# Dynamic Programming

### Recent Articles on Dynamic Programming

**Topics :**

- Ugly numbers
- Fibonacci numbers
- nth Catalan Number
- Bell Numbers (Number of ways to Partition a Set)
- Binomial Coefficient
- Permutation Coefficient
- Tiling Problem
- Gold Mine Problem
- Coin change problem
- Friends Pairing Problem
- Subset Sum Problem
- Subset Sum Problem in O(sum) space
- Subset with sum divisible by m
- Largest divisible pairs subset
- Perfect Sum Problem (Print all subsets with given sum)
- Compute nCr % p
- Choice of area
- Cutting a Rod
- Tiling with Dominoes
- Painting Fence Algorithm
- Newman–Shanks–Williams prime
- Assembly line scheduling
- Golomb sequence
- Moser-de Bruijn Sequence
- Newman-Conway Sequence
- Find maximum length Snake sequence
- Print n terms of Newman-Conway Sequence
- Print Fibonacci sequence using 2 variables
- Print Fibonacci Series in reverse order
- Count even length binary sequences with same sum of first and second half bits
- Sequences of given length where every element is more than or equal to twice of previous
- Longest Common Subsequence
- Longest Repeated Subsequence
- Longest Increasing Subsequence
- A Space Optimized Solution of LCS
- LCS (Longest Common Subsequence) of three strings
- Maximum sum Bi-tonic Sub-sequence
- Maximum Sum Increasing Subsequence
- Maximum product of an increasing subsequence
- Count all subsequences having product less than K
- Maximum subsequence sum such that no three are consecutive
- Longest subsequence such that difference between adjacents is one
- Maximum length subsequence with difference between adjacent elements as either 0 or 1
- Maximum sum increasing subsequence from a prefix and a given element after prefix is must
- Maximum Length Chain of Pairs
- Print Maximum Length Chain of Pairs
- Path with maximum average value
- Maximum games played by winner
- Maximum path sum in a triangle
- Minimum Sum Path in a Triangle
- Maximum sum of a path in a Right Number Triangle
- Size of The Subarray With Maximum Sum
- Maximum sum of pairs with specific difference
- Maximum size square sub-matrix with all 1s
- Maximum number of segments of lengths a, b and c
- Recursively break a number in 3 parts to get maximum sum
- Maximum value with the choice of either dividing or considering as it is
- Maximum weight path ending at any element of last row in a matrix
- Maximum sum in a 2 x n grid such that no two elements are adjacent
- Maximum difference of zeros and ones in binary string | Set 2 (O(n) time)
- Maximum path sum for each position with jumps under divisibility condition
- Maximize the sum of selected numbers from an array to make it empty
- Maximum subarray sum in an array created after repeated concatenation
- Maximum path sum that starting with any cell of 0-th row and ending with any cell of (N-1)-th row
- Min Cost Path
- Minimum number of jumps to reach end
- Minimum cost to fill given weight in a bag
- Minimum sum of multiplications of n numbers
- Minimum removals from array to make max – min <= K
- Minimum steps to minimize n as per given condition
- Minimum number of edits ( operations ) require to convert string 1 to string 2
- Minimum time to write characters using insert, delete and copy operation
- Longest Common Substring
- Longest Common Substring (Space optimized DP solution)
- Sum of all substrings of a string representing a number | Set 1
- Find number of endless points
- Find n-th element from Stern’s Diatomic Series
- Find maximum possible stolen value from houses
- Find number of solutions of a linear equation of n variables
- Count number of ways to reach a given score in a game
- Count ways to reach the nth stair using step 1, 2 or 3
- Count of different ways to express N as the sum of 1, 3 and 4
- Count ways to build street under given constraints
- Count Balanced Binary Trees of Height h
- Counting pairs when a person can form pair with at most one
- Counts paths from a point to reach Origin
- Count number of ways to cover a distance
- Count of arrays having consecutive element with different values
- Count ways to divide circle using N non-intersecting chords
- Count the number of ways to tile the floor of size n x m using 1 x m size tiles
- Count all possible paths from top left to bottom right of a mXn matrix
- Count number of ways to fill a “n x 4” grid using “1 x 4” tiles
- Largest Sum Contiguous Subarray
- Smallest sum contiguous subarray
- Size of array after repeated deletion of LIS
- Remove array end element to maximize the sum of product
- Convert to Strictly increasing array with minimum changes
- Longest alternating (positive and negative) subarray starting at every index
- Ways to sum to N using array elements with repetition allowed
- Unique paths in a Grid with Obstacles
- Number of n-digits non-decreasing integers
- Number of ways to arrange N items under given constraints
- Probability of reaching a point with 2 or 3 steps at a time
- Value of continuous floor function : F(x) = F(floor(x/2)) + x
- Number of decimal numbers of length k, that are strict monotone
- Different ways to sum n using numbers greater than or equal to m

