Maximum number of segments of lengths a, b and c
Given a positive integer N, find the maximum number of segments of lengths a, b and c that can be formed from N .
Examples :
Input : N = 7, a = 5, b, = 2, c = 5
Output : 2
N can be divided into 2 segments of lengths
2 and 5. For the second example,
Input : N = 17, a = 2, b = 1, c = 3
Output : 17
N can be divided into 17 segments of 1 or 8
segments of 2 and 1 segment of 1. But 17 segments
of 1 is greater than 9 segments of 2 and 1.
To understand any DP problem clearly, we need to write first of all its recursive code and then go for optimization.
Recursion-Based Solution:
Here for any value of n, we have 3 possibilities, for making the maximum segment count
if (n >= a) we can make 1 segment of length a + another possible segment from the length of n - a
if (n >= b) we can make 1 segment of length b + another possible segment from the length of n - b
if (n >= c) we can make 1 segment of length c + another possible segment from the length of n - c
so now we have to take the maximum possible segment above in 3 condition
Below is an implementation for the same.
C++
#include <bits/stdc++.h>
using namespace std;
int maximumSegments( int n, int a, int b, int c)
{
if (n == 0) {
return 0;
}
int maxa = INT_MIN;
if (n >= a) {
maxa = max(maxa,
1 + maximumSegments(n - a, a, b, c));
}
if (n >= b) {
maxa = max(maxa,
1 + maximumSegments(n - b, a, b, c));
}
if (n >= c) {
maxa = max(maxa,
1 + maximumSegments(n - c, a, b, c));
}
return maxa;
}
int main()
{
int n = 7, a = 5, b = 2, c = 5;
cout << maximumSegments(n, a, b, c);
return 0;
}
|
Java
class GFG {
static int INT_MIN = - 1000000000 ;
static int maximumSegments( int n, int a, int b, int c)
{
if (n == 0 ) {
return 0 ;
}
int maxa = INT_MIN;
if (n >= a) {
maxa = Math.max(maxa, 1 + maximumSegments(n - a, a, b, c));
}
if (n >= b) {
maxa = Math.max(maxa, 1 + maximumSegments(n - b, a, b, c));
}
if (n >= c) {
maxa = Math.max(maxa, 1 + maximumSegments(n - c, a, b, c));
}
return maxa;
}
public static void main(String[] args) {
int n = 7 , a = 5 , b = 2 , c = 5 ;
System.out.println(maximumSegments(n, a, b, c));
}
}
|
Python
def maximumSegments(n, a, b, c):
if n = = 0 :
return 0
maxa = float ( '-inf' )
if n > = a:
maxa = max (maxa,
1 + maximumSegments(n - a, a, b, c))
if n > = b:
maxa = max (maxa,
1 + maximumSegments(n - b, a, b, c))
if n > = c:
maxa = max (maxa,
1 + maximumSegments(n - c, a, b, c))
return maxa
if __name__ = = '__main__' :
n = 7
a = 5
b = 2
c = 5
print (maximumSegments(n, a, b, c))
|
C#
using System;
namespace ConsoleApp {
class Program {
static void Main( string [] args)
{
int n = 7, a = 5, b = 2, c = 5;
Console.WriteLine(maximumSegments(n, a, b, c));
}
static int maximumSegments( int n, int a, int b, int c)
{
if (n == 0) {
return 0;
}
int maxa = int .MinValue;
if (n >= a) {
maxa = Math.Max(
maxa, 1 + maximumSegments(n - a, a, b, c));
}
if (n >= b) {
maxa = Math.Max(
maxa, 1 + maximumSegments(n - b, a, b, c));
}
if (n >= c) {
maxa = Math.Max(
maxa, 1 + maximumSegments(n - c, a, b, c));
}
return maxa;
}
}
}
|
Javascript
function maximumSegments(n, a, b, c) {
if (n === 0) {
return 0;
}
let maxa = Number.MIN_SAFE_INTEGER;
if (n >= a) {
maxa = Math.max(maxa,
1 + maximumSegments(n - a, a, b, c));
}
if (n >= b) {
maxa = Math.max(maxa,
1 + maximumSegments(n - b, a, b, c));
}
if (n >= c) {
maxa = Math.max(maxa,
1 + maximumSegments(n - c, a, b, c));
}
return maxa;
}
let n = 7, a = 5, b = 2, c = 5;
console.log(maximumSegments(n, a, b, c));
|
Time Complexity: O(3n)
Auxiliary Space : O(n)
Optimized Approach : The approach used is Dynamic Programming. The base dp0 will be 0 as initially it has no segments. After that, iterate from 1 to n, and for each of the 3 states i.e, dpi+a, dpi+b and dpi+c, store the maximum value obtained by either using or not using the a, b or c segment.
