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Data Structures and Algorithms

The course teaches algorithm analysis, linear data structures, trees, graphs, and sorting algorithms, enabling students to implement, evaluate, and choose optimal solutions for computational problems.

Who Should Take This

It is ideal for undergraduate computer science majors, software engineering interns, and self‑taught programmers who have basic programming experience and want to deepen their understanding of data structures and algorithmic efficiency. Learners will gain the ability to perform asymptotic analysis, design robust implementations, and select the most appropriate techniques for real‑world challenges.

What's Included in AccelaStudy® AI

Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
20 Activity Formats

Course Outline

46 learning goals
1 Algorithm Analysis
1 topic

Asymptotic Notation and Complexity

  • Define Big-O, Big-Omega, and Big-Theta notation and classify common complexity classes (O(1), O(log n), O(n), O(n log n), O(n^2), O(2^n), O(n!)) by their growth rates and practical implications.
  • Analyze the time complexity of code with nested loops, sequential statements, and conditional branches, determining the tight asymptotic bound by identifying the dominant term.
  • Analyze space complexity by counting auxiliary memory usage (excluding input), distinguishing between in-place algorithms (O(1) extra space) and algorithms that require O(n) or O(n^2) auxiliary structures.
  • Solve recurrence relations using the Master theorem (T(n) = aT(n/b) + f(n)) and the recursion tree method to determine the time complexity of divide-and-conquer algorithms.
  • Explain amortized analysis and apply the aggregate method to show that a sequence of n operations on a dynamic array costs O(n) total despite individual operations costing O(n) in the worst case.
2 Linear Data Structures
3 topics

Arrays and Linked Lists

  • Compare static arrays and dynamic arrays (amortized resizing), explaining O(1) random access, O(n) insertion/deletion at arbitrary positions, and the amortized O(1) append operation with doubling strategy.
  • Implement singly and doubly linked lists with insert, delete, search, and reverse operations, analyzing the O(1) insertion/deletion at known positions versus O(n) search tradeoff compared to arrays.
  • Evaluate when to use arrays versus linked lists for a given problem based on access patterns (random vs sequential), insertion/deletion frequency, cache locality, and memory overhead considerations.

Stacks and Queues

  • Implement stacks (LIFO) and queues (FIFO) using both arrays and linked lists, performing push/pop and enqueue/dequeue operations in O(1) time, and explain the deque (double-ended queue) abstraction.
  • Apply stacks to solve problems including balanced parentheses checking, expression evaluation (postfix/infix), and function call simulation, and apply queues to solve BFS traversal and scheduling problems.
  • Implement a monotonic stack to solve next-greater-element, stock span, and largest-rectangle-in-histogram problems, explaining how the monotonic invariant reduces time complexity from O(n^2) to O(n).

Hash Tables

  • Explain hash table operations (insert, search, delete), hash functions, and collision resolution strategies (separate chaining, linear probing, quadratic probing, double hashing), analyzing expected O(1) vs worst-case O(n) performance.
  • Analyze load factor, resizing strategies, and their impact on hash table performance, and explain the requirements for a good hash function (uniform distribution, deterministic, efficient to compute).
  • Apply hash tables to solve frequency counting, two-sum, grouping/bucketing, and duplicate detection problems, choosing hash-based solutions when O(1) average lookup outweighs the overhead of hashing.
3 Trees
3 topics

Binary Trees and BSTs

  • Define binary tree terminology (root, parent, child, leaf, height, depth, level) and implement the four standard traversals (in-order, pre-order, post-order, level-order) both recursively and iteratively.
  • Implement binary search tree insert, search, delete (including the three cases: leaf, one child, two children with in-order successor), and explain why BST operations are O(h) where h is the tree height.
  • Explain how balanced BSTs (AVL trees and red-black trees) maintain O(log n) height through rotations after insertions and deletions, comparing their balancing guarantees and constant-factor overhead.

Heaps and Priority Queues

  • Implement a binary min-heap and max-heap using an array representation, performing insert (sift-up) and extract-min/max (sift-down) in O(log n) and build-heap in O(n) time.
  • Use priority queues (heaps) to solve top-k elements, merge k sorted lists, median maintenance, and task scheduling problems, choosing between min-heap and max-heap based on the access pattern.

