Dynamic programming is a technique where an algorithmic problem is broken down into subproblems.
Dynamic programming is a computer programming technique that involves breaking down an algorithmic problem into smaller sub-problems, saving their results, and then optimizing these sub-problems to find the overall solution. This approach is typically used to find the maximum or minimum range of an algorithmic query. This article provides a detailed examination of how dynamic programming works, complete with examples.
What Is Dynamic Programming?
Dynamic programming is a computer programming technique that breaks down an algorithmic problem into smaller sub-problems, saves their results, and then optimizes these sub-problems to find the overall solution. This technique is often used to determine the maximum or minimum range of an algorithmic query.
Richard Bellman introduced the concept of dynamic programming in the 1950s. It is a method for mathematical optimization and a technique in computer programming, applicable to problems that can be divided into overlapping subproblems or optimal substructures.
In the case of overlapping subproblems, a large set of equations is broken down into smaller groups. These smaller equations are reused multiple times to arrive at a solution. Optimal substructures, on the other hand, involve finding the best solution to a subproblem and using these solutions to build the best overall result. When a complex problem is divided into smaller parts, a computer algorithm determines the most desirable solutions for these parts, which are then combined to solve the original, more complex problem.
This technique solves problems by breaking them into smaller, overlapping subproblems and storing the results in a table to avoid redundant computations. For example, when using dynamic programming to calculate all possible results from a set of numbers, the initial results are saved and reused in subsequent calculations, thus saving time and reducing computational effort.
The dynamic programming algorithm aims to find the shortest path to a solution by working from the top down or the bottom up. The top-down method breaks equations into smaller ones and reuses the solutions when needed. The bottom-up approach starts with the smallest subproblems, solves them, and uses these solutions to tackle larger subproblems.
Dynamic programming is more efficient than trial-and-error methods but is only useful for problems that can be decomposed into smaller, reusable subproblems.
Recursion vs. dynamic programming
In computer science, recursion is a fundamental concept where the solution to a problem depends on solutions to its smaller subproblems.
Dynamic programming, on the other hand, is an optimization technique for recursive solutions. It is particularly useful for solving recursive functions that make repeated calls to the same inputs. A function is considered recursive if it calls itself during execution. This process can continue multiple times until the solution is computed, and it can potentially run indefinitely if it lacks a base case to terminate the recursion.
However, not all recursive problems can be solved using dynamic programming. Dynamic programming is applicable only when the subproblem solutions overlap. If the subproblems do not overlap, the recursive solution can only be achieved using a divide-and-conquer approach.
For instance, algorithms like merge sort and quick sort are not considered dynamic programming problems because they involve combining the best solutions to subproblems that do not overlap.
Drawbacks of recursion
Recursion uses memory space less efficiently because repeated function calls create entries for all the variables and constants in the function stack. These values remain in the stack until the function returns, and since stack space is limited, this can lead to inefficient memory usage. Moreover, if a recursive function demands more memory than is available in the stack, a stack overflow error will occur.
Recursion is also relatively slow compared to iteration, which uses loops. Each function call in recursion incurs the overhead of allocating space for the function and its data in the stack, causing a slight delay in recursive functions.
Where should dynamic programming be used?
Dynamic programming is utilized when a problem can be broken down into smaller subproblems, which can be further divided into even smaller subproblems. These subproblems often overlap, meaning they require previously computed values to be recalculated. By storing these computed values, dynamic programming reduces the need for repeated calculations, thereby saving time and providing faster solutions.
How Does Dynamic Programming Work?
Dynamic programming works by breaking down complex problems into simpler subproblems and then finding optimal solutions for these subproblems. Memoization is a technique that stores the outcomes of these subproblems, so their solutions do not need to be recomputed when needed again. This approach saves time by avoiding the re-computation of subproblems that have already been solved.