- Lobb Number
- Eulerian Number
- Delannoy Number
- Entringer Number
- Rencontres Number
- Jacobsthal and Jacobsthal-Lucas numbers
- Super Ugly Number (Number whose prime factors are in given set)
- Floyd Warshall Algorithm
- Bellman–Ford Algorithm
- 0-1 Knapsack Problem
- Printing Items in 0/1 Knapsack
- Unbounded Knapsack (Repetition of items allowed)
- Temple Offerings
- Egg Dropping Puzzle
- Dice Throw Problem
- Word Break Problem
- Vertex Cover Problem
- Tile Stacking Problem
- Box-Stacking Problem
- Highway Billboard Problem
- Largest Independent Set Problem
- Partition Problem
- Print equal sum sets of array (Partition problem) | Set 1
- Print equal sum sets of array (Partition Problem) | Set 2
- High-effort vs. Low-effort Tasks Problem
- Travelling Salesman Problem | Set 1 (Naive and Dynamic Programming)
- Longest Bitonic Subsequence
- Printing Longest Bitonic Subsequence
- Longest Palindromic Subsequence
- Print Longest Palindromic Subsequence
- Longest palindrome subsequence with O(n) space
- Count All Palindromic Subsequence in a given String
- Longest Palindromic Substring | Set 1
- Count All Palindrome Sub-Strings in a String | Set 1
- Number of palindromic subsequences of length k
- Count of Palindromic substrings in an Index range
- Shortest Common Supersequence
- Maximum sum alternating subsequence
- Longest alternating subsequence
- Shortest Uncommon Subsequence
- Longest Repeating Subsequence
- Count Distinct Subsequences
- Count distinct occurrences as a subsequence
- Longest Common Increasing Subsequence (LCS + LIS)
- Variations of LIS
- LCS formed by consecutive segments of at least length K
- Printing Maximum Sum Increasing Subsequence
- Longest Increasing Odd Even Subsequence
- Count number of increasing subsequences of size k
- Printing longest Increasing consecutive subsequence
- Construction of Longest Increasing Subsequence using Dynamic Programming
- Longest Zig-Zag Subsequence
- Largest sum Zigzag sequence in a matrix
- Find all distinct subset (or subsequence) sums of an array
- Print all longest common sub-sequences in lexicographical order
- Printing Longest Common Subsequence | Set 2 (Printing All)
- Length of Longest Balanced Subsequence
- Non-decreasing subsequence of size k with minimum sum
- Longest Common Subsequence with at most k changes allowed
- Weighted job scheduling
- Weighted Job Scheduling | Set 2 (Using LIS)
- Weighted Job Scheduling in O(n Log n) time
- Number of paths with exactly k coins
- Minimum number of coins that make a given value
- Collect maximum coins before hitting a dead end
- Coin game winner where every player has three choices
- Probability of getting at least K heads in N tosses of Coins
- Count all increasing subsequences
- Count number of paths with at-most k turns
- Count possible ways to construct buildings
- Count number of ways to jump to reach end
- Count number of ways to reach destination in a Maze
- Count all triplets whose sum is equal to a perfect cube
- Count number of binary strings without consecutive 1’s
- Count number of subsets having a particular XOR value
- Count Possible Decodings of a given Digit Sequence
- Count number of ways to partition a set into k subsets
- Count of n digit numbers whose sum of digits equals to given sum
- Count ways to assign unique cap to every person
- Count binary strings with k times appearing adjacent two set bits
- Count of strings that can be formed using a, b and c under given constraints
- Count digit groupings of a number with given constraints
- Count all possible walks from a source to a destination with exactly k edges
- Count Derangements (Permutation such that no element appears in its original position)
- Count total number of N digit numbers such that the difference between sum of even and odd digits is 1
- Maximum Product Cutting
- Maximum profit from sale of wines
- Maximum size subset with given sum
- Maximum difference of zeros and ones in binary string
- Maximum and Minimum Values of an Algebraic Expression
- Maximum average sum partition of an array
- Maximize array elements upto given number
- Maximum subarray sum in O(n) using prefix sum
- Maximum sum subarray removing at most one element
- K maximum sums of non-overlapping contiguous sub-arrays
- Maximum Product Subarray | Added negative product case
- Find maximum sum array of length less than or equal to m
- Find Maximum dot product of two arrays with insertion of 0’s
- Choose maximum weight with given weight and value ratio
- Maximum sum subsequence with at-least k distant elements
- Maximum profit by buying and selling a share at most twice
