datafusion_physical_expr/intervals/

cp_solver.rs

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// distributed with this work for additional information
// regarding copyright ownership.  The ASF licenses this file
// to you under the Apache License, Version 2.0 (the
// "License"); you may not use this file except in compliance
// with the License.  You may obtain a copy of the License at
//
//   http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing,
// software distributed under the License is distributed on an
// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, either express or implied.  See the License for the
// specific language governing permissions and limitations
// under the License.

//! Constraint propagator/solver for custom [`PhysicalExpr`] graphs.
//!
//! The constraint propagator/solver in DataFusion uses interval arithmetic to
//! perform mathematical operations on intervals, which represent a range of
//! possible values rather than a single point value. This allows for the
//! propagation of ranges through mathematical operations, and can be used to
//! compute bounds for a complicated expression. The key idea is that by
//! breaking down a complicated expression into simpler terms, and then
//! combining the bounds for those simpler terms, one can obtain bounds for the
//! overall expression.
//!
//! This way of using interval arithmetic to compute bounds for a complex
//! expression by combining the bounds for the constituent terms within the
//! original expression allows us to reason about the range of possible values
//! of the expression. This information later can be used in range pruning of
//! the provably unnecessary parts of `RecordBatch`es.
//!
//! # Example
//!
//! For example, consider a mathematical expression such as `x^2 + y = 4` \[1\].
//! Since this expression would be a binary tree in [`PhysicalExpr`] notation,
//! this type of an hierarchical computation is well-suited for a graph based
//! implementation. In such an implementation, an equation system `f(x) = 0` is
//! represented by a directed acyclic expression graph (DAEG).
//!
//! In order to use interval arithmetic to compute bounds for this expression,
//! one would first determine intervals that represent the possible values of
//! `x` and `y`` Let's say that the interval for `x` is `[1, 2]` and the interval
//! for `y` is `[-3, 1]`. In the chart below, you can see how the computation
//! takes place.
//!
//! # References
//!
//! 1. Kabak, Mehmet Ozan. Analog Circuit Start-Up Behavior Analysis: An Interval
//!    Arithmetic Based Approach, Chapter 4. Stanford University, 2015.
//! 2. Moore, Ramon E. Interval analysis. Vol. 4. Englewood Cliffs: Prentice-Hall, 1966.
//! 3. F. Messine, "Deterministic global optimization using interval constraint
//!    propagation techniques," RAIRO-Operations Research, vol. 38, no. 04,
//!    pp. 277-293, 2004.
//!
//! # Illustration
//!
//! ## Computing bounds for an expression using interval arithmetic
//!
//! ```text
//!             +-----+                         +-----+
//!        +----|  +  |----+               +----|  +  |----+
//!        |    |     |    |               |    |     |    |
//!        |    +-----+    |               |    +-----+    |
//!        |               |               |               |
//!    +-----+           +-----+       +-----+           +-----+
//!    |   2 |           |  y  |       |   2 | [1, 4]    |  y  |
//!    |[.]  |           |     |       |[.]  |           |     |
//!    +-----+           +-----+       +-----+           +-----+
//!       |                               |
//!       |                               |
//!     +---+                           +---+
//!     | x | [1, 2]                    | x | [1, 2]
//!     +---+                           +---+
//!
//!  (a) Bottom-up evaluation: Step 1 (b) Bottom up evaluation: Step 2
//!
//!                                      [1 - 3, 4 + 1] = [-2, 5]
//!             +-----+                         +-----+
//!        +----|  +  |----+               +----|  +  |----+
//!        |    |     |    |               |    |     |    |
//!        |    +-----+    |               |    +-----+    |
//!        |               |               |               |
//!    +-----+           +-----+       +-----+           +-----+
//!    |   2 |[1, 4]     |  y  |       |   2 |[1, 4]     |  y  |
//!    |[.]  |           |     |       |[.]  |           |     |
//!    +-----+           +-----+       +-----+           +-----+
//!       |              [-3, 1]          |              [-3, 1]
//!       |                               |
//!     +---+                           +---+
//!     | x | [1, 2]                    | x | [1, 2]
//!     +---+                           +---+
//!
//!  (c) Bottom-up evaluation: Step 3 (d) Bottom-up evaluation: Step 4
//! ```
//!
//! ## Top-down constraint propagation using inverse semantics
//!
//! ```text
//!    [-2, 5] ∩ [4, 4] = [4, 4]               [4, 4]
//!            +-----+                         +-----+
//!       +----|  +  |----+               +----|  +  |----+
//!       |    |     |    |               |    |     |    |
//!       |    +-----+    |               |    +-----+    |
//!       |               |               |               |
//!    +-----+           +-----+       +-----+           +-----+
//!    |   2 | [1, 4]    |  y  |       |   2 | [1, 4]    |  y  | [0, 1]*
//!    |[.]  |           |     |       |[.]  |           |     |
//!    +-----+           +-----+       +-----+           +-----+
//!      |              [-3, 1]          |
//!      |                               |
//!    +---+                           +---+
//!    | x | [1, 2]                    | x | [1, 2]
//!    +---+                           +---+
//!
//!  (a) Top-down propagation: Step 1 (b) Top-down propagation: Step 2
//!
//!                                     [1 - 3, 4 + 1] = [-2, 5]
//!            +-----+                         +-----+
//!       +----|  +  |----+               +----|  +  |----+
//!       |    |     |    |               |    |     |    |
//!       |    +-----+    |               |    +-----+    |
//!       |               |               |               |
//!    +-----+           +-----+       +-----+           +-----+
//!    |   2 |[3, 4]**   |  y  |       |   2 |[3, 4]     |  y  |
//!    |[.]  |           |     |       |[.]  |           |     |
//!    +-----+           +-----+       +-----+           +-----+
//!      |              [0, 1]           |              [-3, 1]
//!      |                               |
//!    +---+                           +---+
//!    | x | [1, 2]                    | x | [sqrt(3), 2]***
//!    +---+                           +---+
//!
//!  (c) Top-down propagation: Step 3  (d) Top-down propagation: Step 4
//!
//!    * [-3, 1] ∩ ([4, 4] - [1, 4]) = [0, 1]
//!    ** [1, 4] ∩ ([4, 4] - [0, 1]) = [3, 4]
//!    *** [1, 2] ∩ [sqrt(3), sqrt(4)] = [sqrt(3), 2]
//! ```

use std::collections::HashSet;
use std::fmt::{Display, Formatter};
use std::mem::{size_of, size_of_val};
use std::sync::Arc;

use super::utils::{
    convert_duration_type_to_interval, convert_interval_type_to_duration, get_inverse_op,
};
use crate::expressions::Literal;
use crate::utils::{build_dag, ExprTreeNode};
use crate::PhysicalExpr;

use arrow::datatypes::{DataType, Schema};
use datafusion_common::{internal_err, Result};
use datafusion_expr::interval_arithmetic::{apply_operator, satisfy_greater, Interval};
use datafusion_expr::Operator;

use petgraph::graph::NodeIndex;
use petgraph::stable_graph::{DefaultIx, StableGraph};
use petgraph::visit::{Bfs, Dfs, DfsPostOrder, EdgeRef};
use petgraph::Outgoing;

/// This object implements a directed acyclic expression graph (DAEG) that
/// is used to compute ranges for expressions through interval arithmetic.
#[derive(Clone, Debug)]
pub struct ExprIntervalGraph {
    graph: StableGraph<ExprIntervalGraphNode, usize>,
    root: NodeIndex,
}

/// This object encapsulates all possible constraint propagation results.
#[derive(PartialEq, Debug)]
pub enum PropagationResult {
    CannotPropagate,
    Infeasible,
    Success,
}

/// This is a node in the DAEG; it encapsulates a reference to the actual
/// [`PhysicalExpr`] as well as an interval containing expression bounds.
#[derive(Clone, Debug)]
pub struct ExprIntervalGraphNode {
    expr: Arc<dyn PhysicalExpr>,
    interval: Interval,
}

impl PartialEq for ExprIntervalGraphNode {
    fn eq(&self, other: &Self) -> bool {
        self.expr.eq(&other.expr)
    }
}

impl Display for ExprIntervalGraphNode {
    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.expr)
    }
}

impl ExprIntervalGraphNode {
    /// Constructs a new DAEG node with an `[-∞, ∞]` range.
    pub fn new_unbounded(expr: Arc<dyn PhysicalExpr>, dt: &DataType) -> Result<Self> {
        Interval::make_unbounded(dt)
            .map(|interval| ExprIntervalGraphNode { expr, interval })
    }

