From c11d60ba038b9c1d5ce8988e2d742c88299e059d Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 13:15:27 +0200
Subject: [PATCH 01/12] benchmark loop

---
 CMakeLists.txt           |  1 +
 benchmark/CMakeLists.txt | 18 ++++++++++++++
 benchmark/benchmark.cpp  | 51 ++++++++++++++++++++++++++++++++++++++++
 3 files changed, 70 insertions(+)
 create mode 100644 benchmark/CMakeLists.txt
 create mode 100644 benchmark/benchmark.cpp

diff --git a/CMakeLists.txt b/CMakeLists.txt
index 417d49f..cbb700b 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -11,6 +11,7 @@ add_subdirectory(main)
 add_subdirectory(LRU)
 add_subdirectory(LFU)
 add_subdirectory(perfect)
+add_subdirectory(benchmark)
 
 
 include(CTest)
diff --git a/benchmark/CMakeLists.txt b/benchmark/CMakeLists.txt
new file mode 100644
index 0000000..953ba69
--- /dev/null
+++ b/benchmark/CMakeLists.txt
@@ -0,0 +1,18 @@
+add_executable(benchmark)
+target_sources(benchmark
+  PRIVATE
+    benchmark.cpp
+)
+
+target_compile_options(benchmark
+  PRIVATE
+    $<$<CXX_COMPILER_ID:MSVC>:/W4 /WX>
+    $<$<CXX_COMPILER_ID:GNU,Clang>:-Wall -Wextra -Werror>
+)
+
+target_link_libraries(benchmark
+  PRIVATE
+    LRU 
+    LFU
+    perfect
+)
diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
new file mode 100644
index 0000000..9185050
--- /dev/null
+++ b/benchmark/benchmark.cpp
@@ -0,0 +1,51 @@
+#include "LFU.hpp"
+#include "LRU.hpp"
+#include "perfect.hpp"
+#include <algorithm>
+#include <cassert>
+#include <cstdlib>
+#include <iostream>
+#include <map>
+#include <string>
+#include <utility>
+
+int slow_get_page(int key) { return key; }
+
+std::vector<int> generate_rnd_keys(size_t n, size_t mod = 100) {
+  std::vector<int> keys(n);
+  std::generate(keys.begin(), keys.end(), [&]() { return rand() % mod; });
+  return keys;
+}
+
+int main() {
+  size_t input_sz = 1'000'000;
+  std::vector<int> keys = generate_rnd_keys(input_sz, 100);
+
+  std::map<size_t, std::map<std::string, int>> cache_sizes_vs_strategy_hits;
+  for (size_t cache_sz = 10; cache_sz < 101; cache_sz += 10) {
+    std::map<std::string, int> cache_hits;
+    cache::perfect_t<int> perfect_cache(cache_sz, keys);
+    cache::LRU_t<int> lru{cache_sz};
+    cache::LFU_t<int> lfu{cache_sz};
+
+    for (const auto key : keys) {
+      if (perfect_cache.lookup_update(key, slow_get_page))
+        cache_hits["perfect_cache"] += 1;
+
+      if (lru.lookup_update(key, slow_get_page))
+        cache_hits["LRU"] += 1;
+
+      if (lfu.lookup_update(key, slow_get_page))
+        cache_hits["LFU"] += 1;
+    }
+    cache_sizes_vs_strategy_hits[cache_sz] = cache_hits;
+
+    std::cout << "Cache type\tN hits\n";
+    for (auto [cache_type, nhits] : cache_hits) {
+      std::cout << cache_type << "\t" << nhits << std::endl;
+    }
+
+  }
+
+  return 0;
+}

From 43a62798fbdd4dd148feacf37748002b05318151 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:05:57 +0200
Subject: [PATCH 02/12] statistics with gnuplot

---
 benchmark/benchmark.cpp            |  26 ++++++++++++++++++++++----
 benchmark/cache_hit_statistics.png | Bin 0 -> 22785 bytes
 benchmark/cache_hits.dat           |  13 +++++++++++++
 benchmark/script.gp                |  11 +++++++++++
 4 files changed, 46 insertions(+), 4 deletions(-)
 create mode 100644 benchmark/cache_hit_statistics.png
 create mode 100644 benchmark/cache_hits.dat
 create mode 100644 benchmark/script.gp

diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index 9185050..bdb756d 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -4,6 +4,7 @@
 #include <algorithm>
 #include <cassert>
 #include <cstdlib>
+#include <fstream>
 #include <iostream>
 #include <map>
 #include <string>
@@ -18,10 +19,12 @@ std::vector<int> generate_rnd_keys(size_t n, size_t mod = 100) {
 }
 
