diff --git a/src/query/plan/operator.cpp b/src/query/plan/operator.cpp
index bf751c012..2065d9916 100644
--- a/src/query/plan/operator.cpp
+++ b/src/query/plan/operator.cpp
@@ -931,7 +931,9 @@ class STShortestPathCursor : public query::plan::Cursor {
   STShortestPathCursor(const ExpandVariable &self,
                        database::GraphDbAccessor *dba,
                        utils::MemoryResource *mem)
-      : self_(self), input_cursor_(self_.input()->MakeCursor(dba, mem)) {
+      : mem_(mem),
+        self_(self),
+        input_cursor_(self_.input()->MakeCursor(dba, mem)) {
     CHECK(self_.common_.existing_node)
         << "s-t shortest path algorithm should only "
            "be used when `existing_node` flag is "
@@ -981,11 +983,15 @@ class STShortestPathCursor : public query::plan::Cursor {
   void Reset() override { input_cursor_->Reset(); }
 
  private:
+  utils::MemoryResource *mem_;
   const ExpandVariable &self_;
   UniqueCursorPtr input_cursor_;
 
-  using VertexEdgeMapT =
-      std::unordered_map<VertexAccessor, std::optional<EdgeAccessor>>;
+  using VertexEdgeMapT = std::unordered_map<
+      VertexAccessor, std::optional<EdgeAccessor>, std::hash<VertexAccessor>,
+      std::equal_to<>,
+      utils::Allocator<
+          std::pair<const VertexAccessor, std::optional<EdgeAccessor>>>>;
 
   void ReconstructPath(const VertexAccessor &midpoint,
                        const VertexEdgeMapT &in_edge,
@@ -1041,18 +1047,22 @@ class STShortestPathCursor : public query::plan::Cursor {
 
     // Holds vertices at the current level of expansion from the source
     // (sink).
-    std::vector<VertexAccessor> source_frontier;
-    std::vector<VertexAccessor> sink_frontier;
+    std::vector<VertexAccessor, utils::Allocator<VertexAccessor>>
+        source_frontier(mem_);
+    std::vector<VertexAccessor, utils::Allocator<VertexAccessor>> sink_frontier(
+        mem_);
 
     // Holds vertices we can expand to from `source_frontier`
     // (`sink_frontier`).
-    std::vector<VertexAccessor> source_next;
-    std::vector<VertexAccessor> sink_next;
+    std::vector<VertexAccessor, utils::Allocator<VertexAccessor>> source_next(
+        mem_);
+    std::vector<VertexAccessor, utils::Allocator<VertexAccessor>> sink_next(
+        mem_);
 
     // Maps each vertex we visited expanding from the source (sink) to the
     // edge used. Necessary for path reconstruction.
-    VertexEdgeMapT in_edge;
-    VertexEdgeMapT out_edge;
+    VertexEdgeMapT in_edge(mem_);
+    VertexEdgeMapT out_edge(mem_);
 
     size_t current_length = 0;
 
@@ -1164,7 +1174,11 @@ class SingleSourceShortestPathCursor : public query::plan::Cursor {
   SingleSourceShortestPathCursor(const ExpandVariable &self,
                                  database::GraphDbAccessor *db,
                                  utils::MemoryResource *mem)
-      : self_(self), input_cursor_(self_.input()->MakeCursor(db, mem)) {
+      : self_(self),
+        input_cursor_(self_.input()->MakeCursor(db, mem)),
+        processed_(mem),
+        to_visit_current_(mem),
+        to_visit_next_(mem) {
     CHECK(!self_.common_.existing_node)
         << "Single source shortest path algorithm "
            "should not be used when `existing_node` "
@@ -1313,10 +1327,18 @@ class SingleSourceShortestPathCursor : public query::plan::Cursor {
   // maps vertices to the edge they got expanded from. it is an optional
   // edge because the root does not get expanded from anything.
   // contains visited vertices as well as those scheduled to be visited.
-  std::unordered_map<VertexAccessor, std::optional<EdgeAccessor>> processed_;
+  std::unordered_map<VertexAccessor, std::optional<EdgeAccessor>,
+                     std::hash<VertexAccessor>, std::equal_to<>,
+                     utils::Allocator<std::pair<const VertexAccessor,
+                                                std::optional<EdgeAccessor>>>>
+      processed_;
   // edge/vertex pairs we have yet to visit, for current and next depth
-  std::vector<std::pair<EdgeAccessor, VertexAccessor>> to_visit_current_;
-  std::vector<std::pair<EdgeAccessor, VertexAccessor>> to_visit_next_;
+  std::vector<std::pair<EdgeAccessor, VertexAccessor>,
+              utils::Allocator<std::pair<EdgeAccessor, VertexAccessor>>>
+      to_visit_current_;
+  std::vector<std::pair<EdgeAccessor, VertexAccessor>,
+              utils::Allocator<std::pair<EdgeAccessor, VertexAccessor>>>
+      to_visit_next_;
 };
 
