rest of results chapter

6 files changed, 247 insertions, 116 deletions

diff --git a/Tasks.org b/Tasks.org
index bc7bd59..b46518f 100644
--- a/Tasks.org
+++ b/Tasks.org
@@ -40,7 +40,7 @@
 
 *** DONE Lead to next chapter
 
-** TODO Design
+** DONE Design
 
 *** DONE Usage Example
 
@@ -66,17 +66,13 @@
 
 **** DONE Segmentation
 
-**** TODO Limitations w/ pre-benchmark steps
-
 *** DONE Selection process & adaptive containers
 
 **** DONE Selection process
 
-**** TODO Adaptive container detection
-
-**** TODO Code generation
+**** DONE Adaptive container detection
 
-** TODO Implementation
+** DONE Implementation
 
 *** DONE Modifications to Primrose
 
@@ -94,7 +90,7 @@
 
 **** DONE Chosen benchmarks
 
-**** TODO Fitting
+**** DONE Fitting
 
 *** DONE Profiling wrapper
 
@@ -110,21 +106,7 @@
 
 **** DONE Generated code (opaque types)
 
-*** TODO Misc Concerns
-
-**** TODO Explain cargo's role in rust projects & how it is integrated
-
-**** TODO Caching and stuff
-
-**** TODO Ease of use
-
-**** TODO Integration w/ Cargo
-
-***** TODO Metadata fetching
-
-***** TODO Caching of build dependencies
-
-** TODO Results & Analysis
+** DONE Results & Analysis
 
 *** DONE Testing setup, benchmarking rationale
 
@@ -150,18 +132,18 @@
 
 **** DONE Summarise predicted versus actual
 
-**** TODO Evaluate performance
+**** DONE Evaluate performance
 
-**** TODO Comment on distribution of best implementation
+**** DONE Comment on distribution of best implementation
 
-**** TODO Surprising ones / Explain failures
+**** DONE Surprising ones / Explain failures
 
-*** TODO Performance of adaptive containers
+*** DONE Performance of adaptive containers
 
-**** TODO Find where adaptive containers get suggested
+**** DONE Find where adaptive containers get suggested
 
-**** TODO Comment on relative performance speedup
+**** DONE Comment on relative performance speedup
 
-**** TODO Suggest future improvements?
+**** DONE Suggest future improvements?
 