 int main() {
-  size_t input_sz = 1'000'000;
-  std::vector<int> keys = generate_rnd_keys(input_sz, 100);
+  size_t input_sz = 100'000;
+  size_t rnd_mod = 100;
+  std::vector<int> keys = generate_rnd_keys(input_sz, rnd_mod);
 
   std::map<size_t, std::map<std::string, int>> cache_sizes_vs_strategy_hits;
+  std::cout << "Cache type\tN hits\n";
   for (size_t cache_sz = 10; cache_sz < 101; cache_sz += 10) {
     std::map<std::string, int> cache_hits;
     cache::perfect_t<int> perfect_cache(cache_sz, keys);
@@ -40,11 +43,26 @@ int main() {
     }
     cache_sizes_vs_strategy_hits[cache_sz] = cache_hits;
 
-    std::cout << "Cache type\tN hits\n";
+  std::cout << "cache size: " << cache_sz << '\n';
     for (auto [cache_type, nhits] : cache_hits) {
-      std::cout << cache_type << "\t" << nhits << std::endl;
+      std::cout << cache_type << "\t" << nhits << '\n';
     }
+    std::cout << std::endl;
+  }
 
+  // output to a file
+  std::ofstream file("benchmark/cache_hits.dat");
+  if (file.is_open()) {
+    file << "#Number of keys = " << input_sz << "\n" ;
+    file << "#input keys: uniform distribution [0:" << rnd_mod << ")\n" ;
+    file << "#Cache_size\tLRU\tLFU\tBelady\n";
+    for(auto &[sz, cache_stats]: cache_sizes_vs_strategy_hits){
+      file << sz <<'\t'
+        << cache_stats["LRU"] << '\t'
+        << cache_stats["LFU"] << '\t'
+        << cache_stats["perfect_cache"] << '\n';
+    }
+    file.close();
   }
 
   return 0;
diff --git a/benchmark/cache_hit_statistics.png b/benchmark/cache_hit_statistics.png
new file mode 100644
index 0000000000000000000000000000000000000000..fead7ddd6180c82284009af43327d7d0326a4b24
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zNy6@4y|OlR`kS^Spf5!KP_9pt($K#enQPS3;eK4Uz++{Tk7$JktUeP~zYD$^nd7eD
zI?=|2-!c!Y8anXz6oyzCHn6v?6h$q^asm0ENN<N*IuDvb@Rs#xY4_bp@`R<CrMT>*
zW*D{Ay(!Te_ob0WwCTFv5<>)L8kN{hbidRC&VCgO%PTh-d@lQJ%2r49-KZPyx1*N{
zfPv+1BdSO-(c*u}C~0E58ZKX!pVJNhIk{`3Qmd$Kka(sbV)b=QcGI!EqDK|Tt33#$
zQs>*3)D*4+gQ@LR<Ev^1&Gq>8SRUZZr2ZEt%m04}T@N|exS*k??VR@yB5P^rs#n6T
GU;PKBFQ>Ht

literal 0
HcmV?d00001

diff --git a/benchmark/cache_hits.dat b/benchmark/cache_hits.dat
new file mode 100644
index 0000000..f601acc
--- /dev/null
+++ b/benchmark/cache_hits.dat
@@ -0,0 +1,13 @@
+#Number of keys = 100000
+#input keys: uniform distribution [0:100)
+#Cache_size	LRU	LFU	Belady
+10	10015	9828	35205
+20	19947	19849	51756
+30	29887	29907	63218
+40	39921	39898	72054
+50	49944	49914	79207
+60	59924	59805	85109
+70	69960	69747	90046
+80	79974	79703	94157
+90	89939	89828	97468
+100	99900	99900	99900
diff --git a/benchmark/script.gp b/benchmark/script.gp
new file mode 100644
index 0000000..ae1645a
--- /dev/null
+++ b/benchmark/script.gp
@@ -0,0 +1,11 @@
+set terminal pngcairo size 800,600 enhanced font "Helvetica,12"
+set output 'cache_hit_statistics.png'
+
+set title "Cache statistics for uniform random integers [0:100)"
+set xlabel "Cache size"
+set ylabel "Cache hits"
+set key bottom right
+
+plot "cache_hits.dat" u 1:2 title "LRU",\
+      "" u 1:3 title "LFU",\
+      "" u 1:4 title "Belady"\

