Árbol indexado binario: actualizaciones de rango y consultas de puntos
Dada una matriz arr[0..n-1]. Es necesario realizar las siguientes operaciones.
Inicialmente, todos los elementos de la matriz son 0. Las consultas pueden realizarse en cualquier orden, es decir, puede haber muchas actualizaciones antes de la consulta puntual.
Ejemplo:
Input: arr = {0 0 0 0 0} Queries: update : l = 0 r = 4 val = 2 getElement : i = 3 update : l = 3 r = 4 val = 3 getElement : i = 3 Output: Element at 3 is 2 Element at 3 is 5 Explanation: Array after first update becomes {2 2 2 2 2} Array after second update becomes {2 2 2 5 5} Método 1 [actualización: O(n) getElement(): O(1)]
La complejidad temporal en el peor de los casos es O(q*n) donde q es el número de consultas y n es el número de elementos.
Método 2 [actualización: O(1) getElement(): O(n)]
¡Podemos evitar actualizar todos los elementos y solo podemos actualizar 2 índices de la matriz!
arr[l] = arr[l] + val arr[r+1] = arr[r+1] - val
Analicemos la consulta de actualización. ¿Por qué agregar val a l? th ¿índice? Agregando val a l th índice significa que todos los elementos después de l aumentan en val ya que calcularemos la suma del prefijo para cada elemento. Por qué restar val de (r+1) th ¿índice? Se requirió una actualización de rango de [lr] pero lo que hemos actualizado es [l n-1] por lo que necesitamos eliminar val de todos los elementos después de r, es decir, restar val de (r+1) th índice. Por lo tanto, el valor se agrega al rango [lr]. A continuación se muestra la implementación del enfoque anterior.
C++ // C++ program to demonstrate Range Update // and Point Queries Without using BIT #include using namespace std ; // Updates such that getElement() gets an increased // value when queried from l to r. void update ( int arr [] int l int r int val ) { arr [ l ] += val ; arr [ r + 1 ] -= val ; } // Get the element indexed at i int getElement ( int arr [] int i ) { // To get ith element sum of all the elements // from 0 to i need to be computed int res = 0 ; for ( int j = 0 ; j <= i ; j ++ ) res += arr [ j ]; return res ; } // Driver program to test above function int main () { int arr [] = { 0 0 0 0 0 }; int n = sizeof ( arr ) / sizeof ( arr [ 0 ]); int l = 2 r = 4 val = 2 ; update ( arr l r val ); //Find the element at Index 4 int index = 4 ; cout < < 'Element at index ' < < index < < ' is ' < < getElement ( arr index ) < < endl ; l = 0 r = 3 val = 4 ; update ( arr l r val ); //Find the element at Index 3 index = 3 ; cout < < 'Element at index ' < < index < < ' is ' < < getElement ( arr index ) < < endl ; return 0 ; }
Java // Java program to demonstrate Range Update // and Point Queries Without using BIT class GfG { // Updates such that getElement() gets an increased // value when queried from l to r. static void update ( int arr [] int l int r int val ) { arr [ l ] += val ; if ( r + 1 < arr . length ) arr [ r + 1 ] -= val ; } // Get the element indexed at i static int getElement ( int arr [] int i ) { // To get ith element sum of all the elements // from 0 to i need to be computed int res = 0 ; for ( int j = 0 ; j <= i ; j ++ ) res += arr [ j ] ; return res ; } // Driver program to test above function public static void main ( String [] args ) { int arr [] = { 0 0 0 0 0 }; int n = arr . length ; int l = 2 r = 4 val = 2 ; update ( arr l r val ); //Find the element at Index 4 int index = 4 ; System . out . println ( 'Element at index ' + index + ' is ' + getElement ( arr index )); l = 0 ; r = 3 ; val = 4 ; update ( arr l r val ); //Find the element at Index 3 index = 3 ; System . out . println ( 'Element at index ' + index + ' is ' + getElement ( arr index )); } }
