Штампање најдуже битонске секвенце
Проблем најдуже битонске поднизове је пронаћи најдужи подниз дате секвенце тако да се прво повећава, а затим опада. Секвенца сортирана у растућем редоследу сматра се битонском са опадајућим делом као празним. Слично опадајућа секвенца се сматра битонском са растућим делом као празним. Примери:
Input: [1 11 2 10 4 5 2 1] Output: [1 2 10 4 2 1] OR [1 11 10 5 2 1] OR [1 2 4 5 2 1] Input: [12 11 40 5 3 1] Output: [12 11 5 3 1] OR [12 40 5 3 1] Input: [80 60 30 40 20 10] Output: [80 60 30 20 10] OR [80 60 40 20 10]
Ин претходни пост о коме смо расправљали о проблему најдуже битонске подсеквенце. Међутим, пост је покривао само код који се односи на проналажење максималног збира растуће подниз, али не и на конструкцију подниза. У овом посту ћемо разговарати о томе како конструисати саму најдужу битонску подсеквенцу. Нека је арр[0..н-1] улазни низ. Дефинишемо вектор ЛИС тако да је ЛИС[и] сам вектор који чува најдужу растућу подниз од арр[0..и] који се завршава са арр[и]. Стога за индекс и ЛИС[и] се може рекурзивно написати као -
LIS[0] = {arr[O]} LIS[i] = {Max(LIS[j])} + arr[i] where j < i and arr[j] < arr[i] = arr[i] if there is no such j Такође дефинишемо вектор ЛДС тако да је ЛДС[и] сам вектор који чува најдужу опадајућу подсеквенцу арр[и..н] која почиње са арр[и]. Стога се за индекс и ЛДС[и] може рекурзивно написати као -
LDS[n] = {arr[n]} LDS[i] = arr[i] + {Max(LDS[j])} where j > i and arr[j] < arr[i] = arr[i] if there is no such j На пример за низ [1 11 2 10 4 5 2 1]
LIS[0]: 1 LIS[1]: 1 11 LIS[2]: 1 2 LIS[3]: 1 2 10 LIS[4]: 1 2 4 LIS[5]: 1 2 4 5 LIS[6]: 1 2 LIS[7]: 1
LDS[0]: 1 LDS[1]: 11 10 5 2 1 LDS[2]: 2 1 LDS[3]: 10 5 2 1 LDS[4]: 4 2 1 LDS[5]: 5 2 1 LDS[6]: 2 1 LDS[7]: 1
Стога најдужа битонска подсеквенца може бити
LIS[1] + LDS[1] = [1 11 10 5 2 1] OR LIS[3] + LDS[3] = [1 2 10 5 2 1] OR LIS[5] + LDS[5] = [1 2 4 5 2 1]
Испод је имплементација горње идеје -
C++ /* Dynamic Programming solution to print Longest Bitonic Subsequence */ #include using namespace std ; // Utility function to print Longest Bitonic // Subsequence void print ( vector < int >& arr int size ) { for ( int i = 0 ; i < size ; i ++ ) cout < < arr [ i ] < < ' ' ; } // Function to construct and print Longest // Bitonic Subsequence void printLBS ( int arr [] int n ) { // LIS[i] stores the length of the longest // increasing subsequence ending with arr[i] vector < vector < int >> LIS ( n ); // initialize LIS[0] to arr[0] LIS [ 0 ]. push_back ( arr [ 0 ]); // Compute LIS values from left to right for ( int i = 1 ; i < n ; i ++ ) { // for every j less than i for ( int j = 0 ; j < i ; j ++ ) { if (( arr [ j ] < arr [ i ]) && ( LIS [ j ]. size () > LIS [ i ]. size ())) LIS [ i ] = LIS [ j ]; } LIS [ i ]. push_back ( arr [ i ]); } /* LIS[i] now stores Maximum Increasing Subsequence of arr[0..i] that ends with arr[i] */ // LDS[i] stores the length of the longest // decreasing subsequence starting with arr[i] vector < vector < int >> LDS ( n ); // initialize LDS[n-1] to arr[n-1] LDS [ n - 1 ]. push_back ( arr [ n - 1 ]); // Compute LDS values from right to left for ( int i = n - 2 ; i >= 0 ; i -- ) { // for every j greater than i for ( int j = n - 1 ; j > i ; j -- ) { if (( arr [ j ] < arr [ i ]) && ( LDS [ j ]. size () > LDS [ i ]. size ())) LDS [ i ] = LDS [ j ]; } LDS [ i ]. push_back ( arr [ i ]); } // reverse as vector as we're inserting at end for ( int i = 0 ; i < n ; i ++ ) reverse ( LDS [ i ]. begin () LDS [ i ]. end ()); /* LDS[i] now stores Maximum Decreasing Subsequence of arr[i..n] that starts with arr[i] */ int max = 0 ; int maxIndex = -1 ; for ( int i = 0 ; i < n ; i ++ ) { // Find maximum value of size of LIS[i] + size // of LDS[i] - 1 if ( LIS [ i ]. size () + LDS [ i ]. size () - 1 > max ) { max = LIS [ i ]. size () + LDS [ i ]. size () - 1 ; maxIndex = i ; } } // print all but last element of LIS[maxIndex] vector print ( LIS [ maxIndex ] LIS [ maxIndex ]. size () - 1 ); // print all elements of LDS[maxIndex] vector print ( LDS [ maxIndex ] LDS [ maxIndex ]. size ()); } // Driver program int main () { int arr [] = { 1 11 2 10 4 5 2 1 }; int n = sizeof ( arr ) / sizeof ( arr [ 0 ]); printLBS ( arr n ); return 0 ; }
Java /* Dynamic Programming solution to print Longest Bitonic Subsequence */ import java.util.* ; class GFG { // Utility function to print Longest Bitonic // Subsequence static void print ( Vector < Integer > arr int size ) { for ( int i = 0 ; i < size ; i ++ ) System . out . print ( arr . elementAt ( i ) + ' ' ); } // Function to construct and print Longest // Bitonic Subsequence static void printLBS ( int [] arr int n ) { // LIS[i] stores the length of the longest // increasing subsequence ending with arr[i] @SuppressWarnings ( 'unchecked' ) Vector < Integer >[] LIS = new Vector [ n ] ; for ( int i = 0 ; i < n ; i ++ ) LIS [ i ] = new Vector <> (); // initialize LIS[0] to arr[0] LIS [ 0 ] . add ( arr [ 0 ] ); // Compute LIS values from left to right for ( int i = 1 ; i < n ; i ++ ) { // for every j less than i for ( int j = 0 ; j < i ; j ++ ) { if (( arr [ i ] > arr [ j ] ) && LIS [ j ] . size () > LIS [ i ] . size ()) { for ( int k : LIS [ j ] ) if ( ! LIS [ i ] . contains ( k )) LIS [ i ] . add ( k ); } } LIS [ i ] . add ( arr [ i ] ); } /* * LIS[i] now stores Maximum Increasing Subsequence * of arr[0..i] that ends with arr[i] */ // LDS[i] stores the length of the longest // decreasing subsequence starting with arr[i] @SuppressWarnings ( 'unchecked' ) Vector < Integer >[] LDS = new Vector [ n ] ; for ( int i = 0 ; i < n ; i ++ ) LDS [ i ] = new Vector <> (); // initialize LDS[n-1] to arr[n-1] LDS [ n - 1 ] . add ( arr [ n - 1 ] ); // Compute LDS values from right to left for ( int i = n - 2 ; i >= 0 ; i -- ) { // for every j greater than i for ( int j = n - 1 ; j > i ; j -- ) { if ( arr [ j ] < arr [ i ] && LDS [ j ] . size () > LDS [ i ] . size ()) for ( int k : LDS [ j ] ) if ( ! LDS [ i ] . contains ( k )) LDS [ i ] . add ( k ); } LDS [ i ] . add ( arr [ i ] ); } // reverse as vector as we're inserting at end for ( int i = 0 ; i < n ; i ++ ) Collections . reverse ( LDS [ i ] ); /* * LDS[i] now stores Maximum Decreasing Subsequence * of arr[i..n] that starts with arr[i] */ int max = 0 ; int maxIndex = - 1 ; for ( int i = 0 ; i < n ; i ++ ) { // Find maximum value of size of // LIS[i] + size of LDS[i] - 1 if ( LIS [ i ] . size () + LDS [ i ] . size () - 1 > max ) { max = LIS [ i ] . size () + LDS [ i ] . size () - 1 ; maxIndex = i ; } } // print all but last element of LIS[maxIndex] vector print ( LIS [ maxIndex ] LIS [ maxIndex ] . size () - 1 ); // print all elements of LDS[maxIndex] vector print ( LDS [ maxIndex ] LDS [ maxIndex ] . size ()); } // Driver Code public static void main ( String [] args ) { int [] arr = { 1 11 2 10 4 5 2 1 }; int n = arr . length ; printLBS ( arr n ); } } // This code is contributed by // sanjeev2552
