mz_timely_util/

reclock.rs

// Copyright Materialize, Inc. and contributors. All rights reserved.
//
// Licensed 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 in the LICENSE file at the
// root of this repository, or online 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.

//! ## Notation
//!
//! Collections are represented with capital letters (T, S, R), collection traces as bold letters
//! (𝐓, 𝐒, 𝐑), and difference traces as δ𝐓.
//!
//! Indexing a collection trace 𝐓 to obtain its version at `t` is written as 𝐓(t). Indexing a
//! collection to obtain the multiplicity of a record `x` is written as T\[x\]. These can be combined
//! to obtain the multiplicity of a record `x` at some version `t` as 𝐓(t)\[x\].
//!
//! ## Overview
//!
//! Reclocking transforms a source collection `S` that evolves with some timestamp `FromTime` into
//! a collection `T` that evolves with some other timestamp `IntoTime`. The reclocked collection T
//! contains all updates `u ∈ S` that are not beyond some `FromTime` frontier R(t). The collection
//! `R` is called the remap collection.
//!
//! More formally, for some arbitrary time `t` of `IntoTime` and some arbitrary record `x`, the
//! reclocked collection `T(t)[x]` is defined to be the `sum{δ𝐒(s)[x]: !(𝐑(t) ⪯ s)}`. Since this
//! holds for any record we can write the definition of Reclock(𝐒, 𝐑) as:
//!
//! > Reclock(𝐒, 𝐑) ≜ 𝐓: ∀ t ∈ IntoTime : 𝐓(t) = sum{δ𝐒(s): !(𝐑(t) ⪯ s)}
//!
//! In order for the reclocked collection `T` to have a sensible definition of progress we require
//! that `t1 ≤ t2 ⇒ 𝐑(t1) ⪯ 𝐑(t2)` where the first `≤` is the partial order of `IntoTime` and the
//! second one the partial order of `FromTime` antichains.
//!
//! ## Total order simplification
//!
//! In order to simplify the implementation we will require that `IntoTime` is a total order. This
//! limitation can be lifted in the future but further elaboration on the mechanics of reclocking
//! is required to ensure a correct implementation.
//!
//! ## The difference trace
//!
//! By the definition of difference traces we have:
//!
//! ```text
//!     δ𝐓(t) = T(t) - sum{δ𝐓(s): s < t}
//! ```
//!
//! Due to the total order assumption we only need to consider two cases.
//!
//! **Case 1:** `t` is the minimum timestamp
//!
//! In this case `sum{δ𝐓(s): s < t}` is the empty set and so we obtain:
//!
//! ```text
//!     δ𝐓(min) = T(min) = sum{δ𝐒(s): !(𝐑(min) ≤ s}
//! ```
//!
//! **Case 2:** `t` is a timestamp with a predecessor `prev`
//!
//! In this case `sum{δ𝐓(s): s < t}` is equal to `T(prev)` because:
//!
//! ```text
//!     sum{δ𝐓(s): s < t} = sum{δ𝐓(s): s ≤ prev} + sum{δ𝐓(s): prev < s < t}
//!                       = T(prev) + ∅
//!                       = T(prev)
//! ```
//!
//! And therefore the difference trace of T is:
//!
//! ```text
//!     δ𝐓(t) = 𝐓(t) - 𝐓(prev)
//!           = sum{δ𝐒(s): !(𝐑(t) ⪯ s)} - sum{δ𝐒(s): !(𝐑(prev) ⪯ s)}
//!           = sum{δ𝐒(s): (𝐑(prev) ⪯ s) ∧ !(𝐑(t) ⪯ s)}
//! ```
//!
//! ## Unique mapping property
//!
//! Given the definition above we can derive the fact that for any source difference δ𝐒(s) there is
//! at most one target timestamp t that it must be reclocked to. This property can be exploited by
//! the implementation of the operator as it can safely discard source updates once a matching
//! δT(t) has been found, making it "stateless" with respect to the source trace. A formal proof of
//! this property is [provided below](#unique-mapping-property-proof).
//!
//! ## Operational description
//!
//! The operator follows a run-to-completion model where on each scheduling it completes all
//! outstanding work that can be completed.
//!
//! ### Unique mapping property proof
//!
//! This section contains the formal proof the unique mapping property. The proof follows the
//! structure proof notation created by Leslie Lamport. Readers unfamiliar with structured proofs
//! can read about them here <https://lamport.azurewebsites.net/pubs/proof.pdf>.
//!
//! #### Statement
//!
//! AtMostOne(X, φ(x)) ≜ ∀ x1, x2 ∈ X : φ(x1) ∧ φ(x2) ⇒ x1 = x2
//!
//! * **THEOREM** UniqueMapping ≜
//!     * **ASSUME**
//!         * **NEW** (FromTime, ⪯) ∈ PartiallyOrderedTimestamps
//!         * **NEW** (IntoTime, ≤) ∈ TotallyOrderedTimestamps
//!         * **NEW** 𝐒 ∈ SetOfCollectionTraces(FromTime)
//!         * **NEW** 𝐑 ∈ SetOfCollectionTraces(IntoTime)
//!         * ∀ t ∈ IntoTime: 𝐑(t) ∈ SetOfAntichains(FromTime)
//!         * ∀ t1, t1 ∈ IntoTime: t1 ≤ t2 ⇒ 𝐑(t1) ⪯ 𝐑(t2)
//!         * **NEW** 𝐓 = Reclock(𝐒, 𝐑)
//!     * **PROVE**  ∀ s ∈ FromTime : AtMostOne(IntoTime, δ𝐒(s) ∈ δ𝐓(x))
//!
//! #### Proof
//!
//! 1. **SUFFICES ASSUME** ∃ s ∈ FromTime: ¬AtMostOne(IntoTime, δ𝐒(s) ∈ δ𝐓(x))
//!     * **PROVE FALSE**
//!     * _By proof by contradiction._
//! 2. **PICK** s ∈ FromTime : ¬AtMostOne(IntoTime, δ𝐒(s) ∈ δ𝐓(x))
//!    * _Proof: Such time exists by <1>1._
//! 3. ∃ t1, t2 ∈ IntoTime : t1 ≠ t2 ∧ δ𝐒(s) ∈ δ𝐓(t1) ∧ δ𝐒(s) ∈ δ𝐓(t2)
//!     1. ¬(∀ x1, x2 ∈ X : (δ𝐒(s) ∈ δ𝐓(x1)) ∧ (δ𝐒(s) ∈ δ𝐓(x2)) ⇒ x1 = x2)
//!         * _Proof: By <1>2 and definition of AtMostOne._
//!     2. Q.E.D
//!         * _Proof: By <2>1, quantifier negation rules, and theorem of propositional logic ¬(P ⇒ Q) ≡ P ∧ ¬Q._
//! 4. **PICK** t1, t2 ∈ IntoTime : t1 < t2 ∧ δ𝐒(s) ∈ δ𝐓(t1) ∧ δ𝐒(s) ∈ δ𝐓(t2)
//!    * _Proof: By <1>3. Assume t1 < t2 without loss of generality._
//! 5. ¬(𝐑(t1) ⪯ s)
//!     1. **CASE** t1 = min(IntoTime)
//!         1. δ𝐓(t1) = sum{δ𝐒(s): !(𝐑(t1)) ⪯ s}
//!             * _Proof: By definition of δ𝐓(min)._
//!         2. δ𝐒(s) ∈ δ𝐓(t1)
//!             * _Proof: By <1>4._
//!         3. Q.E.D
//!             * _Proof: By <3>1 and <3>2._
//!     2. **CASE** t1 > min(IntoTime)
//!         1. **PICK** t1_prev = Predecessor(t1)
//!             * _Proof: Predecessor exists because the set {t: t < t1} is non-empty since it must contain at least min(IntoTime)._
//!         2. δ𝐓(t1) = sum{δ𝐒(s): (𝐑(t1_prev) ⪯ s) ∧ !(𝐑(t1) ⪯ s)}
//!             * _Proof: By definition of δ𝐓(t)._
//!         3. δ𝐒(s) ∈ δ𝐓(t1)
//!             * _Proof: By <1>4._
//!         3. Q.E.D
//!             * _Proof: By <3>2 and <3>3._
//!     3. Q.E.D
//!         * _Proof: From cases <2>1 and <2>2 which are exhaustive_
//! 6. **PICK** t2_prev ∈ IntoTime : t2_prev = Predecessor(t2)
//!    * _Proof: Predecessor exists because by <1>4 the set {t: t < t2} is non empty since it must contain at least t1._
//! 7. t1 ≤ t2_prev
//!    * _Proof: t1 ∈ {t: t < t2} and t2_prev is the maximum element of the set._
//! 8. 𝐑(t2) ⪯ s
//!     1. t2 > min(IntoTime)
//!         * _Proof: By <1>5._
//!     2. **PICK** t2_prev = Predecessor(t2)
//!         * _Proof: Predecessor exists because the set {t: t < t2} is non-empty since it must contain at least min(IntoTime)._
//!     3. δ𝐓(t) = sum{δ𝐒(s): (𝐑(t2_prev) ⪯ s) ∧ !(𝐑(t) ⪯ s)}
//!         * _Proof: By definition of δ𝐓(t)_
//!     4. δ𝐒(s) ∈ δ𝐓(t1)
//!         * _Proof: By <1>4._
//!     5. Q.E.D
//!         * _Proof: By <2>3 and <2>4._
//! 9. 𝐑(t1) ⪯ 𝐑(t2_prev)
//!     * _Proof: By <1>.7 and hypothesis on R_
//! 10. 𝐑(t1) ⪯ s
//!     * _Proof: By <1>8 and <1>9._
//! 11. Q.E.D
//!     * _Proof: By <1>5 and <1>10_