From 4900719714de0ae7bacab30c262264f00a81a803 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:08:58 +0200
Subject: [PATCH 03/12] picture to readme

---
 README.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/README.md b/README.md
index 48cdd45..82d6a45 100644
--- a/README.md
+++ b/README.md
@@ -12,6 +12,12 @@ LRU          3
 LFU          1
 ```
 
+### Benchmark
+input : uniform random integers [0: 100) with input size 100'000
+
+![statistics](./benchmark/cache_hit_statistics.png)
+
+
 
 
 # Tips

From 5156d90b16c8bf645792da7e600b8b9239352251 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:16:35 +0200
Subject: [PATCH 04/12] improvig readme

---
 README.md               | 14 +++++++++-----
 benchmark/benchmark.cpp |  4 ++--
 main/main.cpp           |  2 +-
 3 files changed, 12 insertions(+), 8 deletions(-)

diff --git a/README.md b/README.md
index 82d6a45..1f03ada 100644
--- a/README.md
+++ b/README.md
@@ -8,6 +8,7 @@ How to run
 output is 
 ```bash
 Cache type   N hits
+Belady       3
 LRU          3
 LFU          1
 ```
@@ -17,14 +18,17 @@ input : uniform random integers [0: 100) with input size 100'000
 
 ![statistics](./benchmark/cache_hit_statistics.png)
 
+To reproduce benchmarks one needs to build the project and run
+```bash
+./build/benchmark/benchmark
+cd benchmark
+gnuplot script.gp
+open cache_hit_statistics.png
+```
 
 
 
-# Tips
-
-## CMAKE
-
-
+## How to build: CMAKE
 ```bash
 cmake -S . -B build -G Ninja \
   -DCMAKE_C_COMPILER=gcc-14 \
diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index bdb756d..7ef9707 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -33,7 +33,7 @@ int main() {
 
     for (const auto key : keys) {
       if (perfect_cache.lookup_update(key, slow_get_page))
-        cache_hits["perfect_cache"] += 1;
+        cache_hits["Belady"] += 1;
 
       if (lru.lookup_update(key, slow_get_page))
         cache_hits["LRU"] += 1;
@@ -60,7 +60,7 @@ int main() {
       file << sz <<'\t'
         << cache_stats["LRU"] << '\t'
         << cache_stats["LFU"] << '\t'
-        << cache_stats["perfect_cache"] << '\n';
+        << cache_stats["Belady"] << '\n';
     }
     file.close();
   }
diff --git a/main/main.cpp b/main/main.cpp
index 5400a7a..3bf01f5 100644
--- a/main/main.cpp
+++ b/main/main.cpp
@@ -27,7 +27,7 @@ int main() {
   std::map<std::string, int> cache_hits;
   for (const auto key : keys) {
     if (perfect_cache.lookup_update(key, slow_get_page))
-      cache_hits["perfect_cache"] += 1;
+      cache_hits["Belady"] += 1;
 
     if (lru.lookup_update(key, slow_get_page))
       cache_hits["LRU"] += 1;

From 0fdec937944752652740a7011b531f1db9bb24a1 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:24:32 +0200
Subject: [PATCH 05/12] improve readme

---
 README.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/README.md b/README.md
index 1f03ada..5ca4f7a 100644
--- a/README.md
+++ b/README.md
@@ -1,6 +1,6 @@
-# cpp_proj
+# HWC Caches. LRU, LFU, Belady's optimal cache
 
