Wiki Article · Cinematic

Fenwick Tree

Fenwick tree, jise Binary Indexed Tree (BIT) bhi kehte hain, ek data structure hai jo prefix sum…

Overview

Fenwick Tree — Quick Facts

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Property: Detail

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Full Name: Fenwick Tree / Binary Indexed Tree…

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Inventor: Peter M. Fenwick

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Year: 1994

Topic 1

Infobox

📚 | Property | Detail | |---|---| | Full Name | Fenwick Tree / Binary Indexed Tree (BIT) |…

Topic 2

Introduction

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Update: $O(1)$ — direct index par…

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Query: $O(n)$ — 1 se i tak loop…

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Update: $O(n)$ — ek element change…

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Query: $O(1)$ — direct lookup

Topic 3

Core Intuition

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$i = 12$ (binary: 1100) → LSB = 4…

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$i = 10$ (binary: 1010) → LSB = 2…

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$i = 8$ (binary: 1000) → LSB = 8 →…

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$i = 7$ (binary: 0111) → LSB = 1 →…

Topic 4

Lowest Set Bit (LSB)

⭐ LSB nikalna — yeh Fenwick tree ki sabse critical operation hai. Formula: $$\text{LSB}(i)…

Topic 5

Tree Structure

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Har node $i$ ke "parent" = $i +…

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Har node $i$ ka "previous range" =…

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BIT[1] = a[1] (1 element)

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BIT[2] = a[1] + a[2] (2 elements)

Topic 6

Operations

✨ Point Update Index $i$ ki value me $\delta$ add karna: 1. BIT[$i$] me $\delta$ add karo…

Topic 7

Construction

🌟 Naive Construction — $O(n \log n)$ Empty BIT banao, phir har element ke liye update call…

Topic 8

Range Sum Query

🚀 Prefix sum se range sum easily nikalta hai: $$\text{sum}(l, r) = \text{query}(r) -…

Topic 9

Complexity Comparison

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Implementation simpler (50% less…

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Space efficient (exactly $n$…

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Constant factor smaller (faster in…

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But: less flexible (segment tree…

Topic 10

Fenwick Tree Visualization

💡 `mermaid graph TD subgraph "BIT Array for n=8" B8["BIT[8] = sum(1..8)"] B4["BIT[4] =…

Topic 11

Applications

🎯 Prefix Sum Queries Sabse basic use case — dynamic array me prefix sums efficiently…

Topic 12

Variants

⭐ Range Update BIT Difference array technique ke saath BIT combine karke range update +…

Topic 13

Limitations

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Range update + range query…

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Minimum/Maximum queries…

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Lazy propagation jaise advanced…

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1-indexed hona chahiye — 0-indexed…

Comparison

Complexity Comparison

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Naive Array: $O(1)$

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Prefix Sum Array: $O(n)$

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Fenwick Tree (BIT): $O(\log n)$

Diagram

Visual Flow

📊 Diagram visualization — details in narration

Quiz — Question 1

Fenwick Tree ka sabse sahi definition kya hai?

Quick Quiz

Quiz — Question 2

Fenwick Tree ka 'Full Name' kya hai?

Complete! 🎉

Fenwick Tree Complete!

Aliens Wiki · HIEN · Cinematic Knowledge

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Fenwick Tree Complete

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Fenwick Tree

Fenwick Tree — Quick Facts

Property: Detail

Full Name: Fenwick Tree / Binary Indexed Tree…

Inventor: Peter M. Fenwick

Year: 1994

Infobox

Introduction

Update: $O(1)$ — direct index par…

Query: $O(n)$ — 1 se i tak loop…

Update: $O(n)$ — ek element change…

Query: $O(1)$ — direct lookup

Core Intuition

$i = 12$ (binary: 1100) → LSB = 4…

$i = 10$ (binary: 1010) → LSB = 2…

$i = 8$ (binary: 1000) → LSB = 8 →…

$i = 7$ (binary: 0111) → LSB = 1 →…

Lowest Set Bit (LSB)

Tree Structure

Har node $i$ ke "parent" = $i +…

Har node $i$ ka "previous range" =…

BIT[1] = a[1] (1 element)

BIT[2] = a[1] + a[2] (2 elements)

Operations

Construction

Range Sum Query

Complexity Comparison

Implementation simpler (50% less…

Space efficient (exactly $n$…

Constant factor smaller (faster in…

But: less flexible (segment tree…

Fenwick Tree Visualization

Applications

Variants

Limitations

Range update + range query…

Minimum/Maximum queries…

Lazy propagation jaise advanced…

1-indexed hona chahiye — 0-indexed…

Complexity Comparison

Naive Array: $O(1)$

Prefix Sum Array: $O(n)$

Fenwick Tree (BIT): $O(\log n)$

Visual Flow

See Also

Segment_tree

Prefix_sum

Binary_search

Data_structure

Algorithm_complexity

Bit_manipulation

Quiz — Question 1

Quiz — Question 2

Fenwick Tree Complete!

Fenwick Tree Complete

Segment_tree