payment methods. Person, Company and Trust all can be considered as "Customers" by the shop. In a graph entities with commonalities will be represented by visual annotations like colors in Fig. 2. SQL/PGQ chooses to represents them by labels. Columns of the underlying tables are exposed through properties of labels. Traditionally the labels may be implemented as table inheritance or through tables abstracting commonalities. But it may not be necessary anymore.

Augmented with query constructs in SQL/PGQ they make it easy to write queries, especially analytical. Imagine a query to find all the products paid via credit card. There will be tons of JOIN and UNIONs over those joins. That's almost like "implementing" a logic in SQL. You would ask which parts do I JOIN before UNION, and which parts do I UNION before JOIN and so on. That's against the "imperative" spirit of SQL which should allow you to tell "what" you want and leave "how" for the DBMS to figure out. With SQL/PGQ you tell the DBaaG which paths in the graph to traverse. How to traverse them is the system's responsibility. So the SQL/PGQ query looks like below. Much simpler than joins and unions. In fact, it allows me not to mention edge tables at all in the query.

SELECT distinct name FROM

    GRAPH_TABLE (shop

                MATCHES

                   (o IS Orders)->(py IS Payment WHERE py.type = 'CC')<-(c IS Customer)->(o IS Order)->(p is Product)

                COLUMNS (p.name));

 

I must note that the query looks more like a mathematical equation than SQL which till now followed natural language syntax. But well, there it is.

What those () mean? What about various literals in it? How to specify properties? I am sure I have roused more questions than those answered here. I plan to write more about it in future. This blog has gone longer than I initially intended it to be, but I hope it has aroused your interest in SQL/PGQ nonetheless.

Oh! but before I end, please note that we are working on implementing SQL/PGQ in PostgreSQL. If you are interested and want to contribute, please follow and respond on pgsql-hackers thread.