Count number of ways to cover a distance
Given a distance ‘dist, count total number of ways to cover the distance with 1, 2 and 3 steps.
Examples:
Input: n = 3 Output: 4 Explantion: Below are the four ways 1 step + 1 step + 1 step 1 step + 2 step 2 step + 1 step 3 step Input: n = 4 Output: 7 Explantion: Below are the four ways 1 step + 1 step + 1 step + 1 step 1 step + 2 step + 1 step 2 step + 1 step + 1 step 1 step + 1 step + 2 step 2 step + 2 step 3 step + 1 step 1 step + 3 step
Recursive solution
 Approach: There are n stairs, and a person is allowed to next step, skip one position or skip two positions. So there are n positions. The idea is standing at the ith position the person can move by i+1, i+2, i+3 position. So a recursive function can be formed where at current index i the function is recursively called for i+1, i+2 and i+3 positions.
There is another way of forming the recursive function. To reach position i, a person has to jump either from i1, i2 or i3 position where i is the starting position.  Algorithm:
 Create a recursive function (count(int n)) which takes only one parameter.
 Check the base cases. If the value of n is less than 0 then return 0, and if value of n is equal to zero then return 1 as it is the starting position.
 Call the function recursively with values n1, n2 and n3 and sum up the values that are returned, i.e. sum = count(n1) + count(n2) + count(n3).
 Return the value of sum.

Implementation:
C++
// A naive recursive C++ program to count number of ways to cover
// a distance with 1, 2 and 3 steps
#include<iostream>
using
namespace
std;
// Returns count of ways to cover 'dist'
int
printCountRec(
int
dist)
{
// Base cases
if
(dist<0)
return
0;
if
(dist==0)
return
1;
// Recur for all previous 3 and add the results
return
printCountRec(dist1) +
printCountRec(dist2) +
printCountRec(dist3);
}
// driver program
int
main()
{
int
dist = 4;
cout << printCountRec(dist);
return
0;
}
chevron_rightfilter_noneJava
// A naive recursive Java program to count number
// of ways to cover a distance with 1, 2 and 3 steps
import
java.io.*;
class
GFG
{
// Function returns count of ways to cover 'dist'
static
int
printCountRec(
int
dist)
{
// Base cases
if
(dist<
0
)
return
0
;
if
(dist==
0
)
return
1
;
// Recur for all previous 3 and add the results
return
printCountRec(dist
1
) +
printCountRec(dist
2
) +
printCountRec(dist
3
);
}
// driver program
public
static
void
main (String[] args)
{
int
dist =
4
;
System.out.println(printCountRec(dist));
}
}
// This code is contributed by Pramod Kumar
chevron_rightfilter_nonePython3
# A naive recursive Python3 program
# to count number of ways to cover
# a distance with 1, 2 and 3 steps
# Returns count of ways to
# cover 'dist'
def
printCountRec(dist):
# Base cases
if
dist <
0
:
return
0
if
dist
=
=
0
:
return
1
# Recur for all previous 3 and
# add the results
return
(printCountRec(dist

1
)
+
printCountRec(dist

2
)
+
printCountRec(dist

3
))
# Driver code
dist
=
4
print
(printCountRec(dist))
# This code is contributed by Anant Agarwal.
chevron_rightfilter_noneC#
// A naive recursive C# program to
// count number of ways to cover a
// distance with 1, 2 and 3 steps
using
System;
class
GFG {
// Function returns count of
// ways to cover 'dist'
static
int
printCountRec(
int
dist)
{
// Base cases
if
(dist < 0)
return
0;
if
(dist == 0)
return
1;
// Recur for all previous 3
// and add the results
return
printCountRec(dist  1) +
printCountRec(dist  2) +
printCountRec(dist  3);
}
// Driver Code
public
static
void
Main ()
{
int
dist = 4;
Console.WriteLine(printCountRec(dist));
}
}
// This code is contributed by Sam007.
chevron_rightfilter_nonePHP
<?php
// A naive recursive PHP program to
// count number of ways to cover
// a distance with 1, 2 and 3 steps
// Returns count of ways to cover 'dist'
function
printCountRec(
$dist
)
{
// Base cases
if
(
$dist
<0)
return
0;
if
(
$dist
==0)
return
1;
// Recur for all previous 3
// and add the results
return
printCountRec(
$dist
 1) +
printCountRec(
$dist
 2) +
printCountRec(
$dist
 3);
}
// Driver Code
$dist
= 4;
echo
printCountRec(
$dist
);
// This code is contributed by anuj_67.
?>
chevron_rightfilter_none
Output:7
 Complexity Analysis:
 Time Compelxity: O(3^{n}).
The time complexity of the above solution is exponential, a close upper bound is O(3^{n}). From each state 3, a recursive function is called. So the upper bound for n states is O(3^{n}).  Space complexity: O(1).
No extra space is required.
 Time Compelxity: O(3^{n}).
Efficient solution
 Approach: The idea is similar, but it can be observed that there are n states but the recursive function is called 3 ^ n times. That means that some states are called repeatedly. So the idea is to store the value of states. This can be done in two ways.
 The first way is to keep the recursive structure intact and just store the value in a HashMap and whenever the function is called, return the value store without computing (TopDown Approach).
 The second way is to take an extra space of size n and start computing values of states from 1, 2 .. to n, i.e. compute values of i, i+1, i+2 and then use them to calculate the value of i+3 (BottomUp Approach).
To know more about dynamic programming follow these articles:
 Algorithm:
 Create an array of size n + 1 and initilize the first 3 variables with 1, 1, 2. The base cases.
 Run a loop from 3 to n.
 For each index i, computer value of ith position as dp[i] = dp[i1] + dp[i2] + dp[i3].
 Print the value of dp[n], as the Count of number of ways to cover a distance.

