Design and Analysis of Algorithms

Apply divide and conquer to the defective coin problem, using recursive partitioning and balance checks to show the method's efficiency and recursive nature.

Explore divide-and-conquer in practice by applying binary search to arrays in non-decreasing order, using iterative and recursive approaches to locate a key by narrowing the search range.

Explore the merge sort algorithm, a divide and conquer sorting technique that splits a sequence at the middle, recursively sorts halves, and merges them into a single sorted list.

Explore the quicksort algorithm, a divide and conquer sorting method that uses a pivot to partition data, swaps, and recursively sorts subarrays to achieve a sorted sequence.

Explore the time complexities of merge sort and quick sort, derive T(n)=2T(n/2)+Cn, and show merge sort is n log n while quick sort can be n^2 in the worst case.

Compare merge sort and quick sort through divide-and-conquer principles. Merge sort uses an auxiliary area for merging; quick sort rearranges in place with variable partitioning and possible worst-case when sorted.

Explore Strassen's divide-and-conquer approach to matrix multiplication, reducing multiplications to seven and beating the cubic complexity, with padding to power-of-two sizes and recurrence analysis.

Explore the design and analysis of algorithms by mastering disjoint sets, set operations, and three representations—tree, data, and parent representations—for efficient set management.

Explore set operations in disjoint set unions, including union and find, and compare simple and weighted unions with path compression in tree representations.

Explore graph definitions with vertices and edges, and master depth-first search traversal by visiting adjacent nodes, backtracking, and avoiding cycles to map connections efficiently.

Explore how breadth-first search traverses a graph from a starting node by enqueueing unexplored neighbors and avoiding cycles until all vertices are visited.

Learn about bi connected components and articulation points in graphs, and how deleting a vertex separates a graph into two or more components for understanding connected components.

learn how to identify articulation points in a connected component with a depth-first search and low values. explore constructing the dfs tree, back edges, and root and non-root conditions.

Master the greedy method and its control abstraction with a change-making example. Learn to distinguish feasible from optimal solutions and apply to knapsack, job sequencing with deadlines, and shortest path.

The lecture introduces the fractional knapsack problem and shows how to maximize profit by selecting items in descending profit-to-weight ratio and filling remaining capacity with a fractional part.

Learn how to maximize profit by scheduling jobs with deadlines using a greedy approach: sort by profit, assign jobs to available time slots, and understand feasible solutions.

Prim's algorithm builds a minimum cost spanning tree by starting from an arbitrary vertex and adding the smallest edge to an unvisited vertex, avoiding cycles in a weighted graph.

Kruskal's algorithm constructs a minimum cost spanning tree by repeatedly selecting the smallest edges, skipping those that form cycles, and using union and find operations to unite components.

Explore how to compute the shortest paths from a single source to all vertices using Dijkstra's algorithm on weighted graphs.

Explore dynamic programming and the principle of optimality through a level-wise shortest-path example, comparing with greedy methods and showing how to build optimal solutions step by step.

Learn to solve the 0/1 knapsack problem with dynamic programming to maximize profit under capacity. Choose or skip items, no fractional parts, and determine included objects from the DP steps.

use dynamic programming to find the optimal order of multiplying a chain of matrices, minimizing the number of scalar multiplications and computing the cost table to guide optimal parenthesization.

Explore the travelling salesman problem: start at vertex 1, visit every vertex exactly once, return to 1, and minimize total cost using dynamic programming on a cost matrix.

Explore reliability design using dynamic programming to build a multi-device system with series and parallel configurations, optimizing reliability under a cost constraint.

Use dynamic programming to solve the optimal binary search tree problem by minimizing the expected search cost based on probabilities of successful and unsuccessful searches.

Learn backtracking as a search method to solve problems, illustrated by a queen arrangement without attacks. The lecture explains implicit and explicit constraints, state space, and leaf nodes.

Explain n-queen problem on an 8x8 chessboard, solving it with backtracking by placing queens so no two share a row, column, or diagonal using row minus and row plus constraints.

Explores the graph coloring problem using backtracking, defines chromatic number, and demonstrates coloring graphs with at most m colors via a boolean adjacency matrix.

Explore the Hamiltonian cycle problem through backtracking on graphs, visiting every vertex exactly once and returning to the start, with a concrete tracing example.

Explore the 0/1 knapsack problem solved with lc branch and bound, using branching and bounding to maximize profit under capacity, with upper and lower bounds and an intelligent ranking function.

Learn to solve the 0/1 knapsack problem using FIFO branch-and-bound. Build node bounds, include or exclude items, and prune branches to maximize profit under a capacity constraint.