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La subseqüència més llarga de manera que la diferència entre adjacents és d'un

Prova-ho a GfG Practice ' title=

Donada una a array arr[] de mida n la tasca és trobar el posterior més llarga tal que el diferència absoluta entre elements adjacents és 1.

Exemples: 

Entrada: arr[] = [10 9 4 5 4 8 6]
Sortida: 3
Explicació: Les tres possibles subseqüències de la longitud 3 són [10 9 8] [4 5 4] i [4 5 6] on els elements adjacents tenen una diferència absoluta d'1. No es podria formar cap subseqüència vàlida de major longitud.

Entrada: arr[] = [1 2 3 4 5]
Sortida: 5
Explicació: Tots els elements es poden incloure a la subseqüència vàlida.



Utilitzant la recursència - O(2^n) Temps i O(n) Espai

Per al enfocament recursiu tindrem en compte dos casos a cada pas:

  • Si l'element compleix la condició (el diferència absoluta entre elements adjacents és 1) nosaltres incloure en la subseqüència i passar a la següent element.
  • sinó nosaltres saltar el actual element i passar al següent.

Matemàticament el relació de recurrència tindrà el següent aspecte:

ordenant la llista en java
  • longestSubseq(arr idx anterior) = max(longestSubseq(arr idx + 1 anterior) 1 + longestSubseq(arr idx + 1 idx))

Cas base:

  • Quan idx == arr.size() tenim assolit el final de la matriu així retorn 0 (ja que no es poden incloure més elements).
C++ 
// C++ program to find the longest subsequence such that // the difference between adjacent elements is one using // recursion. #include    using namespace std; int subseqHelper(int idx int prev vector<int>& arr) {  // Base case: if index reaches the end of the array  if (idx == arr.size()) {  return 0;  }  // Skip the current element and move to the next index  int noTake = subseqHelper(idx + 1 prev arr);  // Take the current element if the condition is met  int take = 0;  if (prev == -1 || abs(arr[idx] - arr[prev]) == 1) {    take = 1 + subseqHelper(idx + 1 idx arr);  }  // Return the maximum of the two options  return max(take noTake); } // Function to find the longest subsequence int longestSubseq(vector<int>& arr) {    // Start recursion from index 0   // with no previous element  return subseqHelper(0 -1 arr); } int main() {  vector<int> arr = {10 9 4 5 4 8 6};  cout << longestSubseq(arr);  return 0; }
Java 
// Java program to find the longest subsequence such that // the difference between adjacent elements is one using // recursion. import java.util.ArrayList; class GfG {  // Helper function to recursively find the subsequence  static int subseqHelper(int idx int prev   ArrayList<Integer> arr) {  // Base case: if index reaches the end of the array  if (idx == arr.size()) {  return 0;  }  // Skip the current element and move to the next index  int noTake = subseqHelper(idx + 1 prev arr);  // Take the current element if the condition is met  int take = 0;  if (prev == -1 || Math.abs(arr.get(idx)   - arr.get(prev)) == 1) {    take = 1 + subseqHelper(idx + 1 idx arr);  }  // Return the maximum of the two options  return Math.max(take noTake);  }  // Function to find the longest subsequence  static int longestSubseq(ArrayList<Integer> arr) {  // Start recursion from index 0   // with no previous element  return subseqHelper(0 -1 arr);  }  public static void main(String[] args) {  ArrayList<Integer> arr = new ArrayList<>();  arr.add(10);  arr.add(9);  arr.add(4);  arr.add(5);  arr.add(4);  arr.add(8);  arr.add(6);  System.out.println(longestSubseq(arr));  } }
Python 
# Python program to find the longest subsequence such that # the difference between adjacent elements is one using # recursion. def subseq_helper(idx prev arr): # Base case: if index reaches the end of the array if idx == len(arr): return 0 # Skip the current element and move to the next index no_take = subseq_helper(idx + 1 prev arr) # Take the current element if the condition is met take = 0 if prev == -1 or abs(arr[idx] - arr[prev]) == 1: take = 1 + subseq_helper(idx + 1 idx arr) # Return the maximum of the two options return max(take no_take) def longest_subseq(arr): # Start recursion from index 0  # with no previous element return subseq_helper(0 -1 arr) if __name__ == '__main__': arr = [10 9 4 5 4 8 6] print(longest_subseq(arr))
C# 
// C# program to find the longest subsequence such that // the difference between adjacent elements is one using // recursion. using System; using System.Collections.Generic; class GfG {  // Helper function to recursively find the subsequence  static int SubseqHelper(int idx int prev   List<int> arr) {  // Base case: if index reaches the end of the array  if (idx == arr.Count) {  return 0;  }  // Skip the current element and move to the next index  int noTake = SubseqHelper(idx + 1 prev arr);  // Take the current element if the condition is met  int take = 0;  if (prev == -1 || Math.Abs(arr[idx] - arr[prev]) == 1) {    take = 1 + SubseqHelper(idx + 1 idx arr);  }  // Return the maximum of the two options  return Math.Max(take noTake);  }  // Function to find the longest subsequence  static int LongestSubseq(List<int> arr) {  // Start recursion from index 0   // with no previous element  return SubseqHelper(0 -1 arr);  }  static void Main(string[] args) {    List<int> arr   = new List<int> { 10 9 4 5 4 8 6 };  Console.WriteLine(LongestSubseq(arr));  } }
JavaScript 
// JavaScript program to find the longest subsequence  // such that the difference between adjacent elements  // is one using recursion. function subseqHelper(idx prev arr) {  // Base case: if index reaches the end of the array  if (idx === arr.length) {  return 0;  }  // Skip the current element and move to the next index  let noTake = subseqHelper(idx + 1 prev arr);  // Take the current element if the condition is met  let take = 0;  if (prev === -1 || Math.abs(arr[idx] - arr[prev]) === 1) {  take = 1 + subseqHelper(idx + 1 idx arr);  }  // Return the maximum of the two options  return Math.max(take noTake); } function longestSubseq(arr) {  // Start recursion from index 0   // with no previous element  return subseqHelper(0 -1 arr); } const arr = [10 9 4 5 4 8 6]; console.log(longestSubseq(arr));

