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

# Edit distance and its applications

The **edit distance** between two strings is the minimum number of letter insertions, letter deletions, and letter substitutions required to transform one string into the other. For example, the edit distance between ${\color{Red} FOOD}$ and ${\color{Red} MONEY}$ is at most four:

${\color{Red} \underset{-}{F}OOD } \to {\color{Red}MO\underset{-}{O}D} \to 
{\color{Red} MON\underset{\wedge}{}D} \to
 {\color{Red} MONE\underset{-}D} \to
 {\color{Red} MONEY}$

This distance function was independently proposed by Vladimir Levenshtein in 1965 (working on coding theory), Taras Vintsyuk in 1968 (working on speech recognition), and Stanislaw Ulam in 1972 (working with biological sequences). For this reason, edit distance is sometimes called **Levenshtein distance** or **Ulam distance** (but strangely, never “Vintsyuk distance”).

We can visualize this editing process by aligning the strings one above the other, with a gap in the first word for each insertion and a gap in the second word for each deletion. Columns with two different characters correspond to substitutions. In this representation, the number of editing steps is just the number of columns that do not contain the same character twice.

$\hspace{6cm}{\color{Red} F\hspace{0.3cm} O \hspace{0.3cm}O \hspace{0.87cm} D} \\
 \hspace{5.9cm}{\color{Red} M\hspace{0.275cm} O \hspace{0.27cm} N \hspace{0.3cm} \underset{.}E \hspace{0.3cm} Y}$

It’s fairly obvious that we can’t transform ${\color{Red} FOOD}$ into ${\color{Red} MONEY}$ in three steps, so the edit distance between ${\color{Red} FOOD}$ and ${\color{Red} MONEY}$ is exactly four. Unfortunately, it’s not so easy in general to tell when a sequence of edits is as short as possible. For example, the following alignment shows that the distance between the strings ${\color{Red} ALGORITHM}$ and ${\color{Red} ALTRUISTIC}$ is at most 6. Is that the best we can do?

$\hspace{5.94cm}{\color{Red} A\hspace{0.3cm} L \hspace{0.3cm}G \hspace{0.3cm} O \hspace{0.3cm} R \hspace{0.9cm} I \hspace{0.84cm} T \hspace{0.3cm} H \hspace{0.3cm} M} \\
 \hspace{5.9cm}{\color{Red} A\hspace{0.275cm} L \hspace{0.94cm} T \hspace{0.3cm} R\hspace{0.3cm} U \hspace{0.3cm} I \hspace{0.3cm} S \hspace{0.3cm} T \hspace{0.35cm} I \hspace{0.45cm} C}$

# Recursive structure

To develop a dynamic programming algorithm to compute edit distance, we first need to formulate the problem recursively. Our alignment representation for edit sequences has a crucial “optimal substructure” property. Suppose we have the gap representation for the shortest edit sequence for two strings. If we remove the last column, the remaining columns must represent the shortest edit sequence for the remaining prefixes. We can easily prove this observation by contradiction: If the prefixes had a shorter edit sequence, gluing the last column back on would give us a shorter edit sequence for the original strings. So once we figure out what should happen in the last column, the Recursion Fairy can figure out the rest of the optimal gap representation.

Said differently, the alignment we are looking for represents a sequence of editing operations, ordered (for no particular reason) from right to left. Solving the edit distance problem requires making a sequence of decisions, one for each column in the output alignment. In the middle of this sequence of decisions, we have already aligned a suffix of one string with a suffix of the other.

Because the cost of an alignment is just the number of mismatched columns, our remaining decisions don’t depend on the editing operations we’ve already chosen; they only depend on the prefixes we haven’t aligned yet.


Thus, for any two input strings $A[1 .. m]$ and $B[1 .. n]$, we can formulate the edit distance problem recursively as follows: For any indices $i$ and $j$, let $Edit(i, j)$ denote the edit distance between the prefixes $A[1 .. i]$ and $B[1 .. j]$. We need to compute $Edit(m, n)$.

# Recurrence

When $i$ and $j$ are both positive, there are exactly three possibilities for the last column in the optimal alignment of $A[1 .. i]$ and $B[1 .. j]$:

- **Insertion:** The last entry in the top row is empty. In this case, the edit distance is equal to $Edit(i, j − 1) + 1$. The $+1$ is the cost of the final insertion, and the recursive expression gives the minimum cost for the remaining alignment.

Get to know the edit distance problem and its solution using dynamic programming.

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

Edit Distance