dsa4 min read
Segment Trees & BIT (Fenwick Tree) — Complete Guide
Master Segment Trees and Binary Indexed Trees: range sum/min/max queries, point updates, lazy propagation, and 5-language implementations.
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Binary-indexed-tree
16 articles
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Range Sum Query Mutable — BIT / Segment Tree
Support point updates and range sum queries. BIT (Fenwick Tree) gives O(log n) both operations with elegant implementation.
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Count of Smaller Numbers After Self — BIT + Coordinate Compress
Count smaller elements to the right of each element. Process from right to left; BIT tracks seen elements. Coordinate compress values to BIT indices.
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Reverse Pairs — Merge Sort / BIT
Count pairs (i,j) where i<j and nums[i] > 2*nums[j]. Modified merge sort: during merge, count pairs across left/right halves.
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My Calendar I — Sorted Dict / Segment Tree
Book events without double-booking. Use sorted dictionary to find previous and next events; check overlap with both.
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Range Module — Segment Tree / SortedDict
Track coverage of half-open intervals. addRange, removeRange, queryRange. Sorted dictionary of disjoint intervals with merge/split logic.
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Falling Squares — Coordinate Compress + Segment Tree
Simulate falling squares, track tallest stack height. Coordinate compress x-coordinates, use segment tree with lazy propagation for range max.
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Count of Range Sum — Merge Sort / BIT
Count subarray sums within [lower, upper]. Use prefix sums and merge sort: during merge count pairs of prefix sums with difference in range.
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Number of Longest Increasing Subsequences — DP + BIT
Count all LIS of maximum length. DP: track length and count. Segment tree on values enables O(n log n) solution.
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Shifting Letters II — Difference Array
Apply range shift operations to letters. Difference array: mark +1 at start and -1 at end+1, prefix sum gives net shift at each position.
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Range Sum Query 2D Immutable — 2D Prefix Sum
Sum of rectangle region. Precompute 2D prefix sums: prefix[i][j] = sum of all cells from (0,0) to (i-1,j-1). Inclusion-exclusion for query.
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My Calendar III — Difference Array / Seg Tree
Track maximum booking overlaps at any time. Difference array: +1 at start, -1 at end; running max of prefix sum. Lazy segment tree for dynamic range.
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Max Sum Rectangle No Larger than K — BIT + Prefix
Find max rectangle sum <= k in 2D matrix. Fix row boundaries, compress to 1D array, use prefix sums + BIT/sorted set to find best subarray sum <= k.
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Longest Subarray with Sum ≤ K — Monotonic Queue + Prefix
Find longest subarray with sum <= k. Build prefix sums, use monotonic deque for decreasing prefix sums to efficiently find left boundaries.
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Product of Array Except Self — Prefix + Suffix Products
For each element compute product of all other elements. Two-pass: left prefix products then multiply with right suffix products.
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Segment Trees & BIT — Master Recap & Cheatsheet
Complete Segment Tree and BIT cheatsheet: templates for BIT, segment tree, lazy propagation, 2D BIT, and problem index.
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