Maximum sum alternating subsequence
Given an array, the task is to find sum of maximum sum alternating subsequence starting with first element. Here alternating sequence means first decreasing, then increasing, then decreasing, … For example 10, 5, 14, 3 is an alternating sequence.
Note that the reverse type of sequence (increasing – decreasing – increasing -…) is not considered alternating here.
Examples:
Input : arr[] = {4, 3, 8, 5, 3, 8}
Output : 28
Explanation:
The alternating subsequence (starting with first element)
that has above maximum sum is {4, 3, 8, 5, 8}
Input : arr[] = {4, 8, 2, 5, 6, 8}
Output : 14
The alternating subsequence (starting with first element)
that has above maximum sum is {4, 2, 8}
This problem is similar to Longest Increasing Subsequence (LIS) problem. and can be solved using Dynamic Programming.
Create two empty array that store result of maximum
sum of alternate sub-sequence
inc[] : inc[i] stores results of maximum sum alternating
subsequence ending with arr[i] such that arr[i]
is greater than previous element of the subsequence
dec[] : dec[i] stores results of maximum sum alternating
subsequence ending with arr[i] such that arr[i]
is less than previous element of the subsequence
Include first element of 'arr' in both inc[] and dec[]
inc[0] = dec[0] = arr[0]
// Maintain a flag i.e. it will makes the greater
// elements count only if the first decreasing element
// is counted.
flag = 0
Traversal two loops
i goes from 1 to n-1
j goes 0 to i-1
IF arr[j] > arr[i]
dec[i] = max(dec[i], inc[j] + arr[i])
// Denotes first decreasing is found
flag = 1
ELSE IF arr[j] < arr[i] && flag == 1
inc[i] = max(inc[i], dec[j]+arr[i]);
Final Last Find maximum value inc[] and dec[] .
Below is implementation of above idea.
C/C++
// C++ program to find sum of maximum // sum alternating sequence starting with // first element. #include<bits/stdc++.h> using namespace std; // Return sum of maximum sum alternating // sequence starting with arr[0] and is first // decreasing. int maxAlternateSum(int arr[], int n) { if (n == 1) return arr[0]; // create two empty array that store result of // maximum sum of alternate sub-sequence // stores sum of decreasing and increasing // sub-sequence int dec[n]; memset(dec, 0, sizeof(dec)); // store sum of increasing and decreasing sun-sequence int inc[n]; memset(inc, 0, sizeof(inc)); // As per question, first element must be part // of solution. dec[0] = inc[0] = arr[0]; int flag = 0 ; // Traverse remaining elements of array for (int i=1; i<n; i++) { for (int j=0; j<i; j++) { // IF current sub-sequence is decreasing the // update dec[j] if needed. dec[i] by current // inc[j] + arr[i] if (arr[j] > arr[i]) { dec[i] = max(dec[i], inc[j]+arr[i]); // Revert the flag , if first decreasing // is found flag = 1; } // If next element is greater but flag should be 1 // i.e. this element should be counted after the // first decreasing element gets counted else if (arr[j] < arr[i] && flag == 1) // If current sub-sequence is increasing // then update inc[i] inc[i] = max(inc[i], dec[j]+arr[i]); } } // find maximum sum in b/w inc[] and dec[] int result = INT_MIN; for (int i = 0 ; i < n; i++) { if (result < inc[i]) result = inc[i]; if (result < dec[i]) result = dec[i]; } // return maximum sum alternate sun-sequence return result; } //Driver program int main() { int arr[]= {8, 2, 3, 5, 7, 9, 10}; int n = sizeof(arr)/sizeof(arr[0]); cout << "Maximum sum = " << maxAlternateSum(arr , n ) << endl; return 0; } |
Java
// Java program to find sum of maximum // sum alternating sequence starting with // first element. public class GFG { // Return sum of maximum sum alternating // sequence starting with arr[0] and is first // decreasing. static int maxAlternateSum(int arr[], int n) { if (n == 1) return arr[0]; // create two empty array that store result of // maximum sum of alternate sub-sequence // stores sum of decreasing and increasing // sub-sequence int dec[] = new int[n]; // store sum of increasing and decreasing sun-sequence int inc[] = new int[n]; // As per question, first element must be part // of solution. dec[0] = inc[0] = arr[0]; int flag = 0 ; // Traverse remaining elements of array for (int i=1; i<n; i++) { for (int j=0; j<i; j++) { // IF current sub-sequence is decreasing the // update dec[j] if needed. dec[i] by current // inc[j] + arr[i] if (arr[j] > arr[i]) { dec[i] = Math.max(dec[i], inc[j]+arr[i]); // Revert the flag , if first decreasing // is found flag = 1; } // If next element is greater but flag should be 1 // i.e. this element should be counted after the // first decreasing element gets counted else if (arr[j] < arr[i] && flag == 1) // If current sub-sequence is increasing // then update inc[i] inc[i] = Math.max(inc[i], dec[j]+arr[i]); } } // find maximum sum in b/w inc[] and dec[] int result = Integer.MIN_VALUE; for (int i = 0 ; i < n; i++) { if (result < inc[i]) result = inc[i]; if (result < dec[i]) result = dec[i]; } // return maximum sum alternate sun-sequence return result; } // Driver Method public static void main(String[] args) { int arr[]= {8, 2, 3, 5, 7, 9, 10}; System.out.println("Maximum sum = " + maxAlternateSum(arr , arr.length)); } } |
