Understand what the interviewer is asking for by using test cases and questions about the problem.
- Established a set (2-3) of test cases to verify their own solution later.
- Established a set (1-2) of edge cases to verify their solution handles complexities.
- Have fully understood the problem and have no clarifying questions.
- Have you verified any Time/Space Constraints for this problem?
HAPPY CASE
Input: graph = [[1,2],[2,3],[5],[0],[5],[],[]]
Output: [2,4,5,6]
Explanation: The given graph is shown above.
Nodes 5 and 6 are terminal nodes as there are no outgoing edges from either of them.
Every path starting at nodes 2, 4, 5, and 6 all lead to either node 5 or 6.
EDGE CASE
Input: graph = [[1,2,3,4],[1,2],[3,4],[0,4],[]]
Output: [4]
Explanation:
Only node 4 is a terminal node, and every path starting at node 4 leads to node 4.Match what this problem looks like to known categories of problems, e.g. Linked List or Dynamic Programming, and strategies or patterns in those categories.
For graph problems, some things we want to consider are:
grid[0][0] to grid[n-1][n-1]. We can take grid [[0, 1], [2, 3]] as an example: We can treat this as a graph with 4 vertices 0,1,2,3 and there are 4 edges. {0, 1} with weight 1, {0, 2} with weight 2, {1, 3} with weight 3 and {2, 3} with weight 3 respectively. Each time we choose the smallest edge {m, n} to simulate at time max(m, n) and rain falls. So we first take edge {0, 1} then {0, 2}. When we are processing edge {1, 3}, the original result is 2(getting from {0, 2}), but since original time is less than we currently need, we need time 3 to reach vertex 3. For another case like grid [[3, 2], [1, 0]], the first part is the same. To reach from vertex 3 to vertex 1, we need time 3. Then to reach from vertex 1 to vertex 0, we also only need time 3, since we can move infinite distance in one move. We can see this like at time 3, water can move from vertex 3 to vertex 1 then vertex 0 directly. From 2 cases above, we can get the most important thing in this problem that the time needs to reach grid[i][j] is max(grid[i][j], grid[i'][j']), where grid[i'][j'] is the cell to get to grid[i][j].[0,0] and if we happen to touch [n-1,m-1], then we can say that a path exists. This is general Depth First Traversal, but we have another constraint that we cannot move to a height > t, so we include a constraint that we can move in any of the four directions, if and only if, the height of the building in that direction is less than t.Plan the solution with appropriate visualizations and pseudocode.
General Idea: In a directed graph, we start at some node and every turn, walk along a directed edge of the graph. If we reach a node that is terminal (that is, it has no outgoing directed edges), we stop. Now, say our starting node is eventually safe if and only if we must eventually walk to a terminal node. More specifically, there exists a natural number K so that for any choice of where to walk, we must have stopped at a terminal node in less than K steps. The directed graph has N nodes with labels 0, 1, ..., N-1, where N is the length of graph. The graph is given in the following form: graph[i] is a list of labels j such that (i, j) is a directed edge of the graph. NOTE that we may do some pruning optimization.
To know if the path is not safe, we check if it goes back to the ongoing path. Therefore, I have a set visiting to store nodes that are in the current DFS. I make sure to remove the node once DFS finishes.
1. When we visit a node, the only possibilities are that we’ve marked the entire subtree black (which must be eventually safe), or it has a cycle and we have only marked the members of that cycle gray. So the invariant that gray nodes are always part of a cycle, and black nodes are always eventually safe is maintained.
2. In order to exit our search quickly when we find a cycle (and not paint other nodes erroneously), we’ll say the result of visiting a node is true if it is eventually safe, otherwise false. This allows information that we’ve reached a cycle to propagate up the call stack so that we can terminate our search early.⚠️ Common Mistakes
We start with nodes that have no outgoing edges - those are eventually safe. Now, we can update any nodes which only point to eventually safe nodes - those are also eventually safe. Then, we can update again, and so on. However, we'll need a good algorithm to make sure our updates are efficient.
Implement the code to solve the algorithm.
class Solution:
def eventualSafeNodes(self, graph: List[List[int]]) -> List[int]:
dt = defaultdict(list)
status = defaultdict(int) # if visiting this node change status to 1 else 0
res = []
for i, val in enumerate(graph):
dt[i] = val
def dfs(num):
if dt[num] == []: return True
if status[num] == 1: return False
status[num] = 1
for val in dt[num]:
if not dfs(val):
return False
dt[num] = []
status[num] = 0
return True
for num in range(len(graph)):
if dfs(num):
res.append(num)
return resclass Solution {
public List<Integer> eventualSafeNodes(int[][] graph) {
List<Integer> res = new ArrayList<>();
int n = graph.length; //number of nodes
int[] visited = new int[n]; //3 status, 0: unvisited, 1: visiting, 2: visited
for (int i = 0; i < n; i++) // try all nodes
if (DFS(graph, i, visited))
res.add(i);
return res;
}
//return true if node i is safe
private boolean DFS(int[][] graph, int i, int[] visited) {
if (visited[i] == 1) //node i is being visited, so cycle
return false;
visited[i] = 1; //visiting
for (int neighbor : graph[i]) {
int status = visited[neighbor]; //status of neighbor
if (status == 1) //neighbor is a node we are visiting, so cycle. All node in a cycle is not safe
return false;
if (status == 2) //already visited
continue;
//otherwise, status == 0, meaning ok to visit
if (!DFS(graph, neighbor, visited))
return false;
}
visited[i] = 2; //visited
return true;
}
}Review the code by running specific example(s) and recording values (watchlist) of your code's variables along the way.
Evaluate the performance of your algorithm and state any strong/weak or future potential work.
Time Complexity - O(V + E)
Space Complexity - O(V + E)