-How to run
+## How to run
 
 ```bash
 ./build/main/main <<< "2 6 3 3 1 2 1 2" # cache_size Nelem elem1 elem2 ...

From 8dae6dda25beb7ce33c26434bb5131f496c8d769 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:36:38 +0200
Subject: [PATCH 06/12] int to sizet for rnd

---
 benchmark/benchmark.cpp | 19 +++++++++----------
 1 file changed, 9 insertions(+), 10 deletions(-)

diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index 7ef9707..76989d7 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -14,7 +14,8 @@ int slow_get_page(int key) { return key; }
 
 std::vector<int> generate_rnd_keys(size_t n, size_t mod = 100) {
   std::vector<int> keys(n);
-  std::generate(keys.begin(), keys.end(), [&]() { return rand() % mod; });
+  std::generate(keys.begin(), keys.end(),
+                [&]() { return static_cast<size_t>(rand() % mod); });
   return keys;
 }
 
@@ -23,7 +24,7 @@ int main() {
   size_t rnd_mod = 100;
   std::vector<int> keys = generate_rnd_keys(input_sz, rnd_mod);
 
-  std::map<size_t, std::map<std::string, int>> cache_sizes_vs_strategy_hits;
+  std::map<size_t, std::map<std::string, size_t>> cache_sizes_vs_strategy_hits;
   std::cout << "Cache type\tN hits\n";
   for (size_t cache_sz = 10; cache_sz < 101; cache_sz += 10) {
     std::map<std::string, int> cache_hits;
@@ -43,7 +44,7 @@ int main() {
     }
     cache_sizes_vs_strategy_hits[cache_sz] = cache_hits;
 
-  std::cout << "cache size: " << cache_sz << '\n';
+    std::cout << "cache size: " << cache_sz << '\n';
     for (auto [cache_type, nhits] : cache_hits) {
       std::cout << cache_type << "\t" << nhits << '\n';
     }
@@ -53,14 +54,12 @@ int main() {
   // output to a file
   std::ofstream file("benchmark/cache_hits.dat");
   if (file.is_open()) {
-    file << "#Number of keys = " << input_sz << "\n" ;
-    file << "#input keys: uniform distribution [0:" << rnd_mod << ")\n" ;
+    file << "#Number of keys = " << input_sz << "\n";
+    file << "#input keys: uniform distribution [0:" << rnd_mod << ")\n";
     file << "#Cache_size\tLRU\tLFU\tBelady\n";
-    for(auto &[sz, cache_stats]: cache_sizes_vs_strategy_hits){
-      file << sz <<'\t'
-        << cache_stats["LRU"] << '\t'
-        << cache_stats["LFU"] << '\t'
-        << cache_stats["Belady"] << '\n';
+    for (auto &[sz, cache_stats] : cache_sizes_vs_strategy_hits) {
+      file << sz << '\t' << cache_stats["LRU"] << '\t' << cache_stats["LFU"]
+           << '\t' << cache_stats["Belady"] << '\n';
     }
     file.close();
   }

From 51b62c1349c4cef0ac6807afc21e7fd46a9e0e5b Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:37:57 +0200
Subject: [PATCH 07/12] sizet to int for rnd

---
 benchmark/benchmark.cpp | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index 76989d7..78d261f 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -15,7 +15,7 @@ int slow_get_page(int key) { return key; }
 std::vector<int> generate_rnd_keys(size_t n, size_t mod = 100) {
   std::vector<int> keys(n);
   std::generate(keys.begin(), keys.end(),
-                [&]() { return static_cast<size_t>(rand() % mod); });
+                [&]() { return static_cast<int>(rand() % mod); });
   return keys;
 }
 

From 1a76a7e2b8c87f71cd81d67c03195093c3d8cad1 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:41:00 +0200
Subject: [PATCH 08/12] remove size_t

---
 benchmark/benchmark.cpp | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index 78d261f..7af0661 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -12,7 +12,7 @@
 
 int slow_get_page(int key) { return key; }
 
-std::vector<int> generate_rnd_keys(size_t n, size_t mod = 100) {
+std::vector<int> generate_rnd_keys(size_t n, int mod = 100) {
   std::vector<int> keys(n);
   std::generate(keys.begin(), keys.end(),
                 [&]() { return static_cast<int>(rand() % mod); });
@@ -24,9 +24,9 @@ int main() {
   size_t rnd_mod = 100;
   std::vector<int> keys = generate_rnd_keys(input_sz, rnd_mod);
 
-  std::map<size_t, std::map<std::string, size_t>> cache_sizes_vs_strategy_hits;
+  std::map<int, std::map<std::string, int>> cache_sizes_vs_strategy_hits;
   std::cout << "Cache type\tN hits\n";
-  for (size_t cache_sz = 10; cache_sz < 101; cache_sz += 10) {
+  for (int cache_sz = 10; cache_sz < 101; cache_sz += 10) {