Python3 # Dynamic Programming solution to print Longest # Bitonic Subsequence def _print ( arr : list size : int ): for i in range ( size ): print ( arr [ i ] end = ' ' ) # Function to construct and print Longest # Bitonic Subsequence def printLBS ( arr : list n : int ): # LIS[i] stores the length of the longest # increasing subsequence ending with arr[i] LIS = [ 0 ] * n for i in range ( n ): LIS [ i ] = [] # initialize LIS[0] to arr[0] LIS [ 0 ] . append ( arr [ 0 ]) # Compute LIS values from left to right for i in range ( 1 n ): # for every j less than i for j in range ( i ): if (( arr [ j ] < arr [ i ]) and ( len ( LIS [ j ]) > len ( LIS [ i ]))): LIS [ i ] = LIS [ j ] . copy () LIS [ i ] . append ( arr [ i ]) # LIS[i] now stores Maximum Increasing # Subsequence of arr[0..i] that ends with # arr[i] # LDS[i] stores the length of the longest # decreasing subsequence starting with arr[i] LDS = [ 0 ] * n for i in range ( n ): LDS [ i ] = [] # initialize LDS[n-1] to arr[n-1] LDS [ n - 1 ] . append ( arr [ n - 1 ]) # Compute LDS values from right to left for i in range ( n - 2 - 1 - 1 ): # for every j greater than i for j in range ( n - 1 i - 1 ): if (( arr [ j ] < arr [ i ]) and ( len ( LDS [ j ]) > len ( LDS [ i ]))): LDS [ i ] = LDS [ j ] . copy () LDS [ i ] . append ( arr [ i ]) # reverse as vector as we're inserting at end for i in range ( n ): LDS [ i ] = list ( reversed ( LDS [ i ])) # LDS[i] now stores Maximum Decreasing Subsequence # of arr[i..n] that starts with arr[i] max = 0 maxIndex = - 1 for i in range ( n ): # Find maximum value of size of LIS[i] + size # of LDS[i] - 1 if ( len ( LIS [ i ]) + len ( LDS [ i ]) - 1 > max ): max = len ( LIS [ i ]) + len ( LDS [ i ]) - 1 maxIndex = i # print all but last element of LIS[maxIndex] vector _print ( LIS [ maxIndex ] len ( LIS [ maxIndex ]) - 1 ) # print all elements of LDS[maxIndex] vector _print ( LDS [ maxIndex ] len ( LDS [ maxIndex ])) # Driver Code if __name__ == '__main__' : arr = [ 1 11 2 10 4 5 2 1 ] n = len ( arr ) printLBS ( arr n ) # This code is contributed by # sanjeev2552
C# /* Dynamic Programming solution to print longest Bitonic Subsequence */ using System ; using System.Linq ; using System.Collections.Generic ; class GFG { // Utility function to print longest Bitonic // Subsequence static void print ( List < int > arr int size ) { for ( int i = 0 ; i < size ; i ++ ) Console . Write ( arr [ i ] + ' ' ); } // Function to construct and print longest // Bitonic Subsequence static void printLBS ( int [] arr int n ) { // LIS[i] stores the length of the longest // increasing subsequence ending with arr[i] List < int > [] LIS = new List < int > [ n ]; for ( int i = 0 ; i < n ; i ++ ) LIS [ i ] = new List < int > (); // initialize LIS[0] to arr[0] LIS [ 0 ]. Add ( arr [ 0 ]); // Compute LIS values from left to right for ( int i = 1 ; i < n ; i ++ ) { // for every j less than i for ( int j = 0 ; j < i ; j ++ ) { if (( arr [ i ] > arr [ j ]) && LIS [ j ]. Count > LIS [ i ]. Count ) { foreach ( int k in LIS [ j ]) if ( ! LIS [ i ]. Contains ( k )) LIS [ i ]. Add ( k ); } } LIS [ i ]. Add ( arr [ i ]); } /* * LIS[i] now stores Maximum Increasing Subsequence * of arr[0..i] that ends with arr[i] */ // LDS[i] stores the length of the longest // decreasing subsequence starting with arr[i] List < int > [] LDS = new List < int > [ n ]; for ( int i = 0 ; i < n ; i ++ ) LDS [ i ] = new List < int > (); // initialize LDS[n-1] to arr[n-1] LDS [ n - 1 ]. Add ( arr [ n - 1 ]); // Compute LDS values from right to left for ( int i = n - 2 ; i >= 0 ; i -- ) { // for every j greater than i for ( int j = n - 1 ; j > i ; j -- ) { if ( arr [ j ] < arr [ i ] && LDS [ j ]. Count > LDS [ i ]. Count ) foreach ( int k in LDS [ j ]) if ( ! LDS [ i ]. Contains ( k )) LDS [ i ]. Add ( k ); } LDS [ i ]. Add ( arr [ i ]); } // reverse as vector as we're inserting at end for ( int i = 0 ; i < n ; i ++ ) LDS [ i ]. Reverse (); /* * LDS[i] now stores Maximum Decreasing Subsequence * of arr[i..n] that starts with arr[i] */ int max = 0 ; int maxIndex = - 1 ; for ( int i = 0 ; i < n ; i ++ ) { // Find maximum value of size of // LIS[i] + size of LDS[i] - 1 if ( LIS [ i ]. Count + LDS [ i ]. Count - 1 > max ) { max = LIS [ i ]. Count + LDS [ i ]. Count - 1 ; maxIndex = i ; } } // print all but last element of LIS[maxIndex] vector print ( LIS [ maxIndex ] LIS [ maxIndex ]. Count - 1 ); // print all elements of LDS[maxIndex] vector print ( LDS [ maxIndex ] LDS [ maxIndex ]. Count ); } // Driver Code public static void Main ( String [] args ) { int [] arr = { 1 11 2 10 4 5 2 1 }; int n = arr . Length ; printLBS ( arr n ); } } // This code is contributed by PrinciRaj1992
JavaScript // Function to print the longest bitonic subsequence function _print ( arr size ) { for ( let i = 0 ; i < size ; i ++ ) { process . stdout . write ( arr [ i ] + ' ' ); } } // Function to construct and print the longest bitonic subsequence function printLBS ( arr n ) { // LIS[i] stores the length of the longest increasing subsequence ending with arr[i] let LIS = new Array ( n ); for ( let i = 0 ; i < n ; i ++ ) { LIS [ i ] = []; } // initialize LIS[0] to arr[0] LIS [ 0 ]. push ( arr [ 0 ]); // Compute LIS values from left to right for ( let i = 1 ; i < n ; i ++ ) { // for every j less than i for ( let j = 0 ; j < i ; j ++ ) { if ( arr [ j ] < arr [ i ] && LIS [ j ]. length > LIS [ i ]. length ) { LIS [ i ] = LIS [ j ]. slice (); } } LIS [ i ]. push ( arr [ i ]); } // LIS[i] now stores the Maximum Increasing Subsequence of arr[0..i] that ends with arr[i] // LDS[i] stores the length of the longest decreasing subsequence starting with arr[i] let LDS = new Array ( n ); for ( let i = 0 ; i < n ; i ++ ) { LDS [ i ] = []; } // initialize LDS[n-1] to arr[n-1] LDS [ n - 1 ]. push ( arr [ n - 1 ]); // Compute LDS values from right to left for ( let i = n - 2 ; i >= 0 ; i -- ) { // for every j greater than i for ( let j = n - 1 ; j > i ; j -- ) { if ( arr [ j ] < arr [ i ] && LDS [ j ]. length > LDS [ i ]. length ) { LDS [ i ] = LDS [ j ]. slice (); } } LDS [ i ]. push ( arr [ i ]); } // reverse the LDS vector as we're inserting at the end for ( let i = 0 ; i < n ; i ++ ) { LDS [ i ]. reverse (); } // LDS[i] now stores the Maximum Decreasing Subsequence of arr[i..n] that starts with arr[i] let max = 0 ; let maxIndex = - 1 ; for ( let i = 0 ; i < n ; i ++ ) { // Find maximum value of size of LIS[i] + size of LDS[i] - 1 if ( LIS [ i ]. length + LDS [ i ]. length - 1 > max ) { max = LIS [ i ]. length + LDS [ i ]. length - 1 ; maxIndex = i ; } } // print all but // print all but last element of LIS[maxIndex] array _print ( LIS [ maxIndex ]. slice ( 0 - 1 ) LIS [ maxIndex ]. length - 1 ); // print all elements of LDS[maxIndex] array _print ( LDS [ maxIndex ] LDS [ maxIndex ]. length ); } // Driver program const arr = [ 1 11 2 10 4 5 2 1 ]; const n = arr . length ; printLBS ( arr n );
Излаз:
1 11 10 5 2 1
Временска сложеност од горе наведеног решења за динамичко програмирање је О(н 2 ). Помоћни простор који програм користи је О(н 2 ).
