Maximum sum of Subset having no consecutive elements

Given an array arr[] of size Nthe task is to find the maximum possible sum of a subset of the array such that no two consecutive elements are part of the subset.

Examples:

Input: arr[]= {2, 3, 2, 3, 3, 4}
Output: 9
Explanation: The subset having all the 3s ie {3, 3, 3} have sum = 9.
This is the maximum possible sum of any possible subset of the array following the condition.

Input: arr[] = {2, 3, 4}
Output: 6
Explanation: The subset is {2, 4}. It has sum = 6 which is the maximum possible.

Naive Approach: The naive approach is to generate all the possible subsets and from them check which subsets are following the given condition. Calculate the sum of those subsets and the maximum among them is the required answer.

Time Complexity: O(2N)
Auxiliary Space: O(2N)

Efficient Approach: An efficient approach is to use dynamic programming with the help of the following idea:

For any element X in arr[], the value X-1 cannot be considered but all the elements having value X can be. So for X the maximum possible answer till X is maximum between (maximum possible answer till X-2 + freq(X)*X) and (maximum possible answer till X-1)

Follow the steps mentioned below to solve the problem:

• Use hashing to store the frequency of each element.
• Find the maximum value of the array. (say X)
• Create a dp[] array where dp[i] stores the maximum possible subset sum when elements with value at most i are included in the subset.
• Iterate from i = 2 to X of the array:
• Calculate the value of dp[i] as per the formula from the observation dp[i] = max(dp[i-2] + i*freq(i), dp[i-1]).
• The maximum value from the dp[] array is the answer.

Below is the implementation of the above approach.

C++

 ` ` `#include ` `using` `namespace` `std;` ` ` `int` `MaximiseStockPurchase(vector<``int``>& nums,` `                          ``int` `n)` `{` `    ``int` `maxi = 0;` `    ``for` `(``int` `i = 0; i < n; i++)` `        ``maxi = max(maxi, nums[i]);` ` ` `    ``vector<``int``> freq(maxi + 1, 0);` `    ``vector<``int``> dp(maxi + 1, 0);` ` ` `    ``for` `(``auto` `i : nums)` `        ``freq[i]++;` `    ``dp[1] = freq[1];` ` ` `    ` `    ` `    ``for` `(``int` `i = 2; i <= maxi; i++)` `        ``dp[i] = max(dp[i - 2] + i * freq[i],` `                    ``dp[i - 1]);` ` ` `    ``return` `dp[maxi];` `}` ` ` `int` `main()` `{` `    ``vector<``int``> arr{ 2, 2, 3, 4, 3, 3 };` `    ``int` `N = arr.size();` ` ` `    ``int` `res = MaximiseStockPurchase(arr, N);` `    ``cout << res;` `    ``return` `0;` `}`

Time Complexity: O(M) where M is the maximum element of the array.
Auxiliary Space: O(M)

Alternative Approach: In the above approach the space of the dp[] array can be optimized as below:

As seen from the observation we only need the value of dp[i-1] and dp[i-2] to calculate the value of dp[i]. So instead of using dp[] array use two variables to store the value of the previous two steps.

Below is the implementation of the above approach:

C++

 ` ` `#include ` `using` `namespace` `std;` ` ` `int` `MaximiseStockPurchase(vector<``int``>& nums,` `                          ``int` `n)` `{` `    ``int` `maxNum = INT_MIN;` `    ``for` `(``auto` `i : nums)` `        ``maxNum = max(maxNum, i);` `    ``vector<``int``> freq(maxNum + 1, 0);` `    ``for` `(``auto` `i : nums)` `        ``freq[i]++;` ` ` `    ``int` `curPoints = freq[1], prevPoints = 0;` ` ` `    ` `    ``for` `(``int` `i = 2; i <= maxNum; i++) {` `        ``int` `tmp = curPoints;` `        ``curPoints = max(prevPoints + i * freq[i],` `                        ``curPoints);` `        ``prevPoints = tmp;` `    ``}` `    ``return` `curPoints;` `}` ` ` `int` `main()` `{` `    ``vector<``int``> arr{ 2, 2, 3, 4, 3, 3 };` `    ``int` `N = arr.size();` ` ` `    ``int` `res = MaximiseStockPurchase(arr, N);` `    ``cout << res;` `    ``return` `0;` `}`

Time complexity: O(N + M) where M is the maximum element of the array
Auxiliary Space: O(M).