Tries and Specialized Trees

  • Implement a trie (prefix tree) with insert, search, and startsWith operations, analyzing O(m) time per operation where m is the key length, and apply tries to autocomplete and word dictionary problems.
4 Graphs
3 topics

Graph Representation and Traversal

  • Represent graphs using adjacency lists and adjacency matrices, comparing their space complexity (O(V+E) vs O(V^2)) and operation efficiency for sparse vs dense graphs.
  • Implement breadth-first search (BFS) using a queue and depth-first search (DFS) using recursion or a stack, analyzing their O(V+E) time complexity and applying them to find connected components and shortest paths in unweighted graphs.
  • Detect cycles in directed graphs using DFS with three-color marking (white/gray/black) and in undirected graphs using union-find or DFS with parent tracking.

Shortest Paths and Minimum Spanning Trees

  • Implement Dijkstra's algorithm using a priority queue for single-source shortest paths in non-negative weighted graphs, analyzing its O((V+E) log V) time complexity with a binary heap.
  • Explain the Bellman-Ford algorithm for graphs with negative edge weights, detecting negative weight cycles, and compare its O(VE) complexity with Dijkstra's approach.
  • Implement topological sort for directed acyclic graphs using both Kahn's algorithm (BFS with in-degree tracking) and DFS-based post-order reversal, applying it to task scheduling and dependency resolution.
  • Implement minimum spanning tree algorithms (Kruskal's with union-find and Prim's with a priority queue), explaining the cut property and comparing their O(E log V) time complexity.

Union-Find

  • Implement the union-find (disjoint set) data structure with union by rank and path compression optimizations, achieving near-O(1) amortized operations, and apply it to connected components and Kruskal's algorithm.
5 Sorting Algorithms
2 topics

Comparison-Based Sorting

  • Implement bubble sort, selection sort, and insertion sort, analyzing their O(n^2) worst-case complexity and explaining why insertion sort is efficient for nearly-sorted data (O(n) best case).
  • Implement merge sort using the divide-and-conquer approach, analyzing its O(n log n) guaranteed time complexity and O(n) auxiliary space requirement, and trace the recursive splitting and merging phases.
  • Implement quicksort with Lomuto and Hoare partitioning schemes, analyzing the O(n log n) average and O(n^2) worst case, and explain how randomized pivot selection and median-of-three mitigate worst-case behavior.
  • Implement heapsort using an in-place binary heap, analyzing its O(n log n) guaranteed time and O(1) auxiliary space, and compare it with merge sort and quicksort on stability, cache performance, and practical speed.
  • Explain the O(n log n) lower bound for comparison-based sorting (decision tree argument) and evaluate which sorting algorithm to choose based on data size, distribution, stability requirements, and memory constraints.

Non-Comparison Sorting

  • Implement counting sort and radix sort, explaining how they achieve O(n+k) and O(d(n+k)) time by exploiting integer key properties, and identify when non-comparison sorts outperform comparison-based alternatives.
6 Searching and Binary Search
1 topic

Binary Search and Variants

  • Implement binary search on sorted arrays (iterative and recursive), analyzing O(log n) time complexity, and handle edge cases including empty arrays, single elements, and off-by-one errors in bounds.
  • Apply binary search variants to find the first/last occurrence, lower/upper bound, search in rotated sorted arrays, and find the minimum in rotated arrays, adapting the search predicate for each variant.
  • Apply binary search on the answer space (parametric search) to optimize problems where the feasibility function is monotonic, such as minimizing the maximum or finding the kth smallest element.
7 Dynamic Programming
1 topic

