We are given a matrix that contains different values in its each cell. Our aim is to find the minimal set of positions in the matrix such that entire matrix can be traversed starting from the positions in the set.
We can traverse the matrix under below conditions:
- We can move only to those neighbors that contain value less than or to equal to the current cell’s value. A neighbor of cell is defined as the cell that shares a side with the given cell.
Input : 1 2 3 2 3 1 1 1 1 Output : 1 1 0 2 If we start from 1, 1 we can cover 6 vertices in the order 1, 1 -> 1, 0 -> 2, 0 -> 2, 1 -> 2, 2 -> 1, 2. We cannot cover the entire matrix with this vertex. Remaining vertices can be covered 0, 2 -> 0, 1 -> 0, 0. Input : 3 3 1 1 Output : 0 1 If we start from 0, 1, we can traverse the entire matrix from this single vertex in this order 0, 0 -> 0, 1 -> 1, 1 -> 1, 0. Traversing the matrix in this order satisfies all the conditions stated above.
From the above examples, we can easily identify that in order to use minimum number of positions we have to start from the positions having highest cell value. Therefore we pick the positions that contain the highest value in the matrix. We take the vertices having highest value in separate array. We perform DFS on every vertex starting from the highest value. If we encounter any unvisited vertex during dfs then we have to include this vertex in our set. When all the cells have been processed then the set contains the required vertices.
How does this work?
We need to visit all vertices and to reach largest values we must start with them. If two largest values are not adjacent, then both of them must be picked. If two largest values are adjacent, then any of them can be picked as moving to equal value neighbors is allowed.
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- Maximum and minimum isolated vertices in a graph
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- Count valid pairs in the array satisfying given conditions
- Count all possible paths between two vertices
- Articulation Points (or Cut Vertices) in a Graph
- Longest path between any pair of vertices
- Construct a graph from given degrees of all vertices
- Finding in and out degrees of all vertices in a graph
- Find if there is a path between two vertices in a directed graph
- Largest subset of Graph vertices with edges of 2 or more colors
- Check whether given degrees of vertices represent a Graph or Tree
- Find maximum number of edge disjoint paths between two vertices
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