use std::cmp::{Ordering, Reverse};
use std::collections::VecDeque;
use std::collections::binary_heap::{BinaryHeap, PeekMut};
use std::iter::FromIterator;

use differential_dataflow::difference::Semigroup;
use differential_dataflow::lattice::Lattice;
use differential_dataflow::{AsCollection, ExchangeData, VecCollection, consolidation};
use mz_ore::Overflowing;
use mz_ore::collections::CollectionExt;
use timely::communication::{Pull, Push};
use timely::dataflow::Scope;
use timely::dataflow::channels::pact::Pipeline;
use timely::dataflow::operators::CapabilitySet;
use timely::dataflow::operators::capture::Event;
use timely::dataflow::operators::generic::OutputBuilder;
use timely::dataflow::operators::generic::builder_rc::OperatorBuilder;
use timely::order::{PartialOrder, TotalOrder};
use timely::progress::frontier::{AntichainRef, MutableAntichain};
use timely::progress::{Antichain, Timestamp};

/// Constructs an operator that reclocks a `source` collection varying with some time `FromTime`
/// into the corresponding `reclocked` collection varying over some time `IntoTime` using the
/// provided `remap` collection.
///
/// In order for the operator to read the `source` collection a `Pusher` is returned which can be
/// used with timely's capture facilities to connect a collection from a foreign scope to this
/// operator.
pub fn reclock<G, D, FromTime, IntoTime, R>(
    remap_collection: &VecCollection<G, FromTime, Overflowing<i64>>,
    as_of: Antichain<G::Timestamp>,
) -> (
    Box<dyn Push<Event<FromTime, Vec<(D, FromTime, R)>>>>,
    VecCollection<G, D, R>,
)
where
    G: Scope<Timestamp = IntoTime>,
    D: ExchangeData,
    FromTime: Timestamp,
    IntoTime: Timestamp + Lattice + TotalOrder,
    R: Semigroup + 'static,
{
    let mut scope = remap_collection.scope();
    let mut builder = OperatorBuilder::new("Reclock".into(), scope.clone());
    // Here we create a channel that can be used to send data from a foreign scope into this
    // operator. The channel is associated with this operator's address so that it is activated
    // every time events are available for consumption. This mechanism is similar to Timely's input
    // handles where data can be introduced into a timely scope from an exogenous source.
    let info = builder.operator_info();
    let channel_id = scope.new_identifier();
    let (pusher, mut events) =
        scope.pipeline::<Event<FromTime, Vec<(D, FromTime, R)>>>(channel_id, info.address);

    let mut remap_input = builder.new_input(&remap_collection.inner, Pipeline);
    let (output, reclocked) = builder.new_output();
    let mut output = OutputBuilder::from(output);

    builder.build(move |caps| {
        let mut capset = CapabilitySet::from_elem(caps.into_element());
        capset.downgrade(&as_of.borrow());

