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For more details, please go through - <a href="https://www.geeksforgeeks.org/dsa/overlapping-subproblems-property-in-dynamic-programming-dp-1/">Overlapping Subproblems Property in Dynamic Programming</a>, <a href="https://www.geeksforgeeks.org/dsa/optimal-substructure-property-in-dynamic-programming-dp-2/">Optimal Substructure Property in Dynamic Programming</a>

Dynamic Programming Tabulation.

For more details, please go through - <a href="https://www.geeksforgeeks.org/dsa/optimal-substructure-property-in-dynamic-programming-dp-2/">Optimal Substructure Property in Dynamic Programming</a>, <a href="https://www.geeksforgeeks.org/dsa/overlapping-subproblems-property-in-dynamic-programming-dp-1/">Overlapping Subproblems Property in Dynamic Programming</a>

Dynamic Programming Memoization.

Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack. In other words, given two integer arrays val[0..n-1] and wt[0..n-1] which represent values and weights associated with n items respectively.&nbsp;Also given an integer W which represents knapsack capacity, find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to W. You cannot break an item, either pick the complete item or don’t pick it (0-1 property).Method 1: Recursion by Brute-Force algorithm OR Exhaustive Search. Approach: A simple solution is to consider all subsets of items and calculate the total weight and value of all subsets. Consider the only subsets whose total weight is smaller than W. From all such subsets, pick the maximum value subset. Optimal Sub-structure: To consider all subsets of items, there can be two cases for every item.&nbsp;<ol><li>Case 1: The item is included in the optimal subset.</li><li>Case 2: The item is not included in the optimal set.</li></ol>Therefore, the maximum value that can be obtained from ‘n’ items is the max of the following two values.&nbsp;Dynamic Programming _ Set 10 (0-1 Knapsack Problem): <a href="https://www.geeksforgeeks.org/0-1-knapsack-problem-dp-10/">https://www.geeksforgeeks.org/0-1-knapsack-problem-dp-10/</a>

Knapsack Problem - Approach to write the code (Dynamic Programming)

Given a sequence, find the length of the longest palindromic subsequence in it.As another example, if the given sequence is “BBABCBCAB”, then the output should be 7 as “BABCBAB” is the longest palindromic subsequence in it. “BBBBB” and “BBCBB” are also palindromic subsequences of the given sequence, but not the longest ones. The naive solution for this problem is to generate all subsequences of the given sequence and find the longest palindromic subsequence. This solution is exponential in terms of time complexity.&nbsp;Let us see how this problem possesses both important properties of a Dynamic Programming (DP) Problem and can efficiently be solved using Dynamic Programming. 1) Optimal Substructure:&nbsp; Let X[0..n-1] be the input sequence of length n and L(0, n-1) be the length of the longest palindromic subsequence of X[0..n-1].&nbsp; If last and first characters of X are same, then L(0, n-1) = L(1, n-2) + 2.&nbsp; Else L(0, n-1) = MAX (L(1, n-1), L(0, n-2)).&nbsp;&nbsp;Related Article : <a href="https://www.geeksforgeeks.org/longest-palindromic-subsequence-dp-12/">https://www.geeksforgeeks.org/longest-palindromic-subsequence-dp-12/</a>

Longest Palindromic Subsequence

Given an array of n positive integers. Write a program to find the sum of maximum sum subsequence of the given array such that the integers in the subsequence are sorted in increasing order. For example, if input is {1, 101, 2, 3, 100, 4, 5}, then output should be 106 (1 + 2 + 3 + 100), if the input array is {3, 4, 5, 10}, then output should be 22 (3 + 4 + 5 + 10) and if the input array is {10, 5, 4, 3}, then output should be 10Maximum Sum Increasing Subsequence : <a href="https://www.geeksforgeeks.org/maximum-sum-increasing-subsequence-dp-14/">https://www.geeksforgeeks.org/maximum-sum-increasing-subsequence-dp-14/</a>

Maximum Sum Increasing Subsequence

Given three strings A, B and C. Write a function that checks whether C is an interleaving of A and B. C is said to be interleaving A and B, if it contains all and only characters of A and B and order of all characters in individual strings is preserved.Example:Input: strings: "XXXXZY", "XXY", "XXZ" Output: XXXXZY is interleaved of XXY and XXZ The string XXXXZY can be made by&nbsp; interleaving XXY and XXZ String: &nbsp; &nbsp;XXXXZY String 1: &nbsp; &nbsp;XX Y String 2: &nbsp;XX &nbsp;Z Check whether a given string is an interleaving of two other given strings: <a href="https://www.geeksforgeeks.org/find-if-a-string-is-interleaved-of-two-other-strings-dp-33/">https://www.geeksforgeeks.org/find-if-a-string-is-interleaved-of-two-other-strings-dp-33/</a>

Check whether a given string is an interleaving of two other given strings

Catalan numbers are a sequence of natural numbers that occurs in many interesting counting problems like following.&nbsp;Count the number of expressions containing n pairs of parentheses which are correctly matched. For n = 3, possible expressions are ((())), ()(()), ()()(), (())(), (()()). Count the number of possible Binary Search Trees with n keys (See this) Count the number of full binary trees (A rooted binary tree is full if every vertex has either two children or no children) with n+1 leaves. Given a number n, return the number of ways you can draw n chords in a circle with 2 x n points such that no 2 chords intersect&nbsp;Program for Nth Catalan Number (using Binomial Coefficient) : <a href="https://www.geeksforgeeks.org/program-nth-catalan-number/">https://www.geeksforgeeks.org/program-nth-catalan-number/</a>

Program for Nth Catalan Number (using Binomial Coefficient)