Dynamic programming can be implemented using two approaches:
- Top-down approach:
- In computer science, problems are resolved by recursively formulating solutions and using the answers to the subproblems. If the answers to the subproblems overlap, they can be memoized or stored in a table for later use. The top-down approach employs memoization, which combines recursion and caching. Recursion involves directly calling the function, while caching involves preserving intermediate results. Benefits:
- The top-down approach is easy to understand and implement. Problems are broken down into smaller parts, making it easier to identify the steps needed to solve them. This approach simplifies larger, more complex problems into smaller, more manageable ones, some of which may be reusable.
- It allows subproblems to be solved on demand. Solutions for each part can be queried and reused.
- It is easier to debug, as segmenting problems into small parts allows users to follow the solution and quickly identify where an error might have occurred. Disadvantages:
- The top-down approach uses recursion, which occupies more memory in the call stack, leading to reduced performance. Additionally, deep recursion can cause a stack overflow.
- Bottom-up approach:
- In the bottom-up method, a problem is rewritten by solving smaller subproblems first and using their solutions to solve larger subproblems. This approach eliminates recursion, preventing stack overflow and reducing the overhead from recursive functions. It also saves memory space and decreases the time complexity associated with recalculating values. Benefits:
- It makes decisions about small reusable subproblems and then combines them to solve the larger problem.
- It eliminates recursion, promoting efficient use of memory space and reducing time complexity.
Signs of Dynamic Programming Suitability
Dynamic programming is an effective approach for solving complex problems by breaking them into smaller subproblems using recursion and storing their solutions to avoid redundant computations. However, it is not practical for problems without overlapping subproblems, as storing solutions for non-reusable subproblems is inefficient.
Two main indicators suggest that a problem can be solved with dynamic programming: overlapping subproblems and optimal substructure.
Overlapping Subproblems
When the same subproblems recur multiple times in solving the main problem, they are considered overlapping subproblems. In such cases, storing solutions in a table allows for their reuse, preventing unnecessary recalculations. For example, in the recursive program for calculating Fibonacci numbers, several subproblems overlap. Conversely, a binary search does not have overlapping subproblems, as each subproblem involves a unique array segment, making it unsuitable for dynamic programming.
For instance, finding the nth Fibonacci number involves breaking down the problem F(n) into F(n-1) and F(n-2). Further breaking down F(n-1) involves F(n-2) again, illustrating the reuse of subproblems. Hence, the Fibonacci sequence demonstrates overlapping properties.
Optimal Substructure
A problem exhibits the optimal substructure property if the best solution can be derived from the best solutions of its subproblems. This property is often explained through recursion. While the optimal substructure property is not unique to dynamic programming, many problems with optimal substructures do not have overlapping subproblems and therefore do not qualify as dynamic programming problems.
An example of optimal substructure is finding the shortest path between two points. If node p lies on the shortest path from source node t to destination node w, then the shortest path from t to w is the sum of the shortest paths from t to p and from p to w.
Examples of problems with optimal substructures include the longest increasing subsequence, longest palindromic substring, and longest common subsequence problems. Conversely, problems like the longest path problem and addition-chain exponentiation do not exhibit optimal substructure.
Understanding the Longest Common Subsequence concept in dynamic programming
In dynamic programming, the term “Longest Common Subsequence” (LCS) refers to the longest subsequence that is shared among all given sequences. Unlike the problem of finding the longest common substring, the elements of the LCS do not need to appear consecutively within the original sequences.
The LCS problem exhibits both optimal substructure and overlapping subproblems properties. This means the problem can be broken down into smaller, simpler subproblems that can be solved independently. The solutions to higher-level subproblems are reused in solving lower-level subproblems, hence the term overlapping subproblems.
Because of these properties, solving an LCS problem is more efficient using a dynamic programming approach rather than a purely recursive one. Dynamic programming stores the results of each subproblem in a table for reuse, thereby reducing the need for redundant computations.