        // Received remap updates at times `into_time` greater or equal to `remap_input`'s input
        // frontier. As the input frontier advances, we drop elements out of this priority queue
        // and mint new associations.
        let mut pending_remap: BinaryHeap<Reverse<(IntoTime, FromTime, i64)>> = BinaryHeap::new();
        // A trace of `remap_input` that accumulates correctly for all times that are beyond
        // `remap_since` and not beyond `remap_upper`. The updates in `remap_trace` are maintained
        // in time order. An actual DD trace could be used here at the expense of a more
        // complicated API to traverse it. This is left for future work if the naive trace
        // maintenance implemented in this operator becomes problematic.
        let mut remap_upper = Antichain::from_elem(IntoTime::minimum());
        let mut remap_since = as_of.clone();
        let mut remap_trace = Vec::new();

        // A stash of source updates for which we don't know the corresponding binding yet.
        let mut deferred_source_updates: Vec<ChainBatch<_, _, _>> = Vec::new();
        // The frontier of the `events` input
        let mut source_frontier = MutableAntichain::new_bottom(FromTime::minimum());

        let mut binding_buffer = Vec::new();

        // Accumulation buffer for `remap_input` updates.
        use timely::progress::ChangeBatch;
        let mut remap_accum_buffer: ChangeBatch<(IntoTime, FromTime)> = ChangeBatch::new();

        // The operator drains `remap_input` and organizes new bindings that are not beyond
        // `remap_input`'s frontier into the time ordered `remap_trace`.
        //
        // All received data events can either be reclocked to a time included in the
        // `remap_trace`, or deferred until new associations are minted. Each data event that
        // happens at some `FromTime` is mapped to the first `IntoTime` whose associated antichain
        // is not less or equal to the input `FromTime`.
        //
        // As progress events are received from the `events` input, we can advance our
        // held capability to track the least `IntoTime` a newly received `FromTime` could possibly
        // map to and also compact the maintained `remap_trace` to that time.
        move |frontiers| {
            let Some(cap) = capset.get(0).cloned() else {
                return;
            };
            let mut output = output.activate();
            let mut session = output.session(&cap);

            // STEP 1. Accept new bindings into `pending_remap`.
            // Advance all `into` times by `as_of`, and consolidate all updates at that frontier.
            while let Some((_, data)) = remap_input.next() {
                for (from, mut into, diff) in data.drain(..) {
                    into.advance_by(as_of.borrow());
                    remap_accum_buffer.update((into, from), diff.into_inner());
                }
            }
            // Drain consolidated bindings into the `pending_remap` heap.
            // Only do this once any of the `remap_input` frontier has passed `as_of`.
            // For as long as the input frontier is less-equal `as_of`, we have no finalized times.
            if !PartialOrder::less_equal(&frontiers[0].frontier(), &as_of.borrow()) {
                for ((into, from), diff) in remap_accum_buffer.drain() {
                    pending_remap.push(Reverse((into, from, diff)));
                }
            }

            // STEP 2. Extract bindings not beyond `remap_frontier` and commit them into `remap_trace`.
            let prev_remap_upper =
                std::mem::replace(&mut remap_upper, frontiers[0].frontier().to_owned());
            while let Some(update) = pending_remap.peek_mut() {
                if !remap_upper.less_equal(&update.0.0) {
                    let Reverse((into, from, diff)) = PeekMut::pop(update);
                    remap_trace.push((from, into, diff));
                } else {
                    break;
                }
            }

            // STEP 3. Receive new data updates
            //         The `events` input describes arbitrary progress and data over `FromTime`,
            //         which must be translated to `IntoTime`. Each `FromTime` can be found as the
            //         first `IntoTime` associated with a `[FromTime]` that is not less or equal to
            //         the input `FromTime`. Received events that are not yet associated to an
            //         `IntoTime` are collected, and formed into a "chain batch": a sequence of
            //         chains that results from sorting the updates by `FromTime`, and then
            //         segmenting the sequence at elements where the partial order on `FromTime` is
            //         violated.
            let mut stash = Vec::new();
            // Consolidate progress updates before applying them to `source_frontier`, to avoid quadratic
            // behavior in overload scenarios.
            let mut change_batch = ChangeBatch::<FromTime, 2>::default();
            while let Some(event) = events.pull() {
                match event {
                    Event::Progress(changes) => {
                        change_batch.extend(changes.drain(..));
                    }
                    Event::Messages(_, data) => stash.append(data),
                }
            }
            source_frontier.update_iter(change_batch.drain());
            stash.sort_unstable_by(|(_, t1, _): &(D, FromTime, R), (_, t2, _)| t1.cmp(t2));
            let mut new_source_updates = ChainBatch::from_iter(stash);

            // STEP 4: Reclock new and deferred updates
            //         We are now ready to step through the remap bindings in time order and
            //         perform the following actions:
            //         4.1. Match `new_source_updates` against the entirety of bindings contained
            //              in the trace.
            //         4.2. Match `deferred_source_updates` against the bindings that were just
            //              added in the trace.
            //         4.3. Reclock `source_frontier` to calculate the new since frontier of the
            //              remap trace.
            //
            //         The steps above only make sense to perform if there are any times for which
            //         we can correctly accumulate the remap trace, which is what we check here.
            if remap_since.iter().all(|t| !remap_upper.less_equal(t)) {
                let mut cur_binding = MutableAntichain::new();

                let mut remap = remap_trace.iter().peekable();
                let mut reclocked_source_frontier = remap_upper.clone();

