##### ###### ##### ### # # ### # # ###### ## ## ## ## ## ## ## # # # # # ## ##### #### ##### # # # # # # # #### ## # ## ## ## ## # # # # # ## ## # ###### ## ### # ### # ######
##### ###### ##### ### # # ### # # ###### ## ## ## ## ## ## ## # # # # # ## ##### #### ##### # # # # # # # #### ## # ## ## ## ## # # # # # ## ## # ###### ## ### # ### # ######
Sparse Tables answer range min/max in O(1) after O(n log n) prep. Ideal for static data.
Why Sparse Tables matter
The building blocks
DP construction
Idempotent operations
Avoiding log calls
The classic RMQ problem
Complete implementation
Implement your solution
When overlap trick works
O(log n) fallback
GCD is idempotent
Implement your solution
Euler tour + RMQ
Building the tour
O(1) LCA with preprocessing
Implement your solution
Returning position, not value
Rectangle min/max queries
Choosing the right tool
Many queries, no updates
Storing more information
Implement your solution
Tracking position
For non-idempotent operations
2D rectangle queries
XOR is not idempotent
Prefix XOR approach
2D static queries
2D prefix sums
Debugging tips
When memory matters
The reverse direction
Top-k in a range
Test your understanding
What you learned
Perfect introduction to sparse tables with classic RMQ on static array. Standard O(n log n) preprocessing with O(1) queries for learning fundamentals.
Combines LCA with sparse table and tree distance calculations. Teaches how to find equidistant nodes using binary lifting.
Classic LCA problem using sparse table with path queries on trees. Introduces difference arrays on trees combined with LCA.
Demonstrates sparse table for range GCD queries combined with binary search. Shows how GCD is idempotent and perfect for sparse tables.
Creative range GCD with difference arrays. Transforms finding longest subarray with GCD > 1, teaching problem transformation with sparse tables.
Advanced sparse table on trees storing multiple values per node. Combines LCA with binary lifting while maintaining top-k elements.
Uses binary jumping (sparse table) for ancestor queries on trees. Teaches DSU on tree combined with sparse table for counting relatives.
Combines MST with LCA using sparse table to find maximum edge on paths. Shows range maximum queries for tree path optimization.
Sliding window with sparse table for range maximum queries. Demonstrates two-pointer technique with O(1) RMQ in dynamic windows.
Range frequency queries with Mo's algorithm and sparse table optimizations. Great for understanding offline query processing.
Uses monotonic stack to find previous/next smaller elements (related to RMQ). Teaches how range minimum concepts apply to optimization.
Tree DP with rerooting technique using binary lifting concepts. Shows how sparse table principles extend to DP on trees.
Pure binary lifting implementation, the foundation of sparse tables on trees. Perfect for O(n log n) preprocessing with O(log n) queries.
Classic range maximum solvable with sparse table for O(1) per query. Alternative to deque approach for fixed-size sliding windows.
Offline queries with range operations requiring sorted processing. Demonstrates when sparse table preprocessing concepts apply.
Perfect example of prefix operations for associative functions. Shows when sparse table is overkill versus prefix arrays.
RMQ application finding next smaller elements. Fundamental problem showing how range minimum concepts solve optimization problems.
Range minimum with constraint optimization. Teaches practical RMQ where minimum value in range determines score.
Offline queries with sorted processing (similar to sparse table philosophy). Shows preprocessing for efficient bitwise query answering.
Advanced range query with negative numbers requiring monotonic deque. Demonstrates when simple sparse table fails for non-idempotent operations.
