מצא רצף נחש אורך מקסימאלי
בהינתן רשת מספרים מוצאים רצף נחש אורך מקסימאלי ומדפיס אותו. אם קיימים רצפי נחש מרובים כאשר האורך המרבי מדפיס מישהו מהם.
רצף נחש מורכב ממספרים סמוכים ברשת כך שלכל מספר המספר מימין או למספר שמתחת לו הוא +1 או -1 ערכו. לדוגמה, אם אתה נמצא במיקום (x y) ברשת אתה יכול לנוע ימינה, כלומר (x y+1) אם המספר הזה הוא ± 1 או לנוע למטה, כלומר (x+1 y) אם המספר הזה הוא ± 1.
For example 9 6 5 2 8 7 6 5 7 3 1 6 1 1 1 7 In above grid the longest snake sequence is: (9 8 7 6 5 6 7)
להלן איור מציג את כל הנתיבים האפשריים:
אנו ממליצים בחום למזער את הדפדפן שלך ולנסות זאת בעצמך קודם.
הרעיון הוא להשתמש בתכנות דינאמית. עבור כל תא של המטריצה אנו שומרים על אורך מקסימלי של נחש שמסתיים בתא הנוכחי. לרצף הנחש האורך המרבי יהיה ערך מקסימלי. תא הערך המרבי יתאים לזנב הנחש. על מנת להדפיס את הנחש עלינו למסלול אחורי מהזנב כל הדרך חזרה לראשו של הנחש.
Let T[i][i] represent maximum length of a snake which ends at cell (i j) then for given matrix M the DP relation is defined as T[0][0] = 0 T[i][j] = max(T[i][j] T[i][j - 1] + 1) if M[i][j] = M[i][j - 1] ± 1 T[i][j] = max(T[i][j] T[i - 1][j] + 1) if M[i][j] = M[i - 1][j] ± 1
להלן יישום הרעיון
C++ // C++ program to find maximum length // Snake sequence and print it #include using namespace std ; #define M 4 #define N 4 struct Point { int x y ; }; // Function to find maximum length Snake sequence path // (i j) corresponds to tail of the snake list < Point > findPath ( int grid [ M ][ N ] int mat [ M ][ N ] int i int j ) { list < Point > path ; Point pt = { i j }; path . push_front ( pt ); while ( grid [ i ][ j ] != 0 ) { if ( i > 0 && grid [ i ][ j ] - 1 == grid [ i - 1 ][ j ]) { pt = { i - 1 j }; path . push_front ( pt ); i -- ; } else if ( j > 0 && grid [ i ][ j ] - 1 == grid [ i ][ j - 1 ]) { pt = { i j - 1 }; path . push_front ( pt ); j -- ; } } return path ; } // Function to find maximum length Snake sequence void findSnakeSequence ( int mat [ M ][ N ]) { // table to store results of subproblems int lookup [ M ][ N ]; // initialize by 0 memset ( lookup 0 sizeof lookup ); // stores maximum length of Snake sequence int max_len = 0 ; // store coordinates to snake's tail int max_row = 0 ; int max_col = 0 ; // fill the table in bottom-up fashion for ( int i = 0 ; i < M ; i ++ ) { for ( int j = 0 ; j < N ; j ++ ) { // do except for (0 0) cell if ( i || j ) { // look above if ( i > 0 && abs ( mat [ i - 1 ][ j ] - mat [ i ][ j ]) == 1 ) { lookup [ i ][ j ] = max ( lookup [ i ][ j ] lookup [ i - 1 ][ j ] + 1 ); if ( max_len < lookup [ i ][ j ]) { max_len = lookup [ i ][ j ]; max_row = i max_col = j ; } } // look left if ( j > 0 && abs ( mat [ i ][ j - 1 ] - mat [ i ][ j ]) == 1 ) { lookup [ i ][ j ] = max ( lookup [ i ][ j ] lookup [ i ][ j - 1 ] + 1 ); if ( max_len < lookup [ i ][ j ]) { max_len = lookup [ i ][ j ]; max_row = i max_col = j ; } } } } } cout < < 'Maximum length of Snake sequence is: ' < < max_len < < endl ; // find maximum length Snake sequence path list < Point > path = findPath ( lookup mat max_row max_col ); cout < < 'Snake sequence is:' ; for ( auto it = path . begin (); it != path . end (); it ++ ) cout < < endl < < mat [ it -> x ][ it -> y ] < < ' (' < < it -> x < < ' ' < < it -> y < < ')' ; } // Driver code int main () { int mat [ M ][ N ] = { { 9 6 5 2 } { 8 7 6 5 } { 7 3 1 6 } { 1 1 1 7 } }; findSnakeSequence ( mat ); return 0 ; }
