Asymptotic notation provides a mathematical framework for analyzing the resource consumption (execution time and memory footprint) of an algorithm as input size $N o \infty$:

• Big-O (O): Formal upper bound (f(n) <= c * g(n)). Represents the worst-case scenario.
• Big-Omega (Omega): Formal lower bound (f(n) >= c * g(n)). Represents the best-case scenario.
• Big-Theta (Theta): Asymptotically tight bound (c1 * g(n) <= f(n) <= c2 * g(n)).

While Linked Lists offer theoretical $O(1)$ node insertion, modern CPU hardware memory architectures heavily penalize node pointer chasing due to CPU cache misses. Contiguous dynamic arrays (vectors) vastly outperform linked lists for 99% of production workloads due to L1/L2 cache prefetching!

// Monotonic Decreasing Stack: Finding Next Greater Element in O(N) Time
#include <vector>
#include <stack>

std::vector<int> nextGreaterElements(const std::vector<int>& nums) {
    int n = nums.size();
    std::vector<int> result(n, -1);
    std::stack<int> monoStack; // Stores indices

    for (int i = 0; i < n; ++i) {
        // While current element is greater than stack top, resolve top element!
        while (!monoStack.empty() && nums[i] > nums[monoStack.top()]) {
            result[monoStack.top()] = nums[i];
            monoStack.pop();
        }
        monoStack.push(i);
    }
    return result;
}

Red-Black Trees guarantee $O(\log N)$ search, insertion, and deletion by enforcing 5 strict invariant rules:

1. 1. Every node is either RED or BLACK.
2. 2. The root is always BLACK.
3. 3. Every leaf (NIL) is BLACK.
4. 4. If a node is RED, both its children must be BLACK (No consecutive red nodes).
5. 5. For each node, all simple paths from the node to descendant leaves contain the same number of BLACK nodes.

A Fenwick Tree (Binary Indexed Tree) computes range sum queries and point updates in $O(\log N)$ time with $O(N)$ space using the low-bit equation i & (-i):

// Binary Indexed Tree (Fenwick Tree) Implementation
class FenwickTree {
private:
    std::vector<long long> tree;
    int size;

public:
    explicit FenwickTree(int n) : size(n), tree(n + 1, 0) {}

    void add(int index, long long delta) {
        for (; index <= size; index += index & (-index)) {
            tree[index] += delta;
        }
    }

    long long query(int index) const {
        long long sum = 0;
        for (; index > 0; index -= index & (-index)) {
            sum += tree[index];
        }
        return sum;
    }

    long long queryRange(int left, int right) const {
        return query(right) - query(left - 1);
    }
};

Dijkstra's Algorithm calculates single-source shortest paths on weighted graphs with non-negative edge weights in $O((V + E) \log V)$ time using a min-priority queue.

Modern production runtimes use hybrid sorting algorithms:

• Timsort (Python / Java): Merges pre-sorted natural runs using Insertion Sort for small blocks ($O(N)$ best-case on partially sorted data).
• Introsort (C++ std::sort): Starts as QuickSort, transitions to HeapSort if recursion depth exceeds $2 \log N$ (preventing $O(N^2)$ worst-case), and finishes with Insertion Sort on small subarrays.

Master the core DP archetypes: 0/1 Knapsack, Longest Common Subsequence (LCS), Interval DP, Tree DP, and Bitmask DP (Traveling Salesperson in $O(N^2 2^N)$ time).

Greedy algorithms make locally optimal choices at each step (e.g. Huffman Coding, Kruskal's MST). Backtracking explores state trees systematically with aggressive alpha-beta and branch pruning.

The KMP Algorithm precomputes a Longest Prefix Suffix (LPS) array in $O(M)$ time, enabling linear $O(N + M)$ string pattern searching without backtracking along the input text.

Bloom Filters provide constant-time set membership testing with zero false negatives. HyperLogLog estimates the unique cardinality of billions of stream items in just 1.5 KB of RAM!

How storage engines (RocksDB, Cassandra) use Log-Structured Merge-Trees (LSM-Trees) combining in-memory Red-Black MemTables with disk SSTables and Bloom Filters for high-throughput write performance.