 class ExpandWeightedShortestPathCursor : public query::plan::Cursor {
diff --git a/src/utils/algorithm.hpp b/src/utils/algorithm.hpp
index a5063513b..2ecce1c99 100644
--- a/src/utils/algorithm.hpp
+++ b/src/utils/algorithm.hpp
@@ -112,15 +112,18 @@ inline TVal First(TIterable &&iterable, TVal &&empty_value) {
   return empty_value;
 }
 
-template <typename TElement>
-inline bool Contains(const std::unordered_set<TElement> &iterable,
-                     const TElement &element) {
+template <class TElement, class THash, class TEqual, class TAllocator>
+bool Contains(
+    const std::unordered_set<TElement, THash, TEqual, TAllocator> &iterable,
+    const TElement &element) {
   return iterable.find(element) != iterable.end();
 }
 
-template <typename TKey, typename TValue>
-inline bool Contains(const std::unordered_map<TKey, TValue> &iterable,
-                     const TKey &key) {
+template <class TKey, class TValue, class THash, class TKeyEqual,
+          class TAllocator>
+bool Contains(const std::unordered_map<TKey, TValue, THash, TKeyEqual,
+                                       TAllocator> &iterable,
+              const TKey &key) {
   return iterable.find(key) != iterable.end();
 }
 
diff --git a/tests/benchmark/query/execution.cpp b/tests/benchmark/query/execution.cpp
index a942b976a..adced9e13 100644
--- a/tests/benchmark/query/execution.cpp
+++ b/tests/benchmark/query/execution.cpp
@@ -1,3 +1,4 @@
+#include <random>
 #include <string>
 
 #include <benchmark/benchmark.h>
@@ -34,6 +35,25 @@ static void AddStarGraph(database::GraphDb *db, int spoke_count, int depth) {
   dba.Commit();
 }
 
+static void AddTree(database::GraphDb *db, int vertex_count) {
+  auto dba = db->Access();
+  std::vector<VertexAccessor> vertices;
+  vertices.reserve(vertex_count);
+  auto root = dba.InsertVertex();
+  root.add_label(dba.Label(kStartLabel));
+  vertices.push_back(root);
+  // NOLINTNEXTLINE(cert-msc32-c,cert-msc51-cpp)
+  std::mt19937_64 rg(42);
+  for (int i = 1; i < vertex_count; ++i) {
+    auto v = dba.InsertVertex();
+    std::uniform_int_distribution<> dis(0U, vertices.size() - 1U);
+    auto &parent = vertices.at(dis(rg));
+    dba.InsertEdge(parent, v, dba.EdgeType("Type"));
+    vertices.push_back(v);
+  }
+  dba.Commit();
+}
+
 static query::CypherQuery *ParseCypherQuery(const std::string &query_string,
                                             query::AstStorage *ast) {
   query::frontend::ParsingContext parsing_context;
@@ -114,6 +134,23 @@ BENCHMARK(DistinctLinearAllocator)
     ->Range(1024, 1U << 21U)
     ->Unit(benchmark::kMicrosecond);
 
