Longest Increasing Sequence 2D matrix recursion

Question

I have been presented with a new homework assignment that has been somewhat frustrating to say the least. Basically, I have a create a 2D array of integers as follows:

Answer 1

Update

I learnt dynamic programming recently, and I have found a better algorithm for the question.

The algorithm is simple: find the longest length for every point, and record the result in a 2D array so that we do not need to calculate the longest length for some points again.

int original[m][n] = {...};
int longest[m][n] = {0};

int find() {
    int max = 0;
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < n; j++) {
            int current = findfor(i, j);
            if (current > max) { max = current; }
        }
    }
    return max;
}

int findfor(int i, int j) {
    if (longest[i][j] == 0) {
        int max = 0;
        for (int k = -1; k <= 1; k++) {
            for (int l = -1; l <= 1; l++) {
                if (!(k == 0 && l == 0) &&
                    i + k >= 0 && i + k < m &&
                    j + l >= 0 && j + l < n &&
                    original[i + k][j + l] > original[i][j]
                   )
                    int current = findfor(i + k, j + l);
                    if (current > max) { max = current; }
                }
            }
        }
        longest[i][j] = max + 1;
    }
    return longest[i][j];
}    

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Recursion

1) start with a point (and this step has to be taken for all necessary points)

2) if no surrounding point is greater, then this path ends; else pick a greater surrounding point to continue the path, and go to 2).

2.1) if the (ended) path is longer than recorded longest path, substitute itself as the longest.

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Hint

(less computation but more coding)

For the longest path, the start point of which will be a local minimum point, and the end point of which will be a local maximum point.

Local minimum, less than (or equal to) all (at most) 8 surrounding points.

Local maximum, greater than (or equal to) all (at most) 8 surrounding points.

Proof

If the path does not start with a local minimum, then the start point must be greater than at least a surrounding point, and thus the path can be extended. Reject! Thus, the path must start with a local minimum. Similar for the reason to end with a local maximum.

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pseudo code

for all local minimum
  do a recursive_search

recursive_search (point)
  if point is local maximum
    end, and compare (and substitute if necessary) longest
  else
    for all greater surrounding points
      do a recursive_search