Subarrays, Subsequences, and Subsets in Array

What is a Subarray?

A subarray is a contiguous part of array, i.e., Subarray is an array that is inside another array.

In general, for an array of size n, there are n*(n+1)/2 non-empty subarrays.

For example, Consider the array [1, 2, 3, 4], There are 10 non-empty sub-arrays. The subarrays are:

(1), (2), (3), (4), 
(1,2), (2,3), (3,4), 
(1,2,3), (2,3,4), and 
(1,2,3,4)

What is a Subsequence?

A subsequence is a sequence that can be derived from another sequence by removing zero or more elements, without changing the order of the remaining elements.

More generally, we can say that for a sequence of size n, we can have (2ⁿ – 1) non-empty sub-sequences in total.

For the same above example, there are 15 sub-sequences. They are:

(1), (2), (3), (4), 
(1,2), (1,3),(1,4), (2,3), (2,4), (3,4), 
(1,2,3), (1,2,4), (1,3,4), (2,3,4), 
(1,2,3,4). 

What is a Subset?

If a Set has all its elements belonging to other sets, this set will be known as a subset of the other set.

A Subset is denoted as “⊆“. If set A is a subset of set B, it is represented as A ⊆ B.

For example, Let Set_A = {m, n, o, p, q}, Set_ B = {k, l, m, n, o, p, q, r}

Then, A ⊆ B.

Topics:

Problems on Subarray:

• Easy Problems on Subarray:
  1. Split an array into two equal Sum subarrays
  2. Check if subarray with given product exists in an array
  3. Subarray of size k with given sum
  4. Sort an array where a subarray of a sorted array is in reverse order
  5. Count subarrays with all elements greater than K
  6. Maximum length of the sub-array whose first and last elements are same
  7. Check whether an Array is Subarray of another Array
  8. Find array such that no subarray has xor zero or Y
  9. Maximum subsequence sum such that all elements are K distance apart
  10. Longest sub-array with maximum GCD
  11. Count of subarrays with sum at least K
  12. Length of Smallest subarray in range 1 to N with sum greater than a given value
  13. Sum of all subarrays of size K
  14. Split array into K disjoint subarrays such that sum of each subarray is odd.
  15. Find an array of size N having exactly K subarrays with sum S
  16. Find the subarray of size K with minimum XOR
  17. Length of the longest alternating even odd subarray
  18. Count of subarrays which start and end with the same element
  19. Count of subarrays having exactly K perfect square numbers
  20. Split array into two subarrays such that difference of their maximum is minimum
• Intermediate problems on Subarray:
  1. Print all K digit repeating numbers in a very large number
  2. Length of longest subarray whose sum is not divisible by integer K
  3. Min difference between maximum and minimum element in all Y size subarrays
  4. Longest subarray of non-empty cells after removal of at most a single empty cell
  5. First subarray with negative sum from the given Array
  6. Largest subarray with frequency of all elements same
  7. Bitwise operations on Subarrays of size K
  8. Count subarrays having sum of elements at even and odd positions equal
  9. Longest Subarray consisting of unique elements from an Array
  10. Minimum Decrements on Subarrays required to reduce all Array elements to zero
  11. Split array into two subarrays such that difference of their sum is minimum
  12. Maximize count of non-overlapping subarrays with sum K
  13. Smallest subarray which upon repetition gives the original array
  14. Split array into maximum subarrays such that every distinct element lies in a single subarray
  15. Maximize product of subarray sum with its minimum element
  16. Sum of products of all possible Subarrays
  17. Check if all subarrays contains at least one unique element
  18. Length of smallest subarray to be removed such that the remaining array is sorted
  19. Length of longest subarray having frequency of every element equal to K
  20. Length of the longest increasing subsequence which does not contain a given sequence as Subarray
• Hard Problems on Subarray:
  1. Length of smallest subarray to be removed to make sum of remaining elements divisible by K
  2. Maximum length of same indexed subarrays from two given arrays satisfying the given condition
  3. Count ways to split array into two equal sum subarrays by changing sign of any one array element
  4. Longest subarray in which all elements are smaller than K
  5. Maximize product of a strictly increasing or decreasing subarray
  6. Sum of maximum of all subarrays by adding even frequent maximum twice
  7. Longest subarray of an array which is a subsequence in another array
  8. Count of subarrays having product as a perfect cube
  9. Minimize difference between maximum and minimum array elements by removing a K-length subarray
  10. Maximum sum submatrix
  11. Minimum removal of elements from end of an array required to obtain sum K
  12. Check if any subarray of length M repeats at least K times consecutively or not
  13. Minimize flips on K-length subarrays required to make all array elements equal to 1
  14. Split array into K subarrays such that sum of maximum of all subarrays is maximized
  15. Find minimum subarray sum for each index i in subarray [i, N-1]
  16. Longest subarray with GCD greater than 1
  17. Longest subsegment of ‘1’s formed by changing at most k ‘0’s
  18. Lexicographically smallest Permutation of Array by reversing at most one Subarray
  19. Find a subsequence which upon reversing gives the maximum sum subarray
  20. Minimize steps to make Array elements 0 by reducing same A[i] – X from Subarray