+static query::plan::ExpandVariable MakeExpandVariable(
+    query::EdgeAtom::Type expand_type, query::SymbolTable *symbol_table) {
+  auto input_symbol = symbol_table->CreateSymbol("input", false);
+  auto dest_symbol = symbol_table->CreateSymbol("dest", false);
+  auto edge_symbol = symbol_table->CreateSymbol("edge", false);
+  auto lambda_node_symbol = symbol_table->CreateSymbol("n", false);
+  auto lambda_edge_symbol = symbol_table->CreateSymbol("e", false);
+  query::plan::ExpansionLambda filter_lambda;
+  filter_lambda.inner_node_symbol = lambda_node_symbol;
+  filter_lambda.inner_edge_symbol = lambda_edge_symbol;
+  filter_lambda.expression = nullptr;
+  return query::plan::ExpandVariable(
+      nullptr, input_symbol, dest_symbol, edge_symbol, expand_type,
+      query::EdgeAtom::Direction::OUT, {}, false, nullptr, nullptr, false,
+      filter_lambda, std::nullopt, std::nullopt);
+}
+
 // NOLINTNEXTLINE(google-runtime-references)
 static void ExpandVariableDefaultAllocator(benchmark::State &state) {
   query::AstStorage ast;
@@ -121,22 +158,10 @@ static void ExpandVariableDefaultAllocator(benchmark::State &state) {
   database::GraphDb db;
   AddStarGraph(&db, state.range(0), state.range(1));
   query::SymbolTable symbol_table;
-  auto input_symbol = symbol_table.CreateSymbol("input", false);
-  auto dest_symbol = symbol_table.CreateSymbol("dest", false);
-  auto edge_symbol = symbol_table.CreateSymbol("edge", false);
-  auto lambda_node_symbol = symbol_table.CreateSymbol("n", false);
-  auto lambda_edge_symbol = symbol_table.CreateSymbol("e", false);
+  auto expand_variable =
+      MakeExpandVariable(query::EdgeAtom::Type::DEPTH_FIRST, &symbol_table);
   auto dba = db.Access();
   query::Frame frame(symbol_table.max_position());
-  query::plan::ExpansionLambda filter_lambda;
-  filter_lambda.inner_node_symbol = lambda_node_symbol;
-  filter_lambda.inner_edge_symbol = lambda_edge_symbol;
-  filter_lambda.expression = nullptr;
-  query::plan::ExpandVariable expand_variable(
-      nullptr, input_symbol, dest_symbol, edge_symbol,
-      query::EdgeAtom::Type::DEPTH_FIRST, query::EdgeAtom::Direction::OUT, {},
-      false, nullptr, nullptr, false, filter_lambda, std::nullopt,
-      std::nullopt);
   // Nothing should be used from the EvaluationContext, so leave it empty.
   query::EvaluationContext evaluation_context;
   while (state.KeepRunning()) {
@@ -144,7 +169,7 @@ static void ExpandVariableDefaultAllocator(benchmark::State &state) {
                                               evaluation_context};
     auto cursor = expand_variable.MakeCursor(&dba, utils::NewDeleteResource());
     for (const auto &v : dba.Vertices(dba.Label(kStartLabel), false)) {
-      frame[input_symbol] = v;
+      frame[expand_variable.input_symbol_] = v;
       while (cursor->Pull(frame, execution_context))
         ;
     }
@@ -153,8 +178,8 @@ static void ExpandVariableDefaultAllocator(benchmark::State &state) {
 }
 
 BENCHMARK(ExpandVariableDefaultAllocator)
-    ->Ranges({{1, 1U << 7U}, {512, 1U << 13U}})
-    ->Unit(benchmark::kMillisecond);
+    ->Ranges({{1, 1U << 5U}, {512, 1U << 13U}})
+    ->Unit(benchmark::kMicrosecond);
 