    /// Constructs a new DAEG node with the given range.
    pub fn new_with_interval(expr: Arc<dyn PhysicalExpr>, interval: Interval) -> Self {
        ExprIntervalGraphNode { expr, interval }
    }

    /// Get the interval object representing the range of the expression.
    pub fn interval(&self) -> &Interval {
        &self.interval
    }

    /// This function creates a DAEG node from DataFusion's [`ExprTreeNode`]
    /// object. Literals are created with definite, singleton intervals while
    /// any other expression starts with an indefinite interval (`[-∞, ∞]`).
    pub fn make_node(node: &ExprTreeNode<NodeIndex>, schema: &Schema) -> Result<Self> {
        let expr = Arc::clone(&node.expr);
        if let Some(literal) = expr.as_any().downcast_ref::<Literal>() {
            let value = literal.value();
            Interval::try_new(value.clone(), value.clone())
                .map(|interval| Self::new_with_interval(expr, interval))
        } else {
            expr.data_type(schema)
                .and_then(|dt| Self::new_unbounded(expr, &dt))
        }
    }
}

/// This function refines intervals `left_child` and `right_child` by applying
/// constraint propagation through `parent` via operation. The main idea is
/// that we can shrink ranges of variables x and y using parent interval p.
///
/// Assuming that x,y and p has ranges `[xL, xU]`, `[yL, yU]`, and `[pL, pU]`, we
/// apply the following operations:
/// - For plus operation, specifically, we would first do
///     - `[xL, xU]` <- (`[pL, pU]` - `[yL, yU]`) ∩ `[xL, xU]`, and then
///     - `[yL, yU]` <- (`[pL, pU]` - `[xL, xU]`) ∩ `[yL, yU]`.
/// - For minus operation, specifically, we would first do
///     - `[xL, xU]` <- (`[yL, yU]` + `[pL, pU]`) ∩ `[xL, xU]`, and then
///     - `[yL, yU]` <- (`[xL, xU]` - `[pL, pU]`) ∩ `[yL, yU]`.
/// - For multiplication operation, specifically, we would first do
///     - `[xL, xU]` <- (`[pL, pU]` / `[yL, yU]`) ∩ `[xL, xU]`, and then
///     - `[yL, yU]` <- (`[pL, pU]` / `[xL, xU]`) ∩ `[yL, yU]`.
/// - For division operation, specifically, we would first do
///     - `[xL, xU]` <- (`[yL, yU]` * `[pL, pU]`) ∩ `[xL, xU]`, and then
///     - `[yL, yU]` <- (`[xL, xU]` / `[pL, pU]`) ∩ `[yL, yU]`.
pub fn propagate_arithmetic(
    op: &Operator,
    parent: &Interval,
    left_child: &Interval,
    right_child: &Interval,
) -> Result<Option<(Interval, Interval)>> {
    let inverse_op = get_inverse_op(*op)?;
    match (left_child.data_type(), right_child.data_type()) {
        // If we have a child whose type is a time interval (i.e. DataType::Interval),
        // we need special handling since timestamp differencing results in a
        // Duration type.
        (DataType::Timestamp(..), DataType::Interval(_)) => {
            propagate_time_interval_at_right(
                left_child,
                right_child,
                parent,
                op,
                &inverse_op,
            )
        }
        (DataType::Interval(_), DataType::Timestamp(..)) => {
            propagate_time_interval_at_left(
                left_child,
                right_child,
                parent,
                op,
                &inverse_op,
            )
        }
        _ => {
            // First, propagate to the left:
            match apply_operator(&inverse_op, parent, right_child)?
                .intersect(left_child)?
            {
                // Left is feasible:
                Some(value) => Ok(
                    // Propagate to the right using the new left.
                    propagate_right(&value, parent, right_child, op, &inverse_op)?
                        .map(|right| (value, right)),
                ),
                // If the left child is infeasible, short-circuit.
                None => Ok(None),
            }
        }
    }
}

/// This function refines intervals `left_child` and `right_child` by applying
/// comparison propagation through `parent` via operation. The main idea is
/// that we can shrink ranges of variables x and y using parent interval p.
/// Two intervals can be ordered in 6 ways for a Gt `>` operator:
/// ```text
///                           (1): Infeasible, short-circuit
/// left:   |        ================                                               |
/// right:  |                           ========================                    |
///
///                             (2): Update both interval
/// left:   |              ======================                                   |
/// right:  |                             ======================                    |
///                                          |
///                                          V
/// left:   |                             =======                                   |
/// right:  |                             =======                                   |
///
///                             (3): Update left interval
/// left:   |                  ==============================                       |
/// right:  |                           ==========                                  |
///                                          |
///                                          V
/// left:   |                           =====================                       |
/// right:  |                           ==========                                  |
///
///                             (4): Update right interval
/// left:   |                           ==========                                  |
/// right:  |                   ===========================                         |
///                                          |
///                                          V
/// left:   |                           ==========                                  |
/// right   |                   ==================                                  |
///
///                                   (5): No change
/// left:   |                       ============================                    |
/// right:  |               ===================                                     |
///
///                                   (6): No change
/// left:   |                                    ====================               |
/// right:  |                ===============                                        |
///
///         -inf --------------------------------------------------------------- +inf
/// ```
pub fn propagate_comparison(
    op: &Operator,
    parent: &Interval,
    left_child: &Interval,
    right_child: &Interval,
) -> Result<Option<(Interval, Interval)>> {
    if parent == &Interval::CERTAINLY_TRUE {
        match op {
            Operator::Eq => left_child.intersect(right_child).map(|result| {
                result.map(|intersection| (intersection.clone(), intersection))
            }),
            Operator::Gt => satisfy_greater(left_child, right_child, true),
            Operator::GtEq => satisfy_greater(left_child, right_child, false),
            Operator::Lt => satisfy_greater(right_child, left_child, true)
                .map(|t| t.map(reverse_tuple)),
            Operator::LtEq => satisfy_greater(right_child, left_child, false)
                .map(|t| t.map(reverse_tuple)),
            _ => internal_err!(
                "The operator must be a comparison operator to propagate intervals"
            ),
        }
    } else if parent == &Interval::CERTAINLY_FALSE {
        match op {
            Operator::Eq => {
                // TODO: Propagation is not possible until we support interval sets.
                Ok(None)
            }
            Operator::Gt => satisfy_greater(right_child, left_child, false),
            Operator::GtEq => satisfy_greater(right_child, left_child, true),
            Operator::Lt => satisfy_greater(left_child, right_child, false)
                .map(|t| t.map(reverse_tuple)),
            Operator::LtEq => satisfy_greater(left_child, right_child, true)
                .map(|t| t.map(reverse_tuple)),
            _ => internal_err!(
                "The operator must be a comparison operator to propagate intervals"
            ),
        }
    } else {
        // Uncertainty cannot change any end-point of the intervals.
        Ok(None)
    }
}

impl ExprIntervalGraph {
    pub fn try_new(expr: Arc<dyn PhysicalExpr>, schema: &Schema) -> Result<Self> {
        // Build the full graph:
        let (root, graph) =
            build_dag(expr, &|node| ExprIntervalGraphNode::make_node(node, schema))?;
        Ok(Self { graph, root })
    }

    pub fn node_count(&self) -> usize {
        self.graph.node_count()
    }

    /// Estimate size of bytes including `Self`.
    pub fn size(&self) -> usize {
        let node_memory_usage = self.graph.node_count()
            * (size_of::<ExprIntervalGraphNode>() + size_of::<NodeIndex>());
        let edge_memory_usage =
            self.graph.edge_count() * (size_of::<usize>() + size_of::<NodeIndex>() * 2);