Python3 # Python3 program to demonstrate Range # Update and PoQueries Without using BIT # Updates such that getElement() gets an # increased value when queried from l to r. def update ( arr l r val ): arr [ l ] += val if r + 1 < len ( arr ): arr [ r + 1 ] -= val # Get the element indexed at i def getElement ( arr i ): # To get ith element sum of all the elements # from 0 to i need to be computed res = 0 for j in range ( i + 1 ): res += arr [ j ] return res # Driver Code if __name__ == '__main__' : arr = [ 0 0 0 0 0 ] n = len ( arr ) l = 2 r = 4 val = 2 update ( arr l r val ) # Find the element at Index 4 index = 4 print ( 'Element at index' index 'is' getElement ( arr index )) l = 0 r = 3 val = 4 update ( arr l r val ) # Find the element at Index 3 index = 3 print ( 'Element at index' index 'is' getElement ( arr index )) # This code is contributed by PranchalK
C# // C# program to demonstrate Range Update // and Point Queries Without using BIT using System ; class GfG { // Updates such that getElement() // gets an increased value when // queried from l to r. static void update ( int [] arr int l int r int val ) { arr [ l ] += val ; if ( r + 1 < arr . Length ) arr [ r + 1 ] -= val ; } // Get the element indexed at i static int getElement ( int [] arr int i ) { // To get ith element sum of all the elements // from 0 to i need to be computed int res = 0 ; for ( int j = 0 ; j <= i ; j ++ ) res += arr [ j ]; return res ; } // Driver code public static void Main ( String [] args ) { int [] arr = { 0 0 0 0 0 }; int n = arr . Length ; int l = 2 r = 4 val = 2 ; update ( arr l r val ); //Find the element at Index 4 int index = 4 ; Console . WriteLine ( 'Element at index ' + index + ' is ' + getElement ( arr index )); l = 0 ; r = 3 ; val = 4 ; update ( arr l r val ); //Find the element at Index 3 index = 3 ; Console . WriteLine ( 'Element at index ' + index + ' is ' + getElement ( arr index )); } } // This code is contributed by PrinciRaj1992
PHP // PHP program to demonstrate Range Update // and Point Queries Without using BIT // Updates such that getElement() gets an // increased value when queried from l to r. function update ( & $arr $l $r $val ) { $arr [ $l ] += $val ; if ( $r + 1 < sizeof ( $arr )) $arr [ $r + 1 ] -= $val ; } // Get the element indexed at i function getElement ( & $arr $i ) { // To get ith element sum of all the elements // from 0 to i need to be computed $res = 0 ; for ( $j = 0 ; $j <= $i ; $j ++ ) $res += $arr [ $j ]; return $res ; } // Driver Code $arr = array ( 0 0 0 0 0 ); $n = sizeof ( $arr ); $l = 2 ; $r = 4 ; $val = 2 ; update ( $arr $l $r $val ); // Find the element at Index 4 $index = 4 ; echo ( 'Element at index ' . $index . ' is ' . getElement ( $arr $index ) . ' n ' ); $l = 0 ; $r = 3 ; $val = 4 ; update ( $arr $l $r $val ); // Find the element at Index 3 $index = 3 ; echo ( 'Element at index ' . $index . ' is ' . getElement ( $arr $index )); // This code is contributed by Code_Mech ?>
JavaScript //JavaScript program to demonstrate Range Update // and Point Queries Without using BIT // Updates such that getElement() gets an increased // value when queried from l to r. function update ( arr l r val ) { arr [ l ] += val ; arr [ r + 1 ] -= val ; } // Get the element indexed at i function getElement ( rr i ) { // To get ith element sum of all the elements // from 0 to i need to be computed let res = 0 ; for ( let j = 0 ; j <= i ; j ++ ) res += arr [ j ]; return res ; } // Driver program to test above function let arr = [ 0 0 0 0 0 ]; let n = arr . length ; let l = 2 r = 4 val = 2 ; update ( arr l r val ); // Find the element at Index 4 let index = 4 ; console . log ( 'Element at index ' index ' is ' getElement ( arr index )); l = 0 r = 3 val = 4 ; update ( arr l r val ); // Find the element at Index 3 index = 3 ; console . log ( 'Element at index ' index ' is ' getElement ( arr index )); // This code is contributed by vikkycirus
Producción:
Element at index 4 is 2 Element at index 3 is 6
Complejidad del tiempo : O(q*n) donde q es el número de consultas.
Espacio Auxiliar: En)
Método 3 (usando un árbol indexado binario)
En el método 2 hemos visto que el problema se puede reducir a consultas de actualización y suma de prefijos. hemos visto eso BIT se puede utilizar para realizar consultas de suma de prefijos y actualizaciones en tiempo O (Logn). A continuación se muestra la implementación.