Python3
# Python3 program to find sum of maximum # sum alternating sequence starting with # first element. # Return sum of maximum sum alternating # sequence starting with arr[0] and is # first decreasing. def maxAlternateSum(arr, n): if (n == 1): return arr[0] # Create two empty array that # store result of maximum sum # of alternate sub-sequence # Stores sum of decreasing and # increasing sub-sequence dec = [0 for i in range(n + 1)] # store sum of increasing and # decreasing sun-sequence inc = [0 for i in range(n + 1)] # As per question, first element # must be part of solution. dec[0] = inc[0] = arr[0] flag = 0 # Traverse remaining elements of array for i in range(1, n): for j in range(i): # IF current sub-sequence is decreasing the # update dec[j] if needed. dec[i] by current # inc[j] + arr[i] if (arr[j] > arr[i]): dec[i] = max(dec[i], inc[j] + arr[i]) # Revert the flag, if first # decreasing is found flag = 1 # If next element is greater but flag should be 1 # i.e. this element should be counted after the # first decreasing element gets counted elif (arr[j] < arr[i] and flag == 1): # If current sub-sequence is # increasing then update inc[i] inc[i] = max(inc[i], dec[j] + arr[i]) # Find maximum sum in b/w inc[] and dec[] result = -2147483648 for i in range(n): if (result < inc[i]): result = inc[i] if (result < dec[i]): result = dec[i] # Return maximum sum # alternate sun-sequence return result # Driver program arr = [8, 2, 3, 5, 7, 9, 10] n = len(arr) print("Maximum sum = ", maxAlternateSum(arr , n )) # This code is contributed by Anant Agarwal. |
C#
// C# program to find sum of maximum // sum alternating sequence starting with // first element. using System; class GFG { // Return sum of maximum // sum alternating // sequence starting with // arr[0] and is first // decreasing. static int maxAlternateSum(int []arr, int n) { if (n == 1) return arr[0]; // create two empty array that // store result of maximum sum // of alternate sub-sequence // stores sum of decreasing // and increasing sub-sequence int []dec = new int[n]; // store sum of increasing and // decreasing sun-sequence int []inc = new int[n]; // As per question, first // element must be part // of solution. dec[0] = inc[0] = arr[0]; int flag = 0 ; // Traverse remaining elements of array for (int i = 1; i < n; i++) { for (int j = 0; j < i; j++) { // IF current sub-sequence // is decreasing the // update dec[j] if needed. // dec[i] by current // inc[j] + arr[i] if (arr[j] > arr[i]) { dec[i] = Math.Max(dec[i], inc[j] + arr[i]); // Revert the flag , if // first decreasing // is found flag = 1; } // If next element is greater // but flag should be 1 // i.e. this element should // be counted after the // first decreasing element // gets counted else if (arr[j] < arr[i] && flag == 1) // If current sub-sequence // is increasing then update // inc[i] inc[i] = Math.Max(inc[i], dec[j] + arr[i]); } } // find maximum sum in b/w // inc[] and dec[] int result = int.MinValue; for (int i = 0 ; i < n; i++) { if (result < inc[i]) result = inc[i]; if (result < dec[i]) result = dec[i]; } // return maximum sum // alternate sun-sequence return result; } // Driver Method public static void Main() { int []arr= {8, 2, 3, 5, 7, 9, 10}; Console.Write("Maximum sum = " + maxAlternateSum(arr , arr.Length)); } } // This code is contributed by Nitin Mittal. |
PHP
<?php // PHP program to find sum of maximum // sum alternating sequence starting // with first element. // Return sum of maximum sum alternating // sequence starting with arr[0] and is // first decreasing. function maxAlternateSum($arr, $n) { if ($n == 1) return $arr[0]; // create two empty array that store result // of maximum sum of alternate sub-sequence // stores sum of decreasing and // increasing sub-sequence $dec = array_fill(0, $n, 0); // store sum of increasing and // decreasing sun-sequence $inc = array_fill(0, $n, 0); // As per question, first element // must be part of solution. $dec[0] = $inc[0] = $arr[0]; $flag = 0; // Traverse remaining elements of array for ($i = 1; $i < $n; $i++) { for ($j = 0; $j < $i; $j++) { // IF current sub-sequence is decreasing // the update dec[j] if needed. dec[i] // by current inc[j] + arr[i] if ($arr[$j] > $arr[$i]) { $dec[$i] = max($dec[$i], $inc[$j] + $arr[$i]); // Revert the flag , if first // decreasing is found $flag = 1; } // If next element is greater but flag // should be 1 i.e. this element should // be counted after the first decreasing // element gets counted else if ($arr[$j] < $arr[$i] && $flag == 1) // If current sub-sequence is increasing // then update inc[i] $inc[$i] = max($inc[$i], $dec[$j] + $arr[$i]); } } // find maximum sum in b/w inc[] and dec[] $result = -(PHP_INT_MAX - 1); for ($i = 0 ; $i < $n; $i++) { if ($result < $inc[$i]) $result = $inc[$i]; if ($result < $dec[$i]) $result = $dec[$i]; } // return maximum sum alternate sun-sequence return $result; } // Driver Code $arr = array(8, 2, 3, 5, 7, 9, 10); $n = sizeof($arr); echo "Maximum sum = ", maxAlternateSum($arr, $n ); // This code is contributed by Ryuga ?> |
Output:
Maximum sum = 25
Time Complexity : O(n2)
Auxiliary Space : O(n)
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