        size_of_val(self) + node_memory_usage + edge_memory_usage
    }

    // Sometimes, we do not want to calculate and/or propagate intervals all
    // way down to leaf expressions. For example, assume that we have a
    // `SymmetricHashJoin` which has a child with an output ordering like:
    //
    // ```text
    // PhysicalSortExpr {
    //     expr: BinaryExpr('a', +, 'b'),
    //     sort_option: ..
    // }
    // ```
    //
    // i.e. its output order comes from a clause like `ORDER BY a + b`. In such
    // a case, we must calculate the interval for the `BinaryExpr(a, +, b)`
    // instead of the columns inside this `BinaryExpr`, because this interval
    // decides whether we prune or not. Therefore, children `PhysicalExpr`s of
    // this `BinaryExpr` may be pruned for performance. The figure below
    // explains this example visually.
    //
    // Note that we just remove the nodes from the DAEG, do not make any change
    // to the plan itself.
    //
    // ```text
    //
    //                                  +-----+                                          +-----+
    //                                  | GT  |                                          | GT  |
    //                         +--------|     |-------+                         +--------|     |-------+
    //                         |        +-----+       |                         |        +-----+       |
    //                         |                      |                         |                      |
    //                      +-----+                   |                      +-----+                   |
    //                      |Cast |                   |                      |Cast |                   |
    //                      |     |                   |             --\      |     |                   |
    //                      +-----+                   |       ----------     +-----+                   |
    //                         |                      |             --/         |                      |
    //                         |                      |                         |                      |
    //                      +-----+                +-----+                   +-----+                +-----+
    //                   +--|Plus |--+          +--|Plus |--+                |Plus |             +--|Plus |--+
    //                   |  |     |  |          |  |     |  |                |     |             |  |     |  |
    //  Prune from here  |  +-----+  |          |  +-----+  |                +-----+             |  +-----+  |
    //  ------------------------------------    |           |                                    |           |
    //                   |           |          |           |                                    |           |
    //                +-----+     +-----+    +-----+     +-----+                              +-----+     +-----+
    //                | a   |     |  b  |    |  c  |     |  2  |                              |  c  |     |  2  |
    //                |     |     |     |    |     |     |     |                              |     |     |     |
    //                +-----+     +-----+    +-----+     +-----+                              +-----+     +-----+
    //
    // ```

    /// This function associates stable node indices with [`PhysicalExpr`]s so
    /// that we can match `Arc<dyn PhysicalExpr>` and NodeIndex objects during
    /// membership tests.
    pub fn gather_node_indices(
        &mut self,
        exprs: &[Arc<dyn PhysicalExpr>],
    ) -> Vec<(Arc<dyn PhysicalExpr>, usize)> {
        let graph = &self.graph;
        let mut bfs = Bfs::new(graph, self.root);
        // We collect the node indices (usize) of [PhysicalExpr]s in the order
        // given by argument `exprs`. To preserve this order, we initialize each
        // expression's node index with usize::MAX, and then find the corresponding
        // node indices by traversing the graph.
        let mut removals = vec![];
        let mut expr_node_indices = exprs
            .iter()
            .map(|e| (Arc::clone(e), usize::MAX))
            .collect::<Vec<_>>();
        while let Some(node) = bfs.next(graph) {
            // Get the plan corresponding to this node:
            let expr = &graph[node].expr;
            // If the current expression is among `exprs`, slate its children
            // for removal:
            if let Some(value) = exprs.iter().position(|e| expr.eq(e)) {
                // Update the node index of the associated `PhysicalExpr`:
                expr_node_indices[value].1 = node.index();
                for edge in graph.edges_directed(node, Outgoing) {
                    // Slate the child for removal, do not remove immediately.
                    removals.push(edge.id());
                }
            }
        }
        for edge_idx in removals {
            self.graph.remove_edge(edge_idx);
        }
        // Get the set of node indices reachable from the root node:
        let connected_nodes = self.connected_nodes();
        // Remove nodes not connected to the root node:
        self.graph
            .retain_nodes(|_, index| connected_nodes.contains(&index));
        expr_node_indices
    }

    /// Returns the set of node indices reachable from the root node via a
    /// simple depth-first search.
    fn connected_nodes(&self) -> HashSet<NodeIndex> {
        let mut nodes = HashSet::new();
        let mut dfs = Dfs::new(&self.graph, self.root);
        while let Some(node) = dfs.next(&self.graph) {
            nodes.insert(node);
        }
        nodes
    }

    /// Updates intervals for all expressions in the DAEG by successive
    /// bottom-up and top-down traversals.
    pub fn update_ranges(
        &mut self,
        leaf_bounds: &mut [(usize, Interval)],
        given_range: Interval,
    ) -> Result<PropagationResult> {
        self.assign_intervals(leaf_bounds);
        let bounds = self.evaluate_bounds()?;
        // There are three possible cases to consider:
        // (1) given_range ⊇ bounds => Nothing to propagate
        // (2) ∅ ⊂ (given_range ∩ bounds) ⊂ bounds => Can propagate
        // (3) Disjoint sets => Infeasible
        if given_range.contains(bounds)? == Interval::CERTAINLY_TRUE {
            // First case:
            Ok(PropagationResult::CannotPropagate)
        } else if bounds.contains(&given_range)? != Interval::CERTAINLY_FALSE {
            // Second case:
            let result = self.propagate_constraints(given_range);
            self.update_intervals(leaf_bounds);
            result
        } else {
            // Third case:
            Ok(PropagationResult::Infeasible)
        }
    }

    /// This function assigns given ranges to expressions in the DAEG.
    /// The argument `assignments` associates indices of sought expressions
    /// with their corresponding new ranges.
    pub fn assign_intervals(&mut self, assignments: &[(usize, Interval)]) {
        for (index, interval) in assignments {
            let node_index = NodeIndex::from(*index as DefaultIx);
            self.graph[node_index].interval = interval.clone();
        }
    }

    /// This function fetches ranges of expressions from the DAEG. The argument
    /// `assignments` associates indices of sought expressions with their ranges,
    /// which this function modifies to reflect the intervals in the DAEG.
    pub fn update_intervals(&self, assignments: &mut [(usize, Interval)]) {
        for (index, interval) in assignments.iter_mut() {
            let node_index = NodeIndex::from(*index as DefaultIx);
            *interval = self.graph[node_index].interval.clone();
        }
    }

    /// Computes bounds for an expression using interval arithmetic via a
    /// bottom-up traversal.
    ///
    /// # Examples
    ///
    /// ```
    /// use arrow::datatypes::DataType;
    /// use arrow::datatypes::Field;
    /// use arrow::datatypes::Schema;
    /// use datafusion_common::ScalarValue;
    /// use datafusion_expr::interval_arithmetic::Interval;
    /// use datafusion_expr::Operator;
    /// use datafusion_physical_expr::expressions::{BinaryExpr, Column, Literal};
    /// use datafusion_physical_expr::intervals::cp_solver::ExprIntervalGraph;
    /// use datafusion_physical_expr::PhysicalExpr;
    /// use std::sync::Arc;
    ///
    /// let expr = Arc::new(BinaryExpr::new(
    ///     Arc::new(Column::new("gnz", 0)),
    ///     Operator::Plus,
    ///     Arc::new(Literal::new(ScalarValue::Int32(Some(10)))),
    /// ));
    ///
    /// let schema = Schema::new(vec![Field::new("gnz".to_string(), DataType::Int32, true)]);
    ///
    /// let mut graph = ExprIntervalGraph::try_new(expr, &schema).unwrap();
    /// // Do it once, while constructing.
    /// let node_indices = graph
    ///     .gather_node_indices(&[Arc::new(Column::new("gnz", 0))]);
    /// let left_index = node_indices.get(0).unwrap().1;
    ///
    /// // Provide intervals for leaf variables (here, there is only one).
    /// let intervals = vec![(
    ///     left_index,
    ///     Interval::make(Some(10), Some(20)).unwrap(),
    /// )];
    ///
    /// // Evaluate bounds for the composite expression:
    /// graph.assign_intervals(&intervals);
    /// assert_eq!(
    ///     graph.evaluate_bounds().unwrap(),
    ///     &Interval::make(Some(20), Some(30)).unwrap(),
    /// )
    /// ```
    pub fn evaluate_bounds(&mut self) -> Result<&Interval> {
        let mut dfs = DfsPostOrder::new(&self.graph, self.root);
        while let Some(node) = dfs.next(&self.graph) {
            let neighbors = self.graph.neighbors_directed(node, Outgoing);
            let mut children_intervals = neighbors
                .map(|child| self.graph[child].interval())
                .collect::<Vec<_>>();
            // If the current expression is a leaf, its interval should already
            // be set externally, just continue with the evaluation procedure:
            if !children_intervals.is_empty() {
                // Reverse to align with `PhysicalExpr`'s children:
                children_intervals.reverse();
                self.graph[node].interval =
                    self.graph[node].expr.evaluate_bounds(&children_intervals)?;
            }
        }
        Ok(self.graph[self.root].interval())
    }

