Context

Join Query : $Q(x_1,\dots ,x_n)=\underset{i=1}{\overset{k}{\bigwedge }}{R}_i(\overrightarrow{z_i})$

where $\overrightarrow{z_i}$ is a tuple over $X=\{x_1,\dots ,x_n\}$

Example: $Q(city,country,name,id)=People(id,name,city)\wedge Capitals(city,country)$

Direct Access

We want to access the $k$-th element of $Q(\mathbb{D})$ for a given order.

Make $Q(\mathbb{D})$ an array, sort it and then we have direct access?

Direct Access

We want to access the $k$-th element of $Q(\mathbb{D})$ for a given order.

Make $Q(\mathbb{D})$ an array, sort it and then we have direct access?

Direct Access

We want to access the $k$-th element of $Q(\mathbb{D})$ for a given order.

Make $Q(\mathbb{D})$ an array, sort it and then we have direct access?

We need another way to represent the data

Acyclic Queries

Central class of queries because of their tractability

$Q=R_1(x,y,z)\wedge R_2(x,z,u)\wedge R_3(x,y,t)\wedge R_4(y,t)\wedge R_5(y,v)$

Acyclicity and elimination order

An $\alpha$-leaf in a query $Q$ is a variable $x$ such that the neighbourhood $N(x)$ of $x$ is covered by an atom.

Acyclicity in Queries

In the previous example, 1, 2, 3, 4 is an $\alpha$-elimination order

Direct Access on acyclic queries

Algorithm schema

1. construct a join tree compatible with an $\alpha$-elimination order ;
2. load data into the join tree & annotate the tuples ;
3. top-down induction to answer $Q(\mathbb{D})[k]$.

Construct a join tree

$Q=R_1(x,y,z)\wedge R_2(x,z,u)\wedge R_3(x,y,t)\wedge R_4(y,t)\wedge R_5(y,v)$

The order used here is $(x,y,z,t,u,v)$.

Load data and annotate the tuples

Answer DA tasks

$Q(\mathbb{D})[3]$ must set $x\leftarrow 2,y\leftarrow 1,z\leftarrow 0$, then proceed down in the tree.

Recap

We want to access the $k$-th solution to a query for a given database

Definition

Signed Query: $Q(x_1,\dots ,x_n)=\underset{i=1}{\overset{k}{\bigwedge }}{R}_i(\overrightarrow{z_i})$ $\underset{j=1}{\overset{l}{\bigwedge }}\mathrm{\neg }{S}_i(\overrightarrow{z_j})$

positively encoding $\mathrm{\neg }S(\overrightarrow{z})$ on a domain $D$ requires $(D^{|\overrightarrow{z}|}-\mathrm{\#}R)$ tuples

$Q$ being tractable for every $\mathbb{D}$ does not imply that any $Q^{\prime }\subseteq Q$ is tractable.

Example

Let $Q$ be any query and consider $Q^{\prime }=Q(x_1,\dots ,x_n)\wedge \mathrm{\neg }R(x_1,\dots ,x_n)$.

For any database $\mathbb{D}$ such that $R^{\mathbb{D}}=\mathrm{\varnothing }$, we have $Q(\mathbb{D})=Q^{\prime }(\mathbb{D})$

$⟹$ if $Q^{\prime }$ is tractable, $Q$ is tractable

But $Q^{\prime }$ is $\alpha$-acyclic and $Q$ is not restricted

$\alpha$-acyclic does not suffice as it is not monotone

$\beta$-acyclicity

Good candidate for another measure of tractability: every $Q^{\prime }\subseteq Q$ is $\alpha$-acyclic.

This is known as $\beta$-acyclicity

Intuition: a $\beta$-elimination order is an order that is an $\alpha$-elimination order for every subquery

$\beta$-acyclicity

A $\beta$-leaf is a variable $x$ such that all the atoms that include $x$ are contained in one another.

Characterisation: A query $Q$ is $\beta$-acyclic iff one can obtain $\mathrm{\varnothing }$ by successively removing $\beta$-leaves in $Q$. This induces an order on $V$ called an $\beta$-elimination order.

Recap

We want to access the $k$-th solution to a query for a given database

Relational Circuits

Relational Circuits

Ordered Relational Circuits

Ordered Relational Circuits

Preprocessing

Idea : for each gate $v$ over $x_i$ and for each domain value $d$

compute the size of the relation where $x_i$ is set to a value $d^{\prime }⩽d$

Preprocessing

Direct Access

Compute the 7^th solution $\to$ 111

Direct Access

Compute the 13^th solution $\to$ 221

From CQ to circuit

$Q$ a CQ and $x_1\prec \cdots \prec x_n$ an order over the variable set

From CQ to circuit

$Q$ a CQ and $x_1\prec \cdots \prec x_n$ an order over the variable set

recursive implementation + cache $⟹$ ordered relational circuit computing $Q(\mathbb{D})$

Compiling Signed Queries

Recap

Recap

Algorithm schema:

Generalisation

This technique generalises to:

1. conjunctive (with $\mathrm{\exists }$ quantifiers) signed queries:
   • project $\mathrm{\exists }$ directly on the circuit
   • as long as the projection is on a suffix
2. Non-acyclic signed conjunctive queries:
   • we can associate a notion of width to elimination orders
   • positive case $\to$ fractional hypertree width
   • corresponds to the width of the worst subquery in the negative case

Next steps

$Q(p,c,g,\mathsf{count}\mathsf{(}\mathsf{)})=\mathsf{Teams}(p,c)\wedge \mathsf{Games}(g,c,\cdot )\wedge \mathsf{Tries}(g,p)$

How should we integrate the aggregation in the lexicographical order?

How does the aggregation fit in to the compiled circuits?

recent works (Zhao, Fan, Ouyang, Koutris, 2023)