Java // Java program to find maximum length // Snake sequence and print it import java.util.* ; class GFG { static int M = 4 ; static int N = 4 ; static class Point { int x y ; public Point ( int x int y ) { this . x = x ; this . y = y ; } }; // Function to find maximum length Snake sequence path // (i j) corresponds to tail of the snake static List < Point > findPath ( int grid [][] int mat [][] int i int j ) { List < Point > path = new LinkedList <> (); Point pt = new Point ( i j ); path . add ( 0 pt ); while ( grid [ i ][ j ] != 0 ) { if ( i > 0 && grid [ i ][ j ] - 1 == grid [ i - 1 ][ j ] ) { pt = new Point ( i - 1 j ); path . add ( 0 pt ); i -- ; } else if ( j > 0 && grid [ i ][ j ] - 1 == grid [ i ][ j - 1 ] ) { pt = new Point ( i j - 1 ); path . add ( 0 pt ); j -- ; } } return path ; } // Function to find maximum length Snake sequence static void findSnakeSequence ( int mat [][] ) { // table to store results of subproblems int [][] lookup = new int [ M ][ N ] ; // initialize by 0 // stores maximum length of Snake sequence int max_len = 0 ; // store coordinates to snake's tail int max_row = 0 ; int max_col = 0 ; // fill the table in bottom-up fashion for ( int i = 0 ; i < M ; i ++ ) { for ( int j = 0 ; j < N ; j ++ ) { // do except for (0 0) cell if ( i != 0 || j != 0 ) { // look above if ( i > 0 && Math . abs ( mat [ i - 1 ][ j ] - mat [ i ][ j ] ) == 1 ) { lookup [ i ][ j ] = Math . max ( lookup [ i ][ j ] lookup [ i - 1 ][ j ] + 1 ); if ( max_len < lookup [ i ][ j ] ) { max_len = lookup [ i ][ j ] ; max_row = i ; max_col = j ; } } // look left if ( j > 0 && Math . abs ( mat [ i ][ j - 1 ] - mat [ i ][ j ] ) == 1 ) { lookup [ i ][ j ] = Math . max ( lookup [ i ][ j ] lookup [ i ][ j - 1 ] + 1 ); if ( max_len < lookup [ i ][ j ] ) { max_len = lookup [ i ][ j ] ; max_row = i ; max_col = j ; } } } } } System . out . print ( 'Maximum length of Snake ' + 'sequence is: ' + max_len + 'n' ); // find maximum length Snake sequence path List < Point > path = findPath ( lookup mat max_row max_col ); System . out . print ( 'Snake sequence is:' ); for ( Point it : path ) System . out . print ( 'n' + mat [ it . x ][ it . y ] + ' (' + it . x + ' ' + it . y + ')' ); } // Driver code public static void main ( String [] args ) { int mat [][] = {{ 9 6 5 2 } { 8 7 6 5 } { 7 3 1 6 } { 1 1 1 7 }}; findSnakeSequence ( mat ); } } // This code is contributed by 29AjayKumar
