**Least Frequently Used (LFU)** is a caching algorithm in which the least frequently used cache block is removed whenever the cache is overflowed. In LFU we check the old page as well as the frequency of that page and if the frequency of the page is larger than the old page we cannot remove it and if all the old pages are having same frequency then take last i.e FIFO method for that and remove that page.

**Min-heap** data structure is a good option to implement this algorithm, as it handles insertion, deletion, and update in logarithmic time complexity. A tie can be resolved by removing the least recently used cache block. The following two containers have been used to solve the problem:

- A vector of integer pairs has been used to represent the cache, where each pair consists of the block number and the number of times it has been used. The vector is ordered in the form of a min-heap, which allows us to access the least frequently used block in constant time.
- A hashmap has been used to store the indices of the cache blocks which allows searching in constant time.

Below is the implementation of the above approach:

// C++ program for LFU cache implementation #include <bits/stdc++.h> using namespace std; // Generic function to swap two pairs void swap(pair<int, int>& a, pair<int, int>& b) { pair<int, int> temp = a; a = b; b = temp; } // Returns the index of the parent node inline int parent(int i) { return (i - 1) / 2; } // Returns the index of the left child node inline int left(int i) { return 2 * i + 1; } // Returns the index of the right child node inline int right(int i) { return 2 * i + 2; } // Self made heap tp Rearranges // the nodes in order to maintain the heap property void heapify(vector<pair<int, int> >& v, unordered_map<int, int>& m, int i, int n) { int l = left(i), r = right(i), minim; if (l < n) minim = ((v[i].second < v[l].second) ? i : l); else minim = i; if (r < n) minim = ((v[minim].second < v[r].second) ? minim : r); if (minim != i) { m[v[minim].first] = i; m[v[i].first] = minim; swap(v[minim], v[i]); heapify(v, m, minim, n); } } // Function to Increment the frequency // of a node and rearranges the heap void increment(vector<pair<int, int> >& v, unordered_map<int, int>& m, int i, int n) { ++v[i].second; heapify(v, m, i, n); } // Function to Insert a new node in the heap void insert(vector<pair<int, int> >& v, unordered_map<int, int>& m, int value, int& n) { if (n == v.size()) { m.erase(v[0].first); cout << "Cache block " << v[0].first << " removed.\n"; v[0] = v[--n]; heapify(v, m, 0, n); } v[n++] = make_pair(value, 1); m.insert(make_pair(value, n - 1)); int i = n - 1; // Insert a node in the heap by swapping elements while (i && v[parent(i)].second > v[i].second) { m[v[i].first] = parent(i); m[v[parent(i)].first] = i; swap(v[i], v[parent(i)]); i = parent(i); } cout << "Cache block " << value << " inserted.\n"; } // Function to refer to the block value in the cache void refer(vector<pair<int, int> >& cache, unordered_map<int, int>& indices, int value, int& cache_size) { if (indices.find(value) == indices.end()) insert(cache, indices, value, cache_size); else increment(cache, indices, indices[value], cache_size); } // Driver Code int main() { int cache_max_size = 4, cache_size = 0; vector<pair<int, int> > cache(cache_max_size); unordered_map<int, int> indices; refer(cache, indices, 1, cache_size); refer(cache, indices, 2, cache_size); refer(cache, indices, 1, cache_size); refer(cache, indices, 3, cache_size); refer(cache, indices, 2, cache_size); refer(cache, indices, 4, cache_size); refer(cache, indices, 5, cache_size); return 0; }

**Output:**

Cache block 1 inserted. Cache block 2 inserted. Cache block 3 inserted. Cache block 4 inserted. Cache block 3 removed. Cache block 5 inserted.

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