    /// Updates/shrinks bounds for leaf expressions using interval arithmetic
    /// via a top-down traversal.
    fn propagate_constraints(
        &mut self,
        given_range: Interval,
    ) -> Result<PropagationResult> {
        // Adjust the root node with the given range:
        if let Some(interval) = self.graph[self.root].interval.intersect(given_range)? {
            self.graph[self.root].interval = interval;
        } else {
            return Ok(PropagationResult::Infeasible);
        }

        let mut bfs = Bfs::new(&self.graph, self.root);

        while let Some(node) = bfs.next(&self.graph) {
            let neighbors = self.graph.neighbors_directed(node, Outgoing);
            let mut children = neighbors.collect::<Vec<_>>();
            // If the current expression is a leaf, its range is now final.
            // So, just continue with the propagation procedure:
            if children.is_empty() {
                continue;
            }
            // Reverse to align with `PhysicalExpr`'s children:
            children.reverse();
            let children_intervals = children
                .iter()
                .map(|child| self.graph[*child].interval())
                .collect::<Vec<_>>();
            let node_interval = self.graph[node].interval();
            let propagated_intervals = self.graph[node]
                .expr
                .propagate_constraints(node_interval, &children_intervals)?;
            if let Some(propagated_intervals) = propagated_intervals {
                for (child, interval) in children.into_iter().zip(propagated_intervals) {
                    self.graph[child].interval = interval;
                }
            } else {
                // The constraint is infeasible, report:
                return Ok(PropagationResult::Infeasible);
            }
        }
        Ok(PropagationResult::Success)
    }

    /// Returns the interval associated with the node at the given `index`.
    pub fn get_interval(&self, index: usize) -> Interval {
        self.graph[NodeIndex::new(index)].interval.clone()
    }
}

/// This is a subfunction of the `propagate_arithmetic` function that propagates to the right child.
fn propagate_right(
    left: &Interval,
    parent: &Interval,
    right: &Interval,
    op: &Operator,
    inverse_op: &Operator,
) -> Result<Option<Interval>> {
    match op {
        Operator::Minus => apply_operator(op, left, parent),
        Operator::Plus => apply_operator(inverse_op, parent, left),
        Operator::Divide => apply_operator(op, left, parent),
        Operator::Multiply => apply_operator(inverse_op, parent, left),
        _ => internal_err!("Interval arithmetic does not support the operator {}", op),
    }?
    .intersect(right)
}

/// During the propagation of [`Interval`] values on an [`ExprIntervalGraph`],
/// if there exists a `timestamp - timestamp` operation, the result would be
/// of type `Duration`. However, we may encounter a situation where a time interval
/// is involved in an arithmetic operation with a `Duration` type. This function
/// offers special handling for such cases, where the time interval resides on
/// the left side of the operation.
fn propagate_time_interval_at_left(
    left_child: &Interval,
    right_child: &Interval,
    parent: &Interval,
    op: &Operator,
    inverse_op: &Operator,
) -> Result<Option<(Interval, Interval)>> {
    // We check if the child's time interval(s) has a non-zero month or day field(s).
    // If so, we return it as is without propagating. Otherwise, we first convert
    // the time intervals to the `Duration` type, then propagate, and then convert
    // the bounds to time intervals again.
    let result = if let Some(duration) = convert_interval_type_to_duration(left_child) {
        match apply_operator(inverse_op, parent, right_child)?.intersect(duration)? {
            Some(value) => {
                let left = convert_duration_type_to_interval(&value);
                let right = propagate_right(&value, parent, right_child, op, inverse_op)?;
                match (left, right) {
                    (Some(left), Some(right)) => Some((left, right)),
                    _ => None,
                }
            }
            None => None,
        }
    } else {
        propagate_right(left_child, parent, right_child, op, inverse_op)?
            .map(|right| (left_child.clone(), right))
    };
    Ok(result)
}

/// During the propagation of [`Interval`] values on an [`ExprIntervalGraph`],
/// if there exists a `timestamp - timestamp` operation, the result would be
/// of type `Duration`. However, we may encounter a situation where a time interval
/// is involved in an arithmetic operation with a `Duration` type. This function
/// offers special handling for such cases, where the time interval resides on
/// the right side of the operation.
fn propagate_time_interval_at_right(
    left_child: &Interval,
    right_child: &Interval,
    parent: &Interval,
    op: &Operator,
    inverse_op: &Operator,
) -> Result<Option<(Interval, Interval)>> {
    // We check if the child's time interval(s) has a non-zero month or day field(s).
    // If so, we return it as is without propagating. Otherwise, we first convert
    // the time intervals to the `Duration` type, then propagate, and then convert
    // the bounds to time intervals again.
    let result = if let Some(duration) = convert_interval_type_to_duration(right_child) {
        match apply_operator(inverse_op, parent, &duration)?.intersect(left_child)? {
            Some(value) => {
                propagate_right(left_child, parent, &duration, op, inverse_op)?
                    .and_then(|right| convert_duration_type_to_interval(&right))
                    .map(|right| (value, right))
            }
            None => None,
        }
    } else {
        apply_operator(inverse_op, parent, right_child)?
            .intersect(left_child)?
            .map(|value| (value, right_child.clone()))
    };
    Ok(result)
}

fn reverse_tuple<T, U>((first, second): (T, U)) -> (U, T) {
    (second, first)
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::expressions::{BinaryExpr, Column};
    use crate::intervals::test_utils::gen_conjunctive_numerical_expr;

    use arrow::array::types::{IntervalDayTime, IntervalMonthDayNano};
    use arrow::datatypes::{Field, TimeUnit};
    use datafusion_common::ScalarValue;

    use itertools::Itertools;
    use rand::rngs::StdRng;
    use rand::{Rng, SeedableRng};
    use rstest::*;

    #[allow(clippy::too_many_arguments)]
    fn experiment(
        expr: Arc<dyn PhysicalExpr>,
        exprs_with_interval: (Arc<dyn PhysicalExpr>, Arc<dyn PhysicalExpr>),
        left_interval: Interval,
        right_interval: Interval,
        left_expected: Interval,
        right_expected: Interval,
        result: PropagationResult,
        schema: &Schema,
    ) -> Result<()> {
        let col_stats = vec![
            (Arc::clone(&exprs_with_interval.0), left_interval),
            (Arc::clone(&exprs_with_interval.1), right_interval),
        ];
        let expected = vec![
            (Arc::clone(&exprs_with_interval.0), left_expected),
            (Arc::clone(&exprs_with_interval.1), right_expected),
        ];
        let mut graph = ExprIntervalGraph::try_new(expr, schema)?;
        let expr_indexes = graph.gather_node_indices(
            &col_stats.iter().map(|(e, _)| Arc::clone(e)).collect_vec(),
        );

        let mut col_stat_nodes = col_stats
            .iter()
            .zip(expr_indexes.iter())
            .map(|((_, interval), (_, index))| (*index, interval.clone()))
            .collect_vec();
        let expected_nodes = expected
            .iter()
            .zip(expr_indexes.iter())
            .map(|((_, interval), (_, index))| (*index, interval.clone()))
            .collect_vec();

        let exp_result =
            graph.update_ranges(&mut col_stat_nodes[..], Interval::CERTAINLY_TRUE)?;
        assert_eq!(exp_result, result);
        col_stat_nodes.iter().zip(expected_nodes.iter()).for_each(
            |((_, calculated_interval_node), (_, expected))| {
                // NOTE: These randomized tests only check for conservative containment,
                // not openness/closedness of endpoints.