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Ús de dalt a baix DP (Memoization )-  O(n^2)  Temps i  O(n^2)  Espai

Si observem amb atenció, podem observar que la solució recursiva anterior té les dues propietats següents de  Programació dinàmica :

1. Subestructura òptima: La solució per trobar la subseqüència més llarga tal que el diferència entre elements adjacents es pot derivar a partir de les solucions òptimes de subproblemes més petits. Concretament per a qualsevol donat idx (índex actual) i anterior (índex anterior a la subseqüència) podem expressar la relació recursiva de la següent manera:

  • subseqHelper(idx anterior) = max(subseqHelper(idx + 1 anterior) 1 + subseqHelper(idx + 1 idx))

2. Subproblemes superposats: En implementar a recursiu enfocament per resoldre el problema observem que molts subproblemes es calculen diverses vegades. Per exemple a l'hora d'informàtica subseqHelper(0 -1) per a una matriu arr = [10 9 4 5] el subproblema subseqHelper(2 -1) es pot calcular múltiples vegades. Per evitar aquesta repetició, utilitzem la memoització per emmagatzemar els resultats dels subproblemes calculats prèviament.

La solució recursiva implica dos paràmetres:

  • idx (l'índex actual de la matriu).
  • anterior (l'índex de l'últim element inclòs a la subseqüència).

Hem de fer un seguiment tots dos paràmetres així que creem un Memo de matriu 2D de mida (n) x (n+1) . Iniciem el Memo de matriu 2D amb -1 per indicar que encara no s'han calculat subproblemes. Abans de calcular un resultat comprovem si el valor a nota[idx][anterior+1] és -1. Si ho és calculem i botiga el resultat. En cas contrari, retornem el resultat emmagatzemat.