C++ // C++ code to demonstrate Range Update and // Point Queries on a Binary Index Tree #include using namespace std ; // Updates a node in Binary Index Tree (BITree) at given index // in BITree. The given value 'val' is added to BITree[i] and // all of its ancestors in tree. void updateBIT ( int BITree [] int n int index int val ) { // index in BITree[] is 1 more than the index in arr[] index = index + 1 ; // Traverse all ancestors and add 'val' while ( index <= n ) { // Add 'val' to current node of BI Tree BITree [ index ] += val ; // Update index to that of parent in update View index += index & ( - index ); } } // Constructs and returns a Binary Indexed Tree for given // array of size n. int * constructBITree ( int arr [] int n ) { // Create and initialize BITree[] as 0 int * BITree = new int [ n + 1 ]; for ( int i = 1 ; i <= n ; i ++ ) BITree [ i ] = 0 ; // Store the actual values in BITree[] using update() for ( int i = 0 ; i < n ; i ++ ) updateBIT ( BITree n i arr [ i ]); // Uncomment below lines to see contents of BITree[] //for (int i=1; i <=n; i++) // cout < < BITree[i] < < ' '; return BITree ; } // SERVES THE PURPOSE OF getElement() // Returns sum of arr[0..index]. This function assumes // that the array is preprocessed and partial sums of // array elements are stored in BITree[] int getSum ( int BITree [] int index ) { int sum = 0 ; // Initialize result // index in BITree[] is 1 more than the index in arr[] index = index + 1 ; // Traverse ancestors of BITree[index] while ( index > 0 ) { // Add current element of BITree to sum sum += BITree [ index ]; // Move index to parent node in getSum View index -= index & ( - index ); } return sum ; } // Updates such that getElement() gets an increased // value when queried from l to r. void update ( int BITree [] int l int r int n int val ) { // Increase value at 'l' by 'val' updateBIT ( BITree n l val ); // Decrease value at 'r+1' by 'val' updateBIT ( BITree n r + 1 - val ); } // Driver program to test above function int main () { int arr [] = { 0 0 0 0 0 }; int n = sizeof ( arr ) / sizeof ( arr [ 0 ]); int * BITree = constructBITree ( arr n ); // Add 2 to all the element from [24] int l = 2 r = 4 val = 2 ; update ( BITree l r n val ); // Find the element at Index 4 int index = 4 ; cout < < 'Element at index ' < < index < < ' is ' < < getSum ( BITree index ) < < ' n ' ; // Add 2 to all the element from [03] l = 0 r = 3 val = 4 ; update ( BITree l r n val ); // Find the element at Index 3 index = 3 ; cout < < 'Element at index ' < < index < < ' is ' < < getSum ( BITree index ) < < ' n ' ; return 0 ; }
Java /* Java code to demonstrate Range Update and * Point Queries on a Binary Index Tree. * This method only works when all array * values are initially 0.*/ class GFG { // Max tree size final static int MAX = 1000 ; static int BITree [] = new int [ MAX ] ; // Updates a node in Binary Index // Tree (BITree) at given index // in BITree. The given value 'val' // is added to BITree[i] and // all of its ancestors in tree. public static void updateBIT ( int n int index int val ) { // index in BITree[] is 1 // more than the index in arr[] index = index + 1 ; // Traverse all ancestors // and add 'val' while ( index <= n ) { // Add 'val' to current // node of BITree BITree [ index ] += val ; // Update index to that // of parent in update View index += index & ( - index ); } } // Constructs Binary Indexed Tree // for given array of size n. public static void constructBITree ( int arr [] int n ) { // Initialize BITree[] as 0 for ( int i = 1 ; i <= n ; i ++ ) BITree [ i ] = 0 ; // Store the actual values // in BITree[] using update() for ( int i = 0 ; i < n ; i ++ ) updateBIT ( n i arr [ i ] ); // Uncomment below lines to // see contents of BITree[] // for (int i=1; i <=n; i++) // cout < < BITree[i] < < ' '; } // SERVES THE PURPOSE OF getElement() // Returns sum of arr[0..index]. This // function assumes that the array is // preprocessed and partial sums of // array elements are stored in BITree[] public static int getSum ( int index ) { int sum = 0 ; //Initialize result // index in BITree[] is 1 more // than the index in arr[] index = index + 1 ; // Traverse ancestors // of BITree[index] while ( index > 0 ) { // Add current element // of BITree to sum sum += BITree [ index ] ; // Move index to parent // node in getSum View index -= index & ( - index ); } // Return the sum return sum ; } // Updates such that getElement() // gets an increased value when // queried from l to r. public static void update ( int l int r int n int val ) { // Increase value at // 'l' by 'val' updateBIT ( n l val ); // Decrease value at // 'r+1' by 'val' updateBIT ( n r + 1 - val ); } // Driver Code public static void main ( String args [] ) { int arr [] = { 0 0 0 0 0 }; int n = arr . length ; constructBITree ( arr n ); // Add 2 to all the // element from [24] int l = 2 r = 4 val = 2 ; update ( l r n val ); int index = 4 ; System . out . println ( 'Element at index ' + index + ' is ' + getSum ( index )); // Add 2 to all the // element from [03] l = 0 ; r = 3 ; val = 4 ; update ( l r n val ); // Find the element // at Index 3 index = 3 ; System . out . println ( 'Element at index ' + index + ' is ' + getSum ( index )); } } // This code is contributed // by Puneet Kumar.