This video is part of the Recursion and Backtracking section under GFG SDE Sheet. Imagine you have a special keyboard with the following keys:&nbsp;Key 1: &nbsp;Prints 'A' on screen Key 2: (Ctrl-A): Select screen Key 3: (Ctrl-C): Copy selection to buffer Key 4: (Ctrl-V): Print buffer on screen appending it after what has already been printed.Find maximum numbers of A's that can be produced by pressing keys on the special keyboard N times.&nbsp; Example 1:Input: N = 3
 Output: 3
 Explanation: Press key 1 three times. Try it out before watching the implementation of the problem in the video. We recommend watching the video, even if you can solve the problem. You may discover something new. All the best!!!Do check out:- Problem: <a href="https://www.geeksforgeeks.org/problems/special-keyboard3018/1">https://www.geeksforgeeks.org/problems/special-keyboard3018/1</a> SDE Sheet Link: <a href="https://www.geeksforgeeks.org/sde-sheet-a-complete-guide-for-sde-preparation/">https://www.geeksforgeeks.org/sde-sheet-a-complete-guide-for-sde-preparation/</a> Article Link: <a href="https://www.geeksforgeeks.org/how-to-print-maximum-number-of-a-using-given-four-keys/">https://www.geeksforgeeks.org/how-to-print-maximum-number-of-a-using-given-four-keys/</a>

SDE Sheet - Special Keyboard

Given a string consisting of opening and closing parenthesis, find the length of the longest valid parenthesis substring.Examples:Input : ((() Output : 2 Explanation : ()Input: )()()) Output : 4 Explanation: ()()Input: &nbsp;()(())))) Output: 6 Explanation: &nbsp;()(()) A Simple Approach is to find all the substrings of given string. For every string, check if it is a valid string or not. If valid and length is more than maximum length so far, then update maximum length. We can check whether a substring is valid or not in linear time using a stack (See this for details). Time complexity of this solution is O(n2)&nbsp;An Efficient Solution can solve this problem in O(n) time. The idea is to store indexes of previous starting brackets in a stack. The first element of the stack is a special element that provides index before the beginning of valid substring (base for next valid string).&nbsp;&nbsp;Related Article : <a href="https://www.geeksforgeeks.org/length-of-the-longest-valid-substring/">https://www.geeksforgeeks.org/length-of-the-longest-valid-substring/</a>

Length of the longest valid substring

Welcome to the daily solving of our PROBLEM OF THE DAY with Karan Mashru We will discuss the entire problem step-by-step and work towards developing an optimized solution. This will not only help you brush up on your concepts of Sliding-window but also build up problem-solving skills. Given two strings s and p. Find the smallest window in the string s consisting of all the characters(including duplicates) of the string p.&nbsp;&nbsp;Return "-1" in case there is no such window present.&nbsp;In case there are multiple such windows of same length, return the one with the least starting index. Note : All characters are in Lowercase alphabets.&nbsp;Examples:Input: s = "timetopractice", p = "toc"
Output: toprac
Explanation: "toprac" is the smallest
substring in which "toc" can be found.Give the problem a try before going through the video. All the best!!! Problem Link: <a href="https://practice.geeksforgeeks.org/problems/smallest-window-in-a-string-containing-all-the-characters-of-another-string-1587115621/1">https://practice.geeksforgeeks.org/problems/smallest-window-in-a-string-containing-all-the-characters-of-another-string-1587115621/1</a>

PROBLEM OF THE DAY : 24/09/2024 | Smallest window in a string containing all the characters of another string

Welcome to the daily solving of our PROBLEM OF THE DAY with Shivam Sinha We will discuss the entire problem step-by-step and work towards developing an optimized solution. This will not only help you brush up on your concepts of Arrays but also build up problem-solving skills.There is an array arr of heights of stone and Geek is standing at the first stone and can jump to one of the following: Stone i+1, i+2, ... i+k stone, where k is the maximum number of steps that can be jumped and cost will be |hi-hj| is incurred, where j is the stone to land on. Find the minimum possible total cost incurred before the Geek reaches the last stone.Example:Input: k = 3, arr[]= [10, 30, 40, 50, 20] Output: 30 Explanation: Geek will follow the path 1-&gt;2-&gt;5, the total cost would be | 10-30| + |30-20| = 30, which is minimumGive the problem a try before going through the video. All the best!!! Problem Link: <a href="https://practice.geeksforgeeks.org/problems/minimal-cost/1">https://practice.geeksforgeeks.org/problems/minimal-cost/1</a>

PROBLEM OF THE DAY : 28/09/2024 | Minimal Cost

This video is part of the Binary Search Tree section under GFG SDE Sheet. Given a string str, find the longest palindromic substring in str. Substring of string str: str[ i . . . . j ] where 0 ≤ i ≤ j &lt; len(str). Return the longest palindromic substring of str.Palindrome string: A string that reads the same backward. More formally, str is a palindrome if reverse(str) = str. In case of conflict, return the substring which occurs first ( with the least starting index).Examples :Input: str = "aaaabbaa"
Output: aabbaa
Explanation: The longest Palindromic substring is "aabbaa".Do check out:- Problem: <a href="https://www.geeksforgeeks.org/problems/longest-palindrome-in-a-string3411/1">https://www.geeksforgeeks.org/problems/longest-palindrome-in-a-string3411/1</a> SDE Sheet Link: <a href="https://www.geeksforgeeks.org/sde-sheet-a-complete-guide-for-sde-preparation/">https://www.geeksforgeeks.org/sde-sheet-a-complete-guide-for-sde-preparation/</a> Article Link: <a href="https://www.geeksforgeeks.org/longest-palindromic-substring-using-dynamic-programming-2/">https://www.geeksforgeeks.org/longest-palindromic-substring-using-dynamic-programming-2/</a>

SDE Sheet - Longest Palindrome in a String