Dynamic programming is an important algorithm design technique often tested in coding interviews. This comprehensive guide will explain dynamic programming concepts, provide Python code examples, and offer tips on applying dynamic programming to ace technical interviews.

## Table of Contents

## Open Table of Contents

## Introduction

Dynamic programming is an optimization technique used to solve complex problems by breaking them down into simpler subproblems. It is applicable when subproblems overlap - that is, when recursive solutions to subproblems are needed multiple times. The key aspects of dynamic programming are:

- Breaking down a problem into smaller sub-problems
- Storing the results of subproblems to avoid recalculating them
- Combining subproblem solutions to arrive at an overall solution

Understanding dynamic programming is key to succeeding in coding interviews, as it demonstrates strong algorithmic and problem-solving skills. This guide will provide a thorough overview of dynamic programming in Python to help prepare for technical interviews.

## Core Concepts and Approach

There are two key components of dynamic programming:

### 1. Optimal Substructure

A problem exhibits optimal substructure if its overall optimal solution can be constructed from optimal solutions to its subproblems. This enables breaking down the problem into smaller parts and solving each part only once.

For example, in the Fibonacci sequence, the nth term is the sum of (n-1)th and (n-2)th terms. So the optimal solution for the overall problem relies on optimal solutions for subproblems.

### 2. Overlapping Subproblems

Overlapping subproblems occur when the space of subproblems is small, so solving a subproblem repeatedly occurs when recursively breaking down the problem. By storing results of solved subproblems, we can avoid recomputing them repeatedly.

For example, to compute the 10th Fibonacci number, we need to calculate the 9th and 8th numbers multiple times recursively. By storing previously computed terms, we can avoid recalculating them.

The top-down approach starts with the full problem and recursively breaks it down into subproblems. The bottom-up approach iteratively solves subproblems first and combines results to reach the overall solution. Both approaches rely on optimal substructure and avoiding recomputation using memoization.

## Python Implementation

Dynamic programming solutions in Python typically follow a bottom-up approach, iteratively building up to the final solution. There are two key steps:

Define a table (usually a dictionary or array) to store subproblem solutions.

Iterate through subproblems, filling up the table by combining prior subproblem solutions. The final table value gives the overall optimal solution.

Let’s look at Python code examples for two common dynamic programming interview problems.

### 1. Fibonacci Sequence

Here is an implementation to calculate the nth Fibonacci number:

```
def fib(n):
# table to store subproblem solutions
table = {0: 0, 1: 1}
for i in range(2, n+1):
table[i] = table[i-1] + table[i-2]
return table[n]
```

This builds up the sequence iteratively, relying on previously computed terms to avoid recomputation.

### 2. 0-1 Knapsack Problem

The 0-1 knapsack problem involves choosing items with given weights and values to maximize value without exceeding a weight capacity. Here is one solution:

```
def knapsack(weights, values, capacity):
# create 2D table to store subproblem solutions
table = [[0 for _ in range(capacity+1)] for _ in range(len(weights)+1)]
for i in range(1, len(weights)+1):
weight, value = weights[i-1], values[i-1]
for c in range(1, capacity+1):
# if item weight exceeds capacity, take previous solution
if weight > c:
table[i][c] = table[i-1][c]
else:
# maximize value by taking max of:
# - excluding item
# - including item
table[i][c] = max(table[i-1][c],
value + table[i-1][c-weight])
return table[-1][-1]
```

This iteratively tries including/excluding items to maximize value within the weight constraint, storing optimal subsolutions to avoid recomputation.

## Tips for Technical Interviews

Here are some key tips for applying dynamic programming in technical interviews:

**Identify optimal substructure**: Determine if the problem can be broken down where optimal solutions to subproblems contribute to the overall optimal solution.**Recognize overlapping subproblems**: See if subproblems are solved repeatedly and can be memoized. This indicates dynamic programming can optimize repeated work.**Choose a state representation**: Define the parameters (state) to represent a subproblem for storing results. This will be the table keys.**Determine recurrence relation**: Decide the formula relating subproblem solutions to the overall problem. This will be used to iteratively fill the table.**Initialize a table**: Set up a table (dict, array etc.) to map subproblem states to solutions.**Iterate & fill table**: Iteratively solve subproblems using the recurrence relation and fill the table. The final solution will be the last table value.**Space/time tradeoff**: Decide whether to optimize for space or time by either storing all subproblem solutions or only those needed for the optimal solution.

## Common Pitfalls to Avoid

Some common mistakes to avoid when applying dynamic programming include:

Attempting to use memoization when optimal substructure does not exist.

Failing to initialize the table correctly to store subproblems.

Not properly defining the state to uniquely identify subproblems.

Complexity issues from unnecessary table values or inefficient state definitions.

Incorrect recursion logic when combining subsolutions.

Solving repeated subproblems instead of reusing prior solutions.

Incomplete solutions from missing edge cases in recurrence relation.

With practice and by applying the steps covered in this guide, these pitfalls can be avoided to implement efficient and correct dynamic programming solutions.

## Conclusion

Dynamic programming is a must-know technique for technical coding interviews. This guide provided an overview of core concepts, Python code examples, tips for applying dynamic programming during interviews, and common mistakes to avoid. By mastering dynamic programming, you can successfully tackle complex algorithm problems and demonstrate strong problem-solving abilities. With practice on sample problems and by internalizing these techniques, dynamic programming can be a valuable asset in acing coding interviews.