The 3 states to deal with are :
dpi+a=max(dpi+1, dpi+a);
dpi+b=max(dpi+1, dpi+b);
dpi+c=max(dpi+1, dpi+c);
Below is the implementation of above idea :
C++
#include <bits/stdc++.h>
using namespace std;
int maximumSegments( int n, int a, int b, int c)
{
int dp[n + 1];
memset (dp, -1, sizeof (dp));
dp[0] = 0;
for ( int i = 0; i < n; i++) {
if (dp[i] != -1) {
if (i + a <= n)
dp[i + a] = max(dp[i] + 1, dp[i + a]);
if (i + b <= n)
dp[i + b] = max(dp[i] + 1, dp[i + b]);
if (i + c <= n)
dp[i + c] = max(dp[i] + 1, dp[i + c]);
}
}
return dp[n];
}
int main()
{
int n = 7, a = 5, b = 2, c = 5;
cout << maximumSegments(n, a, b, c);
return 0;
}
|
Java
import java.util.*;
class GFG
{
static int maximumSegments( int n, int a,
int b, int c)
{
int dp[] = new int [n + 10 ];
Arrays.fill(dp, - 1 );
dp[ 0 ] = 0 ;
for ( int i = 0 ; i < n; i++)
{
if (dp[i] != - 1 )
{
if (i + a <= n )
dp[i + a] = Math.max(dp[i] + 1 ,
dp[i + a]);
if (i + b <= n )
dp[i + b] = Math.max(dp[i] + 1 ,
dp[i + b]);
if (i + c <= n )
dp[i + c] = Math.max(dp[i] + 1 ,
dp[i + c]);
}
}
return dp[n];
}
public static void main(String arg[])
{
int n = 7 , a = 5 , b = 2 , c = 5 ;
System.out.print(maximumSegments(n, a, b, c));
}
}
|
Python3
def maximumSegments(n, a, b, c) :
dp = [ - 1 ] * (n + 10 )
dp[ 0 ] = 0
for i in range ( 0 , n) :
if (dp[i] ! = - 1 ) :
if (i + a < = n ):
dp[i + a] = max (dp[i] + 1 ,
dp[i + a])
if (i + b < = n ):
dp[i + b] = max (dp[i] + 1 ,
dp[i + b])
if (i + c < = n ):
dp[i + c] = max (dp[i] + 1 ,
dp[i + c])
return dp[n]
n = 7
a = 5
b = 2
c = 5
print (maximumSegments(n, a,
b, c))
|
C#
using System;
class GFG
{
static int maximumSegments( int n, int a,
int b, int c)
{
int []dp = new int [n + 10];
for ( int i = 0; i < n + 10; i++)
dp[i]= -1;
dp[0] = 0;
for ( int i = 0; i < n; i++)
{
if (dp[i] != -1)
{
if (i + a <= n )
dp[i + a] = Math.Max(dp[i] + 1,
dp[i + a]);
if (i + b <= n )
dp[i + b] = Math.Max(dp[i] + 1,
dp[i + b]);
if (i + c <= n )
dp[i + c] = Math.Max(dp[i] + 1,
dp[i + c]);
}
}
return dp[n];
}
public static void Main()
{
int n = 7, a = 5, b = 2, c = 5;
Console.Write(maximumSegments(n, a, b, c));
}
}
|
PHP
<?php
function maximumSegments( $n , $a ,
$b , $c )
{
$dp = array ();
for ( $i = 0; $i < $n + 10; $i ++)
$dp [ $i ]= -1;
$dp [0] = 0;
for ( $i = 0; $i < $n ; $i ++)
{
if ( $dp [ $i ] != -1)
{
if ( $i + $a <= $n )
$dp [ $i + $a ] = max( $dp [ $i ] + 1,
$dp [ $i + $a ]);
if ( $i + $b <= $n )
$dp [ $i + $b ] = max( $dp [ $i ] + 1,
$dp [ $i + $b ]);
if ( $i + $c <= $n )
$dp [ $i + $c ] = max( $dp [ $i ] + 1,
$dp [ $i + $c ]);
}
}
return $dp [ $n ];
}
$n = 7; $a = 5;
$b = 2; $c = 5;
echo (maximumSegments( $n , $a ,
$b , $c ));
?>
|
Javascript
<script>
function maximumSegments(n, a, b, c)
{
let dp = [];
for (let i = 0; i < n + 10; i++)
dp[i]= -1;
dp[0] = 0;
for (let i = 0; i < n; i++)
{
if (dp[i] != -1)
{
if (i + a <= n )
dp[i + a] = Math.max(dp[i] + 1,
dp[i + a]);
if (i + b <= n )
dp[i + b] = Math.max(dp[i] + 1,
dp[i + b]);
if (i + c <= n )
dp[i + c] = Math.max(dp[i] + 1,
dp[i + c]);
}
}
return dp[n];
}
let n = 7, a = 5, b = 2, c = 5;
document.write(maximumSegments(n, a, b, c));
</script>
|
Time Complexity: O(N), as we are using a loop to traverse N times.
Auxiliary Space: O(N), as we are using extra space for dp array.
Last Updated :
13 Jan, 2023
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