C# // C# program to find maximum length // Snake sequence and print it using System ; using System.Collections.Generic ; class GFG { static int M = 4 ; static int N = 4 ; public class Point { public int x y ; public Point ( int x int y ) { this . x = x ; this . y = y ; } }; // Function to find maximum length Snake sequence path // (i j) corresponds to tail of the snake static List < Point > findPath ( int [ ] grid int [ ] mat int i int j ) { List < Point > path = new List < Point > (); Point pt = new Point ( i j ); path . Insert ( 0 pt ); while ( grid [ i j ] != 0 ) { if ( i > 0 && grid [ i j ] - 1 == grid [ i - 1 j ]) { pt = new Point ( i - 1 j ); path . Insert ( 0 pt ); i -- ; } else if ( j > 0 && grid [ i j ] - 1 == grid [ i j - 1 ]) { pt = new Point ( i j - 1 ); path . Insert ( 0 pt ); j -- ; } } return path ; } // Function to find maximum length Snake sequence static void findSnakeSequence ( int [ ] mat ) { // table to store results of subproblems int [ ] lookup = new int [ M N ]; // initialize by 0 // stores maximum length of Snake sequence int max_len = 0 ; // store coordinates to snake's tail int max_row = 0 ; int max_col = 0 ; // fill the table in bottom-up fashion for ( int i = 0 ; i < M ; i ++ ) { for ( int j = 0 ; j < N ; j ++ ) { // do except for (0 0) cell if ( i != 0 || j != 0 ) { // look above if ( i > 0 && Math . Abs ( mat [ i - 1 j ] - mat [ i j ]) == 1 ) { lookup [ i j ] = Math . Max ( lookup [ i j ] lookup [ i - 1 j ] + 1 ); if ( max_len < lookup [ i j ]) { max_len = lookup [ i j ]; max_row = i ; max_col = j ; } } // look left if ( j > 0 && Math . Abs ( mat [ i j - 1 ] - mat [ i j ]) == 1 ) { lookup [ i j ] = Math . Max ( lookup [ i j ] lookup [ i j - 1 ] + 1 ); if ( max_len < lookup [ i j ]) { max_len = lookup [ i j ]; max_row = i ; max_col = j ; } } } } } Console . Write ( 'Maximum length of Snake ' + 'sequence is: ' + max_len + 'n' ); // find maximum length Snake sequence path List < Point > path = findPath ( lookup mat max_row max_col ); Console . Write ( 'Snake sequence is:' ); foreach ( Point it in path ) Console . Write ( 'n' + mat [ it . x it . y ] + ' (' + it . x + ' ' + it . y + ')' ); } // Driver code public static void Main ( String [] args ) { int [ ] mat = { { 9 6 5 2 } { 8 7 6 5 } { 7 3 1 6 } { 1 1 1 7 } }; findSnakeSequence ( mat ); } } // This code is contributed by Princi Singh
Python3 def snakesequence ( S m n ): sequence = {} DP = [[ 1 for x in range ( m + 1 )] for x in range ( n + 1 )] a b maximum = 0 0 0 position = [ 0 0 ] for i in range ( 0 n + 1 ): for j in range ( 0 m + 1 ): a b = 0 0 p = 'initial' if ( i > 0 and abs ( S [ i ][ j ] - S [ i - 1 ][ j ]) == 1 ): a = DP [ i - 1 ][ j ] if ( j > 0 and abs ( S [ i ][ j ] - S [ i ][ j - 1 ]) == 1 ): b = DP [ i ][ j - 1 ] if a != 0 and a >= b : p = str ( i - 1 ) + ' ' + str ( j ) elif b != 0 : p = str ( i ) + ' ' + str ( j - 1 ) q = str ( i ) + ' ' + str ( j ) sequence [ q ] = p DP [ i ][ j ] = DP [ i ][ j ] + max ( a b ) if DP [ i ][ j ] >= maximum : maximum = DP [ i ][ j ] position [ 0 ] = i position [ 1 ] = j snakeValues = [] snakePositions = [] snakeValues . append ( S [ position [ 0 ]][ position [ 1 ]]) check = 'found' str_next = str ( position [ 0 ]) + ' ' + str ( position [ 1 ]) findingIndices = sequence [ str_next ] . split () while ( check == 'found' ): if sequence [ str_next ] == 'initial' : snakePositions . insert ( 0 str_next ) check = 'end' continue findingIndices = sequence [ str_next ] . split () g = int ( findingIndices [ 0 ]) h = int ( findingIndices [ 1 ]) snakeValues . insert ( 0 S [ g ][ h ]) snake_position = str ( g ) + ' ' + str ( h ) snakePositions . insert ( 0 str_next ) str_next = sequence [ str_next ] return [ snakeValues snakePositions ] S = [[ 9 6 5 2 ] [ 8 7 6 5 ] [ 7 3 1 6 ] [ 1 1 10 7 ]] m = 3 n = 3 seq = snakesequence ( S m n ) for i in range ( len ( seq [ 0 ])): print ( seq [ 0 ][ i ] '' seq [ 1 ][ i ] . split ())
JavaScript function snakesequence ( S m n ) { let sequence = {} let DP = new Array ( n + 1 ) for ( var i = 0 ; i <= n ; i ++ ) DP [ i ] = new Array ( m + 1 ). fill ( 1 ) let a = 0 b = 0 maximum = 0 let position = [ 0 0 ] for ( var i = 0 ; i <= n ; i ++ ) { for ( var j = 0 ; j <= m ; j ++ ) { a = 0 b = 0 let p = 'initial' if ( i > 0 && Math . abs ( S [ i ][ j ] - S [ i - 1 ][ j ]) == 1 ) a = DP [ i - 1 ][ j ] if ( j > 0 && Math . abs ( S [ i ][ j ] - S [ i ][ j - 1 ]) == 1 ) b = DP [ i ][ j - 1 ] if ( a != 0 && a >= b ) p = String ( i - 1 ) + ' ' + String ( j ) else if ( b != 0 ) p = String ( i ) + ' ' + String ( j - 1 ) let q = String ( i ) + ' ' + String ( j ) sequence [ q ] = p DP [ i ][ j ] = DP [ i ][ j ] + Math . max ( a b ) if ( DP [ i ][ j ] >= maximum ) { maximum = DP [ i ][ j ] position [ 0 ] = i position [ 1 ] = j } } } let snakeValues = [] let snakePositions = [] snakeValues . push ( S [ position [ 0 ]][ position [ 1 ]]) let check = 'found' let String_next = String ( position [ 0 ]) + ' ' + String ( position [ 1 ]) let findingIndices = sequence [ String_next ]. split ( ' ' ) while ( check == 'found' ) { if ( sequence [ String_next ] == 'initial' ) { snakePositions . unshift ( String_next ) check = 'end' continue } findingIndices = sequence [ String_next ]. split ( ' ' ) let g = parseInt ( findingIndices [ 0 ]) let h = parseInt ( findingIndices [ 1 ]) snakeValues . unshift ( S [ g ][ h ]) let snake_position = String ( g ) + ' ' + String ( h ) snakePositions . unshift ( String_next ) String_next = sequence [ String_next ] } return [ snakeValues snakePositions ] } // Driver Code let S = [[ 9 6 5 2 ] [ 8 7 6 5 ] [ 7 3 1 6 ] [ 1 1 10 7 ]] let m = 3 let n = 3 let seq = snakesequence ( S m n ) for ( var i = 0 ; i < seq [ 0 ]. length ; i ++ ) console . log ( seq [ 0 ][ i ] + '' seq [ 1 ][ i ]. split ( ' ' ))
תְפוּקָה
Maximum length of Snake sequence is: 6 Snake sequence is: 9 (0 0) 8 (1 0) 7 (1 1) 6 (1 2) 5 (1 3) 6 (2 3) 7 (3 3)
מורכבות הזמן של הפתרון לעיל היא O (M*n). שטח עזר המשמש את הפתרון לעיל הוא O (M*n). אם איננו נדרשים להדפיס את שטח הנחש ניתן להפחית עוד יותר ל- O (n) מכיוון שאנו משתמשים רק בתוצאה מהשורה האחרונה.