 // NOLINTNEXTLINE(google-runtime-references)
 static void ExpandVariableLinearAllocator(benchmark::State &state) {
@@ -163,22 +188,10 @@ static void ExpandVariableLinearAllocator(benchmark::State &state) {
   database::GraphDb db;
   AddStarGraph(&db, state.range(0), state.range(1));
   query::SymbolTable symbol_table;
-  auto input_symbol = symbol_table.CreateSymbol("input", false);
-  auto dest_symbol = symbol_table.CreateSymbol("dest", false);
-  auto edge_symbol = symbol_table.CreateSymbol("edge", false);
-  auto lambda_node_symbol = symbol_table.CreateSymbol("n", false);
-  auto lambda_edge_symbol = symbol_table.CreateSymbol("e", false);
+  auto expand_variable =
+      MakeExpandVariable(query::EdgeAtom::Type::DEPTH_FIRST, &symbol_table);
   auto dba = db.Access();
   query::Frame frame(symbol_table.max_position());
-  query::plan::ExpansionLambda filter_lambda;
-  filter_lambda.inner_node_symbol = lambda_node_symbol;
-  filter_lambda.inner_edge_symbol = lambda_edge_symbol;
-  filter_lambda.expression = nullptr;
-  query::plan::ExpandVariable expand_variable(
-      nullptr, input_symbol, dest_symbol, edge_symbol,
-      query::EdgeAtom::Type::DEPTH_FIRST, query::EdgeAtom::Direction::OUT, {},
-      false, nullptr, nullptr, false, filter_lambda, std::nullopt,
-      std::nullopt);
   // Nothing should be used from the EvaluationContext, so leave it empty.
   query::EvaluationContext evaluation_context;
   while (state.KeepRunning()) {
@@ -187,7 +200,7 @@ static void ExpandVariableLinearAllocator(benchmark::State &state) {
     utils::MonotonicBufferResource memory(query::kExecutionMemoryBlockSize);
     auto cursor = expand_variable.MakeCursor(&dba, &memory);
     for (const auto &v : dba.Vertices(dba.Label(kStartLabel), false)) {
-      frame[input_symbol] = v;
+      frame[expand_variable.input_symbol_] = v;
       while (cursor->Pull(frame, execution_context))
         ;
     }
@@ -196,7 +209,139 @@ static void ExpandVariableLinearAllocator(benchmark::State &state) {
 }
 