 ** TODO Conclusion
diff --git a/analysis/vis.livemd b/analysis/vis.livemd
index 0aa2711..6a0a7c5 100644
--- a/analysis/vis.livemd
+++ b/analysis/vis.livemd
@@ -245,6 +245,21 @@ graph
 
 ```elixir
 graph =
+  CostModel.plot(
+    cost_models,
+    cost_model_points,
+    ["HashSet", "BTreeSet"],
+    "insert",
+    ns: 1..500//10,
+    # y_domain: [0, 4000],
+    draw_points: true
+  )
+
+graph
+```
+
+```elixir
+graph =
   CostModel.split_plot(
     cost_models,
     cost_model_points,
@@ -457,29 +472,22 @@ DF.n_rows(singular_benchmarks)
 ```
 
 ```elixir
-# Best and predicted best implementation for each container type
-selection_comparison =
-  singular_benchmarks
-  |> DF.explode("using")
-  |> DF.unnest("using")
-  |> DF.group_by(["proj"])
-  |> DF.filter(time == min(time))
-  |> DF.join(
-    cost_estimates
-    |> DF.filter(not contains(impl, "until"))
-    |> DF.group_by(["proj", "ctn"])
-    |> DF.filter(cost == min(cost))
-    |> DF.rename(%{"impl" => "predicted_impl"})
-  )
-  |> DF.select(["proj", "ctn", "impl", "predicted_impl"])
-  |> DF.rename(%{"impl" => "best_impl"})
+display_using = fn using ->
+  using
+  |> Enum.map(fn %{"ctn" => ctn, "impl" => impl} -> ctn <> "=" <> impl end)
+  |> Enum.join(", ")
+end
 ```
 
 ```elixir
 # Tools for printing out latex
 defmodule Latex do
-  def escape_latex(str) do
-    String.replace(str, ~r/(\\|{|}|_|\^|#|&|\$|%|~)/, "\\\\\\1")
+  def escape_latex(val) do
+    if is_number(val) do
+      "$" <> to_string(val) <> "$"
+    else
+      String.replace(to_string(val), ~r/(\\|{|}|_|\^|#|&|\$|%|~)/, "\\\\\\1")
+    end
   end
 
   def table(df) do
@@ -493,14 +501,51 @@ defmodule Latex do
       (DF.to_rows(df)
        |> Enum.map(fn row ->
          cols
-         |> Enum.map(&escape_latex(Kernel.to_string(row[&1])))
+         |> Enum.map(&escape_latex(row[&1]))
          |> Enum.join(" & ")
        end)
        |> Enum.join(" \\\\\n")) <>
       " \\\\\n\\end{tabular}"
   end
 end
+```
+
+```elixir
+singular_benchmarks
+|> DF.group_by("proj")
+|> DF.summarise(max: max(time), min: min(time))
+|> DF.mutate(spread: round((max - min) * ^(10 ** -6), 2), slowdown: round(max / min - 1, 1))
+|> DF.discard(["max", "min"])
+|> DF.sort_by(proj)
+|> DF.rename(%{
+  "proj" => "Project",
+  "spread" => "Maximum slowdown (ms)",
+  "slowdown" => "Maximum relative slowdown"
+})
+|> Latex.table()
+|> IO.puts()
+```
 
+```elixir
+# Best and predicted best implementation for each container type
+selection_comparison =
+  singular_benchmarks
+  |> DF.explode("using")
+  |> DF.unnest("using")
+  |> DF.group_by(["proj"])
+  |> DF.filter(time == min(time))
+  |> DF.join(
+    cost_estimates
+    |> DF.filter(not contains(impl, "until"))
+    |> DF.group_by(["proj", "ctn"])
+    |> DF.filter(cost == min(cost))
+    |> DF.rename(%{"impl" => "predicted_impl"})
+  )
+  |> DF.select(["proj", "ctn", "impl", "predicted_impl"])
+  |> DF.rename(%{"impl" => "best_impl"})
+```
+
+```elixir
 Latex.table(selection_comparison)
 
 selection_comparison
@@ -517,6 +562,8 @@ selection_comparison
   SE.not_equal(selection_comparison["best_impl"], selection_comparison["predicted_impl"])
   |> SE.transform(&if &1, do: "*", else: "")
 )
+|> DF.ungroup()
+|> DF.sort_by(proj)
 |> DF.rename(%{
   "mark" => " ",
   "proj" => "Project",
@@ -531,32 +578,37 @@ selection_comparison
 ## Adaptive Containers
 
 ```elixir
-# Projects where an adaptive container was suggested
-adaptive_projs =
-  (estimated_costs
-   |> DF.to_rows()
-   |> Enum.filter(fn %{"using" => using} ->
-     using
-     |> Enum.map(fn %{"impl" => impl} -> String.contains?(impl, "until") end)
-     |> Enum.any?()
-   end)
-   |> DF.new()
-   |> DF.distinct(["proj"]))["proj"]
+# Container types where an adaptive container was suggested
+adaptive_suggestions =
+  estimated_costs
+  |> DF.explode("using")
+  |> DF.unnest("using")
+  |> DF.filter(contains(impl, "until"))
+  |> DF.distinct(["proj", "ctn", "impl"])
+
+adaptive_suggestions
+# hacky but oh well
+|> DF.mutate(impl: replace(impl, "std::collections::", ""))
+|> DF.mutate(impl: replace(impl, "std::vec::", ""))
+|> DF.mutate(impl: replace(impl, "primrose_library::", ""))
+|> DF.sort_by(asc: proj, asc: ctn)
+|> DF.rename(%{
+  "proj" => "Project",
+  "ctn" => "Container Type",
+  "impl" => "Suggestion"
+})
+|> Latex.table()
+|> IO.puts()
 ```
 
 ```elixir
+adaptive_projs = DF.distinct(adaptive_suggestions, ["proj"])["proj"]
 adaptive_estimated_costs = estimated_costs |> DF.filter(proj in ^adaptive_projs)
 
 adaptive_raw_benchmarks =
   raw_benchmarks
   |> DF.filter(proj in ^adaptive_projs)
 
