Fundamentals of Data Structures and Algorithms

The article introduces the basic concepts of algorithms and data structures, explains complexity functions, and presents methods for evaluating the strengths and weaknesses of various structures, aiming to help readers focus on the trade‑offs of each technique.

Case study 1 examines the selection problem (finding the k‑th largest element) and describes two straightforward solutions: sorting the entire array with a simple algorithm like bubble sort, and a slightly improved method that keeps only the top k elements while scanning the rest.

Case study 2 addresses a word‑search puzzle where words may appear in any direction; two intuitive approaches are outlined, both relying on extensive nested loops, highlighting the performance challenges as the grid size grows.

The fundamentals section defines data as the carrier of information, distinguishes logical relationships (linear, tree, graph structures) from physical storage methods (sequential, linked, indexed, hashed), and classifies logical structures accordingly.

Algorithm basics cover the definition of an algorithm, the importance of time and space resources, and introduce Big‑O notation with common complexity classes such as O(1), O(log N), O(N), O(N log N), O(N²), O(2ᴺ), and the principles for simplifying asymptotic expressions.

Complexity functions are discussed, explaining both time and space complexity, their measurement units, and how optimizing one often impacts the other.

The knowledge‑reserve part explains prime‑number theory as a basis for hash functions, describes common hash algorithms (MD4, MD5, SHA‑1, custom hash), and outlines collision‑resolution techniques like chaining and open addressing.

Tree fundamentals define trees, depth, height, and traversal methods (pre‑order, post‑order), providing a foundation for the subsequent tree‑based structures.

Binary Search Tree (BST) concepts lead into AVL trees, detailing single and double rotations, their O(log N) search time, and the trade‑off of extra rebalancing work.

Red‑Black trees are presented as self‑balancing BSTs with color properties, guaranteeing O(log N) operations and describing insertion cases and rebalancing steps.

B+Tree is described as a disk‑oriented variant of B‑Tree, emphasizing leaf‑node chaining for range queries, its typical depth (2–4), and its O(log N) performance, along with pros such as high storage capacity and cons like write amplification.

SkipList is introduced as a layered linked list offering O(log N) expected time for search, insertion, and deletion, with a brief explanation of its level‑wise structure and random promotion.

Bloom filter algorithm steps are outlined: using k hash functions to set bits in a fixed‑size bit array, checking membership by verifying all corresponding bits, noting its O(k) query time, constant space, inability to delete, and its suitability for cache‑penetration scenarios.

HashTree construction uses consecutive prime numbers to define branching factors, achieving at most ten levels of lookup with O(1) time in practice, and is noted for its simple structure and fast searches, though it lacks ordering.

The article concludes by indicating that additional important data structures will be covered in future installments.