C++ 
// C++ program to find the longest subsequence such that // the difference between adjacent elements is one using // recursion with memoization. #include    using namespace std; // Helper function to recursively find the subsequence int subseqHelper(int idx int prev vector<int>& arr   vector<vector<int>>& memo) {  // Base case: if index reaches the end of the array  if (idx == arr.size()) {  return 0;  }  // Check if the result is already computed  if (memo[idx][prev + 1] != -1) {  return memo[idx][prev + 1];  }  // Skip the current element and move to the next index  int noTake = subseqHelper(idx + 1 prev arr memo);  // Take the current element if the condition is met  int take = 0;  if (prev == -1 || abs(arr[idx] - arr[prev]) == 1) {  take = 1 + subseqHelper(idx + 1 idx arr memo);  }  // Store the result in the memo table  return memo[idx][prev + 1] = max(take noTake); } // Function to find the longest subsequence int longestSubseq(vector<int>& arr) {    int n = arr.size();  // Create a memoization table initialized to -1  vector<vector<int>> memo(n vector<int>(n + 1 -1));  // Start recursion from index 0 with no previous element  return subseqHelper(0 -1 arr memo); } int main() {  // Input array of integers  vector<int> arr = {10 9 4 5 4 8 6};  cout << longestSubseq(arr);  return 0; }
Java 
// Java program to find the longest subsequence such that // the difference between adjacent elements is one using // recursion with memoization. import java.util.ArrayList; import java.util.Arrays; class GfG {  // Helper function to recursively find the subsequence  static int subseqHelper(int idx int prev   ArrayList<Integer> arr   int[][] memo) {  // Base case: if index reaches the end of the array  if (idx == arr.size()) {  return 0;  }  // Check if the result is already computed  if (memo[idx][prev + 1] != -1) {  return memo[idx][prev + 1];  }  // Skip the current element and move to the next index  int noTake = subseqHelper(idx + 1 prev arr memo);  // Take the current element if the condition is met  int take = 0;  if (prev == -1 || Math.abs(arr.get(idx)   - arr.get(prev)) == 1) {  take = 1 + subseqHelper(idx + 1 idx arr memo);  }  // Store the result in the memo table  memo[idx][prev + 1] = Math.max(take noTake);  // Return the stored result  return memo[idx][prev + 1];  }  // Function to find the longest subsequence  static int longestSubseq(ArrayList<Integer> arr) {  int n = arr.size();  // Create a memoization table initialized to -1  int[][] memo = new int[n][n + 1];  for (int[] row : memo) {  Arrays.fill(row -1);  }  // Start recursion from index 0   // with no previous element  return subseqHelper(0 -1 arr memo);  }  public static void main(String[] args) {  ArrayList<Integer> arr = new ArrayList<>();  arr.add(10);  arr.add(9);  arr.add(4);  arr.add(5);  arr.add(4);  arr.add(8);  arr.add(6);  System.out.println(longestSubseq(arr));  } }
Python 