-display_using = fn using ->
-  using
-  |> Enum.map(fn %{"ctn" => ctn, "impl" => impl} -> ctn <> "=" <> impl end)
-  |> Enum.join(", ")
-end
-
 adaptive_raw_benchmarks =
   adaptive_raw_benchmarks
   |> DF.put(
@@ -573,6 +625,10 @@ adaptive_raw_benchmarks =
 ```
 
 ```elixir
+format_dur = fn dur ->
+  String.split(to_string(dur), " ") |> hd
+end
+
 best_usings =
   adaptive_raw_benchmarks
   # get best set of assignments for each project
@@ -586,13 +642,24 @@ best_usings =
   # select adaptive container and the best assignment for each project
   |> DF.join(adaptive_raw_benchmarks)
   |> DF.filter(using == best_using or contains(using, "until"))
-  # summary data point
-  |> DF.mutate(value: cast(mean, :string) <> " +/- " <> cast(stderr, :string))
+
+# summary data point
+
+best_usings =
+  best_usings
+  |> DF.put("mean", SE.transform(best_usings["mean"], format_dur))
+  |> DF.put("stderr", SE.transform(best_usings["stderr"], format_dur))
+  |> DF.mutate(value: mean <> " +/- " <> stderr)
   |> DF.select(["proj", "using", "n", "value"])
 ```
 
