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# Maximum sum alternating subsequence

• Difficulty Level : Medium
• Last Updated : 08 Mar, 2022

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 = dec = arr

// 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.
Note:- For the case where the first element of the array is the smallest element in the array. The output will be the first element only. This is an edge case that need to be checked. Taking a variable and initializing it with the first value of the array and then comparing it with other values will find the min. Check if the min is equal to arr. If it is true then arr is to be returned, because there is no decreasing step available to find an alternating subsequence.

## C++

 `// C++ program to find sum of maximum``// sum alternating sequence starting with``// first element.``#include ``using` `namespace` `std;` `// Return sum of maximum sum alternating``// sequence starting with arr and is first``// decreasing.``int` `maxAlternateSum(``int` `arr[], ``int` `n)``{``    ``if` `(n == 1)``        ``return` `arr;``    ``// handling the edge case``    ``int` `min = arr;``    ``for` `(``int` `i = 1; i < n; i++) {``        ``if` `(min > arr[i])``            ``min = arr[i];``    ``}``    ``if` `(min == arr) {``        ``return` `arr;``    ``}``    ``// 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 sub-sequence``    ``int` `inc[n];``    ``memset``(inc, 0, ``sizeof``(inc));` `    ``// As per question, first element must be part``    ``// of solution.``    ``dec = inc = arr;` `    ``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 sub-sequence``    ``return` `result;``}` `// Driver program``int` `main()``{``    ``int` `arr[] = { 8, 2, 3, 5, 7, 9, 10 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);``    ``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 and is first``    ``// decreasing.``    ``static` `int` `maxAlternateSum(``int` `arr[], ``int` `n)``    ``{``        ``if` `(n == ``1``)``            ``return` `arr[``0``];``        ``// handling the edge case``        ``int` `min = arr[``0``];``        ``for` `(``int` `i = ``1``; i < n; i++) {``            ``if` `(min > arr[i])``                ``min = arr[i];``        ``}``        ``if` `(min == arr[``0``]) {``            ``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``        ``// sub-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 sub-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 and is``# first decreasing.`  `def` `maxAlternateSum(arr, n):` `    ``if` `(n ``=``=` `1``):``        ``return` `arr[``0``]``    ``min` `=` `arr[``0``]``    ``for` `i ``in` `range``(``1``, n):``        ``if``(``min` `> arr[i]):``            ``min` `=` `arr[i]``    ``if``(arr[``0``] ``=``=` `min``):``        ``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 sub-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``            ``else` `if` `(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 sub-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 and is first``    ``// decreasing.``    ``static` `int` `maxAlternateSum(``int``[] arr, ``int` `n)``    ``{``        ``if` `(n == 1)``            ``return` `arr;``        ``// handling the edge case``        ``int` `min = arr;``        ``for` `(``int` `i = 1; i < n; i++) {``            ``if` `(min > arr[i])``                ``min = arr[i];``        ``}``        ``if` `(min == arr) {``            ``return` `arr;``        ``}``        ``// 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 sub-sequence``        ``int``[] inc = ``new` `int``[n];` `        ``// As per question, first``        ``// element must be part``        ``// of solution.``        ``dec = inc = arr;` `        ``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 sub-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

 ` ``\$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 sub-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``?>`

## Javascript

 ``

Output

`Maximum sum = 25`

Time Complexity : O(n2
Auxiliary Space : O(n)
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