# Python program to find the longest subsequence such that # the difference between adjacent elements is one using # recursion with memoization. def subseq_helper(idx prev arr memo): # Base case: if index reaches the end of the array if idx == len(arr): return 0 # Check if the result is already computed if memo[idx][prev + 1] != -1: return memo[idx][prev + 1] # Skip the current element and move to the next index no_take = subseq_helper(idx + 1 prev arr memo) # Take the current element if the condition is met take = 0 if prev == -1 or abs(arr[idx] - arr[prev]) == 1: take = 1 + subseq_helper(idx + 1 idx arr memo) # Store the result in the memo table memo[idx][prev + 1] = max(take no_take) # Return the stored result return memo[idx][prev + 1] def longest_subseq(arr): n = len(arr) # Create a memoization table initialized to -1 memo = [[-1 for _ in range(n + 1)] for _ in range(n)] # Start recursion from index 0 with  # no previous element return subseq_helper(0 -1 arr memo) if __name__ == '__main__': arr = [10 9 4 5 4 8 6] print(longest_subseq(arr))
C# 
// C# program to find the longest subsequence such that // the difference between adjacent elements is one using // recursion with memoization. using System; using System.Collections.Generic; class GfG {  // Helper function to recursively find the subsequence  static int SubseqHelper(int idx int prev  List<int> arr int[] memo) {  // Base case: if index reaches the end of the array  if (idx == arr.Count) {  return 0;  }  // Check if the result is already computed  if (memo[idx prev + 1] != -1) {  return memo[idx prev + 1];  }  // Skip the current element and move to the next index  int noTake = SubseqHelper(idx + 1 prev arr memo);  // Take the current element if the condition is met  int take = 0;  if (prev == -1 || Math.Abs(arr[idx] - arr[prev]) == 1) {  take = 1 + SubseqHelper(idx + 1 idx arr memo);  }  // Store the result in the memoization table  memo[idx prev + 1] = Math.Max(take noTake);  // Return the stored result  return memo[idx prev + 1];  }  // Function to find the longest subsequence  static int LongestSubseq(List<int> arr) {    int n = arr.Count;    // Create a memoization table initialized to -1  int[] memo = new int[n n + 1];  for (int i = 0; i < n; i++) {  for (int j = 0; j <= n; j++) {  memo[i j] = -1;  }  }  // Start recursion from index 0 with no previous element  return SubseqHelper(0 -1 arr memo);  }  static void Main(string[] args) {  List<int> arr   = new List<int> { 10 9 4 5 4 8 6 };  Console.WriteLine(LongestSubseq(arr));  } }
JavaScript 
// JavaScript program to find the longest subsequence  // such that the difference between adjacent elements  // is one using recursion with memoization. function subseqHelper(idx prev arr memo) {  // Base case: if index reaches the end of the array  if (idx === arr.length) {  return 0;  }  // Check if the result is already computed  if (memo[idx][prev + 1] !== -1) {  return memo[idx][prev + 1];  }  // Skip the current element and move to the next index  let noTake = subseqHelper(idx + 1 prev arr memo);  // Take the current element if the condition is met  let take = 0;  if (prev === -1 || Math.abs(arr[idx] - arr[prev]) === 1) {  take = 1 + subseqHelper(idx + 1 idx arr memo);  }  // Store the result in the memoization table  memo[idx][prev + 1] = Math.max(take noTake);  // Return the stored result  return memo[idx][prev + 1]; } function longestSubseq(arr) {  let n = arr.length;    // Create a memoization table initialized to -1  let memo =  Array.from({ length: n } () => Array(n + 1).fill(-1));  // Start recursion from index 0 with no previous element  return subseqHelper(0 -1 arr memo); } const arr = [10 9 4 5 4 8 6]; console.log(longestSubseq(arr));

