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.

Easy Problems on Subarray:

• Split an array into two equal Sum subarrays
• Check if subarray with given product exists in an array
• Subarray of size k with given sum
• Sort an array where a subarray of a sorted array is in reverse order
• Count subarrays with all elements greater than K
• Maximum length of the sub-array whose first and last elements are same
• Check whether an Array is Subarray of another Array
• Find array such that no subarray has xor zero or Y
• Maximum subsequence sum such that all elements are K distance apart
• Longest sub-array with maximum GCD
• Count of subarrays with sum at least K
• Length of Smallest subarray in range 1 to N with sum greater than a given value
• Sum of all subarrays of size K
• Split array into K disjoint subarrays such that sum of each subarray is odd.
• Find an array of size N having exactly K subarrays with sum S
• Find the subarray of size K with minimum XOR
• Length of the longest alternating even odd subarray
• Count of subarrays which start and end with the same element
• Count of subarrays having exactly K perfect square numbers
• Split array into two subarrays such that difference of their maximum is minimum

Medium Problems on Subarray:

• Print all K digit repeating numbers in a very large number
• Length of longest subarray whose sum is not divisible by integer K
• Min difference between maximum and minimum element in all Y size subarrays
• Longest subarray of non-empty cells after removal of at most a single empty cell
• First subarray with negative sum from the given Array
• Largest subarray with frequency of all elements same
• Bitwise operations on Subarrays of size K
• Count subarrays having sum of elements at even and odd positions equal
• Longest Subarray consisting of unique elements from an Array
• Minimum Decrements on Subarrays required to reduce all Array elements to zero
• Split array into two subarrays such that difference of their sum is minimum
• Maximize count of non-overlapping subarrays with sum K
• Smallest subarray which upon repetition gives the original array
• Split array into maximum subarrays such that every distinct element lies in a single subarray
• Maximize product of subarray sum with its minimum element
• Sum of products of all possible Subarrays
• Check if all subarrays contains at least one unique element
• Length of smallest subarray to be removed such that the remaining array is sorted
• Length of longest subarray having frequency of every element equal to K
• Length of the longest increasing subsequence which does not contain a given sequence as Subarray

Hard Problems on Subarray:

• Length of smallest subarray to be removed to make sum of remaining elements divisible by K
• Maximum length of same indexed subarrays from two given arrays satisfying the given condition
• Count ways to split array into two equal sum subarrays by changing sign of any one array element
• Longest subarray in which all elements are smaller than K
• Maximize product of a strictly increasing or decreasing subarray
• Sum of maximum of all subarrays by adding even frequent maximum twice
• Longest subarray of an array which is a subsequence in another array
• Count of subarrays having product as a perfect cube
• Minimize difference between maximum and minimum array elements by removing a K-length subarray
• Maximum sum submatrix
• Minimum removal of elements from end of an array required to obtain sum K
• Check if any subarray of length M repeats at least K times consecutively or not
• Minimize flips on K-length subarrays required to make all array elements equal to 1
• Split array into K subarrays such that sum of maximum of all subarrays is maximized
• Find minimum subarray sum for each index i in subarray [i, N-1]
• Longest subarray with GCD greater than 1
• Longest subsegment of ‘1’s formed by changing at most k ‘0’s
• Lexicographically smallest Permutation of Array by reversing at most one Subarray
• Find a subsequence which upon reversing gives the maximum sum subarray
• Minimize steps to make Array elements 0 by reducing same A[i] – X from Subarray

Easy Problems on Subsequence:

• Longest subsequence having equal numbers of 0 and 1
• Powers of two and subsequences
• Longest Subsequence where index of next element is arr[arr[i] + i]
• Number of subsequences with zero sum
• Longest sub-sequence with maximum GCD
• Maximum Bitwise AND value of subsequence of length K
• Length of the longest subsequence such that xor of adjacent elements is non-decreasing
• Maximum product of bitonic subsequence of size 3
• Length of Smallest Subsequence such that sum of elements is greater than equal to K
• Longest subsequence of even numbers in an Array
• Maximum length Subsequence with alternating sign and maximum Sum
• Count of possible subarrays and subsequences using given length of Array
• Maximum bitwise OR value of subsequence of length K
• Count of subsequences consisting of the same element
• Smallest occurring element in each subsequence
• Length of the longest subsequence consisting of distinct elements
• Minimize elements to be added to a given array such that it contains another given array as its subsequence
• Maximize subsequences having array elements not exceeding length of the subsequence
• Length of longest subsequence consisting of distinct adjacent elements
• Maximum Sum Subsequence

Medium Problems on Subsequence:

• Minimum removals required to make a given array Bitonic
• Check if a non-contiguous subsequence same as the given subarray exists or not
• Minimize the number of strictly increasing subsequences in an array
• Count of unique subsequences from given number which are power of 2
• Minimum number of insertions required such that first K natural numbers can be obtained as sum of a subsequence of the array
• Length of longest subsequence such that prefix sum at every element remains greater than zero
• Check if given Array can be divided into subsequences of K increasing consecutive integers
• Longest Subsequence such that difference between adjacent elements is either A or B
• Count subsequences of Array having single digit integer sum K
• Shortest Subsequence with sum exactly K
• Printing Longest Bitonic Subsequence
• Sorted subsequence of size 3 in linear time using constant space
• Count of subsequences having maximum distinct elements
• Construct array having X subsequences with maximum difference smaller than d
• Print all subsequences of a string using ArrayList
• Longest Subsequence with at least one common digit in every element
• Maximum Sum Subsequence of length k
• Sum of minimum element of all sub-sequences of a sorted array
• Find all combinations of two equal sum subsequences
• Minimum cost of choosing 3 increasing elements in an array of size N

Hard Problems on Subsequence:

• Number of subsequences with positive product
• Longest subsequence having difference atmost K
• Find all subsequences with sum equals to K
• Maximize product of digit sum of consecutive pairs in a subsequence of length K
• Count of subsequences of length atmost K containing distinct prime elements
• Sum of all subsequences of length K
• Minimize sum of smallest elements from K subsequences of length L
• Unique subsequences of length K with given sum
• Smallest subsequence with sum of absolute difference of consecutive elements maximized
• Maximize product of same-indexed elements of same size subsequences
• Longest Increasing Subsequence having sum value atmost K
• Longest subsequence of a number having same left and right rotation
• Maximize length of Non-Decreasing Subsequence by reversing at most one Subarray
• Maximum subsequence sum possible by multiplying each element by its index
• Generate all distinct subsequences of array using backtracking
• Maximum subsequence sum such that no K elements are consecutive
• Print all possible K-length subsequences of first N natural numbers with sum N
• Longest subsequence having difference between the maximum and minimum element equal to K
• Maximize difference between sum of even and odd-indexed elements of a subsequence
• Convert an array into another by repeatedly removing the last element and placing it at any arbitrary index

Easy Problems on Subset:

• Find if there is any subset of size K with 0 sum in an array of -1 and +1
• Sum of sum of all subsets of a set formed by first N natural numbers
• Count of subsets not containing adjacent elements
• Sum of the sums of all possible subsets
• Find whether an array is subset of another array
• Total number of Subsets of size at most K
• Check if it is possible to split given Array into K odd-sum subsets
• Partition a set into two subsets such that difference between max of one and min of other is minimized
• Sum of all possible expressions of a numeric string possible by inserting addition operators
• Check if it’s possible to split the Array into strictly increasing subsets of size at least K
• Largest subset of Array having sum at least 0

Medium Problems on Subset:

• Number of subsets with sum divisible by m
• Fibonacci sum of a subset with all elements <= k
• Number of possible Equivalence Relations on a finite set
• Largest divisible pairs subset
• Recursive program to print all subsets with given sum
• Subset Sum Queries in a Range using Bitset
• Find all distinct subset (or subsequence) sums of an array | Set-2
• Sum of (maximum element – minimum element) for all the subsets of an array
• Count no. of ordered subsets having a particular XOR value
• Sum of subsets of all the subsets of an array
• Perfect Sum Problem
• Count of subsets having sum of min and max element less than K
• Split array into two equal length subsets such that all repetitions of a number lies in a single subset
• Nth Subset of the Sequence consisting of powers of K in increasing order of their Sum
• Largest possible Subset from an Array such that no element is K times any other element in the Subset
• Check if an array can be split into subsets of K consecutive elements

Hard Problems on Subset:

• Minimize count of divisions by D to obtain at least K equal array elements
• Split array into K-length subsets to minimize sum of second smallest element of each subset
• Median of all non-empty subset sums
• Minimum removals required such that sum of remaining array modulo M is X
• Sum of length of two smallest subsets possible from a given array with sum at least K
• Reduce sum of any subset of an array to 1 by multiplying all its elements by any value
• Sum of all subsets whose sum is a Perfect Number from a given array
• Minimize sum of incompatibilities of K equal-length subsets made up of unique elements
• Maximize sum of subsets from two arrays having no consecutive values
• Product of the maximums of all subsets of an array
• Count ways to place ‘+’ and ‘-‘ in front of array elements to obtain sum K
• Count ways to split array into two subsets having difference between their sum equal to K
• Find the subset of Array with given LCM
• Count of subsets whose product is multiple of unique primes
• Minimum count of elements to be inserted in Array to form all values in [1, K] using subset sum
• Maximum subset sum having difference between its maximum and minimum in range [L, R]
• Find all unique subsets of a given set using C++ STL
• Subset sum problem where Array sum is at most N
  

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