                // We go over all the times for which we might need to output data at. These times
                // are restrticted to the times at which there exists an update in `remap_trace`
                // and the minimum timestamp for the case where `remap_trace` is completely empty,
                // in which case the minimum timestamp maps to the empty `FromTime` frontier and
                // therefore all data events map to that minimum timestamp.
                //
                // The approach taken here will take time proportional to the number of elements in
                // `remap_trace`. During development an alternative approach was considered where
                // the updates in `remap_trace` are instead fully materialized into an ordered list
                // of antichains in which every data update can be binary searched into. The are
                // two concerns with this alternative approach that led to preferring this one:
                // 1. Materializing very wide antichains with small differences between them
                //    needs memory proportial to the number of bindings times the width of the
                //    antichain.
                // 2. It locks in the requirement of a totally ordered target timestamp since only
                //    in that case can one binary search a binding.
                // The linear scan is expected to be fine due to the run-to-completion nature of
                // the operator since its cost is amortized among the number of outstanding
                // updates.
                let mut min_time = IntoTime::minimum();
                min_time.advance_by(remap_since.borrow());
                let mut prev_cur_time = None;
                let mut interesting_times = std::iter::once(&min_time)
                    .chain(remap_trace.iter().map(|(_, t, _)| t))
                    .filter(|&v| {
                        let prev = prev_cur_time.replace(v);
                        prev != prev_cur_time
                    });
                let mut frontier_reclocked = false;
                while !(new_source_updates.is_empty()
                    && deferred_source_updates.is_empty()
                    && frontier_reclocked)
                    && let Some(cur_time) = interesting_times.next()
                {
                    // 4.0. Load updates of `cur_time` from the trace into `cur_binding` to
                    //      construct the `[FromTime]` frontier that `cur_time` maps to.
                    while let Some((t_from, _, diff)) = remap.next_if(|(_, t, _)| t == cur_time) {
                        binding_buffer.push((t_from.clone(), *diff));
                    }
                    cur_binding.update_iter(binding_buffer.drain(..));
                    let cur_binding = cur_binding.frontier();

                    // 4.1. Extract updates from `new_source_updates`
                    for (data, _, diff) in new_source_updates.extract(cur_binding) {
                        session.give((data, cur_time.clone(), diff));
                    }

                    // 4.2. Extract updates from `deferred_source_updates`.
                    //      The deferred updates contain all updates that were not able to be
                    //      reclocked with the bindings until `prev_remap_upper`. For this reason
                    //      we only need to reconsider these updates when we start looking at new
                    //      bindings, i.e bindings that are beyond `prev_remap_upper`.
                    if prev_remap_upper.less_equal(cur_time) {
                        deferred_source_updates.retain_mut(|batch| {
                            for (data, _, diff) in batch.extract(cur_binding) {
                                session.give((data, cur_time.clone(), diff));
                            }
                            // Retain non-empty batches
                            !batch.is_empty()
                        })
                    }

                    // 4.3. Reclock `source_frontier`
                    //      If any FromTime in source frontier could possibly be reclocked to this
                    //      binding then we must maintain our capability to emit data at that time
                    //      and not compact past it. Since we iterate over this loop in time order
                    //      and IntoTime is a total order we only need to perform this step once.
                    //      Once a `cur_time` is inserted into `reclocked_source_frontier` no more
                    //      changes can be made to the frontier by inserting times later in the
                    //      loop.
                    if !frontier_reclocked
                        && source_frontier
                            .frontier()
                            .iter()
                            .any(|t| !cur_binding.less_equal(t))
                    {
                        reclocked_source_frontier.insert(cur_time.clone());
                        frontier_reclocked = true;
                    }
                }

                // STEP 5. Downgrade capability and compact remap trace
                capset.downgrade(&reclocked_source_frontier.borrow());
                remap_since = reclocked_source_frontier;
                for (_, t, _) in remap_trace.iter_mut() {
                    t.advance_by(remap_since.borrow());
                }
                consolidation::consolidate_updates(&mut remap_trace);
                remap_trace
                    .sort_unstable_by(|(_, t1, _): &(_, IntoTime, _), (_, t2, _)| t1.cmp(t2));

                // If using less than a quarter of the capacity, shrink the container. To avoid having
                // to resize the container on a subsequent push, shrink to 2x the length, which is
                // what push would grow it to.
                if remap_trace.len() < remap_trace.capacity() / 4 {
                    remap_trace.shrink_to(remap_trace.len() * 2);
                }
            }

            // STEP 6. Tidy up deferred updates
            //         Deferred updates are represented as a list of chain batches where each batch
            //         contains two times the updates of the batch proceeding it. This organization
            //         leads to a logarithmic number of batches with respect to the outstanding
            //         number of updates.
            deferred_source_updates.sort_unstable_by_key(|b| Reverse(b.len()));
            if !new_source_updates.is_empty() {
                deferred_source_updates.push(new_source_updates);
            }
            let dsu = &mut deferred_source_updates;
            while dsu.len() > 1 && (dsu[dsu.len() - 1].len() >= dsu[dsu.len() - 2].len() / 2) {
                let a = dsu.pop().unwrap();
                let b = dsu.pop().unwrap();
                dsu.push(a.merge_with(b));
            }

            // If using less than a quarter of the capacity, shrink the container. To avoid having
            // to resize the container on a subsequent push, shrink to 2x the length, which is
            // what push would grow it to.
            if deferred_source_updates.len() < deferred_source_updates.capacity() / 4 {
                deferred_source_updates.shrink_to(deferred_source_updates.len() * 2);
            }
        }
    });

    (Box::new(pusher), reclocked.as_collection())
}

/// A batch of differential updates that vary over some partial order. This type maintains the data
/// as a set of chains that allows for efficient extraction of batches given a frontier.
#[derive(Debug, PartialEq)]
struct ChainBatch<D, T, R> {
    /// A list of chains (sets of mutually comparable times) sorted by the partial order.
    chains: Vec<VecDeque<(D, T, R)>>,
}

impl<D, T: Timestamp, R> ChainBatch<D, T, R> {
    /// Extracts all updates with time not greater or equal to any time in `upper`.
    fn extract<'a>(
        &'a mut self,
        upper: AntichainRef<'a, T>,
    ) -> impl Iterator<Item = (D, T, R)> + 'a {
        self.chains.retain(|chain| !chain.is_empty());
        self.chains.iter_mut().flat_map(move |chain| {
            // A chain is a sorted list of mutually comparable elements so we keep extracting
            // elements that are not beyond upper.
            std::iter::from_fn(move || {
                let (_, into, _) = chain.front()?;
                if !upper.less_equal(into) {
                    chain.pop_front()
                } else {
                    None
                }
            })
        })
    }