                // Calculated bounds are relaxed by 1 to cover all strict and
                // and non-strict comparison cases since we have only closed bounds.
                let one = ScalarValue::new_one(&expected.data_type()).unwrap();
                assert!(
                    calculated_interval_node.lower()
                        <= &expected.lower().add(&one).unwrap(),
                    "{}",
                    format!(
                        "Calculated {} must be less than or equal {}",
                        calculated_interval_node.lower(),
                        expected.lower()
                    )
                );
                assert!(
                    calculated_interval_node.upper()
                        >= &expected.upper().sub(&one).unwrap(),
                    "{}",
                    format!(
                        "Calculated {} must be greater than or equal {}",
                        calculated_interval_node.upper(),
                        expected.upper()
                    )
                );
            },
        );
        Ok(())
    }

    macro_rules! generate_cases {
        ($FUNC_NAME:ident, $TYPE:ty, $SCALAR:ident) => {
            fn $FUNC_NAME<const ASC: bool>(
                expr: Arc<dyn PhysicalExpr>,
                left_col: Arc<dyn PhysicalExpr>,
                right_col: Arc<dyn PhysicalExpr>,
                seed: u64,
                expr_left: $TYPE,
                expr_right: $TYPE,
            ) -> Result<()> {
                let mut r = StdRng::seed_from_u64(seed);

                let (left_given, right_given, left_expected, right_expected) = if ASC {
                    let left = r.gen_range((0 as $TYPE)..(1000 as $TYPE));
                    let right = r.gen_range((0 as $TYPE)..(1000 as $TYPE));
                    (
                        (Some(left), None),
                        (Some(right), None),
                        (Some(<$TYPE>::max(left, right + expr_left)), None),
                        (Some(<$TYPE>::max(right, left + expr_right)), None),
                    )
                } else {
                    let left = r.gen_range((0 as $TYPE)..(1000 as $TYPE));
                    let right = r.gen_range((0 as $TYPE)..(1000 as $TYPE));
                    (
                        (None, Some(left)),
                        (None, Some(right)),
                        (None, Some(<$TYPE>::min(left, right + expr_left))),
                        (None, Some(<$TYPE>::min(right, left + expr_right))),
                    )
                };

                experiment(
                    expr,
                    (left_col.clone(), right_col.clone()),
                    Interval::make(left_given.0, left_given.1).unwrap(),
                    Interval::make(right_given.0, right_given.1).unwrap(),
                    Interval::make(left_expected.0, left_expected.1).unwrap(),
                    Interval::make(right_expected.0, right_expected.1).unwrap(),
                    PropagationResult::Success,
                    &Schema::new(vec![
                        Field::new(
                            left_col.as_any().downcast_ref::<Column>().unwrap().name(),
                            DataType::$SCALAR,
                            true,
                        ),
                        Field::new(
                            right_col.as_any().downcast_ref::<Column>().unwrap().name(),
                            DataType::$SCALAR,
                            true,
                        ),
                    ]),
                )
            }
        };
    }
    generate_cases!(generate_case_i32, i32, Int32);
    generate_cases!(generate_case_i64, i64, Int64);
    generate_cases!(generate_case_f32, f32, Float32);
    generate_cases!(generate_case_f64, f64, Float64);

    #[test]
    fn testing_not_possible() -> Result<()> {
        let left_col = Arc::new(Column::new("left_watermark", 0));
        let right_col = Arc::new(Column::new("right_watermark", 0));

        // left_watermark > right_watermark + 5
        let left_and_1 = Arc::new(BinaryExpr::new(
            Arc::clone(&left_col) as Arc<dyn PhysicalExpr>,
            Operator::Plus,
            Arc::new(Literal::new(ScalarValue::Int32(Some(5)))),
        ));
        let expr = Arc::new(BinaryExpr::new(
            left_and_1,
            Operator::Gt,
            Arc::clone(&right_col) as Arc<dyn PhysicalExpr>,
        ));
        experiment(
            expr,
            (
                Arc::clone(&left_col) as Arc<dyn PhysicalExpr>,
                Arc::clone(&right_col) as Arc<dyn PhysicalExpr>,
            ),
            Interval::make(Some(10_i32), Some(20_i32))?,
            Interval::make(Some(100), None)?,
            Interval::make(Some(10), Some(20))?,
            Interval::make(Some(100), None)?,
            PropagationResult::Infeasible,
            &Schema::new(vec![
                Field::new(
                    left_col.as_any().downcast_ref::<Column>().unwrap().name(),
                    DataType::Int32,
                    true,
                ),
                Field::new(
                    right_col.as_any().downcast_ref::<Column>().unwrap().name(),
                    DataType::Int32,
                    true,
                ),
            ]),
        )
    }

    macro_rules! integer_float_case_1 {
        ($TEST_FUNC_NAME:ident, $GENERATE_CASE_FUNC_NAME:ident, $TYPE:ty, $SCALAR:ident) => {
            #[rstest]
            #[test]
            fn $TEST_FUNC_NAME(
                #[values(0, 1, 2, 3, 4, 12, 32, 314, 3124, 123, 125, 211, 215, 4123)]
                seed: u64,
                #[values(Operator::Gt, Operator::GtEq)] greater_op: Operator,
                #[values(Operator::Lt, Operator::LtEq)] less_op: Operator,
            ) -> Result<()> {
                let left_col = Arc::new(Column::new("left_watermark", 0));
                let right_col = Arc::new(Column::new("right_watermark", 0));

                // left_watermark + 1 > right_watermark + 11 AND left_watermark + 3 < right_watermark + 33
                let expr = gen_conjunctive_numerical_expr(
                    left_col.clone(),
                    right_col.clone(),
                    (
                        Operator::Plus,
                        Operator::Plus,
                        Operator::Plus,
                        Operator::Plus,
                    ),
                    ScalarValue::$SCALAR(Some(1 as $TYPE)),
                    ScalarValue::$SCALAR(Some(11 as $TYPE)),
                    ScalarValue::$SCALAR(Some(3 as $TYPE)),
                    ScalarValue::$SCALAR(Some(33 as $TYPE)),
                    (greater_op, less_op),
                );
                // l > r + 10 AND r > l - 30
                let l_gt_r = 10 as $TYPE;
                let r_gt_l = -30 as $TYPE;
                $GENERATE_CASE_FUNC_NAME::<true>(
                    expr.clone(),
                    left_col.clone(),
                    right_col.clone(),
                    seed,
                    l_gt_r,
                    r_gt_l,
                )?;
                // Descending tests
                // r < l - 10 AND l < r + 30
                let r_lt_l = -l_gt_r;
                let l_lt_r = -r_gt_l;
                $GENERATE_CASE_FUNC_NAME::<false>(
                    expr, left_col, right_col, seed, l_lt_r, r_lt_l,
                )
            }
        };
    }

    integer_float_case_1!(case_1_i32, generate_case_i32, i32, Int32);
    integer_float_case_1!(case_1_i64, generate_case_i64, i64, Int64);
    integer_float_case_1!(case_1_f64, generate_case_f64, f64, Float64);
    integer_float_case_1!(case_1_f32, generate_case_f32, f32, Float32);

    macro_rules! integer_float_case_2 {
        ($TEST_FUNC_NAME:ident, $GENERATE_CASE_FUNC_NAME:ident, $TYPE:ty, $SCALAR:ident) => {
            #[rstest]
            #[test]
            fn $TEST_FUNC_NAME(
                #[values(0, 1, 2, 3, 4, 12, 32, 314, 3124, 123, 125, 211, 215, 4123)]
                seed: u64,
                #[values(Operator::Gt, Operator::GtEq)] greater_op: Operator,
                #[values(Operator::Lt, Operator::LtEq)] less_op: Operator,
            ) -> Result<()> {
                let left_col = Arc::new(Column::new("left_watermark", 0));
                let right_col = Arc::new(Column::new("right_watermark", 0));

                // left_watermark - 1 > right_watermark + 5 AND left_watermark + 3 < right_watermark + 10
                let expr = gen_conjunctive_numerical_expr(
                    left_col.clone(),
                    right_col.clone(),
                    (
                        Operator::Minus,
                        Operator::Plus,
                        Operator::Plus,
                        Operator::Plus,
                    ),
                    ScalarValue::$SCALAR(Some(1 as $TYPE)),
                    ScalarValue::$SCALAR(Some(5 as $TYPE)),
                    ScalarValue::$SCALAR(Some(3 as $TYPE)),
                    ScalarValue::$SCALAR(Some(10 as $TYPE)),
                    (greater_op, less_op),
                );
                // l > r + 6 AND r > l - 7
                let l_gt_r = 6 as $TYPE;
                let r_gt_l = -7 as $TYPE;
                $GENERATE_CASE_FUNC_NAME::<true>(
                    expr.clone(),
                    left_col.clone(),
                    right_col.clone(),
                    seed,
                    l_gt_r,
                    r_gt_l,
                )?;
                // Descending tests
                // r < l - 6 AND l < r + 7
                let r_lt_l = -l_gt_r;
                let l_lt_r = -r_gt_l;
                $GENERATE_CASE_FUNC_NAME::<false>(
                    expr, left_col, right_col, seed, l_lt_r, r_lt_l,
                )
            }
        };
    }

    integer_float_case_2!(case_2_i32, generate_case_i32, i32, Int32);
    integer_float_case_2!(case_2_i64, generate_case_i64, i64, Int64);
    integer_float_case_2!(case_2_f64, generate_case_f64, f64, Float64);
    integer_float_case_2!(case_2_f32, generate_case_f32, f32, Float32);

    macro_rules! integer_float_case_3 {
        ($TEST_FUNC_NAME:ident, $GENERATE_CASE_FUNC_NAME:ident, $TYPE:ty, $SCALAR:ident) => {
            #[rstest]
            #[test]
            fn $TEST_FUNC_NAME(
                #[values(0, 1, 2, 3, 4, 12, 32, 314, 3124, 123, 125, 211, 215, 4123)]
                seed: u64,
                #[values(Operator::Gt, Operator::GtEq)] greater_op: Operator,
                #[values(Operator::Lt, Operator::LtEq)] less_op: Operator,
            ) -> Result<()> {
                let left_col = Arc::new(Column::new("left_watermark", 0));
                let right_col = Arc::new(Column::new("right_watermark", 0));