- Maximum sum path in a matrix from top to bottom
- Maximum decimal value path in a binary matrix
- Finding the maximum square sub-matrix with all equal elements
- Maximum points collected by two persons allowed to meet once
- Maximum number of trailing zeros in the product of the subsets of size k
- Minimum Sum Path In 3-D Array
- Minimum insertions to sort an array
- Minimum sum submatrix in a given 2D array
- Minimum Initial Points to Reach Destination
- Minimum Cost To Make Two Strings Identical
- Paper Cut into Minimum Number of Squares | Set 2
- Minimum and Maximum values of an expression with * and +
- Minimum insertions to form a palindrome
- Minimum number of deletions to make a string palindrome
- Minimum number of deletions to make a string palindrome | Set 2
- Minimum jumps to reach last building in a matrix
- Sub-tree with minimum color difference in a 2-coloured tree
- Minimum number of deletions to make a sorted sequence
- Minimum number of squares whose sum equals to given number n
- Remove minimum elements from either side such that 2*min becomes more than max
- Minimal moves to form a string by adding characters or appending string itself
- Minimum steps to delete a string after repeated deletion of palindrome substrings
- Clustering/Partitioning an array such that sum of square differences is minimum
- Minimum sum subsequence such that at least one of every four consecutive elements is picked
- Minimum cost to make Longest Common Subsequence of length k
- Minimum cost to make two strings identical by deleting the digits
- Minimum time to finish tasks without skipping two consecutive
- Minimum cells required to reach destination with jumps equal to cell values
- Minimum number of deletions and insertions to transform one string into another
- Find minimum adjustment cost of an array
- Find if string is K-Palindrome or not | Set 1
- Find if string is K-Palindrome or not | Set 2
- Find Jobs involved in Weighted Job Scheduling
- Find the Longest Increasing Subsequence in Circular manner
- Find the longest path in a matrix with given constraints
- Find the minimum cost to reach destination using a train
- Find minimum sum such that one of every three consecutive elements is taken
- Find number of times a string occurs as a subsequence in given string
- Find length of the longest consecutive path from a given starting character
- Find length of longest subsequence of one string which is substring of another string
- Find longest bitonic sequence such that increasing and decreasing parts are from two different arrays
- Wildcard Pattern Matching
- WildCard pattern matching having three symbols ( * , + , ? )
- Dynamic Programming | Wildcard Pattern Matching | Linear Time and Constant Space
- Check if any valid sequence is divisible by M
- Check for possible path in 2D matrix
- Check if possible to cross the matrix with given power
- Check if it is possible to transform one string to another
- Given a large number, check if a subsequence of digits is divisible by 8
- Hosoya’s Triangle
- Optimal Strategy for a game
- Optimal Binary Search Tree
- Number of permutation with K inversions
- Largest divisible pairs subset
- Sum of average of all subsets
- Compute sum of digits in all numbers from 1 to n
- Total number of non-decreasing numbers with n digits
- Non-crossing lines to connect points in a circle
- Dynamic Programming | Building Bridges
- Longest Increasing Path in Matrix
- Prefix Sum of Matrix (Or 2D Array)
- Multistage Graph (Shortest Path)
- Number of n digit stepping numbers
- Number of substrings divisible by 8 but not by 3
- Number of ordered pairs such that (Ai & Aj) = 0
- Number of ways to form a heap with n distinct integers
- Ways to write n as sum of two or more positive integers
- Modify array to maximize sum of adjacent differences
- Sum of products of all combination taken (1 to n) at a time
- Maximize the binary matrix by filpping submatrix once
- Length of the longest substring without repeating characters
- Longest Even Length Substring such that Sum of First and Second Half is same
- Shortest path with exactly k edges in a directed and weighted graph
- Ways to arrange Balls such that adjacent balls are of different types
- Ways of transforming one string to other by removing 0 or more characters
- Balanced expressions such that given positions have opening brackets
- Longest alternating sub-array starting from every index in a Binary Array
- Partition a set into two subsets such that the difference of subset sums is minimum
- Pyramid form (increasing then decreasing) consecutive array using reduce operations

