Problem Description

The problem revolves around a scenario where you have a list of bombs, each with its own location (X and Y coordinates) and a radius that represents the range of its explosion. When a bomb explodes, any bombs within its explosion range will also explode. This chain reaction can lead to a cascade of exploding bombs.

The challenge here is to figure out the maximum number of bombs that can be detonated if you are allowed to detonate only one bomb to start the chain reaction.

The primary task is to detect connections between bombs—understanding which bombs can trigger the explosion of other bombs—and then finding the optimal starting point that results in the highest number of detonations in the chain reaction.

Flowchart Walkthrough

Let's employ the algorithm flowchart to navigate through the solution for the problem presented in Leetcode 2101. Detonate the Maximum Bombs. Here’s a step-by-step analysis using the Flowchart:

Is it a graph?

• Yes: Each bomb can be seen as a node, and there is an edge from one bomb to another if the second bomb is within the blast radius of the first.

Is it a tree?

• No: The graph described by the bombs and their blast radii is not necessarily hierarchical and may contain cycles.

Is the problem related to directed acyclic graphs (DAGs)?

• No: Although the problem does involve directions (i.e., one bomb triggering another), the presence of cycles (bombs triggering each other in a loop) means it’s not acyclic.

Is the problem related to shortest paths?

• No: The main issue is to maximize the number of detonated bombs, not to find a path of minimum length or cost.

Does the problem involve connectivity?

• Yes: The solution requires understanding which bombs can be triggered by others, forming connected subgraphs of reachable nodes.

Is the graph weighted?

• No: The connections (edges) between bombs are simply based on whether one bomb can trigger another, without any weighting system besides the range of effect.

Conclusion: We opt for using Depth-First Search (DFS) from each bomb to explore all reachable bombs and count the maximum number that can be detonated, given the description of the problem as finding the largest connected component in a graph where nodes can trigger one another.

Intuition

To solve this problem, we need to find relationships between bombs—specifically, which bombs can cause others to explode. This is akin to creating a graph where each node is a bomb, and there is a directed edge from one bomb to another if the second bomb is within the first bomb's blast radius.

We can approach the solution by creating such a graph and then using Breadth-First Search (BFS) to simulate the explosion of bombs one by one from a starting bomb. The BFS approach is well-suited here because it allows us to explore all possible chains of detonations level by level, ensuring we count all bombs reachable from the starting bomb.

To explore these chains efficiently, we check for each bomb whether it can trigger other bombs and maintain a list of such connections. Then, for each bomb chosen as the starting point, we apply BFS to count how many bombs would be detonated if we were to start from that bomb. While traversing the graph, we also maintain a visitation status to ensure we do not count any bomb more than once.

Finally, we keep track of the maximum number of detonations from each starting point, and return the highest count as our answer. This approach allows us to consider all possibilities and find the most explosive starting bomb, so to speak.

Learn more about Depth-First Search, Breadth-First Search, Graph and Math patterns.

Solution Approach

The solution provided uses Breadth-First Search (BFS) to traverse the graph constructed from the bombs' connections. Here's a step-by-step breakdown of the implementation:

Data Structures and Algorithm:

• Graph Representation: A dictionary g is used to represent the graph where each key is a bomb index, and the associated value is a list of indices of bombs that can be triggered by this bomb.

• Helper Function (check): A function is defined to determine whether bomb j is within the range of bomb i. This is done by calculating the Euclidean distance between the bombs and comparing it to the radius of bomb i.

• Graph Construction: Two nested loops iterate over all pairs of bombs, and the check function is used to establish the graph's edges, which represent reachable bombs within explosion range.

• BFS Traversal: For each bomb k, we perform a BFS starting from k. A queue q and a visitation list vis are maintained. The queue is initialized with the starting bomb, k, and its corresponding visitation status is marked as True.

• Counting Detonations: While the queue is not empty, bombs are dequeed one by one, each time incrementing a counter cnt. For every dequeued bomb i, we enqueue all unvisited bombs that can be triggered by bomb i based on the graph g, marking them as visited to prevent repeated counting.

• Finding the Maximum: A variable ans keeps track of the maximum number of detonations. After each BFS traversal for a starting bomb k, the counter cnt is compared with ans and updated if it is larger.