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Ús de DP de baix a dalt (tabulació) -   O(n)  Temps i  O(n)  Espai

L'enfocament és similar al recursiu però en lloc de desglossar el problema de forma recursiva, construïm iterativament la solució en a manera de baix a dalt.
En lloc d'utilitzar la recursència, utilitzem a hashmap taula de programació dinàmica basada (dp) per emmagatzemar el longituds de les subseqüències més llargues. Això ens ajuda a calcular i actualitzar de manera eficient posterior longituds per a tots els valors possibles dels elements de la matriu.

Relació de programació dinàmica:

dp[x] representa la longitud de la subseqüència més llarga que acaba amb l'element x.

Per a cada element arr[i] a la matriu: Si arr[i] + 1 o arr[i] - 1 existeix en dp:

  • dp[arr[i]] = 1 + màx(dp[arr[i] + 1] dp[arr[i] - 1]);

Això vol dir que podem ampliar les subseqüències que acaben amb arr[i] + 1 o arr[i] - 1 per inclòs arr[i].

En cas contrari, inicieu una nova subseqüència:

  • dp[arr[i]] = 1;
C++ 
// C++ program to find the longest subsequence such that // the difference between adjacent elements is one using // Tabulation. #include    using namespace std; int longestSubseq(vector<int>& arr) {    int n = arr.size();  // Base case: if the array has only   // one element  if (n == 1) {  return 1;  }  // Map to store the length of the longest subsequence  unordered_map<int int> dp;  int ans = 1;  // Loop through the array to fill the map  // with subsequence lengths  for (int i = 0; i < n; ++i) {    // Check if the current element is adjacent  // to another subsequence  if (dp.count(arr[i] + 1) > 0   || dp.count(arr[i] - 1) > 0) {    dp[arr[i]] = 1 +   max(dp[arr[i] + 1] dp[arr[i] - 1]);  }   else {  dp[arr[i]] = 1;   }    // Update the result with the maximum  // subsequence length  ans = max(ans dp[arr[i]]);  }  return ans; } int main() {    vector<int> arr = {10 9 4 5 4 8 6};  cout << longestSubseq(arr);  return 0; }
Java 
// Java code to find the longest subsequence such that // the difference between adjacent elements  // is one using Tabulation. import java.util.HashMap; import java.util.ArrayList; class GfG {  static int longestSubseq(ArrayList<Integer> arr) {  int n = arr.size();  // Base case: if the array has only one element  if (n == 1) {  return 1;  }  // Map to store the length of the longest subsequence  HashMap<Integer Integer> dp = new HashMap<>();  int ans = 1;  // Loop through the array to fill the map   // with subsequence lengths  for (int i = 0; i < n; ++i) {  // Check if the current element is adjacent   // to another subsequence  if (dp.containsKey(arr.get(i) + 1)   || dp.containsKey(arr.get(i) - 1)) {  dp.put(arr.get(i) 1 +   Math.max(dp.getOrDefault(arr.get(i) + 1 0)   dp.getOrDefault(arr.get(i) - 1 0)));  }   else {  dp.put(arr.get(i) 1);   }  // Update the result with the maximum   // subsequence length  ans = Math.max(ans dp.get(arr.get(i)));  }  return ans;  }  public static void main(String[] args) {  ArrayList<Integer> arr = new ArrayList<>();  arr.add(10);  arr.add(9);  arr.add(4);  arr.add(5);  arr.add(4);  arr.add(8);  arr.add(6);    System.out.println(longestSubseq(arr));  } }
Python 
# Python code to find the longest subsequence such that # the difference between adjacent elements is  # one using Tabulation. def longestSubseq(arr): n = len(arr) # Base case: if the array has only one element if n == 1: return 1 # Dictionary to store the length of the  # longest subsequence dp = {} ans = 1 for i in range(n): # Check if the current element is adjacent to  # another subsequence if arr[i] + 1 in dp or arr[i] - 1 in dp: dp[arr[i]] = 1 + max(dp.get(arr[i] + 1 0)  dp.get(arr[i] - 1 0)) else: dp[arr[i]] = 1 # Update the result with the maximum # subsequence length ans = max(ans dp[arr[i]]) return ans if __name__ == '__main__': arr = [10 9 4 5 4 8 6] print(longestSubseq(arr))
C# 
// C# code to find the longest subsequence such that // the difference between adjacent elements  // is one using Tabulation. using System; using System.Collections.Generic; class GfG {  static int longestSubseq(List<int> arr) {  int n = arr.Count;  // Base case: if the array has only one element  if (n == 1) {  return 1;  }  // Map to store the length of the longest subsequence  Dictionary<int int> dp = new Dictionary<int int>();  int ans = 1;  // Loop through the array to fill the map with   // subsequence lengths  for (int i = 0; i < n; ++i) {  // Check if the current element is adjacent to  // another subsequence  if (dp.ContainsKey(arr[i] + 1) || dp.ContainsKey(arr[i] - 1)) {  dp[arr[i]] = 1 + Math.Max(dp.GetValueOrDefault(arr[i] + 1 0)  dp.GetValueOrDefault(arr[i] - 1 0));  }   else {  dp[arr[i]] = 1;   }  // Update the result with the maximum   // subsequence length  ans = Math.Max(ans dp[arr[i]]);  }  return ans;  }  static void Main(string[] args) {  List<int> arr   = new List<int> { 10 9 4 5 4 8 6 };  Console.WriteLine(longestSubseq(arr));  } }
JavaScript 
// Function to find the longest subsequence such that // the difference between adjacent elements // is one using Tabulation. function longestSubseq(arr) {  const n = arr.length;  // Base case: if the array has only one element  if (n === 1) {  return 1;  }  // Object to store the length of the  // longest subsequence  let dp = {};  let ans = 1;  // Loop through the array to fill the object  // with subsequence lengths  for (let i = 0; i < n; i++) {  // Check if the current element is adjacent to   // another subsequence  if ((arr[i] + 1) in dp || (arr[i] - 1) in dp) {  dp[arr[i]] = 1 + Math.max(dp[arr[i] + 1]  || 0 dp[arr[i] - 1] || 0);  } else {  dp[arr[i]] = 1;  }  // Update the result with the maximum   // subsequence length  ans = Math.max(ans dp[arr[i]]);  }  return ans; } const arr = [10 9 4 5 4 8 6]; console.log(longestSubseq(arr));

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