Algorithm Vision

Dynamic Programming

Solve complex optimization problems by breaking them into simpler overlapping subproblems. Master memoization and bottom-up approaches through classic problems like Fibonacci sequences, Longest Common Subsequence, Knapsack problems, and Matrix Chain Multiplication. Learn how dynamic programming transforms exponential-time solutions into polynomial-time algorithms.

9 algorithms

Matrix Chain Multiplication

Advanced

Finds the optimal way to parenthesize a chain of matrices to minimize the number of scalar multiplications. Classic dynamic programming problem from CLRS Chapter 15.

O(n³)
O(n²)
dynamic-programmingoptimization
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Fibonacci (Dynamic Programming)

Beginner

Computes Fibonacci numbers efficiently using dynamic programming to avoid redundant calculations. Demonstrates the power of memoization in transforming an exponential-time recursive solution into a linear-time algorithm. A classic introduction to dynamic programming concepts.

O(n)
O(n)
dynamic-programmingmemoization
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Longest Common Subsequence (LCS)

Intermediate

Finds the longest subsequence present in two sequences in the same order. Uses dynamic programming to build a solution table. Applications include DNA sequence alignment, file difference tools (diff), and plagiarism detection.

O(m × n)
O(m × n)
dynamic-programmingstrings
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Edit Distance (Levenshtein)

Intermediate

Dynamic programming algorithm that calculates the minimum number of single-character edits (insertions, deletions, substitutions) required to transform one string into another. Fundamental in spell checking, DNA sequence alignment, and natural language processing.

O(m × n)
O(m × n)
dynamic-programmingstrings
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0/1 Knapsack Problem

Intermediate

Classic dynamic programming problem where you must select items with given weights and values to maximize total value while staying within a weight capacity. Each item can only be taken once (0/1). Applications include resource allocation, portfolio optimization, and cargo loading.

O(n × W)
O(n × W)
dynamic-programmingoptimization
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Longest Increasing Subsequence

Intermediate

Find the longest subsequence where all elements are in increasing order. Classic DP problem.

O(n log n)
O(n)
dynamic-programmingbinary-search
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Kadane's Algorithm (Maximum Subarray)

Intermediate

Elegant dynamic programming algorithm that finds the contiguous subarray with maximum sum in O(n) time. Combines greedy and DP principles by deciding at each step whether to extend the current subarray or start a new one. Essential for stock profit optimization, signal processing, and understanding DP optimization techniques.

O(n)
O(1)
dynamic-programminggreedy
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Coin Change Problem

Intermediate

Classic dynamic programming problem that finds the minimum number of coins needed to make a target amount. Demonstrates optimal substructure where the solution depends on solutions to smaller amounts. Widely applied in financial systems, vending machines, and resource allocation optimization.

O(amount × coins)
O(amount)
dynamic-programmingoptimization
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Palindrome Partitioning

Advanced

Dynamic programming problem that finds the minimum number of cuts needed to partition a string into palindromic substrings. Uses DP to precompute which substrings are palindromes, then finds optimal cuts. Applications include text segmentation, DNA sequence analysis, and string optimization problems.

O(n²)
O(n²)
dynamic-programmingstrings
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💡 Learning Tip

Start with the beginner-level algorithms to build your foundation, then progress to intermediate and advanced topics. Each algorithm includes interactive visualizations, complexity analysis, and code examples in multiple languages.