 ```elixir
-best_usings
-|> DF.filter(proj == "aoc_2022_09")
-|> DF.pivot_wider("n", "value")
+for proj <- SE.distinct(best_usings["proj"]) |> SE.to_enum() do
+  best_usings
+  |> DF.filter(proj == ^proj)
+  |> DF.select(["proj", "using", "n", "value"])
+  |> DF.pivot_wider("n", "value")
+  |> Latex.table()
+  |> IO.puts()
+end
 ```
diff --git a/nix/tex-env.nix b/nix/tex-env.nix
index 13137fe..403bf48 100644
--- a/nix/tex-env.nix
+++ b/nix/tex-env.nix
@@ -14,5 +14,6 @@ pkgs.texlive.combine {
     zapfchan
     latex-bin
     latexmk
+    adjustbox
     ;
 }
diff --git a/thesis/assets/insert_small_n.json b/thesis/assets/insert_small_n.json
index e17b175..abf4de8 100644
--- a/thesis/assets/insert_small_n.json
+++ b/thesis/assets/insert_small_n.json
@@ -1 +1 @@
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\ No newline at end of file
diff --git a/thesis/main.tex b/thesis/main.tex
index a4ff0b1..45285e1 100644
--- a/thesis/main.tex
+++ b/thesis/main.tex
@@ -6,6 +6,7 @@
 
 \usepackage[dvipsnames]{xcolor}
 \usepackage{graphicx}
+\usepackage{adjustbox}
 \usepackage{amsmath}
 
 \usepackage{microtype}
diff --git a/thesis/parts/results.tex b/thesis/parts/results.tex
index bd30fcf..6562d38 100644
--- a/thesis/parts/results.tex
+++ b/thesis/parts/results.tex
@@ -34,7 +34,7 @@ The lines correspond to our fitted curves, while the points indicate the raw obs
 
 \begin{figure}[h!]
   \centering
-  \includegraphics[width=15cm]{assets/insert.png}
+  \includegraphics[width=12cm]{assets/insert.png}
   \caption{Estimated cost of insert operation by implementation}
   \label{fig:cm_insert}
 \end{figure}
@@ -82,7 +82,7 @@ This seems inaccurate, and is likely a result of few data points at low n values
 
 \begin{figure}[h!]
   \centering
-  \includegraphics[width=15cm]{assets/insert_small_n.png}
+  \includegraphics[width=12cm]{assets/insert_small_n.png}
   \caption{Estimated cost of insert operation on set implementations, at small n values}
   \label{fig:cm_insert_small_n}
 \end{figure}
@@ -94,7 +94,7 @@ Figure \ref{fig:cm_contains} shows our built cost models, again grouped for read
 
 \begin{figure}[h!]
   \centering
-  \includegraphics[width=15cm]{assets/contains.png}
+  \includegraphics[width=12cm]{assets/contains.png}
   \caption{Estimated cost of \code{contains} operation by implementation}
   \label{fig:cm_contains}
 \end{figure}
@@ -127,10 +127,12 @@ Future improvements could address the overfitting problems some operations had,
 %% * Predictions
 \section{Selections}
 
+We now proceed with end-to-end testing of the system, selecting containers for a selection of programs with varying needs.
+
 \subsection{Benchmarks}
 
 %% ** Chosen benchmarks
-Our test cases broadly fall into two categories: Example cases, which just repeat a few operations many times, and our 'real' cases, which are implementations of common algorithms and solutions to programming puzles.
+Our test cases broadly fall into two categories: Example cases, which repeat a few operations many times, and 'real' cases, which are implementations of common algorithms and solutions to programming puzles.
 We expect the results from our example cases to be relatively unsurprising, while our real cases are more complex and harder to predict.
 
 Most of our real cases are solutions to puzzles from Advent of Code\citep{wastl_advent_2015}, a popular collection of programming puzzles.
@@ -157,77 +159,155 @@ Table \ref{table:test_cases} lists and briefly describes our test cases.
 
 %% ** Effect of selection on benchmarks (spread in execution time)
 Table \ref{table:benchmark_spread} shows the difference in benchmark results between the slowest possible assignment of containers, and the fastest.
-Even in our example projects, we see that the wrong choice of container can slow down our programs substantially.
+Even in our example projects, we see that the wrong choice of container can slow down our programs substantially, with the exception of two of our test cases which were largely unaffected.
 
 \begin{table}[h!]
 \centering
 \begin{tabular}{|c|c|c|}
-Project & worst - best time (seconds) & Maximum slowdown \\
+Project & Maximum slowdown (ms) & Maximum relative slowdown \\
 \hline
-aoc\_2021\_09 & 29.685 & 4.75 \\
-aoc\_2022\_08 & 0.036 & 2.088 \\
-aoc\_2022\_09 & 10.031 & 132.844 \\
-aoc\_2022\_14 & 0.293 & 2.036 \\
-prime\_sieve & 28.408 & 18.646 \\
-example\_mapping & 0.031 & 1.805 \\
-example\_sets & 0.179 & 12.65 \\
-example\_stack & 1.931 & 8.454 \\
+aoc\_2021\_09 & $55206.94$ & $12.0$ \\
+aoc\_2022\_08 & $12161.38$ & $392.5$ \\
+aoc\_2022\_09 & $18.96$ & $0.3$ \\
+aoc\_2022\_14 & $83.82$ & $0.3$ \\
+example\_mapping & $85.88$ & $108.4$ \\
+example\_sets & $1.33$ & $1.6$ \\
+example\_stack & $0.36$ & $19.2$ \\
+prime\_sieve & $26093.26$ & $34.1$ \\
 \end{tabular}
 \caption{Spread in total benchmark results by project}
