If Agile software development is about breaking down large, monolithic applications into small, interconnected microservices, dynamic programming takes a similar approach to complex problems.

Except, dynamic programming isnâ€™t a necessarily a computer programming concept. Since mathematician Richard E. Bellman developed it in the 1950s, dynamic programming has been used to solve complex problems across industries.

In this blog post, we see how you can use the concept and its principles to improve the performance of your software team.

## What is Dynamic Programming?

Dynamic programming refers to breaking down a complex problem into simpler sub-problems in a recursive manner.

It suggests a divide-and-conquer approach, dividing big problems into easy-to-manage parts. By solving the smallest of sub-problems and working your way up, you can combine solutions to arrive at the answer for the original complex problem.

About coining the name, Bellman writes that he chose the word â€˜dynamicâ€™ as it represents something thatâ€™s multi-stage or time-varying. It also has an absolutely precise meaning in the classical physical sense as well as while used as an adjective. He preferred the word â€˜programmingâ€™ as he found it more suitable than planning, decision-making or thinking.

In that sense, dynamic programming is both a method and a tried-and-tested structure.

## The Structure of Dynamic Programming

To effectively use dynamic programming methods, you need to understand two key properties:

### Optimal sub-structure

Optimal sub-structure or optimality is the recursive process of breaking down complex problems into sub-problems which must ensure that the optimal solutions for the smaller ones combine to solve the original one. Optimality stresses the importance of the manner in which you break down your problems.

**The Bellman equation**

The Bellman equation is an important tool that helps build the optimal sub-structure. It breaks down a complex problem into simpler subproblems by expressing the value of a decision/action based on two things:

- The immediate reward of the decision/action
- The discounted value of the next state as a result of that decision/action

Letâ€™s say youâ€™re deciding the best route to take to your office from home. Using dynamic programming, youâ€™d break the journey down into a few milestones. Then, youâ€™d apply the Bellman equation to consider the time it takes to reach a milestone (immediate reward), and the estimated time to reach the next one (discounted value).

By iteratively applying the Bellman equation, you can find the highest value for each state and the best solution for your original problem.

**The Hamilton-Jacobi equation**

The Hamilton-Jacobi equation expands on the Bellman equation by describing the relationship between the value function and the system dynamics. This equation is used for continuous-time problems to directly derive the optimal control law, i.e., the action to take at each state.

**Recurrence relation **

The recurrence relation defines each sequence term in terms of the preceding terms. Using this, you can recursively determine the sequence by first specifying an initial condition and then its relation to each subsequent item.

Consequently, stronger the solution for each sub-problem, more effective the solution for the big problem.

## Overlapping Subproblems and Memoization in Dynamic Programming

Overlapping sub-problems occur when the same problem is part of multiple sub-problemsâ€”being solved repeatedlyâ€”in the process of solving the original problem. Dynamic programming prevents this inefficiency by storing solutions in a table or an array for future reference.

Memoization optimizes goes one step further. It stores the results of expensive functions and reuses them when the same inputs occur again. This prevents redundant calculations, significantly improving the algorithm’s efficiency.

Lazy evaluation, also known as call-by-need, simply postpones the evaluation of an expression until the value is actually needed. This also increases efficiency by avoiding unnecessary calculations and improving performance.

In summary, this is the structure and approach you might take to dynamic programming for solving problems.

**Identify overlapping sub-problems**: With the help of problem statement templates, determine which sub-problems are solved multiple times**Run lazy evaluation**: Make only those evaluations for which values are necessary**Store results**: Use data structures (such as a dictionary, array, or hash table) to store the results of these subproblems**Reuse results**: Before solving a sub-problem, check if its result is already stored. If it is, reuse the stored result. If not, solve the subproblem and store the result for future use

Now that weâ€™ve seen how dynamic programming works in theory, letâ€™s see some of the common algorithms using this technique.

## Common Dynamic Programming Algorithms

The dynamic programming algorithm youâ€™d use depends on the nature of the problem youâ€™re solving. Here are some of the most commonly used algorithms today.

### Floyd-Warshall algorithm

The Floyd-Warshall algorithm is used to find the shortest paths between all pairs of vertices in a weighted graph. It iteratively represents the shortest distance between any two vertices, considering each vertex as an intermediate point.

### Dijkstra’s algorithm

Dijkstraâ€™s algorithm finds the shortest path from a single source node to all other nodes in a weighted graph. It is used in graphs with non-negative edge weights. It takes the greedy approach to make the locally optimal choice at each step to find the overall shortest path.

