Problem Statement
What is Dynamic Programming? Explain its key principles and when to use it.
Explanation
Dynamic programming is an algorithmic technique for solving optimization problems by breaking them down into simpler overlapping subproblems and storing their solutions to avoid redundant computation. It is applicable when a problem exhibits two key properties: optimal substructure and overlapping subproblems.
Optimal substructure means that the optimal solution to a problem can be constructed from optimal solutions to its subproblems. For example, the shortest path from A to C through B is the shortest path from A to B plus the shortest path from B to C. If a problem lacks this property, DP cannot be applied.
Overlapping subproblems means that the same subproblem is solved multiple times during the recursive solution. This is where DP provides its efficiency gain. For example, computing Fibonacci of 5 requires computing Fibonacci of 3 twice through different recursive paths. By storing the result of Fibonacci of 3 the first time, we can reuse it instead of recalculating.
There are two main approaches to dynamic programming. The top-down approach, called memoization, starts with the original problem and uses recursion to break it down into subproblems. It stores results in a cache such as an array or hash map to avoid recomputing. The code is often more intuitive as it follows the natural recursive structure. However, it has recursion overhead and may compute only necessary subproblems.
The bottom-up approach, called tabulation, starts with the smallest subproblems and iteratively builds up to the solution. It fills a DP table in a specific order ensuring dependencies are satisfied. The code uses iteration instead of recursion, avoiding stack overflow. It may compute all subproblems even if some are unnecessary, but often allows better space optimization.
To identify when to use dynamic programming, look for these signals. The problem asks for optimization such as maximum, minimum, longest, or shortest. The problem can be broken into similar smaller subproblems. Solving subproblems independently and combining results gives the solution. A naive recursive solution has exponential time complexity due to repeated calculations. The problem involves counting ways or possibilities. Examples include finding minimum or maximum values, counting paths or arrangements, and problems with choices at each step.
The typical steps to solve a DP problem are: first, define the DP state clearly, identifying what each DP entry represents. Second, establish the recurrence relation showing how to compute a state from previous states. Third, identify base cases where the answer is known without computation. Fourth, determine the evaluation order ensuring dependencies are computed before they are needed. Fifth, implement using either memoization or tabulation. Sixth, optimize space if the solution uses too much memory.
Common DP patterns include linear DP where state depends on previous states in a sequence like Fibonacci or climbing stairs, interval DP where state represents a range of elements like matrix chain multiplication, subset DP where we consider including or excluding elements like knapsack, grid DP where we move through a 2D grid like unique paths, and string DP involving operations on strings like edit distance or LCS.