- Palindrome Partitioning
- Word Wrap Problem
- Mobile Numeric Keypad Problem
- The painter’s partition problem
- Boolean Parenthesization Problem
- Program for Bridge and Torch problem
- A Space Optimized DP solution for 0-1 Knapsack Problem
- Matrix Chain Multiplication
- Printing brackets in Matrix Chain Multiplication Problem
- Number of palindromic paths in a matrix
- Largest rectangular sub-matrix whose sum is 0
- Largest rectangular sub-matrix having sum divisible by k
- Largest area rectangular sub-matrix with equal number of 1’s and 0’s
- Maximum sum bitonic subarray
- Maximum sum rectangle in a 2D matrix
- Maximum Subarray Sum Excluding Certain Elements
- Maximum weight transformation of a given string
- Collect maximum points in a grid using two traversals
- K maximum sums of overlapping contiguous sub-arrays
- How to print maximum number of A’s using given four keys
- Maximize arr[j] – arr[i] + arr[l] – arr[k], such that i < j < k < l
- Maximum profit by buying and selling a share at most k times
- Maximum points from top left of matrix to bottom right and return back
- Check whether row or column swaps produce maximum size binary sub-matrix with all 1s
- Minimum Cost Polygon Triangulation
- Minimum cost to sort strings using reversal operations of different costs
- Find minimum possible size of array with given rules for removing elements
- Minimum number of elements which are not part of Increasing or decreasing subsequence in array
- Count ways to increase LCS length of two strings by one
- Count of AP (Arithmetic Progression) Subsequences in an array
- Count of arrays in which all adjacent elements are such that one of them divide the another
- Number of NGEs to the right
- Longest Arithmetic Progression
- Longest Geometric Progression
- Dynamic Programming on Trees | Set-1
- Dynamic Programming on Trees | Set-2
- All ways to add parenthesis for evaluation
- Shortest possible combination of two strings
- Check if all people can vote on two machines
- Find if a string is interleaved of two other strings
- Longest repeating and non-overlapping substring
- Probability of Knight to remain in the chessboard
- Number of subsequences of the form a^i b^j c^k
- Number of subsequences in a string divisible by n
- Printing Shortest Common Supersequence
- Smallest length string with repeated replacement of two distinct adjacent
- Number of ways to insert a character to increase the LCS by one
- Traversal of tree with k jumps allowed between nodes of same height
- Find all combinations of k-bit numbers with n bits set where 1 <= n <= k in sorted order

- Top 20 Dynamic Programming Interview Questions
- ‘Practice Problems’ on Dynamic Programming
- ‘Quiz’ on Dynamic Programming

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