Surinkite visas monetas minimaliu žingsnių skaičiumi
Atsižvelgiant į daugybę monetų šūsnių, kurios yra išdėstytos greta. Visas šias monetas turime surinkti minimaliu žingsnių skaičiumi, kad vienu žingsniu galėtume surinkti vieną horizontalią monetų eilutę arba vertikalią monetų eilutę, o renkamos monetos turėtų būti ištisinės.
Pavyzdžiai:
Input : height[] = [2 1 2 5 1] Each value of this array corresponds to the height of stack that is we are given five stack of coins where in first stack 2 coins are there then in second stack 1 coin is there and so on. Output : 4 We can collect all above coins in 4 steps which are shown in below diagram. Each step is shown by different color. First we have collected last horizontal line of coins after which stacks remains as [1 0 1 4 0] after that another horizontal line of coins is collected from stack 3 and 4 then a vertical line from stack 4 and at the end a horizontal line from stack 1. Total steps are 4.
Šią problemą galime išspręsti naudodami „skaldyk ir valdyk“ metodą. Matome, kad visada pravartu pašalinti horizontalias linijas iš apačios. Tarkime, kad mes dirbame su krūvomis nuo l indekso iki r indekso rekursijos žingsniu kiekvieną kartą, kai pasirenkame minimalų aukštį, pašaliname daugybę horizontalių linijų, po kurių krūva bus padalinta į dvi dalis nuo l iki minimumo ir minimumo +1 iki r, ir mes iškviesime rekursyviai tose pogrupiuose. Kitas dalykas, monetas galime rinkti ir naudodami vertikalias linijas, todėl rinksimės minimumą tarp rekursinių skambučių rezultato ir (r - l), nes naudodami (r - l) vertikalias linijas visada galime surinkti visas monetas.
Kadangi kiekvieną kartą iškviečiame kiekvieną pogrupį ir randame minimalų viso sprendimo sudėtingumo laiką bus O(N 2 )
// C++ program to find minimum number of // steps to collect stack of coins #include using namespace std ; // recursive method to collect coins from // height array l to r with height h already // collected int minStepsRecur ( int height [] int l int r int h ) { // if l is more than r no steps needed if ( l >= r ) return 0 ; // loop over heights to get minimum height // index int m = l ; for ( int i = l ; i < r ; i ++ ) if ( height [ i ] < height [ m ]) m = i ; /* choose minimum from 1) collecting coins using all vertical lines (total r - l) 2) collecting coins using lower horizontal lines and recursively on left and right segments */ return min ( r - l minStepsRecur ( height l m height [ m ]) + minStepsRecur ( height m + 1 r height [ m ]) + height [ m ] - h ); } // method returns minimum number of step to // collect coin from stack with height in // height[] array int minSteps ( int height [] int N ) { return minStepsRecur ( height 0 N 0 ); } // Driver code to test above methods int main () { int height [] = { 2 1 2 5 1 }; int N = sizeof ( height ) / sizeof ( int ); cout < < minSteps ( height N ) < < endl ; return 0 ; }
Java // Java Code to Collect all coins in // minimum number of steps import java.util.* ; class GFG { // recursive method to collect coins from // height array l to r with height h already // collected public static int minStepsRecur ( int height [] int l int r int h ) { // if l is more than r no steps needed if ( l >= r ) return 0 ; // loop over heights to get minimum height // index int m = l ; for ( int i = l ; i < r ; i ++ ) if ( height [ i ] < height [ m ] ) m = i ; /* choose minimum from 1) collecting coins using all vertical lines (total r - l) 2) collecting coins using lower horizontal lines and recursively on left and right segments */ return Math . min ( r - l minStepsRecur ( height l m height [ m ] ) + minStepsRecur ( height m + 1 r height [ m ] ) + height [ m ] - h ); } // method returns minimum number of step to // collect coin from stack with height in // height[] array public static int minSteps ( int height [] int N ) { return minStepsRecur ( height 0 N 0 ); } /* Driver program to test above function */ public static void main ( String [] args ) { int height [] = { 2 1 2 5 1 }; int N = height . length ; System . out . println ( minSteps ( height N )); } } // This code is contributed by Arnav Kr. Mandal.