    fn merge_with(
        mut self: ChainBatch<D, T, R>,
        mut other: ChainBatch<D, T, R>,
    ) -> ChainBatch<D, T, R>
    where
        D: ExchangeData,
        T: Timestamp,
        R: Semigroup,
    {
        let mut updates1 = self.chains.drain(..).flatten().peekable();
        let mut updates2 = other.chains.drain(..).flatten().peekable();

        let merged = std::iter::from_fn(|| {
            match (updates1.peek(), updates2.peek()) {
                (Some((d1, t1, _)), Some((d2, t2, _))) => {
                    match (t1, d1).cmp(&(t2, d2)) {
                        Ordering::Less => updates1.next(),
                        Ordering::Greater => updates2.next(),
                        // If the same (d, t) pair is found, consolidate their diffs
                        Ordering::Equal => {
                            let (d1, t1, mut r1) = updates1.next().unwrap();
                            while let Some((_, _, r)) =
                                updates1.next_if(|(d, t, _)| (d, t) == (&d1, &t1))
                            {
                                r1.plus_equals(&r);
                            }
                            while let Some((_, _, r)) =
                                updates2.next_if(|(d, t, _)| (d, t) == (&d1, &t1))
                            {
                                r1.plus_equals(&r);
                            }
                            Some((d1, t1, r1))
                        }
                    }
                }
                (Some(_), None) => updates1.next(),
                (None, Some(_)) => updates2.next(),
                (None, None) => None,
            }
        });

        ChainBatch::from_iter(merged.filter(|(_, _, r)| !r.is_zero()))
    }

    /// Returns the number of updates in the batch.
    fn len(&self) -> usize {
        self.chains.iter().map(|chain| chain.len()).sum()
    }

    /// Returns true if the batch contains no updates.
    fn is_empty(&self) -> bool {
        self.len() == 0
    }
}

impl<D, T: Timestamp, R> FromIterator<(D, T, R)> for ChainBatch<D, T, R> {
    /// Computes the chain decomposition of updates according to the partial order `T`.
    fn from_iter<I: IntoIterator<Item = (D, T, R)>>(updates: I) -> Self {
        let mut chains = vec![];
        let mut updates = updates.into_iter();
        if let Some((d, t, r)) = updates.next() {
            let mut chain = VecDeque::new();
            chain.push_back((d, t, r));
            for (d, t, r) in updates {
                let prev_t = &chain[chain.len() - 1].1;
                if !PartialOrder::less_equal(prev_t, &t) {
                    chains.push(chain);
                    chain = VecDeque::new();
                }
                chain.push_back((d, t, r));
            }
            chains.push(chain);
        }
        Self { chains }
    }
}

#[cfg(test)]
mod test {
    use std::sync::atomic::AtomicUsize;
    use std::sync::mpsc::{Receiver, TryRecvError};

    use differential_dataflow::consolidation;
    use differential_dataflow::input::{Input, InputSession};
    use serde::{Deserialize, Serialize};
    use timely::communication::allocator::Thread;
    use timely::dataflow::operators::capture::{Event, Extract};
    use timely::dataflow::operators::unordered_input::UnorderedHandle;
    use timely::dataflow::operators::{ActivateCapability, Capture, UnorderedInput};
    use timely::progress::PathSummary;
    use timely::progress::timestamp::Refines;
    use timely::worker::Worker;

    use crate::capture::PusherCapture;
    use crate::order::Partitioned;

    use super::*;

    type Diff = Overflowing<i64>;
    type FromTime = Partitioned<u64, u64>;
    type IntoTime = u64;
    type BindingHandle<FromTime> = InputSession<IntoTime, FromTime, Diff>;
    type DataHandle<D, FromTime> = (
        UnorderedHandle<FromTime, (D, FromTime, Diff)>,
        ActivateCapability<FromTime>,
    );
    type ReclockedStream<D> = Receiver<Event<IntoTime, Vec<(D, IntoTime, Diff)>>>;

    /// A helper function that sets up a dataflow program to test the reclocking operator. Each
    /// test provides a test logic closure which accepts four arguments:
    ///
    /// * A reference to the worker that allows the test to step the computation
    /// * A [`BindingHandle`] that allows the test to manipulate the remap bindings
    /// * A [`DataHandle`] that allows the test to submit the data to be reclocked
    /// * A [`ReclockedStream`] that allows observing the result of the reclocking process
    ///
    /// Note that the `DataHandle` contains a capability that should be dropped or downgraded before
    /// calling [`step`] to process data at the time.
    fn harness<FromTime, D, F, R>(as_of: Antichain<IntoTime>, test_logic: F) -> R
    where
        FromTime: Timestamp + Refines<()>,
        D: ExchangeData,
        F: FnOnce(
                &mut Worker<Thread>,
                BindingHandle<FromTime>,
                DataHandle<D, FromTime>,
                ReclockedStream<D>,
            ) -> R
            + Send
            + Sync
            + 'static,
        R: Send + 'static,
    {
        timely::execute_directly(move |worker| {
            let (bindings, data, data_cap, reclocked) = worker.dataflow::<(), _, _>(|scope| {
                let (bindings, data_pusher, reclocked) =
                    scope.scoped::<IntoTime, _, _>("IntoScope", move |scope| {
                        let (binding_handle, binding_collection) = scope.new_collection();
                        let (data_pusher, reclocked_collection) =
                            reclock(&binding_collection, as_of);
                        let reclocked_capture = reclocked_collection.inner.capture();
                        (binding_handle, data_pusher, reclocked_capture)
                    });

                let (data, data_cap) = scope.scoped::<FromTime, _, _>("FromScope", move |scope| {
                    let ((handle, cap), data) = scope.new_unordered_input();
                    data.capture_into(PusherCapture(data_pusher));
                    (handle, cap)
                });

                (bindings, data, data_cap, reclocked)
            });

            test_logic(worker, bindings, (data, data_cap), reclocked)
        })
    }

    /// Steps the worker four times which is the required number of times for both data and
    /// frontier updates to propagate across the two scopes and into the probing channels.
    fn step(worker: &mut Worker<Thread>) {
        for _ in 0..4 {
            worker.step();
        }
    }