                // left_watermark - 1 > right_watermark + 5 AND left_watermark - 3 < right_watermark + 10
                let expr = gen_conjunctive_numerical_expr(
                    left_col.clone(),
                    right_col.clone(),
                    (
                        Operator::Minus,
                        Operator::Plus,
                        Operator::Minus,
                        Operator::Plus,
                    ),
                    ScalarValue::$SCALAR(Some(1 as $TYPE)),
                    ScalarValue::$SCALAR(Some(5 as $TYPE)),
                    ScalarValue::$SCALAR(Some(3 as $TYPE)),
                    ScalarValue::$SCALAR(Some(10 as $TYPE)),
                    (greater_op, less_op),
                );
                // l > r + 6 AND r > l - 13
                let l_gt_r = 6 as $TYPE;
                let r_gt_l = -13 as $TYPE;
                $GENERATE_CASE_FUNC_NAME::<true>(
                    expr.clone(),
                    left_col.clone(),
                    right_col.clone(),
                    seed,
                    l_gt_r,
                    r_gt_l,
                )?;
                // Descending tests
                // r < l - 6 AND l < r + 13
                let r_lt_l = -l_gt_r;
                let l_lt_r = -r_gt_l;
                $GENERATE_CASE_FUNC_NAME::<false>(
                    expr, left_col, right_col, seed, l_lt_r, r_lt_l,
                )
            }
        };
    }

    integer_float_case_3!(case_3_i32, generate_case_i32, i32, Int32);
    integer_float_case_3!(case_3_i64, generate_case_i64, i64, Int64);
    integer_float_case_3!(case_3_f64, generate_case_f64, f64, Float64);
    integer_float_case_3!(case_3_f32, generate_case_f32, f32, Float32);

    macro_rules! integer_float_case_4 {
        ($TEST_FUNC_NAME:ident, $GENERATE_CASE_FUNC_NAME:ident, $TYPE:ty, $SCALAR:ident) => {
            #[rstest]
            #[test]
            fn $TEST_FUNC_NAME(
                #[values(0, 1, 2, 3, 4, 12, 32, 314, 3124, 123, 125, 211, 215, 4123)]
                seed: u64,
                #[values(Operator::Gt, Operator::GtEq)] greater_op: Operator,
                #[values(Operator::Lt, Operator::LtEq)] less_op: Operator,
            ) -> Result<()> {
                let left_col = Arc::new(Column::new("left_watermark", 0));
                let right_col = Arc::new(Column::new("right_watermark", 0));

                // left_watermark - 10 > right_watermark - 5 AND left_watermark - 30 < right_watermark - 3
                let expr = gen_conjunctive_numerical_expr(
                    left_col.clone(),
                    right_col.clone(),
                    (
                        Operator::Minus,
                        Operator::Minus,
                        Operator::Minus,
                        Operator::Plus,
                    ),
                    ScalarValue::$SCALAR(Some(10 as $TYPE)),
                    ScalarValue::$SCALAR(Some(5 as $TYPE)),
                    ScalarValue::$SCALAR(Some(3 as $TYPE)),
                    ScalarValue::$SCALAR(Some(10 as $TYPE)),
                    (greater_op, less_op),
                );
                // l > r + 5 AND r > l - 13
                let l_gt_r = 5 as $TYPE;
                let r_gt_l = -13 as $TYPE;
                $GENERATE_CASE_FUNC_NAME::<true>(
                    expr.clone(),
                    left_col.clone(),
                    right_col.clone(),
                    seed,
                    l_gt_r,
                    r_gt_l,
                )?;
                // Descending tests
                // r < l - 5 AND l < r + 13
                let r_lt_l = -l_gt_r;
                let l_lt_r = -r_gt_l;
                $GENERATE_CASE_FUNC_NAME::<false>(
                    expr, left_col, right_col, seed, l_lt_r, r_lt_l,
                )
            }
        };
    }

    integer_float_case_4!(case_4_i32, generate_case_i32, i32, Int32);
    integer_float_case_4!(case_4_i64, generate_case_i64, i64, Int64);
    integer_float_case_4!(case_4_f64, generate_case_f64, f64, Float64);
    integer_float_case_4!(case_4_f32, generate_case_f32, f32, Float32);

    macro_rules! integer_float_case_5 {
        ($TEST_FUNC_NAME:ident, $GENERATE_CASE_FUNC_NAME:ident, $TYPE:ty, $SCALAR:ident) => {
            #[rstest]
            #[test]
            fn $TEST_FUNC_NAME(
                #[values(0, 1, 2, 3, 4, 12, 32, 314, 3124, 123, 125, 211, 215, 4123)]
                seed: u64,
                #[values(Operator::Gt, Operator::GtEq)] greater_op: Operator,
                #[values(Operator::Lt, Operator::LtEq)] less_op: Operator,
            ) -> Result<()> {
                let left_col = Arc::new(Column::new("left_watermark", 0));
                let right_col = Arc::new(Column::new("right_watermark", 0));

                // left_watermark - 10 > right_watermark - 5 AND left_watermark - 30 < right_watermark - 3
                let expr = gen_conjunctive_numerical_expr(
                    left_col.clone(),
                    right_col.clone(),
                    (
                        Operator::Minus,
                        Operator::Minus,
                        Operator::Minus,
                        Operator::Minus,
                    ),
                    ScalarValue::$SCALAR(Some(10 as $TYPE)),
                    ScalarValue::$SCALAR(Some(5 as $TYPE)),
                    ScalarValue::$SCALAR(Some(30 as $TYPE)),
                    ScalarValue::$SCALAR(Some(3 as $TYPE)),
                    (greater_op, less_op),
                );
                // l > r + 5 AND r > l - 27
                let l_gt_r = 5 as $TYPE;
                let r_gt_l = -27 as $TYPE;
                $GENERATE_CASE_FUNC_NAME::<true>(
                    expr.clone(),
                    left_col.clone(),
                    right_col.clone(),
                    seed,
                    l_gt_r,
                    r_gt_l,
                )?;
                // Descending tests
                // r < l - 5 AND l < r + 27
                let r_lt_l = -l_gt_r;
                let l_lt_r = -r_gt_l;
                $GENERATE_CASE_FUNC_NAME::<false>(
                    expr, left_col, right_col, seed, l_lt_r, r_lt_l,
                )
            }
        };
    }

    integer_float_case_5!(case_5_i32, generate_case_i32, i32, Int32);
    integer_float_case_5!(case_5_i64, generate_case_i64, i64, Int64);
    integer_float_case_5!(case_5_f64, generate_case_f64, f64, Float64);
    integer_float_case_5!(case_5_f32, generate_case_f32, f32, Float32);