 \label{table:benchmark_spread}
+
 \end{table}
 
-%% ** Summarise predicted versus actual
 \subsection{Prediction accuracy}
 
-We now compare the implementations suggested by our system, to the selection that is actually best.
+We now compare the implementations suggested by our system to the selection that is actually best, obtained by brute force.
 For now, we ignore suggestions for adaptive containers.
 
 Table \ref{table:predicted_actual} shows the predicted best assignments alongside the actual best assignment, obtained by brute-force.
-In all but two of our test cases (marked with *), we correctly identify the best container.
-
-\todo{but also its just vec/hashset every time, which is kinda boring. we should either get more variety (by adding to the library or adding new test cases), or mention this as a limitation in testing}
+In all but three of our test cases (marked with *), we correctly identify the best container.
 
 \begin{table}[h!]
   \centering
-\begin{tabular}{|c|c|c|c|}
-Project & Container Type & Best implementation & Predicted best \\
-\hline
-aoc\_2022\_09 & Set & HashSet & HashSet \\
-example\_stack & StackCon & Vec & Vec \\
-aoc\_2021\_09 & Set & HashSet & HashSet \\
-aoc\_2021\_09 & Map & HashMap & HashMap \\
-aoc\_2022\_14 & Set & HashSet & HashSet \\
-aoc\_2022\_14 & List & Vec & LinkedList \\
-aoc\_2022\_08 & Map & HashMap & HashMap \\
-example\_sets & Set & HashSet & HashSet \\
-example\_mapping & Map & HashMap & HashMap \\
-prime\_sieve & Primes & HashSet & BTreeSet \\
-prime\_sieve & Sieve & Vec & LinkedList \\
-\end{tabular}
+  \begin{tabular}{|c|c|c|c|c|}
+    Project & Container Type & Best implementation & Predicted best &   \\
+    \hline
+    aoc\_2021\_09 & Set & HashSet & HashSet &  \\
+    aoc\_2021\_09 & Map & HashMap & HashMap &  \\
+    aoc\_2022\_08 & Map & HashMap & HashMap &  \\
+    aoc\_2022\_09 & Set & HashSet & HashSet &  \\
+    aoc\_2022\_14 & Set & HashSet & HashSet &  \\
+    aoc\_2022\_14 & List & Vec & LinkedList & * \\
+    example\_mapping & Map & HashMap & HashMap &  \\
+    example\_sets & Set & HashSet & HashSet &  \\
+    example\_stack & StackCon & Vec & Vec &  \\
+    prime\_sieve & Sieve & Vec & LinkedList & * \\
+    prime\_sieve & Primes & HashSet & BTreeSet & * \\
+  \end{tabular}
   \caption{Actual best vs predicted best implementations}
   \label{table:predicted_actual}
 \end{table}
 
-%% ** Evaluate performance
-\subsection{Evaluation}
+Two of these failures appear to be caused by being overly eager to suggest a \code{LinkedList}.
+From looking at detailed profiling information, it seems that both of these container types had a relatively small amount of items in them.
+Therefore this is likely caused by our cost models being inaccurate at small $n$ values, such as in Figure \ref{fig:cm_insert_small_n}.
 
-%% ** Comment on distribution of best implementation
+Our only other failure comes from suggesting a \code{BTreeSet} instead of a \code{HashSet}.
+Our cost models suggest that a \code{BTreeSet} is more suitable for the \code{prime_sieve} benchmarks with a smaller $n$ value, but not for the larger ones.
+However, because the smaller benchmarks complete in less time, Criterion (the benchmarking framework used) chooses to run them for more iterations.
+This causes the smaller $n$ values to carry more weight than they should.
 
-%% ** Surprising ones / Explain failures
+This could be worked around by adjusting Criterion's settings to run all benchmarks for the same number of iterations, at the cost of the increased accuracy for smaller benchmarks that the existing strategy gives.
+Another method would be to only fix the number of iterations when profiling the application, and to run benchmarks as normal otherwise.
+Whilst this should be possible, Criterion doesn't currently support this.
 
-%% * Performance of adaptive containers
-\section{Adaptive containers}
+Overall, our results show our system is able to suggest the best containers, at least for large $n$ values.
+Unfortunately, these tests are somewhat limited, as the best container seems relatively predictable: \code{Vec} where uniqueness is not important, and \code{Hash*} otherwise.
+Therefore more thorough testing is needed to fully establish the system's effectiveness.
 
-\todo{These also need more work, and better test cases}
+\subsection{Adaptive containers}
+
+We now look at cases where an adaptive container was suggested, and evaluate the result.
+
+Table \ref{table:adaptive_suggestions} shows the container types for which adaptive containers were suggested, along with the inner types and the threshold at which to switch.
 
 %% ** Find where adaptive containers get suggested
+\begin{table}[h]
+  \centering
+  \begin{tabular}{|c|c|c|}
+    Project & Container Type & Suggestion \\
+    \hline