    #[mz_ore::test]
    fn basic_reclocking() {
        let as_of = Antichain::from_elem(IntoTime::minimum());
        harness::<FromTime, _, _, _>(
            as_of,
            |worker, bindings, (mut data, data_cap), reclocked| {
                // Reclock everything at the minimum IntoTime
                bindings.close();
                data.activate()
                    .session(&data_cap)
                    .give(('a', Partitioned::minimum(), Diff::ONE));
                drop(data_cap);
                step(worker);
                let extracted = reclocked.extract();
                let expected = vec![(0, vec![('a', 0, Diff::ONE)])];
                assert_eq!(extracted, expected);
            },
        )
    }

    /// Generates a `Partitioned<u64, u64>` Antichain where all the provided
    /// partitions are at the specified offset and the gaps in between are filled with range
    /// timestamps at offset zero.
    fn partitioned_frontier<I>(items: I) -> Antichain<Partitioned<u64, u64>>
    where
        I: IntoIterator<Item = (u64, u64)>,
    {
        let mut frontier = Antichain::new();
        let mut prev = 0;
        for (pid, offset) in items {
            if prev < pid {
                frontier.insert(Partitioned::new_range(prev, pid - 1, 0));
            }
            frontier.insert(Partitioned::new_singleton(pid, offset));
            prev = pid + 1
        }
        frontier.insert(Partitioned::new_range(prev, u64::MAX, 0));
        frontier
    }

    #[mz_ore::test]
    fn test_basic_usage() {
        let as_of = Antichain::from_elem(IntoTime::minimum());
        harness(
            as_of,
            |worker, mut bindings, (mut data, data_cap), reclocked| {
                // Reclock offsets 1 and 3 to timestamp 1000
                bindings.update_at(Partitioned::minimum(), 0, Diff::ONE);
                bindings.update_at(Partitioned::minimum(), 1000, Diff::MINUS_ONE);
                for time in partitioned_frontier([(0, 4)]) {
                    bindings.update_at(time, 1000, Diff::ONE);
                }
                bindings.advance_to(1001);
                bindings.flush();
                data.activate().session(&data_cap).give_iterator(
                    vec![
                        (1, Partitioned::new_singleton(0, 1), Diff::ONE),
                        (1, Partitioned::new_singleton(0, 1), Diff::ONE),
                        (3, Partitioned::new_singleton(0, 3), Diff::ONE),
                    ]
                    .into_iter(),
                );

                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Messages(
                        0u64,
                        vec![
                            (1, 1000, Diff::ONE),
                            (1, 1000, Diff::ONE),
                            (3, 1000, Diff::ONE)
                        ]
                    ))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(0, -1), (1000, 1)]))
                );

                // Reclock more messages for offsets 3 to the same timestamp
                data.activate().session(&data_cap).give_iterator(
                    vec![
                        (3, Partitioned::new_singleton(0, 3), Diff::ONE),
                        (3, Partitioned::new_singleton(0, 3), Diff::ONE),
                    ]
                    .into_iter(),
                );
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Messages(
                        1000u64,
                        vec![(3, 1000, Diff::ONE), (3, 1000, Diff::ONE)]
                    ))
                );

                // Drop the capability which should advance the reclocked frontier to 1001.
                drop(data_cap);
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(1000, -1), (1001, 1)]))
                );
            },
        );
    }

    #[mz_ore::test]
    fn test_reclock_frontier() {
        let as_of = Antichain::from_elem(IntoTime::minimum());
        harness::<_, (), _, _>(
            as_of,
            |worker, mut bindings, (_data, data_cap), reclocked| {
                // Initialize the bindings such that the minimum IntoTime contains the minimum FromTime
                // frontier.
                bindings.update_at(Partitioned::minimum(), 0, Diff::ONE);
                bindings.advance_to(1);
                bindings.flush();
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(0, -1), (1, 1)]))
                );

                // Mint a couple of bindings for multiple partitions
                bindings.update_at(Partitioned::minimum(), 1000, Diff::MINUS_ONE);
                for time in partitioned_frontier([(1, 10)]) {
                    bindings.update_at(time.clone(), 1000, Diff::ONE);
                    bindings.update_at(time, 2000, Diff::MINUS_ONE);
                }
                for time in partitioned_frontier([(1, 10), (2, 10)]) {
                    bindings.update_at(time, 2000, Diff::ONE);
                }
                bindings.advance_to(2001);
                bindings.flush();

                // The initial frontier should now map to the minimum between the two partitions
                step(worker);
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(1, -1), (1000, 1)]))
                );

                // Downgrade data frontier such that only one of the partitions is advanced
                let mut part1_cap = data_cap.delayed(&Partitioned::new_singleton(1, 9));
                let mut part2_cap = data_cap.delayed(&Partitioned::new_singleton(2, 0));
                let _rest_cap = data_cap.delayed(&Partitioned::new_range(3, u64::MAX, 0));
                drop(data_cap);
                step(worker);
                assert_eq!(reclocked.try_recv(), Err(TryRecvError::Empty));

                // Downgrade the data frontier past the first binding
                part1_cap.downgrade(&Partitioned::new_singleton(1, 10));
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(1000, -1), (2000, 1)]))
                );

                // Downgrade the data frontier past the second binding
                part2_cap.downgrade(&Partitioned::new_singleton(2, 10));
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(2000, -1), (2001, 1)]))
                );

                // Advance the binding frontier and confirm that we get to the next timestamp
                bindings.advance_to(3001);
                bindings.flush();
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(2001, -1), (3001, 1)]))
                );
            },
        );
    }

    #[mz_ore::test]
    fn test_reclock() {
        let as_of = Antichain::from_elem(IntoTime::minimum());
        harness(
            as_of,
            |worker, mut bindings, (mut data, data_cap), reclocked| {
                // Initialize the bindings such that the minimum IntoTime contains the minimum FromTime
                // frontier.
                bindings.update_at(Partitioned::minimum(), 0, Diff::ONE);

                // Setup more precise capabilities for the rest of the test
                let mut part0_cap = data_cap.delayed(&Partitioned::new_singleton(0, 0));
                let rest_cap = data_cap.delayed(&Partitioned::new_range(1, u64::MAX, 0));
                drop(data_cap);

