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Graph Algorithms
Dynamic Programming
Greedy
Divide & Conquer

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Back to Algorithms

Dynamic Programming

Master the art of solving complex problems by breaking them into simpler subproblems. Learn memoization, tabulation, and optimization techniques for efficient algorithms.

Dynamic Programming Fundamentals

Optimal Substructure

Optimal solution can be constructed from optimal solutions of subproblems

Overlapping Subproblems

Same subproblems are solved multiple times in naive recursive approach

Memoization

Store and reuse results of expensive function calls for efficiency

Common DP Patterns

Linear DP

Problems with 1D state space

Fibonacci • House Robber • Climbing Stairs

2D DP

Problems with 2D state space

Knapsack • LCS • Grid Path

Interval DP

Problems on ranges or intervals

Matrix Chain • Burst Balloons • Optimal BST

Tree DP

DP on tree structures

Tree Diameter • House Robber III • Binary Tree Cameras

Memoization vs Tabulation

Memoization (Top-Down)

Recursive approach with caching of results

Advantages:

• Natural recursive thinking
• Computes only needed subproblems
• Easy to implement

Disadvantages:

• Function call overhead
• Stack space usage
• May hit recursion limits

Best For:

Complex state transitions, when not all subproblems needed

Tabulation (Bottom-Up)

Iterative approach building solutions from base cases

Advantages:

• No recursion overhead
• Better space optimization
• Easier to optimize

Disadvantages:

• Less intuitive
• May compute unnecessary subproblems
• Complex initialization

Best For:

When all subproblems are needed, space optimization important

Classic DP Problems

Fibonacci Sequence

Classic introduction to DP concepts with overlapping subproblems

Memoization • Tabulation • Space Optimized

F(n) = F(n-1) + F(n-2)

Learning DP fundamentals, Mathematical sequences

0/1 Knapsack Problem

Select items with maximum value within weight constraint

O(n × W) / O(W)

2D Table • Space Optimized • Backtracking

dp[i][w] = max(dp[i-1][w], dp[i-1][w-weight[i]] + value[i])

Resource allocation, Budget optimization

Longest Common Subsequence

Find longest subsequence common to two sequences

O(m × n) / O(min(m,n))

2D Table • Space Optimized • Reconstruction

If match: dp[i][j] = 1 + dp[i-1][j-1], else: max(dp[i-1][j], dp[i][j-1])

DNA sequence analysis, Diff utilities

Coin Change Problem

Unbounded Knapsack

Find minimum coins needed or number of ways to make amount

Bottom-up DP • Memoization • Space Optimized

dp[amount] = min(dp[amount], dp[amount - coin] + 1)

Currency systems, Change making

Longest Increasing Subsequence

Find length of longest strictly increasing subsequence

DP O(n^2) • Binary Search O(n log n) • Patience Sorting

dp[i] = max(dp[j] + 1) for j < i where arr[j] < arr[i]

Sequence analysis, Stock prices

Edit Distance (Levenshtein)

Minimum operations to transform one string to another

O(m × n) / O(min(m,n))

2D DP • Space Optimized • Operations Tracking

dp[i][j] = min(insert, delete, replace) + cost

Spell checkers, DNA analysis

Matrix Chain Multiplication

Find optimal way to parenthesize matrix multiplications

Interval DP • Memoization • Optimal Parenthesization

dp[i][j] = min(dp[i][k] + dp[k+1][j] + cost(i,k,j))

Compiler optimization, GPU computations

Palindrome Partitioning

Find minimum cuts to partition string into palindromes

O(n^3) / O(n^2)

Preprocessing • Manacher Algorithm • Optimized DP

dp[i] = min(dp[j] + 1) for j < i where s[j+1:i] is palindrome

Text processing, String analysis

Maximum Subarray (Kadane)

Find contiguous subarray with maximum sum

Kadane Algorithm • DP Approach • Divide & Conquer

maxSum = max(arr[i], maxSum + arr[i])

Stock trading, Signal processing

Space Optimization Techniques

Rolling Array

Use only current and previous rows instead of entire 2D table

Single Array Optimization

Update 1D array in place if transitions only depend on previous state

State Compression

Use bitmasks to represent complex states in a single integer

Back to Graph Algorithms Next: Divide & Conquer