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Dynamic Programming
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DP Table Filling
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subproblems
O(2^n)→O(n)
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Dynamic Programming
Loading Dynamic Programming...
DP Table Filling
Master the art of solving complex problems by breaking them into overlapping subproblems. Watch memoization and tabulation transform exponential algorithms into polynomial ones.
Solutions to larger problems depend on optimal solutions to smaller subproblems — the foundation of DP.
The same subproblems are solved multiple times in a naive recursion — memoization eliminates this waste.
Top-down caches results as they're computed; bottom-up fills a table iteratively from the base case up.
Explore classic dynamic programming problems with step-by-step animations
Classic example of memoization transforming exponential time complexity into linear by caching previously computed values in the recursion tree.
O(n)O(n)Find the minimum number of coins needed to make a target amount using bottom-up tabulation. Classic unbounded knapsack pattern.
O(amount × coins)O(amount)2D DP table approach to find the longest subsequence common to two strings. Foundation for diff tools, version control, and DNA analysis.
O(m × n)O(m × n)Classic optimization problem using 2D DP to maximize total value within a weight capacity, where each item can be taken at most once.
O(n × W)O(n × W)Maximize money robbed from a street of houses with the constraint that no two adjacent houses can be robbed. Elegant 1D DP solution.
O(n)O(1)Find the length of the longest strictly increasing subsequence in an array. Can be solved in O(n²) with DP or O(n log n) with patience sorting.
O(n²) / O(n log n)O(n)Minimum number of insertions, deletions, and substitutions to transform one string into another. Foundation of spell checkers and autocorrect.
O(m × n)O(m × n)Find the optimal parenthesization of a matrix chain to minimize scalar multiplications. Classic interval DP — the table fills diagonally by chain length.
O(n³)O(n²)Kadane's algorithm finds the contiguous subarray with maximum sum in linear time — a beautiful example of optimized DP.
O(n)O(1)Learn when to use each approach
Start with the problem, recurse, cache results
Start with the original problem and recursively break it down, storing results to avoid recomputation.
Natural recursive problems, when not all subproblems are needed, tree-like recursion patterns.
Fibonacci, Tree DP, some optimization problems.
Start from base cases, fill a table iteratively
Start with base cases and iteratively build up solutions to larger problems using a table.
When all subproblems are needed, better space optimization, iterative problem patterns.
Coin Change, LCS, Knapsack, Edit Distance.