Key Steps of Dijkstra's Algorithm:

1. Initialize distances: set source to 0, all others to infinity
2. Create a priority queue and add all vertices with their distances
3. While the priority queue is not empty:
   • Extract the vertex with minimum distance (let's call it u)
   • For each adjacent vertex v of u:
     • Calculate distance = distance[u] + weight of edge (u, v)
     • If this distance is less than the current distance[v], update distance[v]
4. Return the array of shortest distances from source to all vertices

Key Steps of Bellman-Ford Algorithm:

1. Initialize distances: set source to 0, all others to infinity
2. Repeat V-1 times (where V is the number of vertices):
   • For each edge (u,v) with weight w in the graph:
     • If distance[u] + w < distance[v], update distance[v]=distance[u] + w
3. Check for negative cycles:
   • For each edge (u,v) with weight w in the graph:
     • If distance[u] + w < distance[v], report negative cycle
4. Return the array of shortest distances from source to all vertices

Key Steps of the Ford-Fulkerson Algorithm in Amazon's Systems:

1. Model the traffic network:
   • Define sources (customer entry points) and sinks (service endpoints)
   • Map intermediate nodes (servers, fulfillment centers, etc.)
   • Establish capacity constraints for each channel
   • Create a directed graph with edge capacities representing maximum traffic flow
2. Initialize flow values:
   • Set initial flow to zero for all edges
   • Create a residual graph tracking remaining capacity
3. Iteratively augment flow:
   • Find an augmenting path from source to sink with available capacity
   • Identify the bottleneck capacity along the path
   • Increase flow along the path by the bottleneck value
   • Update residual capacities in both directions
4. Dynamic rebalancing:
   • Monitor actual traffic patterns in real-time
   • Adjust capacity constraints based on system performance
   • Recalculate optimal flow when conditions change
5. Handle overflow conditions:
   • Implement graceful degradation strategies when demand exceeds total capacity
   • Apply priority rules for critical traffic
   • Dynamically provision additional resources when possible

Key Steps of Skip List Operations in Amazon's Search Systems:

1. Search Operation:
   • Start at the highest level of the skip list from the leftmost node
   • Move horizontally as far as possible without overshooting target
   • Drop down one level and repeat until reaching the bottom level
   • Perform final horizontal scan to locate exact match or insertion point
2. Insertion Operation:
   • Search to find the appropriate position for the new element
   • Randomly determine the element's maximum level (height)
   • Insert the element at the bottom level and all randomly chosen levels above
   • Update relevant pointers to maintain the skip list structure
3. Range Query Operation:
   • Search to locate the starting position of the range
   • Traverse horizontally at the bottom level, collecting elements within range
   • Continue until reaching the upper bound of the query range
   • Apply any additional filters or transformations to the results
4. Concurrent Access Optimization:
   • Implement lock-free or fine-grained locking strategies
   • Use atomic operations for node updates to prevent race conditions
   • Maintain consistent state during concurrent read/write operations
   • Employ optimistic retry mechanisms for contended updates

Key Steps of Dependency Resolution using Kahn's Algorithm:

1. Represent dependencies as a directed graph:
   • Each task/component is a vertex
   • Each dependency is a directed edge
   • If A depends on B, add edge B→A (B must complete before A)
2. Calculate in-degree for each vertex (number of dependencies):
   • Initialize in-degree counter for each vertex
   • For each edge in the graph, increment the in-degree of the destination vertex
3. Identify starting points:
   • Find all vertices with in-degree of 0 (no dependencies)
   • Add these vertices to a queue
4. Process the dependency graph:
   • While the queue is not empty, remove a vertex
   • Add the removed vertex to the result list
   • For each neighbor of the removed vertex, decrement its in-degree
   • If any neighbor's in-degree becomes 0, add it to the queue
5. Verify the resolution:
   • If the result list contains all vertices, a valid topological ordering was found
   • Otherwise, the graph contains a cycle (circular dependency)

Reference: Kahn, A. B. (1962). "Topological sorting of large networks." Communications of the ACM, 5(11), 558-562.

How Amazon Uses MST for Inventory:

1. Model fulfillment centers as a network with transportation costs as connection weights
2. Apply MST algorithms to find the cheapest way to connect all centers
3. Use these connections to plan inventory transfers and rebalancing
4. Update the model when costs change or new centers open

Product Grouping Process using Union-Find:

1. Initialize Disjoint Sets:
   • Create individual sets for each product in the catalog
   • Initialize parent array where each product points to itself
   • Set up rank array for union by rank optimization
2. Similarity Detection:
   • Compare product pairs using multiple similarity metrics
   • Apply threshold-based matching for different product categories
   • Use ML models to enhance similarity detection accuracy
3. Union Operations:
   • Merge sets when products are determined to be similar
   • Apply union by rank to maintain balanced tree structure
   • Update group representatives for efficient set identification
4. Path Compression:
   • Flatten tree paths during Find operations for future efficiency
   • Ensure all nodes point directly to root for O(1) subsequent lookups

LRU Cache Implementation Process:

1. Data Structure Setup:
   • Initialize hash map for O(1) key-to-node mapping
   • Create doubly linked list with head and tail sentinels
   • Set cache capacity limit for memory management
2. Get Operation:
   • Check hash map for key existence
   • If found, move node to front of list (most recent)
   • Return cached value
3. Put Operation:
   • If key exists, update value and move to front
   • If new key and cache full, remove least recent item
   • Add new node to front and update hash map

Product Matching Process using LCS:

1. Text Preprocessing:
   • Tokenize product descriptions into words or features
   • Remove common stop words and normalize text
   • Extract key specifications and attributes
2. LCS Computation:
   • Build DP table comparing two product descriptions
   • Track longest common subsequences of features
   • Calculate similarity score based on LCS length
3. Similarity Analysis:
   • Compute similarity percentage: LCS length / max(len1, len2)
   • Apply thresholds for different product categories
   • Generate confidence scores for matching decisions

Cart Management Process using Stack:

1. Initialize Cart Stack:
   • Create empty stack for cart items
   • Initialize action history stack for undo operations
   • Set up cart state persistence
2. Item Addition:
   • Push new item onto cart stack
   • Record action in history for undo capability
   • Update cart total and item count
3. Item Removal & Undo:
   • Pop item from cart stack for removal
   • Implement undo by reversing last action
   • Maintain action history for multiple undo levels

Robot Navigation Process using A*:

1. Initialize Navigation:
   • Create grid map of warehouse with obstacles marked
   • Set robot start position and destination coordinates
   • Initialize open set with start node (f = h value)
2. Path Search:
   • Select node with lowest f(n) from open set
   • Move to closed set and check if destination reached
   • Evaluate all neighboring nodes for path extension
3. Dynamic Adaptation:
   • Monitor for new obstacles or traffic
   • Recalculate path when conditions change
   • Coordinate with other robots to prevent collisions

Huffman Coding Process for Media:

1. Frequency Analysis:
   • Analyze symbol frequency in media data blocks
   • Create frequency table for characters/coefficients
   • Identify most and least common data patterns
2. Tree Construction:
   • Build binary tree with least frequent symbols at bottom
   • Merge nodes using priority queue (min-heap)
   • Generate optimal prefix-free code assignments
3. Encoding & Integration:
   • Apply Huffman codes within codec frameworks
   • Integrate with quantization and transformation stages
   • Optimize for real-time streaming requirements