Deep Dive into Algorithm Design Paradigms

Explore algorithm design paradigms, including divide and conquer, greedy, dynamic programming, backtracking, branch and bound, randomized and approximation, and master complexity analysis with big O, omega, and theta.

Frame recurrence relations to model time complexity of recursive algorithms and analyze pseudocode, using substitution, recursion tree method, and master's theorem with Fibonacci and binary search examples.

Explore solving a linear recurrence t(n)=t(n-1)+1 with the substitution method, uncovering the pattern t(n)=t(n-k)+k and arriving at time complexity O(n) for the base case n=0.

Apply the master's theorem to divide-and-conquer recurrences, derive theta time complexities from a t(n/b) + f(n) forms, and learn the three cases with practical examples.

Apply the master's theorem to decreasing recurrences, using the three cases based on a to derive theta complexities from f(n).

Learn merge sort's pseudocode and time complexity analysis, including dividing the array with low, high, and mid, merging with two subarrays, and the recurrence t(n)=2 t(n/2)+O(n) yielding n log n.

Explore the quicksort algorithm, a divide-and-conquer sorting method that uses a pivot to partition the list into left and right subarrays and sorts each side.

Explore the quicksort algorithm through its pseudocode and partitioning steps, analyze best, worst, and average cases, and relate time complexity using big O, omega, and theta.

Binary search on sorted lists compares the middle element and halves the search range with low and high pointers, illustrating iterative and recursive approaches.

Compare iterative and recursive binary search: locate the key using mid, low, high pointers, returning the index or zero, with time complexity log n worst case and 1 best case.

Discover how to find the maximum and minimum in an array using the dividend conquer method with a recursive divide-and-conquer approach. Practice with examples, pseudocode, and time complexity analysis.

Compute the maximum and minimum in an array using a straightforward iterative method, initializing max and min to the first element and scanning to update them, achieving O(n) time.

Learn to solve the maximum subarray sum with a divide and conquer approach that combines left, right, and cross midpoint sums, yielding O(n log n) time.

Master the greedy method, a paradigm that makes locally optimal choices to reach a global optimum. Learn its greedy choice property and optimal substructure through illustrative coin examples.

Explore Huffman coding, a greedy, variable-length data compression method that assigns shorter codes to frequent symbols and uses a priority queue to build the Huffman tree for lossless decoding.

Explore the Huffman coding algorithm through its pseudocode, priority queue construction, and tree-building steps, with complexity analysis and applications in file, image, video, audio, and text compression.

Explore how Prim's algorithm builds a minimum spanning tree for a connected weighted undirected graph by greedily selecting the smallest edge to add new vertices, ensuring connectivity without cycles.

Explore three fundamental tree traversals—preorder, inorder, and postorder—detailing the visiting order of root, left subtree, and right subtree to visit every node exactly once.

Master in-order traversal by visiting the left subtree, then the root, then the right subtree, with step-by-step examples and the corresponding pseudocode.

Explore preorder traversal by visiting the root, then the left subtree, then the right subtree, illustrated with an example and its corresponding traversal steps and pseudocode.

Explore postorder traversal by visiting the left subtree, then the right subtree, and finally the root node, to visit all nodes in the tree.

Solve the 0/1 knapsack problem with dynamic programming, building a 2d table to store subproblem results, and choose whole items to maximize profit under capacity.

Master the longest common subsequence problem using dynamic programming, building a table to compute the maximum length and reconstruct the subsequence from two sequences.

Through dynamic programming, this lecture demonstrates matrix chain multiplication with four matrices, computing optimal cost 72 and the exact order, starting with a2 and a3, then a1, then a4.

Explore the floyd-warshall algorithm, a dynamic-programming all-pairs shortest-path method using a distance matrix and intermediate nodes to compute minimal paths for every vertex pair, works for directed and undirected graphs.

Explore the sum of subset problem using backtracking to print subsets that sum to the target sum from a set of non-negative integers, with include-exclude steps and pruning.

Explore the hamiltonian cycle problem on undirected graphs using backtracking to visit each vertex exactly once and return to start. Apply it to traveling salesman and routing via depth-first search.

Use branch and bound on 0/1 knapsack, with fractional knapsack upper bounds and profit-by-weight ratios to prune state-space and pick items 2, 3, and 4 for 57 at capacity 12.

Explore breadth-first search, a level-order traversal for graphs and trees that uses a queue to visit each level's neighbors, enabling shortest paths, web crawling, and puzzle solving with O(V+E).

Explore depth-first search, traversing as far as possible along each branch before backtracking. Its time complexity is O(V+E) and it enables topological sorting, cycle detection, and strongly connected components.