Problems on Subsequence:

• Easy Problems on Subsequence:
  1. Longest subsequence having equal numbers of 0 and 1
  2. Powers of two and subsequences
  3. Longest Subsequence where index of next element is arr[arr[i] + i]
  4. Number of subsequences with zero sum
  5. Longest sub-sequence with maximum GCD
  6. Maximum Bitwise AND value of subsequence of length K
  7. Length of the longest subsequence such that xor of adjacent elements is non-decreasing
  8. Maximum product of bitonic subsequence of size 3
  9. Length of Smallest Subsequence such that sum of elements is greater than equal to K
  10. Longest subsequence of even numbers in an Array
  11. Maximum length Subsequence with alternating sign and maximum Sum
  12. Count of possible subarrays and subsequences using given length of Array
  13. Maximum bitwise OR value of subsequence of length K
  14. Count of subsequences consisting of the same element
  15. Smallest occurring element in each subsequence
  16. Length of the longest subsequence consisting of distinct elements
  17. Minimize elements to be added to a given array such that it contains another given array as its subsequence
  18. Maximize subsequences having array elements not exceeding length of the subsequence
  19. Length of longest subsequence consisting of distinct adjacent elements
  20. Maximum Sum Subsequence
• Intermediate Problems on Subsequence:
  1. Minimum removals required to make a given array Bitonic
  2. Check if a non-contiguous subsequence same as the given subarray exists or not
  3. Minimize the number of strictly increasing subsequences in an array
  4. Count of unique subsequences from given number which are power of 2
  5. Minimum number of insertions required such that first K natural numbers can be obtained as sum of a subsequence of the array
  6. Length of longest subsequence such that prefix sum at every element remains greater than zero
  7. Check if given Array can be divided into subsequences of K increasing consecutive integers
  8. Longest Subsequence such that difference between adjacent elements is either A or B
  9. Count subsequences of Array having single digit integer sum K
  10. Shortest Subsequence with sum exactly K
  11. Printing Longest Bitonic Subsequence
  12. Sorted subsequence of size 3 in linear time using constant space
  13. Count of subsequences having maximum distinct elements
  14. Construct array having X subsequences with maximum difference smaller than d
  15. Print all subsequences of a string using ArrayList
  16. Longest Subsequence with at least one common digit in every element
  17. Maximum Sum Subsequence of length k
  18. Sum of minimum element of all sub-sequences of a sorted array
  19. Find all combinations of two equal sum subsequences
  20. Minimum cost of choosing 3 increasing elements in an array of size N
• Hard Problems on Subsequence:
  1. Number of subsequences with positive product
  2. Longest subsequence having difference atmost K
  3. Find all subsequences with sum equals to K
  4. Maximize product of digit sum of consecutive pairs in a subsequence of length K
  5. Count of subsequences of length atmost K containing distinct prime elements
  6. Sum of all subsequences of length K
  7. Minimize sum of smallest elements from K subsequences of length L
  8. Unique subsequences of length K with given sum
  9. Smallest subsequence with sum of absolute difference of consecutive elements maximized
  10. Maximize product of same-indexed elements of same size subsequences
  11. Longest Increasing Subsequence having sum value atmost K
  12. Longest subsequence of a number having same left and right rotation
  13. Maximize length of Non-Decreasing Subsequence by reversing at most one Subarray
  14. Maximum subsequence sum possible by multiplying each element by its index
  15. Generate all distinct subsequences of array using backtracking
  16. Maximum subsequence sum such that no K elements are consecutive
  17. Print all possible K-length subsequences of first N natural numbers with sum N
  18. Longest subsequence having difference between the maximum and minimum element equal to K
  19. Maximize difference between sum of even and odd-indexed elements of a subsequence
  20. Convert an array into another by repeatedly removing the last element and placing it at any arbitrary index