Implementation:
C++
// A Dynamic Programming based C++ program to count number of ways
// to cover a distance with 1, 2 and 3 steps
#include<iostream>
using
namespace
std;
int
printCountDP(
int
dist)
{
int
count[dist+1];
// Initialize base values. There is one way to cover 0 and 1
// distances and two ways to cover 2 distance
count[0] = 1, count[1] = 1, count[2] = 2;
// Fill the count array in bottom up manner
for
(
int
i=3; i<=dist; i++)
count[i] = count[i1] + count[i2] + count[i3];
return
count[dist];
}
// driver program
int
main()
{
int
dist = 4;
cout << printCountDP(dist);
return
0;
}
chevron_rightfilter_noneJava
// A Dynamic Programming based Java program
// to count number of ways to cover a distance
// with 1, 2 and 3 steps
import
java.io.*;
class
GFG
{
// Function returns count of ways to cover 'dist'
static
int
printCountDP(
int
dist)
{
int
[] count =
new
int
[dist+
1
];
// Initialize base values. There is one way to
// cover 0 and 1 distances and two ways to
// cover 2 distance
count[
0
] =
1
;
count[
1
] =
1
;
count[
2
] =
2
;
// Fill the count array in bottom up manner
for
(
int
i=
3
; i<=dist; i++)
count[i] = count[i
1
] + count[i
2
] + count[i
3
];
return
count[dist];
}
// driver program
public
static
void
main (String[] args)
{
int
dist =
4
;
System.out.println(printCountDP(dist));
}
}
// This code is contributed by Pramod Kumar
chevron_rightfilter_nonePython3
# A Dynamic Programming based on Python3
# program to count number of ways to
# cover a distance with 1, 2 and 3 steps
def
printCountDP(dist):
count
=
[
0
]
*
(dist
+
1
)
# Initialize base values. There is
# one way to cover 0 and 1 distances
# and two ways to cover 2 distance
count[
0
]
=
1
count[
1
]
=
1
count[
2
]
=
2
# Fill the count array in bottom
# up manner
for
i
in
range
(
3
, dist
+
1
):
count[i]
=
(count[i

1
]
+
count[i

2
]
+
count[i

3
])
return
count[dist];
# driver program
dist
=
4
;
print
( printCountDP(dist))
# This code is contributed by Sam007.
chevron_rightfilter_noneC#
// A Dynamic Programming based C# program
// to count number of ways to cover a distance
// with 1, 2 and 3 steps
using
System;
class
GFG {
// Function returns count of ways
// to cover 'dist'
static
int
printCountDP(
int
dist)
{
int
[] count =
new
int
[dist + 1];
// Initialize base values. There is one
// way to cover 0 and 1 distances
// and two ways to cover 2 distance
count[0] = 1;
count[1] = 1;
count[2] = 2;
// Fill the count array
// in bottom up manner
for
(
int
i = 3; i <= dist; i++)
count[i] = count[i  1] +
count[i  2] +
count[i  3];
return
count[dist];
}
// Driver Code
public
static
void
Main ()
{
int
dist = 4;
Console.WriteLine(printCountDP(dist));
}
}
// This code is contributed by Sam007.
chevron_rightfilter_nonePHP
<?php
// A Dynamic Programming based PHP program
// to count number of ways to cover a
// distance with 1, 2 and 3 steps
function
printCountDP(
$dist
)
{
$count
=
array
();
// Initialize base values. There is
// one way to cover 0 and 1 distances
// and two ways to cover 2 distance
$count
[0] = 1;
$count
[1] = 1;
$count
[2] = 2;
// Fill the count array
// in bottom up manner
for
(
$i
= 3;
$i
<=
$dist
;
$i
++)
$count
[
$i
] =
$count
[
$i
 1] +
$count
[
$i
 2] +
$count
[
$i
 3];
return
$count
[
$dist
];
}
// Driver Code
$dist
= 4;
echo
printCountDP(
$dist
);
// This code is contributed by anuj_67.
?>
chevron_rightfilter_none
Output :7

Complexity Analysis:
 Time Compelxity: O(n).
Only one traversal of the array is needed. So Time Complexity is O(n)  Space complexity: O(n).
To store the values in a DP O(n) extra space is needed.
 Time Compelxity: O(n).
This article is contributed by Vignesh Venkatesan. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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