Linear Probing
Linear probing is a cache-conscious collision resolution algorithm used in open-addressing hash tables. When a hash collision occurs at index h, the algorithm sequentially probes subsequent indices (h + 1, h + 2, …, mod capacity) until an empty slot or the matching key is found. While highly efficient due to contiguous memory allocation and hardware prefetching, it suffers from primary clustering under high load factors.
HOW-TO IMPLEMENT & CONFIGURE
- Modern compiler supporting C++17 or equivalent systems language with direct memory control
- Understanding of modular arithmetic and prime/power-of-two capacity sizing
- Profiler tools such as valgrind, perf, or sanitizers (ASan)
Architectural Setup & Memory Allocation
Initialize a contiguous vector or raw memory buffer with a fixed capacity. Track the current number of active elements and explicitly mark sentinel states for empty and deleted slots.
#include <vector>
#include <optional>
#include <cstdint>
template <typename K, typename V>
class LinearProbingHashMap {
private:
enum class State : uint8_t { EMPTY, OCCUPIED, TOMBSTONE };
struct Entry {
K key;
V value;
State state = State::EMPTY;
};
std::vector<Entry> table;
size_t capacity;
size_t size;
const float max_load_factor = 0.7f;
public:
explicit LinearProbingHashMap(size_t cap = 16) : capacity(cap), size(0) {
table.resize(capacity);
}
};Core Algorithm Implementation: Insertion, Search, and Deletion
Implement the core probing loops for insertion, lookup, and lazy deletion via tombstones to preserve probe chains.
bool insert(const K& key, const V& value) {
if (static_cast<float>(size + 1) / capacity > max_load_factor) {
resize(capacity * 2);
}
size_t hash_val = std::hash<K>{}(key);
size_t idx = hash_val % capacity;
size_t start_idx = idx;
while (table[idx].state == State::OCCUPIED) {
if (table[idx].key == key) {
table[idx].value = value;
return true;
}
idx = (idx + 1) % capacity;
if (idx == start_idx) return false;
}
table[idx].key = key;
table[idx].value = value;
table[idx].state = State::OCCUPIED;
++size;
return true;
}Verification & Unit Testing
Write rigorous unit tests covering edge cases like table saturation, duplicate key updates, and collision chains.
#c++ unit test snippet
LinearProbingHashMap<int, std::string> map(4);
assert(map.insert(1, "alpha"));
assert(map.insert(5, "beta")); // Collides with 1 if capacity is 4
std::string val;
assert(map.search(5, val) && val == "beta");
assert(map.remove(1));
assert(!map.search(1, val));Production Hardening & Resizing Strategy
Handle rehashing when load factor thresholds are breached. Purge tombstones during re-allocation to compress cluster chains.
void resize(size_t new_capacity) {
std::vector<Entry> old_table = std::move(table);
capacity = new_capacity;
table.clear();
table.resize(capacity);
size = 0;
for (const auto& entry : old_table) {
if (entry.state == State::OCCUPIED) {
insert(entry.key, entry.value);
}
}
}Algorithmic Mechanics & Primary Clustering
Linear probing resolves collisions by checking the next sequential slot. The mathematical formulation for the i-th probe is h(k, i) = (h(k) + i) mod m. While this maximizes spatial locality and CPU cache efficiency, it induces primary clustering: contiguous blocks of occupied slots grow larger as the load factor increases, degrading probe times from O(1) expected to O(n) worst-case.
Tombstone Management and Memory Leaks
Because open addressing tables terminate searches upon encountering an empty slot, deleting an element cannot simply clear the slot without breaking probe chains. Standard implementations use 'tombstones' (marked deletion flags). However, an excessive accumulation of tombstones degrades lookup performance, requiring periodic full table rehashes to reclaim space.
📚 STANDARDS & ACADEMIC RFC CITATIONS
- Knuth, D. E. (1998). The Art of Computer Programming, Volume 3: Sorting and Searching (2nd ed.). Addison-Wesley.
- Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.