Problems on Subset:

• Easy Problems on Subset:
  1. Find if there is any subset of size K with 0 sum in an array of -1 and +1
  2. Sum of sum of all subsets of a set formed by first N natural numbers
  3. Count of subsets not containing adjacent elements
  4. Sum of the sums of all possible subsets
  5. Find whether an array is subset of another array
  6. Total number of Subsets of size at most K
  7. Check if it is possible to split given Array into K odd-sum subsets
  8. Partition a set into two subsets such that difference between max of one and min of other is minimized
  9. Sum of all possible expressions of a numeric string possible by inserting addition operators
  10. Check if it’s possible to split the Array into strictly increasing subsets of size at least K
  11. Largest subset of Array having sum at least 0
• Intermediate Problems on Subset:
  1. Number of subsets with sum divisible by m
  2. Fibonacci sum of a subset with all elements <= k
  3. Number of possible Equivalence Relations on a finite set
  4. Largest divisible pairs subset
  5. Recursive program to print all subsets with given sum
  6. Subset Sum Queries in a Range using Bitset
  7. Find all distinct subset (or subsequence) sums of an array | Set-2
  8. Sum of (maximum element – minimum element) for all the subsets of an array
  9. Count no. of ordered subsets having a particular XOR value
  10. Sum of subsets of all the subsets of an array
  11. Perfect Sum Problem
  12. Count of subsets having sum of min and max element less than K
  13. Split array into two equal length subsets such that all repetitions of a number lies in a single subset
  14. Nth Subset of the Sequence consisting of powers of K in increasing order of their Sum
  15. Largest possible Subset from an Array such that no element is K times any other element in the Subset
  16. Check if an array can be split into subsets of K consecutive elements
• Hard Problems on Subset:
  1. Minimize count of divisions by D to obtain at least K equal array elements
  2. Split array into K-length subsets to minimize sum of second smallest element of each subset
  3. Median of all non-empty subset sums
  4. Minimum removals required such that sum of remaining array modulo M is X
  5. Sum of length of two smallest subsets possible from a given array with sum at least K
  6. Reduce sum of any subset of an array to 1 by multiplying all its elements by any value
  7. Sum of all subsets whose sum is a Perfect Number from a given array
  8. Minimize sum of incompatibilities of K equal-length subsets made up of unique elements
  9. Maximize sum of subsets from two arrays having no consecutive values
  10. Product of the maximums of all subsets of an array
  11. Count ways to place ‘+’ and ‘-‘ in front of array elements to obtain sum K
  12. Count ways to split array into two subsets having difference between their sum equal to K
  13. Find the subset of Array with given LCM
  14. Count of subsets whose product is multiple of unique primes
  15. Minimum count of elements to be inserted in Array to form all values in [1, K] using subset sum
  16. Maximum subset sum having difference between its maximum and minimum in range [L, R]
  17. Find all unique subsets of a given set using C++ STL
  18. Subset sum problem where Array sum is at most N

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