### Bellman-Ford algorithm

The Bellman-Ford algorithm finds the shortest paths from a single source vertex to all other vertices in a weighted graph, even if it contains negative-weight edges. It works by iteratively updating the shortest known distance to each vertex by considering each edge in the graph and improving the path by finding a shorter one.

### Binary search algorithm

The binary search algorithm finds the position of a target value in a sorted array. It starts with the search range of the entire array and repeatedly divides the search interval in half.

The algorithm compares the target value to the middle element of the array. If the target value is equal to the middle element, the search is complete. If itâ€™s less than, the search continues on the left half of the array. If itâ€™s more than, it does so on the right half. This process repeats until you find the target value or the empty search range.

Letâ€™s look at some of the examples and real-world applications of dynamic programming.

## Examples of Dynamic Programming Algorithms

### Tower of Hanoi

Even if you didnâ€™t know the name, youâ€™d most likely have seen the Tower of Hanoi. It is a puzzle where youâ€™re expected to move a pile of disks from one rod to another, one at a time, always ensuring there is no larger disk on top of a smaller one.

Dynamic programming solves this problem by:

- Breaking it down into moving nâˆ’1 disks to an auxiliary rod
- Moving the nth disk to the target rod
- Moving the nâˆ’1 disks from the auxiliary rod to the target rod

By storing the number of moves required for each sub-problem (i.e., the minimum number of moves for nâˆ’1 disks), dynamic programming ensures that each of them is only solved once, thus reducing the overall computation time. It uses a table to store the previously calculated values for the minimum number of moves for each sub-problem.

### Matrix chain multiplication

Matrix chain multiplication describes the problem of the most efficient way to multiply a sequence of matrices. The goal is to determine the order of multiplications that minimizes the number of scalar multiplications.

The dynamic programming approach helps break the problem into sub-problems, calculating the cost of multiplying smaller chains of matrices and combining their results. It iteratively solves for chains of increasing lengths, the algorithm ensures that each sub-problem is only solved once.

### Longest common subsequence problem

The longest common subsequence (LCS) problem aims to find the longest sub-sequence common to two given sequences. Dynamic programming solves this problem by constructing a table where each entry represents the length of the LCS.

By iteratively filling in the table, dynamic programming efficiently computes the length of the LCS, with the table ultimately providing the solution to the original problem.

## Real-World Applications of Dynamic Programming

Though dynamic programming is an advanced mathematical theory, it is widely used in software engineering for a number of applications.

**DNA sequence alignment**: In bioinformatics, researchers use dynamic programming for a number of use cases, such as identifying genetic similarities, predicting protein structures, and understanding evolutionary relationships.

By breaking down the alignment problem into smaller sub-problems and storing the solutions in a matrix, the algorithm calculates the best match between sequences. This framework makes otherwise computationally infeasible tasks practical.

**Airline scheduling and routing**: Representing the airports as nodes and flights as directed edges, planners use the Ford-Fulkerson method, to find the optimal routing of passengers through the network.

By iteratively augmenting paths with available capacity, these algorithms ensure efficient resource allocation, utilization, and balance between demand and availability, increasing efficiency and reducing costs.

**Portfolio optimization in finance**: Investment bankers solve the problem of asset allocation across various investments to maximize returns while minimizing risk using dynamic programming.

By breaking down the investment period into stages, dynamic programming evaluates the optimal asset allocation for each stage, considering the returns and risks of different assets. The iterative process involves updating the allocation strategy based on new information and market conditions continuously refining the portfolio.

This approach ensures that the investment strategy adapts over time, leading to a balanced and optimized portfolio that aligns with the investor’s risk tolerance and financial goals.

**Urban transportation network planning**: To find the shortest paths in urban transportation networks, planners use the graph and path theory, which utilizes dynamic programming.

For instance, in a city’s public transit system, stations are represented as nodes and routes as edges with weights corresponding to travel times or distances.

The Floyd-Warshall algorithm optimizes travel routes by iteratively updating the shortest paths using the relationship between direct and indirect routes, reducing overall travel time and enhancing the transportation system’s efficiency.

Despite its many applications, dynamic programming isnâ€™t without challenges.

## Challenges in Dynamic Programming

Unlike the Brute force search approach, where you try every possible solution until you find the correct one, dynamic programming offers the most optimized solution for a large problem. While doing so, here are some key factors to keep in mind.