+    aoc\_2022\_08 & Map & SortedVecMap until n=1664, then HashMap \\
+    aoc\_2022\_09 & Set & HashSet until n=185540, then BTreeSet \\
+    example\_mapping & Map & VecMap until n=225, then HashMap \\
+    prime\_sieve & Primes & BTreeSet until n=34, then HashSet \\
+    prime\_sieve & Sieve & LinkedList until n=747, then Vec \\
+  \end{tabular}
+  \caption{Suggestions for adaptive containers}
+  \label{table:adaptive_suggestions}
+\end{table}
+
+The suggested containers for both \code{aoc_2022_08} and \code{example_mapping} are unsurprising.
+Since hashing incurs a roughly constant cost, it makes sense that below a certain $n$ value, simply searching through a list is more effective.
+The suggestion of \code{SortedVecMap} vs \code{VecMap} likely has to do with the relative frequency of \code{insert} operations compared to others.
+
+The suggestion to start with a \code{LinkedList} for \code{prime_sieve / Sieve} is likely due to the same issues that cause a \code{LinkedList} to be suggested in the non-adaptive case.
+This may also be the case for the suggestion of \code{BTreeSet} for \code{prime_sieve / Primes}.
+
+The suggestion of \code{BTreeSet} for \code{aoc_2022_09} is most surprising.
+As the $n$ threshold after which we switch is outside the range we benchmark our implementations at, this suggestion is based on our model attempting to generalise far outside the range it has seen before.
 
 %% ** Comment on relative performance speedup
+Table \ref{table:adaptive_perfcomp} compares our adaptive container suggestions with the fastest non-adaptive implementation.
+Since we must select an implementation for all containers before selecting a project, we show all possible combinations of adaptive and non-adaptive container selections.
+
+Note that the numbered columns indicate the benchmark 'size', not the actual size that the container reaches within that benchmark.
+The exact definition of this varies by benchmark.
+
+\begin{table}[h]
+  \centering
+  \begin{adjustbox}{angle=90}
+  \begin{tabular}{|c|c|c|c|c|c|}
+    \hline
+    Project & Assignment & 100 & 1000 & 2000 & \\
+    \hline
+    aoc\_2022\_09 & Set=HashSet & 1ms $\pm$ 5us & 13ms $\pm$ 828us & 27ms $\pm$ 1ms & \\
+    aoc\_2022\_09 & Set=Adaptive & 1ms $\pm$ 2us & 11ms $\pm$ 17us & 39ms $\pm$ 684us & \\
+    \hline
+     &  & 100 & 200 & & \\
+    \hline
+    aoc\_2022\_08 & Map=HashMap & 1ms $\pm$ 9us & 5ms $\pm$ 66us & &\\
+    aoc\_2022\_08 & Map=Adaptive & 1ms $\pm$ 6us & 5ms $\pm$ 41us & & \\
+    \hline
+     &  & 50 & 150 & 2500 & 7500 \\
+    \hline
+    example\_mapping & Map=HashMap & 3us $\pm$ 7ns & 11us $\pm$ 49ns & 185us $\pm$ 2us & 591us $\pm$ 1us \\
+    example\_mapping & Map=Adaptive & 4us $\pm$ 7ns & 33us $\pm$ 55ns & 187us $\pm$ 318ns & 595us $\pm$ 1us \\
+    \hline
+     &  & 50 & 500 & 50000 & \\
+    \hline
+    prime\_sieve & Sieve=Vec, Primes=HashSet & 1us $\pm$ 3ns & 78us $\pm$ 1us & 766ms $\pm$ 1ms & \\
+    prime\_sieve & Sieve=Vec, Primes=Adaptive & 1us $\pm$ 3ns & 84us $\pm$ 138ns & 785ms $\pm$ 730us & \\
+    prime\_sieve & Sieve=Adaptive, Primes=HashSet & 2us $\pm$ 6ns & 208us $\pm$ 568ns & 763ms $\pm$ 1ms & \\
+    prime\_sieve & Sieve=Adaptive, Primes=Adaptive & 2us $\pm$ 4ns & 205us $\pm$ 434ns & 762ms $\pm$ 2ms & \\
+    \hline
+  \end{tabular}
+  \end{adjustbox}
+  \caption{Adaptive containers vs the best single container, by size of benchmark}
+  \label{table:adaptive_perfcomp}
+\end{table}
+
+In all but one project, the non-adaptive containers are as fast or faster than the adaptive containers at all sizes of benchmarks.
+In the \code{aoc_2022_09} project, the adaptive container is marginally faster until the benchmark size reaches 2000, at which point it is significantly slower.
 
 %% ** Suggest future improvements?
+This shows that adaptive containers as we have implemented them are not effective in practice.
+Even in cases where we never reach the size threshold, the presence of adaptive containers has an overhead which slows down the program 3x in the worst case (\code{example_mapping}, size = 150).
 
-%% * Selection time / developer experience
-%% \section{Selection time}
+One explanation for this could be that every operation now requires checking which inner implementation we are using, resulting in branching overhead.
+More work could be done to minimise the overhead introduced, such as by using indirect jumps rather than branching instructions.
 
-%% ** Mention speedup versus naive brute force
+It is also unclear if the threshold values that we suggest are the optimal ones.
+Currently, we decide our threshold by picking a value between two partitions with different best containers.
+Future work could take a more complex approach that finds the best threshold value based on our cost models, and takes the overhead of all operations into account.