Python 3 # Python 3 program to find # minimum number of steps # to collect stack of coins # recursive method to collect # coins from height array l to # r with height h already # collected def minStepsRecur ( height l r h ): # if l is more than r # no steps needed if l >= r : return 0 ; # loop over heights to # get minimum height index m = l for i in range ( l r ): if height [ i ] < height [ m ]: m = i # choose minimum from # 1) collecting coins using # all vertical lines (total r - l) # 2) collecting coins using # lower horizontal lines and # recursively on left and # right segments return min ( r - l minStepsRecur ( height l m height [ m ]) + minStepsRecur ( height m + 1 r height [ m ]) + height [ m ] - h ) # method returns minimum number # of step to collect coin from # stack with height in height[] array def minSteps ( height N ): return minStepsRecur ( height 0 N 0 ) # Driver code height = [ 2 1 2 5 1 ] N = len ( height ) print ( minSteps ( height N )) # This code is contributed # by ChitraNayal
C# // C# Code to Collect all coins in // minimum number of steps using System ; class GFG { // recursive method to collect coins from // height array l to r with height h already // collected public static int minStepsRecur ( int [] height int l int r int h ) { // if l is more than r no steps needed if ( l >= r ) return 0 ; // loop over heights to // get minimum height index int m = l ; for ( int i = l ; i < r ; i ++ ) if ( height [ i ] < height [ m ]) m = i ; /* choose minimum from 1) collecting coins using all vertical lines (total r - l) 2) collecting coins using lower horizontal lines and recursively on left and right segments */ return Math . Min ( r - l minStepsRecur ( height l m height [ m ]) + minStepsRecur ( height m + 1 r height [ m ]) + height [ m ] - h ); } // method returns minimum number of step to // collect coin from stack with height in // height[] array public static int minSteps ( int [] height int N ) { return minStepsRecur ( height 0 N 0 ); } /* Driver program to test above function */ public static void Main () { int [] height = { 2 1 2 5 1 }; int N = height . Length ; Console . Write ( minSteps ( height N )); } } // This code is contributed by nitin mittal
PHP // PHP program to find minimum number of // steps to collect stack of coins // recursive method to collect // coins from height array l to // r with height h already // collected function minStepsRecur ( $height $l $r $h ) { // if l is more than r // no steps needed if ( $l >= $r ) return 0 ; // loop over heights to // get minimum height // index $m = $l ; for ( $i = $l ; $i < $r ; $i ++ ) if ( $height [ $i ] < $height [ $m ]) $m = $i ; /* choose minimum from 1) collecting coins using all vertical lines (total r - l) 2) collecting coins using lower horizontal lines and recursively on left and right segments */ return min ( $r - $l minStepsRecur ( $height $l $m $height [ $m ]) + minStepsRecur ( $height $m + 1 $r $height [ $m ]) + $height [ $m ] - $h ); } // method returns minimum number of step to // collect coin from stack with height in // height[] array function minSteps ( $height $N ) { return minStepsRecur ( $height 0 $N 0 ); } // Driver Code $height = array ( 2 1 2 5 1 ); $N = sizeof ( $height ); echo minSteps ( $height $N ) ; // This code is contributed by nitin mittal. ?>
JavaScript < script > // Javascript Code to Collect all coins in // minimum number of steps // recursive method to collect coins from // height array l to r with height h already // collected function minStepsRecur ( height l r h ) { // if l is more than r no steps needed if ( l >= r ) return 0 ; // loop over heights to get minimum height // index let m = l ; for ( let i = l ; i < r ; i ++ ) if ( height [ i ] < height [ m ]) m = i ; /* choose minimum from 1) collecting coins using all vertical lines (total r - l) 2) collecting coins using lower horizontal lines and recursively on left and right segments */ return Math . min ( r - l minStepsRecur ( height l m height [ m ]) + minStepsRecur ( height m + 1 r height [ m ]) + height [ m ] - h ); } // method returns minimum number of step to // collect coin from stack with height in // height[] array function minSteps ( height N ) { return minStepsRecur ( height 0 N 0 ); } /* Driver program to test above function */ let height = [ 2 1 2 5 1 ]; let N = height . length ; document . write ( minSteps ( height N )); // This code is contributed by avanitrachhadiya2155 < /script>
Išvestis:
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Laiko sudėtingumas: Šio algoritmo sudėtingumas laike yra O(N^2), kur N yra elementų skaičius aukščio masyve.
Erdvės sudėtingumas: Šio algoritmo erdvės sudėtingumas yra O(N) dėl rekursinių iškvietimų, atliekamų aukščio masyve.