 BENCHMARK(ExpandVariableLinearAllocator)
-    ->Ranges({{1, 1U << 7U}, {512, 1U << 13U}})
-    ->Unit(benchmark::kMillisecond);
+    ->Ranges({{1, 1U << 5U}, {512, 1U << 13U}})
+    ->Unit(benchmark::kMicrosecond);
+
+// NOLINTNEXTLINE(google-runtime-references)
+static void ExpandBfsDefaultAllocator(benchmark::State &state) {
+  query::AstStorage ast;
+  query::Parameters parameters;
+  database::GraphDb db;
+  AddTree(&db, state.range(0));
+  query::SymbolTable symbol_table;
+  auto expand_variable =
+      MakeExpandVariable(query::EdgeAtom::Type::BREADTH_FIRST, &symbol_table);
+  auto dba = db.Access();
+  query::Frame frame(symbol_table.max_position());
+  // Nothing should be used from the EvaluationContext, so leave it empty.
+  query::EvaluationContext evaluation_context;
+  while (state.KeepRunning()) {
+    query::ExecutionContext execution_context{&dba, symbol_table,
+                                              evaluation_context};
+    auto cursor = expand_variable.MakeCursor(&dba, utils::NewDeleteResource());
+    for (const auto &v : dba.Vertices(dba.Label(kStartLabel), false)) {
+      frame[expand_variable.input_symbol_] = v;
+      while (cursor->Pull(frame, execution_context))
+        ;
+    }
+  }
+  state.SetItemsProcessed(state.iterations());
+}
+
+BENCHMARK(ExpandBfsDefaultAllocator)
+    ->Range(512, 1U << 19U)
+    ->Unit(benchmark::kMicrosecond);
+
+// NOLINTNEXTLINE(google-runtime-references)
+static void ExpandBfsLinearAllocator(benchmark::State &state) {
+  query::AstStorage ast;
+  query::Parameters parameters;
+  database::GraphDb db;
+  AddTree(&db, state.range(0));
+  query::SymbolTable symbol_table;
+  auto expand_variable =
+      MakeExpandVariable(query::EdgeAtom::Type::BREADTH_FIRST, &symbol_table);
+  auto dba = db.Access();
+  query::Frame frame(symbol_table.max_position());
+  // Nothing should be used from the EvaluationContext, so leave it empty.
+  query::EvaluationContext evaluation_context;
+  while (state.KeepRunning()) {
+    query::ExecutionContext execution_context{&dba, symbol_table,
+                                              evaluation_context};
+    utils::MonotonicBufferResource memory(query::kExecutionMemoryBlockSize);
+    auto cursor = expand_variable.MakeCursor(&dba, &memory);
+    for (const auto &v : dba.Vertices(dba.Label(kStartLabel), false)) {
+      frame[expand_variable.input_symbol_] = v;
+      while (cursor->Pull(frame, execution_context))
+        ;
+    }
+  }
+  state.SetItemsProcessed(state.iterations());
+}
+
+BENCHMARK(ExpandBfsLinearAllocator)
+    ->Range(512, 1U << 19U)
+    ->Unit(benchmark::kMicrosecond);
+
+// NOLINTNEXTLINE(google-runtime-references)
+static void ExpandShortestDefaultAllocator(benchmark::State &state) {
+  query::AstStorage ast;
+  query::Parameters parameters;
+  database::GraphDb db;
+  AddTree(&db, state.range(0));
+  query::SymbolTable symbol_table;
+  auto expand_variable =
+      MakeExpandVariable(query::EdgeAtom::Type::BREADTH_FIRST, &symbol_table);
+  expand_variable.common_.existing_node = true;
+  auto dest_symbol = expand_variable.common_.node_symbol;
+  auto dba = db.Access();
+  query::Frame frame(symbol_table.max_position());
+  // Nothing should be used from the EvaluationContext, so leave it empty.
+  query::EvaluationContext evaluation_context;
+  while (state.KeepRunning()) {
+    query::ExecutionContext execution_context{&dba, symbol_table,
+                                              evaluation_context};
+    auto cursor = expand_variable.MakeCursor(&dba, utils::NewDeleteResource());
+    for (const auto &v : dba.Vertices(dba.Label(kStartLabel), false)) {
+      frame[expand_variable.input_symbol_] = v;
+      for (const auto &dest : dba.Vertices(false)) {
+        frame[dest_symbol] = dest;
+        while (cursor->Pull(frame, execution_context))
+          ;
+      }
+    }
+  }
+  state.SetItemsProcessed(state.iterations());
+}
+
+BENCHMARK(ExpandShortestDefaultAllocator)
+    ->Range(512, 1U << 20U)
+    ->Unit(benchmark::kMicrosecond);
+
+// NOLINTNEXTLINE(google-runtime-references)
+static void ExpandShortestLinearAllocator(benchmark::State &state) {
+  query::AstStorage ast;
+  query::Parameters parameters;
+  database::GraphDb db;
+  AddTree(&db, state.range(0));
+  query::SymbolTable symbol_table;
+  auto expand_variable =
+      MakeExpandVariable(query::EdgeAtom::Type::BREADTH_FIRST, &symbol_table);
+  expand_variable.common_.existing_node = true;
+  auto dest_symbol = expand_variable.common_.node_symbol;
+  auto dba = db.Access();
+  query::Frame frame(symbol_table.max_position());
+  // Nothing should be used from the EvaluationContext, so leave it empty.
+  query::EvaluationContext evaluation_context;
+  while (state.KeepRunning()) {
+    query::ExecutionContext execution_context{&dba, symbol_table,
+                                              evaluation_context};
+    utils::MonotonicBufferResource memory(query::kExecutionMemoryBlockSize);
+    auto cursor = expand_variable.MakeCursor(&dba, &memory);
+    for (const auto &v : dba.Vertices(dba.Label(kStartLabel), false)) {
+      frame[expand_variable.input_symbol_] = v;
+      for (const auto &dest : dba.Vertices(false)) {
+        frame[dest_symbol] = dest;
+        while (cursor->Pull(frame, execution_context))
+          ;
+      }
+    }
+  }
+  state.SetItemsProcessed(state.iterations());
+}
+
+BENCHMARK(ExpandShortestLinearAllocator)
+    ->Range(512, 1U << 20U)
+    ->Unit(benchmark::kMicrosecond);
 
 BENCHMARK_MAIN();