     std::map<std::string, int> cache_hits;
     cache::perfect_t<int> perfect_cache(cache_sz, keys);
     cache::LRU_t<int> lru{cache_sz};

From 0b2860f714778280db64ef472b2d24c1d770f86d Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:43:34 +0200
Subject: [PATCH 09/12] cace size is size_t

---
 benchmark/benchmark.cpp | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index 7af0661..480a9e1 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -24,9 +24,9 @@ int main() {
   size_t rnd_mod = 100;
   std::vector<int> keys = generate_rnd_keys(input_sz, rnd_mod);
 
-  std::map<int, std::map<std::string, int>> cache_sizes_vs_strategy_hits;
+  std::map<size_t, std::map<std::string, int>> cache_sizes_vs_strategy_hits;
   std::cout << "Cache type\tN hits\n";
-  for (int cache_sz = 10; cache_sz < 101; cache_sz += 10) {
+  for (size_t cache_sz = 10; cache_sz < 101; cache_sz += 10) {
     std::map<std::string, int> cache_hits;
     cache::perfect_t<int> perfect_cache(cache_sz, keys);
     cache::LRU_t<int> lru{cache_sz};

From f7fe863e8d78bf97f127d7a603774b1f82121466 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:45:50 +0200
Subject: [PATCH 10/12] release flag in CI/CD

---
 .github/workflows/cmake-multi-platform.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/cmake-multi-platform.yml b/.github/workflows/cmake-multi-platform.yml
index 42c677e..d2840a2 100644
--- a/.github/workflows/cmake-multi-platform.yml
+++ b/.github/workflows/cmake-multi-platform.yml
@@ -24,7 +24,7 @@ jobs:
       # To add more build types (Release, Debug, RelWithDebInfo, etc.) customize the build_type list.
       matrix:
         os: [ubuntu-latest, windows-latest]
-        build_type: [Debug]
+        build_type: [Release]
         c_compiler: [gcc, clang, cl]
         include:
           - os: windows-latest

From c335019599731de0a8760a3cd048bd0c7d06dd8c Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:47:38 +0200
Subject: [PATCH 11/12] size_t problem in benchmark.cpp

---
 benchmark/benchmark.cpp | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/benchmark/benchmark.cpp b/benchmark/benchmark.cpp
index 480a9e1..ec258ab 100644
--- a/benchmark/benchmark.cpp
+++ b/benchmark/benchmark.cpp
@@ -21,7 +21,7 @@ std::vector<int> generate_rnd_keys(size_t n, int mod = 100) {
 
 int main() {
   size_t input_sz = 100'000;
-  size_t rnd_mod = 100;
+  int rnd_mod = 100;
   std::vector<int> keys = generate_rnd_keys(input_sz, rnd_mod);
 
   std::map<size_t, std::map<std::string, int>> cache_sizes_vs_strategy_hits;

From 163912fe86901dd585916c7726b5fee4bdb9f9a0 Mon Sep 17 00:00:00 2001
From: Your Name <you@example.com>
Date: Thu, 25 Jun 2026 14:50:50 +0200
Subject: [PATCH 12/12] rm unused var

---
 tests/LFU_test.cpp     | 2 --
 tests/LRU_test.cpp     | 2 --
 tests/perfect_test.cpp | 2 --
 3 files changed, 6 deletions(-)

diff --git a/tests/LFU_test.cpp b/tests/LFU_test.cpp
index 1e20f86..8d1d145 100644
--- a/tests/LFU_test.cpp
+++ b/tests/LFU_test.cpp
@@ -87,9 +87,7 @@ TEST(LFU, eviction_LRU) {
 
 size_t nhits(const std::vector<int> &input) {
   size_t cache_sz = input[0];
-  size_t nelem = input[1];
   std::vector<int> page_ids(input.begin() + 2, input.end());
-  assert(page_ids.size() == nelem);
 
   cache::LFU_t<page_t> lfu{cache_sz};
   size_t hits{};
diff --git a/tests/LRU_test.cpp b/tests/LRU_test.cpp
index 0d12d15..cdab383 100644
--- a/tests/LRU_test.cpp
+++ b/tests/LRU_test.cpp
@@ -67,9 +67,7 @@ TEST(LRU, eviction) {
 
 size_t nhits(const std::vector<int> &input) {
   size_t cache_sz = input[0];
-  size_t nelem = input[1];
   std::vector<int> page_ids(input.begin() + 2, input.end());
-  assert(page_ids.size() == nelem);
 
   cache::LRU_t<page_t> lru{cache_sz};
   size_t hits{};
diff --git a/tests/perfect_test.cpp b/tests/perfect_test.cpp
index 1304617..260d82a 100644
--- a/tests/perfect_test.cpp
+++ b/tests/perfect_test.cpp
@@ -86,9 +86,7 @@ TEST(Perfect_cache, eviction_2) {
 
 size_t nhits(const std::vector<int> &input) {
   size_t cache_sz = input[0];
-  size_t nelem = input[1];
   std::vector<int> page_ids(input.begin() + 2, input.end());
-  assert(page_ids.size() == nelem);
 
   cache::perfect_t<page_t> perf_cache{cache_sz, page_ids};
   size_t hits{};