    #[test]
    fn test_gather_node_indices_dont_remove() -> Result<()> {
        // Expression: a@0 + b@1 + 1 > a@0 - b@1, given a@0 + b@1.
        // Do not remove a@0 or b@1, only remove edges since a@0 - b@1 also
        // depends on leaf nodes a@0 and b@1.
        let left_expr = Arc::new(BinaryExpr::new(
            Arc::new(BinaryExpr::new(
                Arc::new(Column::new("a", 0)),
                Operator::Plus,
                Arc::new(Column::new("b", 1)),
            )),
            Operator::Plus,
            Arc::new(Literal::new(ScalarValue::Int32(Some(1)))),
        ));

        let right_expr = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Minus,
            Arc::new(Column::new("b", 1)),
        ));
        let expr = Arc::new(BinaryExpr::new(left_expr, Operator::Gt, right_expr));
        let mut graph = ExprIntervalGraph::try_new(
            expr,
            &Schema::new(vec![
                Field::new("a", DataType::Int32, true),
                Field::new("b", DataType::Int32, true),
            ]),
        )
        .unwrap();
        // Define a test leaf node.
        let leaf_node = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Plus,
            Arc::new(Column::new("b", 1)),
        ));
        // Store the current node count.
        let prev_node_count = graph.node_count();
        // Gather the index of node in the expression graph that match the test leaf node.
        graph.gather_node_indices(&[leaf_node]);
        // Store the final node count.
        let final_node_count = graph.node_count();
        // Assert that the final node count is equal the previous node count.
        // This means we did not remove any node.
        assert_eq!(prev_node_count, final_node_count);
        Ok(())
    }

    #[test]
    fn test_gather_node_indices_remove() -> Result<()> {
        // Expression: a@0 + b@1 + 1 > y@0 - z@1, given a@0 + b@1.
        // We expect to remove two nodes since we do not need a@ and b@.
        let left_expr = Arc::new(BinaryExpr::new(
            Arc::new(BinaryExpr::new(
                Arc::new(Column::new("a", 0)),
                Operator::Plus,
                Arc::new(Column::new("b", 1)),
            )),
            Operator::Plus,
            Arc::new(Literal::new(ScalarValue::Int32(Some(1)))),
        ));

        let right_expr = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("y", 0)),
            Operator::Minus,
            Arc::new(Column::new("z", 1)),
        ));
        let expr = Arc::new(BinaryExpr::new(left_expr, Operator::Gt, right_expr));
        let mut graph = ExprIntervalGraph::try_new(
            expr,
            &Schema::new(vec![
                Field::new("a", DataType::Int32, true),
                Field::new("b", DataType::Int32, true),
                Field::new("y", DataType::Int32, true),
                Field::new("z", DataType::Int32, true),
            ]),
        )
        .unwrap();
        // Define a test leaf node.
        let leaf_node = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Plus,
            Arc::new(Column::new("b", 1)),
        ));
        // Store the current node count.
        let prev_node_count = graph.node_count();
        // Gather the index of node in the expression graph that match the test leaf node.
        graph.gather_node_indices(&[leaf_node]);
        // Store the final node count.
        let final_node_count = graph.node_count();
        // Assert that the final node count is two less than the previous node
        // count; i.e. that we did remove two nodes.
        assert_eq!(prev_node_count, final_node_count + 2);
        Ok(())
    }

    #[test]
    fn test_gather_node_indices_remove_one() -> Result<()> {
        // Expression: a@0 + b@1 + 1 > a@0 - z@1, given a@0 + b@1.
        // We expect to remove one nodesince we still need a@ but not b@.
        let left_expr = Arc::new(BinaryExpr::new(
            Arc::new(BinaryExpr::new(
                Arc::new(Column::new("a", 0)),
                Operator::Plus,
                Arc::new(Column::new("b", 1)),
            )),
            Operator::Plus,
            Arc::new(Literal::new(ScalarValue::Int32(Some(1)))),
        ));

        let right_expr = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Minus,
            Arc::new(Column::new("z", 1)),
        ));
        let expr = Arc::new(BinaryExpr::new(left_expr, Operator::Gt, right_expr));
        let mut graph = ExprIntervalGraph::try_new(
            expr,
            &Schema::new(vec![
                Field::new("a", DataType::Int32, true),
                Field::new("b", DataType::Int32, true),
                Field::new("z", DataType::Int32, true),
            ]),
        )
        .unwrap();
        // Define a test leaf node.
        let leaf_node = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Plus,
            Arc::new(Column::new("b", 1)),
        ));
        // Store the current node count.
        let prev_node_count = graph.node_count();
        // Gather the index of node in the expression graph that match the test leaf node.
        graph.gather_node_indices(&[leaf_node]);
        // Store the final node count.
        let final_node_count = graph.node_count();
        // Assert that the final node count is one less than the previous node
        // count; i.e. that we did remove two nodes.
        assert_eq!(prev_node_count, final_node_count + 1);
        Ok(())
    }

    #[test]
    fn test_gather_node_indices_cannot_provide() -> Result<()> {
        // Expression: a@0 + 1 + b@1 > y@0 - z@1 -> provide a@0 + b@1
        // TODO: We expect nodes a@0 and b@1 to be pruned, and intervals to be provided from the a@0 + b@1 node.
        // However, we do not have an exact node for a@0 + b@1 due to the binary tree structure of the expressions.
        // Pruning and interval providing for BinaryExpr expressions are more challenging without exact matches.
        // Currently, we only support exact matches for BinaryExprs, but we plan to extend support beyond exact matches in the future.
        let left_expr = Arc::new(BinaryExpr::new(
            Arc::new(BinaryExpr::new(
                Arc::new(Column::new("a", 0)),
                Operator::Plus,
                Arc::new(Literal::new(ScalarValue::Int32(Some(1)))),
            )),
            Operator::Plus,
            Arc::new(Column::new("b", 1)),
        ));

        let right_expr = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("y", 0)),
            Operator::Minus,
            Arc::new(Column::new("z", 1)),
        ));
        let expr = Arc::new(BinaryExpr::new(left_expr, Operator::Gt, right_expr));
        let mut graph = ExprIntervalGraph::try_new(
            expr,
            &Schema::new(vec![
                Field::new("a", DataType::Int32, true),
                Field::new("b", DataType::Int32, true),
                Field::new("y", DataType::Int32, true),
                Field::new("z", DataType::Int32, true),
            ]),
        )
        .unwrap();
        // Define a test leaf node.
        let leaf_node = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Plus,
            Arc::new(Column::new("b", 1)),
        ));
        // Store the current node count.
        let prev_node_count = graph.node_count();
        // Gather the index of node in the expression graph that match the test leaf node.
        graph.gather_node_indices(&[leaf_node]);
        // Store the final node count.
        let final_node_count = graph.node_count();
        // Assert that the final node count is equal the previous node count (i.e., no node was pruned).
        assert_eq!(prev_node_count, final_node_count);
        Ok(())
    }

    #[test]
    fn test_propagate_constraints_singleton_interval_at_right() -> Result<()> {
        let expression = BinaryExpr::new(
            Arc::new(Column::new("ts_column", 0)),
            Operator::Plus,
            Arc::new(Literal::new(ScalarValue::new_interval_mdn(0, 1, 321))),
        );
        let parent = Interval::try_new(
            // 15.10.2020 - 10:11:12.000_000_321 AM
            ScalarValue::TimestampNanosecond(Some(1_602_756_672_000_000_321), None),
            // 16.10.2020 - 10:11:12.000_000_321 AM
            ScalarValue::TimestampNanosecond(Some(1_602_843_072_000_000_321), None),
        )?;
        let left_child = Interval::try_new(
            // 10.10.2020 - 10:11:12 AM
            ScalarValue::TimestampNanosecond(Some(1_602_324_672_000_000_000), None),
            // 20.10.2020 - 10:11:12 AM
            ScalarValue::TimestampNanosecond(Some(1_603_188_672_000_000_000), None),
        )?;
        let right_child = Interval::try_new(
            // 1 day 321 ns
            ScalarValue::IntervalMonthDayNano(Some(IntervalMonthDayNano {
                months: 0,
                days: 1,
                nanoseconds: 321,
            })),
            // 1 day 321 ns
            ScalarValue::IntervalMonthDayNano(Some(IntervalMonthDayNano {
                months: 0,
                days: 1,
                nanoseconds: 321,
            })),
        )?;
        let children = vec![&left_child, &right_child];
        let result = expression
            .propagate_constraints(&parent, &children)?
            .unwrap();

        assert_eq!(
            vec![
                Interval::try_new(
                    // 14.10.2020 - 10:11:12 AM
                    ScalarValue::TimestampNanosecond(
                        Some(1_602_670_272_000_000_000),
                        None
                    ),
                    // 15.10.2020 - 10:11:12 AM
                    ScalarValue::TimestampNanosecond(
                        Some(1_602_756_672_000_000_000),
                        None
                    ),
                )?,
                Interval::try_new(
                    // 1 day 321 ns in Duration type
                    ScalarValue::IntervalMonthDayNano(Some(IntervalMonthDayNano {
                        months: 0,
                        days: 1,
                        nanoseconds: 321,
                    })),
                    // 1 day 321 ns in Duration type
                    ScalarValue::IntervalMonthDayNano(Some(IntervalMonthDayNano {
                        months: 0,
                        days: 1,
                        nanoseconds: 321,
                    })),
                )?
            ],
            result
        );