### Managing multiple sub-problems

**Challenge**: Dynamic programming requires managing numerous sub-problems to arrive at a solution to the larger problem. This means that you must:

- Carefully consider the organization of intermediate results to avoid redundant computations
- Identify, solve, and store each subproblem in a structured format like a table or memoization array
- Efficiently manage memory when the scale of sub-problems increases
- Accurately calculate and retrieve each subproblem

**Solution**: To do all this and more, you need a robust project management software like ClickUp. ClickUp Tasks enables you to create indefinite sub-tasks to manage dynamic programming sequences. You can also set custom statuses, add custom fields and program management system that suits your needs.

### Problem definition

**Challenge**: Complex problems can be a huge challenge for teams to understand, delineate, and break down into meaningful sub-problems.

**Solution**: Bring the team together and brainstorm possibilities. ClickUp Whiteboard is a great virtual canvas for ideating and debating the problem as well as dynamic programming techniques you employ. You can also employ a problem-solving software to help.

### Debugging and testing

**Challenge**: Debugging and testing dynamic programming solutions can be complex due to the interdependence of sub-problems. Errors in one sub-problem can affect the entire solution.

For example, an incorrect recurrence relation in the edit distance problem can lead to incorrect overall results, making it difficult to pinpoint the exact source of the error.

**Solutions**

- Conduct code reviews
- Follow pair programming to have other team members review the code or work together on the implementation, catching mistakes and providing different perspectives
- Use root cause analysis tools to identify the origin of the mistakes to avoid them from occurring again

### Poor workload management

**Challenge**: When different team members are responsible for different parts of the algorithm, there can be inconsistencies in understanding base cases, sub-problem definitions, and uneven workload management, all leading to incorrect results.

**Solutions**: Overcome this challenge by implementing effective resources scheduling with ClickUpâ€™s Workload view.

### Coordination and collaboration

**Challenge**: Complex problems require deep understanding and precise implementation. Ensuring all team members are on the same page regarding the problem formulation, recurrence relations, and overall strategy is a huge task.

**Solution**: Set up a unified collaboration platform like ClickUp. The ClickUp chat view consolidates all messages, allowing you to manage all conversations in one place. You can tag your team members and add comments without moving different tools.

### Performance optimization

**Challenge**: Optimizing the performance of a dynamic programming solution requires careful consideration of both time and space complexity. It is common that while one part of the team optimizes the time complexity, another inadvertently increases the space complexity, leading to suboptimal overall performance.

**Solution**: ClickUp Dashboard comes to the rescue. It gives real-time insights into the performance of the overall project, with which you can measure, adjust, and optimize the dynamic program tasks to get higher efficiency.

### Documentation and knowledge transfer

**Challenge**: Agile teams prioritize working software over documentation. This can present a unique challenge. For instance, if the recurrence relations are not well-documented, new team members may struggle to understand and build upon the existing solution.

**Solution**: Create an operations strategy that strikes a balance between documentation and working code.. Use ClickUp Docs to create, edit, and manage documentation about why and how certain decisions were designed.

## Solve Complex Problems With Dynamic Programming on ClickUp

Modern-day problems are, by their very definition, complex. Especially given the depth and sophistication of todayâ€™s software, the problems that engineering teams face are immense.

Dynamic programming offers an efficient and effective approach to problem-solving. It reduces redundant computations and uses iterative processes to strengthen results while optimizing capacity and performance.

However, managing dynamic programming initiatives end-to-end requires effective project management and capacity planning.

ClickUp for software teams is the ideal choice. It enables you to handle interconnected tasks, document thought-processes, and manage outcomes, all in one place. Donâ€™t take our word for it.

## Common FAQs

### 1. What is meant by dynamic programming?

The term dynamic programming refers to the process of algorithmically solving complex problems by breaking them into simpler sub-problems. The method prioritizes solving each sub-problem just once and storing its solution, typically in a table, to avoid redundant computations.

### 2. What is an example of a dynamic programming algo?

You can use dynamic programming to determine the optimal strategy in anything from the Fibonacci sequence to spatial mapping.

One of the examples of dynamic programming is the Knapsack problem. Here, you have a set of items, each with a weight and a value, and a knapsack with a maximum weight capacity. The goal is to determine the maximum value you can carry in the knapsack without exceeding the weight capacity.

Dynamic programming solves this problem by breaking it down into sub-problems and storing the results of these subproblems in a table. It then uses these results to build the optimal solution to the overall problem.

### 3. What is the basic idea of dynamic programming?

The basic idea is to approach dynamic programming problems by breaking them down into simpler sub-problems, solving each of them once, rolling up to the solution to the larger problem.