                // Reclock offsets 1 and 2 to timestamp 1000
                data.activate().session(&part0_cap).give_iterator(
                    vec![
                        (1, Partitioned::new_singleton(0, 1), Diff::ONE),
                        (2, Partitioned::new_singleton(0, 2), Diff::ONE),
                    ]
                    .into_iter(),
                );

                part0_cap.downgrade(&Partitioned::new_singleton(0, 3));
                bindings.update_at(Partitioned::minimum(), 1000, Diff::MINUS_ONE);
                bindings.update_at(part0_cap.time().clone(), 1000, Diff::ONE);
                bindings.update_at(rest_cap.time().clone(), 1000, Diff::ONE);
                bindings.advance_to(1001);
                bindings.flush();
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Messages(
                        0,
                        vec![(1, 1000, Diff::ONE), (2, 1000, Diff::ONE)]
                    ))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(0, -1), (1000, 1)]))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(1000, -1), (1001, 1)]))
                );

                // Reclock offsets 3 and 4 to timestamp 2000
                data.activate().session(&part0_cap).give_iterator(
                    vec![
                        (3, Partitioned::new_singleton(0, 3), Diff::ONE),
                        (3, Partitioned::new_singleton(0, 3), Diff::ONE),
                        (4, Partitioned::new_singleton(0, 4), Diff::ONE),
                    ]
                    .into_iter(),
                );
                bindings.update_at(part0_cap.time().clone(), 2000, Diff::MINUS_ONE);
                part0_cap.downgrade(&Partitioned::new_singleton(0, 5));
                bindings.update_at(part0_cap.time().clone(), 2000, Diff::ONE);
                bindings.advance_to(2001);
                bindings.flush();
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Messages(
                        1001,
                        vec![
                            (3, 2000, Diff::ONE),
                            (3, 2000, Diff::ONE),
                            (4, 2000, Diff::ONE)
                        ]
                    ))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(1001, -1), (2000, 1)]))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(2000, -1), (2001, 1)]))
                );
            },
        );
    }

    #[mz_ore::test]
    fn test_reclock_gh16318() {
        let as_of = Antichain::from_elem(IntoTime::minimum());
        harness(
            as_of,
            |worker, mut bindings, (mut data, data_cap), reclocked| {
                // Initialize the bindings such that the minimum IntoTime contains the minimum FromTime
                // frontier.
                bindings.update_at(Partitioned::minimum(), 0, Diff::ONE);
                // First mint bindings for 0 at timestamp 1000
                bindings.update_at(Partitioned::minimum(), 1000, Diff::MINUS_ONE);
                for time in partitioned_frontier([(0, 50)]) {
                    bindings.update_at(time, 1000, Diff::ONE);
                }
                // Then only for 1 at timestamp 2000
                for time in partitioned_frontier([(0, 50)]) {
                    bindings.update_at(time, 2000, Diff::MINUS_ONE);
                }
                for time in partitioned_frontier([(0, 50), (1, 50)]) {
                    bindings.update_at(time, 2000, Diff::ONE);
                }
                // Then again only for 0 at timestamp 3000
                for time in partitioned_frontier([(0, 50), (1, 50)]) {
                    bindings.update_at(time, 3000, Diff::MINUS_ONE);
                }
                for time in partitioned_frontier([(0, 100), (1, 50)]) {
                    bindings.update_at(time, 3000, Diff::ONE);
                }
                bindings.advance_to(3001);
                bindings.flush();

                // Reclockng (0, 50) must ignore the updates on the FromTime frontier that happened at
                // timestamp 2000 since those are completely unrelated
                data.activate().session(&data_cap).give((
                    50,
                    Partitioned::new_singleton(0, 50),
                    Diff::ONE,
                ));
                drop(data_cap);
                step(worker);
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Messages(0, vec![(50, 3000, Diff::ONE),]))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(0, -1), (1000, 1)]))
                );
                assert_eq!(
                    reclocked.try_recv(),
                    Ok(Event::Progress(vec![(1000, -1), (3001, 1)]))
                );
            },
        );
    }

    /// Test that compact(reclock(remap, source)) == reclock(compact(remap), source)
    #[mz_ore::test]
    fn test_compaction() {
        let mut remap = vec![];
        remap.push((Partitioned::minimum(), 0, Diff::ONE));
        // Reclock offsets 1 and 2 to timestamp 1000
        remap.push((Partitioned::minimum(), 1000, Diff::MINUS_ONE));
        for time in partitioned_frontier([(0, 3)]) {
            remap.push((time, 1000, Diff::ONE));
        }
        // Reclock offsets 3 and 4 to timestamp 2000
        for time in partitioned_frontier([(0, 3)]) {
            remap.push((time, 2000, Diff::MINUS_ONE));
        }
        for time in partitioned_frontier([(0, 5)]) {
            remap.push((time, 2000, Diff::ONE));
        }

        let source_updates = vec![
            (1, Partitioned::new_singleton(0, 1), Diff::ONE),
            (2, Partitioned::new_singleton(0, 2), Diff::ONE),
            (3, Partitioned::new_singleton(0, 3), Diff::ONE),
            (4, Partitioned::new_singleton(0, 4), Diff::ONE),
        ];

        let since = Antichain::from_elem(1500);

        // Compute reclock(remap, source)
        let as_of = Antichain::from_elem(IntoTime::minimum());
        let remap1 = remap.clone();
        let source_updates1 = source_updates.clone();
        let reclock_remap = harness(
            as_of,
            move |worker, mut bindings, (mut data, data_cap), reclocked| {
                for (from_ts, into_ts, diff) in remap1 {
                    bindings.update_at(from_ts, into_ts, diff);
                }
                bindings.close();
                data.activate()
                    .session(&data_cap)
                    .give_iterator(source_updates1.iter().cloned());
                drop(data_cap);
                step(worker);
                reclocked.extract()
            },
        );
        // Compute compact(reclock(remap, source))
        let mut compact_reclock_remap = reclock_remap;
        for (t, updates) in compact_reclock_remap.iter_mut() {
            t.advance_by(since.borrow());
            for (_, t, _) in updates.iter_mut() {
                t.advance_by(since.borrow());
            }
        }