        Ok(())
    }

    #[test]
    fn test_propagate_constraints_column_interval_at_left() -> Result<()> {
        let expression = BinaryExpr::new(
            Arc::new(Column::new("interval_column", 1)),
            Operator::Plus,
            Arc::new(Column::new("ts_column", 0)),
        );
        let parent = Interval::try_new(
            // 15.10.2020 - 10:11:12 AM
            ScalarValue::TimestampMillisecond(Some(1_602_756_672_000), None),
            // 16.10.2020 - 10:11:12 AM
            ScalarValue::TimestampMillisecond(Some(1_602_843_072_000), None),
        )?;
        let right_child = Interval::try_new(
            // 10.10.2020 - 10:11:12 AM
            ScalarValue::TimestampMillisecond(Some(1_602_324_672_000), None),
            // 20.10.2020 - 10:11:12 AM
            ScalarValue::TimestampMillisecond(Some(1_603_188_672_000), None),
        )?;
        let left_child = Interval::try_new(
            // 2 days in millisecond
            ScalarValue::IntervalDayTime(Some(IntervalDayTime {
                days: 0,
                milliseconds: 172_800_000,
            })),
            // 10 days in millisecond
            ScalarValue::IntervalDayTime(Some(IntervalDayTime {
                days: 0,
                milliseconds: 864_000_000,
            })),
        )?;
        let children = vec![&left_child, &right_child];
        let result = expression
            .propagate_constraints(&parent, &children)?
            .unwrap();

        assert_eq!(
            vec![
                Interval::try_new(
                    // 2 days in millisecond
                    ScalarValue::IntervalDayTime(Some(IntervalDayTime {
                        days: 0,
                        milliseconds: 172_800_000,
                    })),
                    // 6 days
                    ScalarValue::IntervalDayTime(Some(IntervalDayTime {
                        days: 0,
                        milliseconds: 518_400_000,
                    })),
                )?,
                Interval::try_new(
                    // 10.10.2020 - 10:11:12 AM
                    ScalarValue::TimestampMillisecond(Some(1_602_324_672_000), None),
                    // 14.10.2020 - 10:11:12 AM
                    ScalarValue::TimestampMillisecond(Some(1_602_670_272_000), None),
                )?
            ],
            result
        );

        Ok(())
    }

    #[test]
    fn test_propagate_comparison() -> Result<()> {
        // In the examples below:
        // `left` is unbounded: [?, ?],
        // `right` is known to be [1000,1000]
        // so `left` < `right` results in no new knowledge of `right` but knowing that `left` is now < 1000:` [?, 999]
        let left = Interval::make_unbounded(&DataType::Int64)?;
        let right = Interval::make(Some(1000_i64), Some(1000_i64))?;
        assert_eq!(
            (Some((
                Interval::make(None, Some(999_i64))?,
                Interval::make(Some(1000_i64), Some(1000_i64))?,
            ))),
            propagate_comparison(
                &Operator::Lt,
                &Interval::CERTAINLY_TRUE,
                &left,
                &right
            )?
        );

        let left =
            Interval::make_unbounded(&DataType::Timestamp(TimeUnit::Nanosecond, None))?;
        let right = Interval::try_new(
            ScalarValue::TimestampNanosecond(Some(1000), None),
            ScalarValue::TimestampNanosecond(Some(1000), None),
        )?;
        assert_eq!(
            (Some((
                Interval::try_new(
                    ScalarValue::try_from(&DataType::Timestamp(
                        TimeUnit::Nanosecond,
                        None
                    ))
                    .unwrap(),
                    ScalarValue::TimestampNanosecond(Some(999), None),
                )?,
                Interval::try_new(
                    ScalarValue::TimestampNanosecond(Some(1000), None),
                    ScalarValue::TimestampNanosecond(Some(1000), None),
                )?
            ))),
            propagate_comparison(
                &Operator::Lt,
                &Interval::CERTAINLY_TRUE,
                &left,
                &right
            )?
        );

        let left = Interval::make_unbounded(&DataType::Timestamp(
            TimeUnit::Nanosecond,
            Some("+05:00".into()),
        ))?;
        let right = Interval::try_new(
            ScalarValue::TimestampNanosecond(Some(1000), Some("+05:00".into())),
            ScalarValue::TimestampNanosecond(Some(1000), Some("+05:00".into())),
        )?;
        assert_eq!(
            (Some((
                Interval::try_new(
                    ScalarValue::try_from(&DataType::Timestamp(
                        TimeUnit::Nanosecond,
                        Some("+05:00".into()),
                    ))
                    .unwrap(),
                    ScalarValue::TimestampNanosecond(Some(999), Some("+05:00".into())),
                )?,
                Interval::try_new(
                    ScalarValue::TimestampNanosecond(Some(1000), Some("+05:00".into())),
                    ScalarValue::TimestampNanosecond(Some(1000), Some("+05:00".into())),
                )?
            ))),
            propagate_comparison(
                &Operator::Lt,
                &Interval::CERTAINLY_TRUE,
                &left,
                &right
            )?
        );

        Ok(())
    }

    #[test]
    fn test_propagate_or() -> Result<()> {
        let expr = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::Or,
            Arc::new(Column::new("b", 1)),
        ));
        let parent = Interval::CERTAINLY_FALSE;
        let children_set = vec![
            vec![&Interval::CERTAINLY_FALSE, &Interval::UNCERTAIN],
            vec![&Interval::UNCERTAIN, &Interval::CERTAINLY_FALSE],
            vec![&Interval::CERTAINLY_FALSE, &Interval::CERTAINLY_FALSE],
            vec![&Interval::UNCERTAIN, &Interval::UNCERTAIN],
        ];
        for children in children_set {
            assert_eq!(
                expr.propagate_constraints(&parent, &children)?.unwrap(),
                vec![Interval::CERTAINLY_FALSE, Interval::CERTAINLY_FALSE],
            );
        }

        let parent = Interval::CERTAINLY_FALSE;
        let children_set = vec![
            vec![&Interval::CERTAINLY_TRUE, &Interval::UNCERTAIN],
            vec![&Interval::UNCERTAIN, &Interval::CERTAINLY_TRUE],
        ];
        for children in children_set {
            assert_eq!(expr.propagate_constraints(&parent, &children)?, None,);
        }

        let parent = Interval::CERTAINLY_TRUE;
        let children = vec![&Interval::CERTAINLY_FALSE, &Interval::UNCERTAIN];
        assert_eq!(
            expr.propagate_constraints(&parent, &children)?.unwrap(),
            vec![Interval::CERTAINLY_FALSE, Interval::CERTAINLY_TRUE]
        );

        let parent = Interval::CERTAINLY_TRUE;
        let children = vec![&Interval::UNCERTAIN, &Interval::UNCERTAIN];
        assert_eq!(
            expr.propagate_constraints(&parent, &children)?.unwrap(),
            // Empty means unchanged intervals.
            vec![]
        );

        Ok(())
    }

    #[test]
    fn test_propagate_certainly_false_and() -> Result<()> {
        let expr = Arc::new(BinaryExpr::new(
            Arc::new(Column::new("a", 0)),
            Operator::And,
            Arc::new(Column::new("b", 1)),
        ));
        let parent = Interval::CERTAINLY_FALSE;
        let children_and_results_set = vec![
            (
                vec![&Interval::CERTAINLY_TRUE, &Interval::UNCERTAIN],
                vec![Interval::CERTAINLY_TRUE, Interval::CERTAINLY_FALSE],
            ),
            (
                vec![&Interval::UNCERTAIN, &Interval::CERTAINLY_TRUE],
                vec![Interval::CERTAINLY_FALSE, Interval::CERTAINLY_TRUE],
            ),
            (
                vec![&Interval::UNCERTAIN, &Interval::UNCERTAIN],
                // Empty means unchanged intervals.
                vec![],
            ),
            (
                vec![&Interval::CERTAINLY_FALSE, &Interval::UNCERTAIN],
                vec![],
            ),
        ];
        for (children, result) in children_and_results_set {
            assert_eq!(
                expr.propagate_constraints(&parent, &children)?.unwrap(),
                result
            );
        }

        Ok(())
    }
}