Python3 # Python3 code to demonstrate Range Update and # PoQueries on a Binary Index Tree # Updates a node in Binary Index Tree (BITree) at given index # in BITree. The given value 'val' is added to BITree[i] and # all of its ancestors in tree. def updateBIT ( BITree n index val ): # index in BITree[] is 1 more than the index in arr[] index = index + 1 # Traverse all ancestors and add 'val' while ( index <= n ): # Add 'val' to current node of BI Tree BITree [ index ] += val # Update index to that of parent in update View index += index & ( - index ) # Constructs and returns a Binary Indexed Tree for given # array of size n. def constructBITree ( arr n ): # Create and initialize BITree[] as 0 BITree = [ 0 ] * ( n + 1 ) # Store the actual values in BITree[] using update() for i in range ( n ): updateBIT ( BITree n i arr [ i ]) return BITree # SERVES THE PURPOSE OF getElement() # Returns sum of arr[0..index]. This function assumes # that the array is preprocessed and partial sums of # array elements are stored in BITree[] def getSum ( BITree index ): sum = 0 # Initialize result # index in BITree[] is 1 more than the index in arr[] index = index + 1 # Traverse ancestors of BITree[index] while ( index > 0 ): # Add current element of BITree to sum sum += BITree [ index ] # Move index to parent node in getSum View index -= index & ( - index ) return sum # Updates such that getElement() gets an increased # value when queried from l to r. def update ( BITree l r n val ): # Increase value at 'l' by 'val' updateBIT ( BITree n l val ) # Decrease value at 'r+1' by 'val' updateBIT ( BITree n r + 1 - val ) # Driver code arr = [ 0 0 0 0 0 ] n = len ( arr ) BITree = constructBITree ( arr n ) # Add 2 to all the element from [24] l = 2 r = 4 val = 2 update ( BITree l r n val ) # Find the element at Index 4 index = 4 print ( 'Element at index' index 'is' getSum ( BITree index )) # Add 2 to all the element from [03] l = 0 r = 3 val = 4 update ( BITree l r n val ) # Find the element at Index 3 index = 3 print ( 'Element at index' index 'is' getSum ( BITree index )) # This code is contributed by mohit kumar 29
C# using System ; /* C# code to demonstrate Range Update and * Point Queries on a Binary Index Tree. * This method only works when all array * values are initially 0.*/ public class GFG { // Max tree size public const int MAX = 1000 ; public static int [] BITree = new int [ MAX ]; // Updates a node in Binary Index // Tree (BITree) at given index // in BITree. The given value 'val' // is added to BITree[i] and // all of its ancestors in tree. public static void updateBIT ( int n int index int val ) { // index in BITree[] is 1 // more than the index in arr[] index = index + 1 ; // Traverse all ancestors // and add 'val' while ( index <= n ) { // Add 'val' to current // node of BITree BITree [ index ] += val ; // Update index to that // of parent in update View index += index & ( - index ); } } // Constructs Binary Indexed Tree // for given array of size n. public static void constructBITree ( int [] arr int n ) { // Initialize BITree[] as 0 for ( int i = 1 ; i <= n ; i ++ ) { BITree [ i ] = 0 ; } // Store the actual values // in BITree[] using update() for ( int i = 0 ; i < n ; i ++ ) { updateBIT ( n i arr [ i ]); } // Uncomment below lines to // see contents of BITree[] // for (int i=1; i <=n; i++) // cout < < BITree[i] < < ' '; } // SERVES THE PURPOSE OF getElement() // Returns sum of arr[0..index]. This // function assumes that the array is // preprocessed and partial sums of // array elements are stored in BITree[] public static int getSum ( int index ) { int sum = 0 ; //Initialize result // index in BITree[] is 1 more // than the index in arr[] index = index + 1 ; // Traverse ancestors // of