Multi-source BFS was the right move here, seed all warehouse cells at once and let it fan out.
Clarify the grid representation and define input/output formats, then propose a multi-source BFS from all warehouses simultaneously to compute shortest distances to all cells. Justify using a queue for BFS and a distance matrix for O(1) lookups, and discuss handling obstacles and unreachable customers.
Pro tip: Mention that multi-source BFS is optimal for unweighted grids and that you can early-terminate once all customers are reached, which is crucial for DoorDash's large-scale delivery scenarios.
Define the grid format (e.g., 2D array of characters: 'W' for warehouse, 'C' for customer, 'O' for obstacle, '.' for road) and output as a 2D array of integers with distances or -1. Confirm movement allowed in 4 directions and that each step costs 1.
Use a queue for BFS to process cells in order of distance, and a 2D distance array initialized to -1 to store results and track visited cells. Justify: BFS guarantees shortest path in unweighted graphs, and the distance array provides O(1) access and avoids revisiting.
Enqueue all warehouse cells with distance 0, marking them visited. This treats all warehouses as sources, ensuring each cell gets the minimum distance from any warehouse.
While queue is not empty, dequeue a cell, explore its 4 neighbors. If a neighbor is within bounds, not an obstacle, and unvisited, set its distance to current distance + 1, enqueue it, and if it's a customer, record the distance.
After BFS, return the distance array (or only customer distances). Discuss early termination when all customers are reached, and analyze time/space complexity: O(R*C) time and space.
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I talked about not materializing the full output grid if most cells are roads, maybe a sparse map for results.
Start by clarifying the problem constraints and the current solution's bottlenecks, then propose a scalable architecture that addresses memory and time separately. Discuss trade-offs between different strategies (e.g., partitioning, streaming, approximation) and justify your choices based on the specific requirements.
Pro tip: Quantify the scale: estimate memory and time for millions of cells to show you understand the magnitude, and mention that you'd validate with back-of-the-envelope calculations before committing to a design.
Ask about the grid's characteristics (sparse vs. dense, access patterns, update frequency) and the acceptable latency/throughput. This determines whether you need exact or approximate results.
Analyze where the current solution fails: memory (storing entire grid), time (O(n^2) operations), or both. This focuses your scaling efforts.
Suggest techniques like sparse representations, compression, partitioning (sharding), or using disk-based storage with caching. Discuss trade-offs (e.g., speed vs. memory).
Consider parallelization (e.g., MapReduce, GPU), incremental computation, indexing, or approximation algorithms. Explain how these reduce time complexity.
Compare alternatives (e.g., exact vs. approximate, in-memory vs. distributed) and recommend a solution based on constraints. Mention how you'd test and monitor performance.
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Straightforward pivot to Dijkstra with a min-heap.
First, clarify that the problem shifts from BFS to a weighted shortest-path problem, so Dijkstra's algorithm (or A* with an admissible heuristic) is the natural adaptation. Then, explain how to modify the graph representation and algorithm to handle non-uniform costs, and discuss trade-offs like time complexity and potential optimizations.
Pro tip: Mention that if costs are small positive integers, you can use Dial's algorithm (bucket queue) for O(V+E) time, showing you know when to avoid a full heap-based Dijkstra. Also, emphasize that you'd confirm whether costs are static or dynamic, as that affects algorithm choice.
Recognize that non-uniform movement costs turn the grid into a weighted graph, so BFS no longer guarantees shortest paths. State that you need a shortest-path algorithm for weighted graphs.
Select Dijkstra's algorithm for non-negative weights, or A* with a consistent heuristic for better performance. Mention alternatives like Bellman-Ford if negative weights are possible (though unlikely for road costs).
Explain how to model the grid as a graph where each cell is a node and edges have weights equal to the cost of moving into the neighboring cell. Discuss whether to use an implicit graph (compute neighbors on the fly) or explicit adjacency list.
Compare time and space complexity: Dijkstra with binary heap is O((V+E) log V), while A* can be faster with a good heuristic. Discuss potential optimizations like bidirectional search or Dial's algorithm for small integer weights.
Address assumptions: are costs static? Are they positive? What about large grids? Mention that if costs are dynamic, you might need to recompute or use incremental algorithms.
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Explain that to return actual paths, you need to modify the shortest path algorithm to track predecessors for each node, then reconstruct paths by backtracking from the destination. For multiple customer cells, run the algorithm from the source to all destinations or vice versa, storing parent pointers for each node. Finally, reconstruct each path individually using the stored predecessors.
Pro tip: Mention that for large graphs, storing full paths for all nodes can be memory-intensive, so consider storing only parent pointers and reconstructing paths on demand. Also, discuss how to handle multiple sources or destinations efficiently, such as running a single-source shortest path from the depot and reconstructing paths to each customer.
Select an algorithm like Dijkstra or BFS (for unweighted graphs) that can compute shortest distances from a source to all nodes. Ensure it can be augmented to track predecessors.
During relaxation, whenever you update the shortest distance to a node, also record the predecessor node that provided the better path. This builds a predecessor tree.
If there are multiple customer cells, decide whether to run from the depot to all customers or from each customer to the depot, based on graph size and directionality. Typically, one run from the depot suffices if paths are needed from depot to customers.
For each customer cell, backtrack from the customer to the source using the predecessor pointers, then reverse the sequence to get the path from source to customer.
If memory is a concern, avoid storing full paths for all nodes; instead, store only predecessors and reconstruct paths on demand. Consider early termination if only a subset of destinations is needed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.