Gain insights into essential algorithms for effective problem-solving in Java. Delve into recursion, dynamic programming, and graph algorithms to enhance your coding and tackle complex problems confidently.

As a developer, mastering the concepts of algorithms and being proficient in implementing them is essential to improving problem-solving skills. This course aims to equip you with an in-depth understanding of algorithms and how they can be utilized for problem solving in Java.

Starting with the basics, you'll gain a foundational understanding of what algorithms are, with topics ranging from simple multiplication algorithms to analyzing algorithms. Then, you’ll delve into more advanced topics like recursion, backtracking, dynamic programming, and greedy algorithms. You'll also learn about basic graph algorithms, depth-first search, shortest paths, minimum spanning trees, and all-pairs shortest paths. 

By the end of this course, you'll have acquired a wide range of skills that will significantly enhance your ability to solve problems efficiently in Java. Through this course, not only will you improve your coding skills, but you will also gain confidence in tackling complex problems.

Mastering Algorithms for Problem Solving in Java

# Task
The algorithms described in this chapter can also be modified to return an explicit description of some negative cycle in the input graph $G$, if one exists, instead of only reporting whether or not $G$ contains a negative cycle. Analyze the provided algorithm and then provide its Java implementation in the coding workspace provided below.

# Logic building
Here's an algorithm for the modified version of Johnson's algorithm that returns either the array of all shortest-path distances or a negative cycle.

**Algorithm**



- Let $G = (V, E)$ be the input graph, with vertices $V$ and edges $E$.
- Add a new vertex $s$ to $G$ and add zero-weight edges from $s$ to all vertices in $V$.
- Run the Bellman-Ford algorithm with $s$ as the source vertex to compute a new set of edge weights $h: E -> R$ that satisfy the following property: for any edge $(u, v)$ in $E$, $h(u, v) <= h(u, w) + c(w, v)$ for all $w$ in $V$.
- If the Bellman-Ford algorithm detects a negative cycle, then $G$ contains a negative cycle and the algorithm terminates, returning that negative cycle.
- Otherwise, for each vertex $u$ in $V$, set $h(u, v) = h(u, v) + h(u) - h(v)$, where $h(u)$ is the weight of the path from $s$ to $u$ in the Bellman-Ford shortest path tree.
- For each vertex $u$ in $V$, run Dijkstra's algorithm with $h$ as the edge weights to compute the shortest paths from $u$ to all other vertices in $V$.
- Return the resulting distances or the negative cycle detected in step 4.
- This algorithm first transforms the input graph by adding a new vertex and edges with zero weight, and then runs Bellman-Ford to detect negative cycles. If no negative cycles are detected, it adjusts the edge weights based on the computed $h$ values and runs Dijkstra's algorithm to find the shortest paths from each vertex to all other vertices. If a negative cycle is detected, the algorithm terminates and returns that negative cycle.


Discuss the solution to the problem related to all-pairs shortest paths.

Getting Started

Introduction to Algorithm

Recursion

Backtracking

Dynamic Programming

Greedy Algorithms

Prove Your Skills: A Five-Chapter Assessment

Basic Graph Algorithms

Depth-First Search

Minimum Spanning Trees

Shortest Paths

All-Pairs Shortest Paths

Pushing Your Limits: A Comprehensive Assessment

Wrapping up

Solution: All-Pairs Shortest Paths

Task

Logic building