BITree[index] while ( index > 0 ) { // Add current element // of BITree to sum sum += BITree [ index ]; // Move index to parent // node in getSum View index -= index & ( - index ); } // Return the sum return sum ; } // Updates such that getElement() // gets an increased value when // queried from l to r. public static void update ( int l int r int n int val ) { // Increase value at // 'l' by 'val' updateBIT ( n l val ); // Decrease value at // 'r+1' by 'val' updateBIT ( n r + 1 - val ); } // Driver Code public static void Main ( string [] args ) { int [] arr = new int [] { 0 0 0 0 0 }; int n = arr . Length ; constructBITree ( arr n ); // Add 2 to all the // element from [24] int l = 2 r = 4 val = 2 ; update ( l r n val ); int index = 4 ; Console . WriteLine ( 'Element at index ' + index + ' is ' + getSum ( index )); // Add 2 to all the // element from [03] l = 0 ; r = 3 ; val = 4 ; update ( l r n val ); // Find the element // at Index 3 index = 3 ; Console . WriteLine ( 'Element at index ' + index + ' is ' + getSum ( index )); } } // This code is contributed by Shrikant13
JavaScript // Updates a node in Binary Index Tree (BITree) at given index // in BITree. The given value 'val' is added to BITree[i] and // all of its ancestors in tree. function updateBIT ( BITree n index val ) { // index in BITree[] is 1 more than the index in arr[] index = index + 1 ; // Traverse all ancestors and add 'val' while ( index <= n ) { // Add 'val' to current node of BI Tree BITree [ index ] += val ; // Update index to that of parent in update View index += index & ( - index ); } } // Constructs and returns a Binary Indexed Tree for given // array of size n. function constructBITree ( arr n ) { // Create and initialize BITree[] as 0 let BITree = new Array ( n + 1 ). fill ( 0 ); // Store the actual values in BITree[] using update() for ( let i = 0 ; i < n ; i ++ ) { updateBIT ( BITree n i arr [ i ]); } return BITree ; } // SERVES THE PURPOSE OF getElement() // Returns sum of arr[0..index]. This function assumes // that the array is preprocessed and partial sums of // array elements are stored in BITree[] function getSum ( BITree index ) { let sum = 0 ; // Initialize result // index in BITree[] is 1 more than the index in arr[] index = index + 1 ; // Traverse ancestors of BITree[index] while ( index > 0 ) { // Add current element of BITree to sum sum += BITree [ index ]; // Move index to parent node in getSum View index -= index & ( - index ); } return sum ; } // Updates such that getElement() gets an increased // value when queried from l to r. function update ( BITree l r n val ) { // Increase value at 'l' by 'val' updateBIT ( BITree n l val ); // Decrease value at 'r+1' by 'val' updateBIT ( BITree n r + 1 - val ); } // Test the functions let arr = [ 0 0 0 0 0 ]; let n = arr . length ; let BITree = constructBITree ( arr n ); // Add 2 to all the element from [24] let l = 2 r = 4 val = 2 ; update ( BITree l r n val ); // Find the element at Index 4 let index = 4 ; console . log ( `Element at index ${ index } is ${ getSum ( BITree index ) } ` ); // Add 2 to all the element from [03] l = 0 r = 3 val = 4 ; update ( BITree l r n val ); // Find the element at Index 3 index = 3 ; console . log ( `Element at index ${ index } is ${ getSum ( BITree index ) } ` );
Producción:
Element at index 4 is 2 Element at index 3 is 6
Complejidad del tiempo: O(q * log n) + O(n * log n) donde q es el número de consultas.
Espacio Auxiliar: En)
El método 1 es eficiente cuando la mayoría de las consultas son getElement(), el método 2 es eficiente cuando la mayoría de las consultas son actualizaciones() y se prefiere el método 3 cuando hay una combinación de ambas consultas.
Artículos Más Populares
Categoría
Artículos De Interés