For example, consider the sequences “MNOP” and “MONMP.” They share five length-2 common subsequences: “MN,” “MO,” “MP,” “NP,” and “OP.” They also share two length-3 common subsequences: “MNP” and “MOP.” Thus, “MNP” and “MOP” are the longest common subsequences. The LCS concept can be applied in bioinformatics, such as in the process of genome sequencing.
Dynamic Programming Algorithms
Dynamic programming algorithms solve problems by breaking them into smaller parts and then combining the solutions to these subproblems to find the overall solution. They typically aim to find the shortest path. Some primary dynamic programming algorithms include:
- Greedy Algorithms
Greedy algorithms, although different, are sometimes used in conjunction with dynamic programming. They solve problems by finding the local optimum solution at each step, hoping to find a global optimum. However, greedy algorithms do not always guarantee a globally optimal solution, which can be costly in the long run. - Floyd-Warshall Algorithm
The Floyd-Warshall algorithm uses dynamic programming to find the shortest paths between all pairs of vertices in a weighted graph. It works for both directed and undirected weighted graphs. The algorithm compares all possible paths through the graph, gradually optimizing an estimate of the shortest path between two vertices. This method can also be modified to reconstruct the paths.
- Behavior with Negative Cycles: The Floyd-Warshall algorithm can detect negative cycles by checking the diagonal of the path matrix for negative numbers. If a negative cycle exists, it means the graph contains a negative cycle, making it impossible to find a shortest path between any pair of vertices.
- Time Complexity: The algorithm has a time complexity of O(n³) due to its three nested loops, where n is the number of vertices in the graph.
- Bellman-Ford Algorithm
The Bellman-Ford algorithm finds the shortest path from a single source vertex to all other vertices in a weighted digraph. Unlike Dijkstra’s algorithm, the Bellman-Ford algorithm can handle graphs with negative edge weights and still produce a correct result. However, it is slower than Dijkstra’s algorithm. The Bellman-Ford algorithm works through a process called relaxation, which iteratively improves the approximate distances between vertices until the optimal solution is found. If the algorithm detects a negative cycle, it terminates, making it useful for cycle-canceling techniques in network flow analysis.
These dynamic programming algorithms optimize solutions by breaking problems into manageable subproblems, storing intermediate results, and combining these results to achieve an efficient and optimal overall solution.
Examples of Dynamic Programming
Here are a few examples of how dynamic programming can be applied:
- Counting the Number of Ways to Cover a Distance
In some recursive functions, the recursion technique is invoked multiple times, indicating the overlapping subproblem characteristic essential for dynamic programming. Using the top-down approach, store values in a HashMap while retaining the recursive structure, and then return the stored values instead of recalculating them each time the function is invoked. Alternatively, using the bottom-up method, utilize an extra space of dimension n and compute the values of states starting with 1, 2, …, n. Compute the values of i+1 and i+2 and then apply them to determine the value of i+3. - Identifying the Optimal Strategy for a Game
To identify the optimal strategy in a game, consider the “coins in a line” game. Use memoization to compute the maximum value of coins taken by player A for coins numbered h to k, assuming player B plays optimally (Mh,k). To determine each player’s strategy, assign values to the coins they pick and the value of the opponent’s coins. After computation, the optimal strategy for the game is determined by observing the Mh,k value for both players if player A chooses coin h or k. - Counting the Number of Possible Outcomes of a Die Roll
Given an integer M, the goal is to determine the number of ways to obtain the sum M by repeatedly tossing dice. The partial recursion tree for M=8 reveals overlapping subproblems when using the recursion method. By applying dynamic programming, one can optimize the recursive method. Use an array to store computed values for reuse. This approach significantly reduces the runtime, with a time complexity of O(t * n * m), where t is the number of faces, n is the number of dice, and m is the target sum.
Dynamic programming optimizes these problems by breaking them down into manageable subproblems, storing intermediate results, and combining these results to achieve efficient and optimal solutions.
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