DP Foundations and Techniques

  • Identify problems exhibiting overlapping subproblems and optimal substructure, and explain the difference between memoization (top-down recursive with cache) and tabulation (bottom-up iterative table filling).
  • Formulate DP recurrences for classic 1D problems (Fibonacci, climbing stairs, house robber, coin change minimum, longest increasing subsequence) and implement both memoized and tabulated solutions.
  • Formulate and implement 2D DP solutions for problems including 0/1 knapsack, longest common subsequence, edit distance, and unique paths in a grid, constructing the state transition table and tracing back the optimal solution.
  • Optimize DP space complexity by reducing a 2D table to two 1D arrays or a single array when the recurrence only depends on the previous row, and identify when this optimization is applicable.
8 Greedy Algorithms and Problem-Solving Patterns
2 topics

Greedy Algorithms

  • Explain the greedy choice property and optimal substructure as prerequisites for a greedy approach, and distinguish problems solvable by greedy algorithms from those requiring dynamic programming.
  • Implement greedy solutions for activity selection, interval scheduling, fractional knapsack, and Huffman coding, informally arguing correctness using exchange arguments.

Common Problem-Solving Patterns

  • Apply the two-pointer technique (converging pointers, same-direction pointers) to solve sorted-array problems (two-sum sorted, container with most water, remove duplicates) in O(n) time.
  • Apply the sliding window technique (fixed-size and variable-size windows) to solve substring and subarray optimization problems (maximum sum subarray of size k, longest substring without repeating characters) in O(n) time.
  • Apply the backtracking pattern to generate permutations, combinations, and subsets, and solve constraint satisfaction problems (N-queens, Sudoku), using pruning to reduce the search space.

Scope

Included Topics

  • Fundamental data structures: arrays (static and dynamic), singly and doubly linked lists, stacks, queues, deques, priority queues (heaps), hash tables (chaining and open addressing), and their time/space complexity characteristics.
  • Tree data structures: binary trees, binary search trees (BST), balanced BSTs (AVL trees, red-black trees conceptually), B-trees conceptually, tries (prefix trees), and tree traversals (in-order, pre-order, post-order, level-order).
  • Graph data structures: adjacency matrix and adjacency list representations, directed and undirected graphs, weighted graphs, and graph properties (connectivity, cycles, bipartiteness, topological ordering).
  • Sorting algorithms: bubble sort, selection sort, insertion sort, merge sort, quicksort (with pivot selection strategies), heapsort, counting sort, radix sort, and their time/space complexity analysis including best, average, and worst cases.
  • Searching algorithms: linear search, binary search (iterative and recursive), interpolation search, and search in specialized structures (BST search, hash table lookup).
  • Graph algorithms: breadth-first search (BFS), depth-first search (DFS), Dijkstra's shortest path, Bellman-Ford, topological sort (Kahn's and DFS-based), minimum spanning trees (Kruskal's and Prim's), and cycle detection.
  • Dynamic programming: overlapping subproblems and optimal substructure properties, memoization (top-down) vs tabulation (bottom-up), classic problems (Fibonacci, knapsack 0/1, longest common subsequence, coin change, edit distance, longest increasing subsequence).
  • Greedy algorithms: greedy choice property, activity selection, Huffman coding, fractional knapsack, and proving correctness via exchange arguments.
  • Algorithm analysis: Big-O, Big-Omega, and Big-Theta notation, amortized analysis (aggregate, accounting, potential methods), space complexity analysis, and recurrence relations (Master theorem, recursion tree method).
  • Recursion and divide-and-conquer: recursive problem decomposition, base cases, recursive complexity analysis, merge sort and quicksort as divide-and-conquer exemplars, and binary search as divide-and-conquer.
  • Problem-solving patterns: two pointers, sliding window, fast/slow pointers, monotonic stack, backtracking, and union-find (disjoint set).

Not Covered

  • Advanced algorithmic topics: approximation algorithms, randomized algorithms, NP-completeness proofs, and computational geometry beyond conceptual awareness.
  • Language-specific standard library implementations (e.g., Java Collections, C++ STL, Python's heapq) beyond illustrative examples.
  • Concurrent data structures, lock-free algorithms, and parallel algorithms.
  • Database-specific indexing structures (B+ trees, LSM trees) beyond the conceptual B-tree overview.
  • Machine learning algorithms, numerical methods, and cryptographic algorithms.
  • Formal proofs of algorithm correctness beyond informal correctness arguments and invariant reasoning.

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