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[Coding Made Simple] Maximum Sum Increasing Subsequence

Given an array, find maximum sum increasing subsequence in this array.

Solution 1. Recursion

For arr[0....n], the max sum increasing subsequence can be computed from first computing its subarrays' max sum increasing subsequence.

MaxSumIncreSubseq(n) = max {arr[n], max of {MaxSumIncreSubseq(i) + arr[n], for i from 0 to n - 1}};

 1 public int getMaxSumIncreSubseqRecursion(int[] arr) {
 2     if(arr == null || arr.length == 0) {
 3         return 0;
 4     }
 5     int maxSum = Integer.MIN_VALUE;
 6     for(int i = 0; i < arr.length; i++) {
 7         int sum = recursiveHelper(arr, i);
 8         maxSum = Math.max(maxSum, sum);
 9     }
10     return maxSum;
11 }
12 private int recursiveHelper(int[] arr, int endIdx) {
13     int max = arr[endIdx];
14     for(int i = 0; i < endIdx; i++) {
15         if(arr[i] < arr[endIdx]) {
16             max = Math.max(max, recursiveHelper(arr, i) + arr[endIdx]);
17         }
18     }
19     return max;
20 }

Solution 2. Dynamic Programming

State: T[i]: the max sum of increasing subsequence that ends with arr[i].

Function: T[i] = max of T[j] + arr[i], for j from 0 to i - 1, if arr[j] < arr[i];

Init: T[i] = arr[i], as we always know one of the increasing subsequence that ends with arr[i] is itself without other elements.

Answer: max of T[i].

 1 import java.util.ArrayList;
 2 
 3 public class MaxSumIncreasingSubseq {
 4     private ArrayList<Integer> maxSumSubseq;
 5     public int getMaxSumIncreSubseqDp(int[] arr) {
 6         if(arr == null || arr.length == 0) {
 7             return 0;
 8         }
 9         int[] T = new int[arr.length];
10         int[] seqIdx = new int[arr.length];
11         for(int i = 0; i < T.length; i++) {
12             T[i] = arr[i];
13             seqIdx[i] = i;
14         }
15         for(int i = 0; i < T.length; i++) {
16             for(int j = 0; j < i; j++) {
17                 if(arr[j] < arr[i]) {
18                     if(T[j] + arr[i] > T[i]) {
19                         T[i] = T[j] + arr[i];
20                         seqIdx[i] = j;
21                     }
22                 }
23             }
24         }
25         int max = Integer.MIN_VALUE;
26         int maxIdx = -1;
27         for(int i = 0; i < arr.length; i++) {
28             if(T[i] > max) {
29                 max = T[i];
30                 maxIdx = i;
31             }
32         }
33         maxSumSubseq = new ArrayList<Integer>();
34         int currIdx = maxIdx;
35         while(currIdx != seqIdx[currIdx]) {
36             maxSumSubseq.add(currIdx);
37             currIdx = seqIdx[currIdx];
38         }
39         maxSumSubseq.add(currIdx);
40         return max;
41     }
42 }