        // Compute compact(remap)
        let mut compact_remap = remap;
        for (_, t, _) in compact_remap.iter_mut() {
            t.advance_by(since.borrow());
        }
        consolidation::consolidate_updates(&mut compact_remap);
        // Compute reclock(compact(remap), source)
        let reclock_compact_remap = harness(
            since,
            move |worker, mut bindings, (mut data, data_cap), reclocked| {
                for (from_ts, into_ts, diff) in compact_remap {
                    bindings.update_at(from_ts, into_ts, diff);
                }
                bindings.close();
                data.activate()
                    .session(&data_cap)
                    .give_iterator(source_updates.iter().cloned());
                drop(data_cap);
                step(worker);
                reclocked.extract()
            },
        );

        let expected = vec![(
            1500,
            vec![
                (1, 1500, Diff::ONE),
                (2, 1500, Diff::ONE),
                (3, 2000, Diff::ONE),
                (4, 2000, Diff::ONE),
            ],
        )];
        assert_eq!(expected, reclock_compact_remap);
        assert_eq!(expected, compact_reclock_remap);
    }

    #[mz_ore::test]
    fn test_chainbatch_merge() {
        let a = ChainBatch::from_iter([('a', 0, 1)]);
        let b = ChainBatch::from_iter([('a', 0, -1), ('a', 1, 1)]);
        assert_eq!(a.merge_with(b), ChainBatch::from_iter([('a', 1, 1)]));
    }

    #[mz_ore::test]
    #[cfg_attr(miri, ignore)] // too slow
    fn test_binding_consolidation() {
        use std::sync::atomic::Ordering;

        #[derive(Debug, PartialEq, Eq, PartialOrd, Ord, Hash, Serialize, Deserialize)]
        struct Time(u64);

        // A counter of the number of active Time instances
        static INSTANCES: AtomicUsize = AtomicUsize::new(0);

        impl Time {
            fn new(time: u64) -> Self {
                INSTANCES.fetch_add(1, Ordering::Relaxed);
                Self(time)
            }
        }

        impl Clone for Time {
            fn clone(&self) -> Self {
                INSTANCES.fetch_add(1, Ordering::Relaxed);
                Self(self.0)
            }
        }

        impl Drop for Time {
            fn drop(&mut self) {
                INSTANCES.fetch_sub(1, Ordering::Relaxed);
            }
        }

        impl Timestamp for Time {
            type Summary = ();

            fn minimum() -> Self {
                Time::new(0)
            }
        }

        impl PathSummary<Time> for () {
            fn results_in(&self, src: &Time) -> Option<Time> {
                Some(src.clone())
            }

            fn followed_by(&self, _other: &()) -> Option<Self> {
                Some(())
            }
        }

        impl Refines<()> for Time {
            fn to_inner(_: ()) -> Self {
                Self::minimum()
            }
            fn to_outer(self) -> () {}
            fn summarize(_path: ()) {}
        }

        impl PartialOrder for Time {
            fn less_equal(&self, other: &Self) -> bool {
                self.0.less_equal(&other.0)
            }
        }

        let as_of = 1000;

        // Test that supplying a single big batch of unconsolidated bindings gets
        // consolidated after a single worker step.
        harness::<Time, u64, _, _>(
            Antichain::from_elem(as_of),
            move |worker, mut bindings, _, _| {
                step(worker);
                let instances_before = INSTANCES.load(Ordering::Relaxed);
                for ts in 0..as_of {
                    if ts > 0 {
                        bindings.update_at(Time::new(ts - 1), ts, Diff::MINUS_ONE);
                    }
                    bindings.update_at(Time::new(ts), ts, Diff::ONE);
                }
                bindings.advance_to(as_of);
                bindings.flush();
                step(worker);
                let instances_after = INSTANCES.load(Ordering::Relaxed);
                // The extra instances live in a ChangeBatch which considers compaction when more
                // than 32 elements are inside.
                assert!(instances_after - instances_before < 32);
            },
        );

        // Test that a slow feed of uncompacted bindings over multiple steps never leads to an
        // excessive number of bindings held in memory.
        harness::<Time, u64, _, _>(
            Antichain::from_elem(as_of),
            move |worker, mut bindings, _, _| {
                step(worker);
                let instances_before = INSTANCES.load(Ordering::Relaxed);
                for ts in 0..as_of {
                    if ts > 0 {
                        bindings.update_at(Time::new(ts - 1), ts, Diff::MINUS_ONE);
                    }
                    bindings.update_at(Time::new(ts), ts, Diff::ONE);
                    bindings.advance_to(ts + 1);
                    bindings.flush();
                    step(worker);
                    let instances_now = INSTANCES.load(Ordering::Relaxed);
                    // The extra instances live in a ChangeBatch which considers compaction when
                    // more than 32 elements are inside.
                    assert!(instances_now - instances_before < 32);
                }
            },
        );
    }

    #[cfg(feature = "count-allocations")]
    #[mz_ore::test]
    #[cfg_attr(miri, ignore)] // too slow
    fn test_shrinking() {
        let as_of = 1000_u64;

        // This workflow accumulates updates in remap_trace, advances the source frontier,
        // and validates that memory was reclaimed.  To avoid errant test failures due to
        // optimizations, this only validates that memory is reclaimed, not how much.
        harness::<FromTime, u64, _, _>(
            Antichain::from_elem(0),
            move |worker, mut bindings, (_data, mut data_cap), _| {
                let info1 = allocation_counter::measure(|| {
                    step(worker);
                    for ts in 0..as_of {
                        if ts > 0 {
                            bindings.update_at(
                                Partitioned::new_singleton(0, ts - 1),
                                ts,
                                Diff::MINUS_ONE,
                            );
                        }
                        bindings.update_at(Partitioned::new_singleton(0, ts), ts, Diff::ONE);
                        bindings.advance_to(ts + 1);
                        bindings.flush();
                        step(worker);
                    }
                });
                println!("info = {info1:?}");

                let info2 = allocation_counter::measure(|| {
                    data_cap.downgrade(&Partitioned::new_singleton(0, as_of));
                    step(worker);
                });
                println!("info = {info2:?}");
                assert!(info2.bytes_current < 0);
            },
        );
    }
}