Number of pair of positions in matrix which are not accessible - GeeksforGeeks
Given a positive integer N. Consider a matrix of N X N. No cell can be accessible from any other cell, except the given pair cell in the form of (x1, y1), (x2, y2) i.e there is a path (accessible) between (x2, y2) to (x1, y1). The task is to find the count of pairs (a1, b1), (a2, b2) such that cell (a2, b2) is not accessible from (a1, b1).
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Given a positive integer N. Consider a matrix of N X N. No cell can be accessible from any other cell, except the given pair cell in the form of (x1, y1), (x2, y2) i.e there is a path (accessible) between (x2, y2) to (x1, y1). The task is to find the count of pairs (a1, b1), (a2, b2) such that cell (a2, b2) is not accessible from (a1, b1).
Consider each cell as a node, numbered from 1 to N*N. Each cell (x, y) can be map to number using (x – 1)*N + y. Now, consider each given allowed path as an edge between nodes. This will form a disjoint set of the connected component. Now, using Depth First Traversal or Breadth First Traversal, we can easily find the number of nodes or size of a connected component, say x. Now, count of non-accessible paths are x*(N*N – x). This way we can find non-accessible paths for each connected path.
Time Complexity : O(N*N).
// Counts number of vertices connected in a component
// containing x. Stores the count in k.
void
dfs(vector<
int
> graph[],
bool
visited[],
int
x,
int
*k)
{
for
(
int
i = 0; i < graph[x].size(); i++)
{
if
(!visited[graph[x][i]])
{
// Incrementing the number of node in
// a connected component.
(*k)++;
visited[graph[x][i]] =
true
;
dfs(graph, visited, graph[x][i], k);
}
}
}
// Return the number of count of non-accessible cells.
int
countNonAccessible(vector<
int
> graph[],
int
N)
{
bool
visited[N*N + N];
memset
(visited,
false
,
sizeof
(visited));
int
ans = 0;
for
(
int
i = 1; i <= N*N; i++)
{
if
(!visited[i])
{
visited[i] =
true
;
// Initialize count of connected
// vertices found by DFS starting
// from i.
int
k = 1;
dfs(graph, visited, i, &k);
// Update result
ans += k * (N*N - k);
}
}
return
ans;
}
// Inserting the edge between edge.
void
insertpath(vector<
int
> graph[],
int
N,
int
x1,
int
y1,
int
x2,
int
y2)
{
// Mapping the cell coordinate into node number.
int
a = (x1 - 1) * N + y1;
int
b = (x2 - 1) * N + y2;
// Inserting the edge.
graph[a].push_back(b);
graph[b].push_back(a);
}