BFS in Action:

BFS is an ideal choice for this problem because it ensures that all bombs that can be detonated by one bomb are exhausted before moving to bombs that they trigger. This level-by-level approach guarantees we account for all potential detonations in the sequence they would occur in reality.

Final Result:

After iterating over each bomb as a potential starting point and performing the BFS traversal, the value of ans at the end of iteration would be the maximum number of bombs that can be detonated by triggering just one bomb. This value is then returned as the final answer.

The use of the BFS algorithm and data structures such as the graph (dictionary), queue (deque), and the visitation list (vis) forms the backbone of this solution, allowing effective simulation and counting of the resulting explosions.

Example Walkthrough

To illustrate the solution approach, let's assume we have the following small set of bombs:

• Bomb A at (0,0) with a radius of 3
• Bomb B at (4,0) with a radius of 2
• Bomb C at (2,2) with a radius of 3
• Bomb D at (6,1) with a radius of 2

We want to see which bomb to start with to detonate the most bombs in total.

Step 1: Create the Graph First, we represent the bombs as nodes in a graph. To establish directed edges, we calculate if one bomb is within the explosion radius of another by using the Euclidean distance formula. For instance, to check if bomb B is within the radius of bomb A, we calculate the distance between (0,0) and (4,0), which is 4. Since 4 is greater than bomb A's radius, bomb A cannot trigger bomb B.

After checking all pairs, our graph (let's call it g) might look something like this:

• A: [C] because bomb A can trigger bomb C which is in its radius.
• B: [] because bomb B cannot trigger any other bombs.
• C: [A, D] because bomb C can trigger both A and D.
• D: [] because bomb D cannot trigger any other bombs.

Step 2: Perform BFS We then perform BFS starting from each bomb:

• For bomb A:
  • We visit bomb C since it's reachable, so our count cnt for A's chain is 2 (including A itself).
• For bomb B:
  • No other bombs can be triggered, so the count is 1.
• For bomb C:
  • We visit bomb A and from A we can visit bomb C itself again, but it's already been counted. We also visit bomb D. Thus, cnt for C's chain is 3 (including C itself).
• For bomb D:
  • No other bombs can be triggered, so the count is 1.

Step 3: Find the Maximum Finally, we find the maximum of our counts:

• Maximum detonations are 3, starting with bomb C which causes a chain reaction A->C->D.

This is the final result for this example. The number of bombs that can be detonated by triggering just one bomb, in this case, is three when starting with bomb C.

Solution Implementation

Time and Space Complexity

The given Python code defines a Solution class with a method maximumDetonation that computes the largest number of bombs that can be detonated by a single bomb. It involves graph theory concepts where each bomb is a node in the graph, and there is an edge from bomb i to bomb j if bomb i can detonate bomb j. Below is the analysis of time and space complexity:

Time Complexity

1. The check function checks whether bomb i can detonate bomb j. It runs in constant time, O(1), as it performs a few arithmetic operations and a comparison.
2. The main part of the code initializes a directed graph g using adjacency lists, which is essentially a dictionary with each bomb's index mapping to a list of other bomb indices it can detonate. The double for loop constructs the graph and runs in O(n^2) time, where n is the total number of bombs.
3. For each bomb k, the code performs a Breadth-First Search (BFS) to find all bombs that can be detonated, starting with bomb k. The BFS for each bomb can visit every other bomb and edge at most once, leading to a time complexity of O(n + e), where e is the number of edges in the graph.
4. Since the graph is dense, and in the worst case each bomb can detonate every other bomb, e can be O(n^2). So the complexity for each BFS becomes O(n^2).
5. The outer loop runs BFS for each bomb, so we multiply the complexity of BFS by the number of bombs to get O(n^3).

Therefore, the overall time complexity of the code is O(n^2) + O(n) * O(n^2), which simplifies to O(n^3).

Space Complexity

1. The graph g can hold up to n^2 edges if every bomb can detonate every other bomb, so it requires O(n^2) space.
2. The vis array is of size n, used during each BFS to keep track of the visited bombs, resulting in O(n) space.
3. The BFS queue q could, in the worst case, hold all bombs if they were all interconnected in a way that allows sequential detonation. This also would take O(n) space.
4. Counting the space used by ans and other variables would only add a constant amount of space, O(1).

In summary, the space complexity is dominated by the graph's size, so the overall space complexity is O(n^2).

Learn more about how to find time and space complexity quickly using problem constraints.