Gain insights into algorithms and their implementation for problem-solving in Python. Learn about recursion, dynamic programming, greedy algorithms, and graph algorithms to enhance coding proficiency and problem-solving skills.

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 Python.

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 Python. 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 Python

# Stack: depth-first
If we implement the “bag” using a stack, we recover our original depth-first search algorithm. Stacks support insertions (push) and deletions (pop) in $O(1)$ time each, so the algorithm runs in $O(V + E)$ time. The spanning tree formed by the parent edges is called a **depth-first spanning tree**. The exact shape of the tree depends on the start vertex and on the order that neighbors are visited inside the loop $(†$, as explained in the algorithm in the previous lesson$)$, but in general, depth-first spanning trees are long and skinny. We will consider several important properties and applications of depth-first search later.
# Queue: breadth-first
If we implement the “bag” using a queue, we get a different graph-traversal algorithm called **breadth-first search**. Queues support insertions (push) and deletions (pull) in $O(1)$ time each, so the algorithm runs in $O(V + E)$ time. In this case, the **breadth-first spanning tree** formed by the parent edges contains the shortest paths from the start vertex $s$ to every other vertex in its component; we will consider the shortest paths in detail later. Again, the exact shape of a breadth-first spanning tree depends on the start vertex and on the order that neighbors are visited in the for loop $(†)$, but in general, breadth-first spanning trees are short and bushy.

# Priority queue: best-first
Finally, if we implement the “bag” using a priority queue, we get yet another family of algorithms called **best-first search**. Because the priority queue stores at most one copy of each edge, inserting an edge or extracting the minimum-priority edge requires $O(\log E)$ time, which implies that best-first search runs in $O(V + E\log E)$ time.

We describe best-first search as a “family of algorithms” rather than a single algorithm because there are different methods to assign priorities to the edges, and these choices lead to different algorithmic behavior. We’ll describe three well-known variants below, but there are many others. In all three examples, we assume that every edge $uv$ or $u\rightarrow v$ in the input graph has a non-negative weight $w(uv)$ or $w(u\rightarrow v).$

First, if the input graph is undirected and we use the weight of each edge as its priority, best-first search constructs the **minimum spanning tree** of the component of $s$. Surprisingly, as long as all the edge weights are distinct, the resulting tree does not depend on the start vertex or the order that neighbors are visited; in this case, the minimum spanning tree is actually unique. This instantiation of best-first search is commonly (but, as usual, incorrectly) known as Prim’s algorithm; we’ll discuss this and other minimum spanning trees in more detail later.

The algorithm runs as follows. Define the length of a path to be the sum of the weights of its edges. We can also compute **shortest paths** in weighted graphs using best-first search, as follows. Every marked vertex $v$ stores a distance $dist(v)$. Initially, we set $dist(s) = 0$. For every other vertex $v$, when we set $parent(v) ← p$, we also set $dist(v) ← dist(p) + w(p\rightarrow v)$, and when we insert the edge $v\rightarrow w$ into the priority queue, we use the priority $dist(v) + w(v\rightarrow w)$. Assuming all edge weights are positive, $dist(v)$ is the length of the shortest path from $s$ to $v$. This instantiation of best-first search is commonly (but, as usual, strictly speaking, incorrectly) known as Dijkstra’s algorithm; we’ll see this algorithm again later.


Finally, define the width of a path to be the minimum weight of any edge in the path. A simple modification of “Dijkstra’s” best-first search algorithm computes **widest paths** from $s$ to every other reachable vertex; widest paths are also called **bottleneck shortest paths**. Every marked vertex $v$ stores a value $width(v)$. Initially, we set $width(s) = ∞$. For every other vertex $v$, when we set $parent(v) ← p$, we also set $width(v) ← min\left\{width(p),w(p\rightarrow v)\right\}$, and when we insert the edge $v\rightarrow w$ into the priority queue, we use the priority $min\left\{width(v), w(v\rightarrow 
w)\right\}$. Widest paths are useful in algorithms for computing maximum flows, which (also) we’ll consider later.
# Disconnected graphs
$WhateverFirstSearch(s)$ only visits the vertices reachable from a single start vertex $s$. To visit every vertex in $G$, we can use the following simple “wrapper” function.

Understand the applications of basic graph algorithms in problem-solving.

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

Important Variants