Algorithm Fenwick Tree And Segment Tree Intro Complete Guide

 Last Update:2025-06-22T00:00:00     .NET School AI Teacher - SELECT ANY TEXT TO EXPLANATION.    8 mins read      Difficulty-Level: beginner

Understanding the Core Concepts of Algorithm Fenwick Tree and Segment Tree Intro

Algorithm: Fenwick Tree (Binary Indexed Tree) and Segment Tree Intro

Fenwick Tree (Binary Indexed Tree)

Definition: Fenwick Tree is a full binary tree that allows you to compute the prefix sum of any element in the array in O(log n) time and update any element in O(log n) time. The structure is efficient and requires just O(n) space to store the values.

Key Features:

  • Memory Efficiency: Uses O(n) memory for storing n elements.
  • Fast Updates and Queries: Both update and prefix sum operations can be done in O(log n) time.
  • Simplicity: Relatively easy to implement compared to other data structures like Segment Trees.

Construction:

  • Fenwick Tree is constructed from bottom to top.
  • Each node in the tree represents an interval in the input array, and it stores the sum of all numbers within that interval.
  • Each leaf node corresponds to an element in the input array.

Important Information:

  1. Index Mapping: Indexing in Fenwick Tree starts from 1 (not 0). This simplifies certain bit manipulations.
  2. Bit Trick: Fenwick Tree utilizes a clever bit manipulation trick to determine the parent or child nodes. For example, if the index i of a node in the tree is odd, the parent of the node is at index (i + (i & (-i))) / 2, and if i is even, the parent is at index i / 2.
  3. Prefix Sum Calculation: To calculate the prefix sum up to an index i, move backwards from i by reducing it with i & (-i) until it reaches 0.
  4. Update Operation: When updating an element at index i with a value val, increment all nodes that represent intervals containing i. Use i += i & (-i) to find these nodes.
  5. Zero-based Arrays: If the input array uses zero-based indexing, a common practice is to shift every index by one in the Fenwick Tree implementation.

Example Code: Here’s a simple implementation of a Fenwick Tree in Python for calculating prefix sums and updating:

class FenwickTree:
    def __init__(self, size):
        self.size = size
        self.tree = [0] * (size + 1)

    def update(self, idx, delta):
        # Increase idx by 1 to use 1-based indexing
        idx += 1
        while idx <= self.size:
            self.tree[idx] += delta
            idx += idx & -idx

    def prefix_sum(self, idx):
        # Increase idx by 1 to use 1-based indexing
        idx += 1
        sum_ = 0
        while idx > 0:
            sum_ += self.tree[idx]
            idx -= idx & -idx
        return sum_

# Example usage
arr = [1, 2, 3, 4, 5]
fenwick_tree = FenwickTree(len(arr))
for i, val in enumerate(arr):
    fenwick_tree.update(i, val)

# Query prefix sums
print(fenwick_tree.prefix_sum(2))  # Output: 6 (prefix sum from index 0 to 2 is 1+2+3)

Segment Tree

Definition: Segment Tree is a binary tree where each node contains information about a segment (or range) of the input array. It is particularly useful for range queries and range updates.

Key Features:

  • Range Queries: Can handle point updates and range queries in logarithmic time.
  • Range Updates: More sophisticated implementations can support range updates in logarithmic time as well.
  • Flexibility: More flexible than Fenwick Tree and can be adapted to solve more complex problems.

Construction:

  • Segment Trees are built top-down, with each node being the parent of two sub-nodes.
  • Typically, the tree has a height of log n, and the total number of nodes is approximately 2n.

Important Information:

  1. Tree Representation: Segment Trees can be represented using arrays, where the root of the tree occupies position 1, the left child of position k occupies 2k, and the right child occupies 2k+1.
  2. Building the Tree: To build the tree, recursively divide the array into segments and store the information (like sum, max, min) about each segment in the corresponding node.
  3. Point Update: Updating an element in the original array requires updating all nodes in the tree that correspond to intervals containing that element. This can be done in O(log n) time.
  4. Range Queries: Perform range queries using the precomputed information stored in the nodes. Range queries can also be done in O(log n) time.
  5. Lazy Propagation: To efficiently handle range updates, lazy propagation can be employed. This technique defers updates and combines them when necessary, further optimizing the range update operation.

Example Code: Here is a simple implementation of a Segment Tree for calculating sum over a range and updating elements:

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Step-by-Step Guide: How to Implement Algorithm Fenwick Tree and Segment Tree Intro

Fenwick Tree (Binary Indexed Tree) Introduction

A Fenwick Tree, also known as a Binary Indexed Tree (BIT), is a data structure that provides efficient methods for computing the prefix sums of an array of numbers. This is useful in scenarios where you need to perform both range sum queries and updates to an array.

Structure

  • Array Representation: The Fenwick Tree is represented as an array BIT[], where BIT[i] stores the prefix sum of a specific subset of the original array.
  • Indexing: BIT uses 1-based indexing, which simplifies calculations.

Operations

  • Sum Query: Computing the sum of elements from the beginning of the array to a specific index i.
  • Update: Adding a value to a specific index i and updating the relevant entries in the BIT.

Step-by-Step Example

1. Initialize the Fenwick Tree Assume we have an array arr = [1, 2, 3, 4, 5].

def initialize(n):
    return [0] * (n + 1)  # 1-based indexing

n = 5
BIT = initialize(n)

2. Build the Fenwick Tree To build the BIT, we need to insert each element of the array into the BIT.

def build(arr):
    n = len(arr)
    BIT = initialize(n)
    for i in range(1, n + 1):  # Note: BIT uses 1-based indexing
        update(BIT, i, arr[i - 1])  # Convert back to 0-based indexing for arr
    return BIT

def update(BIT, i, val):
    while i < len(BIT):
        BIT[i] += val
        i += (i & -i)  # Move to the parent

arr = [1, 2, 3, 4, 5]
BIT = build(arr)
print(BIT)  # Output: [0, 1, 3, 3, 10, 5]

3. Query the Fenwick Tree To get the prefix sum up to index i, we traverse the BIT.

def query(BIT, i):
    s = 0
    while i > 0:
        s += BIT[i]
        i -= (i & -i)  # Move to the parent
    return s

print(query(BIT, 5))  # Output: 15 (Sum of arr[0] to arr[4])
print(query(BIT, 3))  # Output: 6 (Sum of arr[0] to arr[2])

4. Update the Fenwick Tree To add a value to index i, we update the BIT accordingly.

update(BIT, 4, 5)  # Add 5 to index 3 (0-based indexing, so 4 in BIT is 3 in arr)
print(query(BIT, 5))  # Output: 20 (Updated sum)

Segment Tree Introduction

A Segment Tree is a data structure that divides an array into segments and helps in answering range queries and point updates effectively. It is primarily used for solving problems involving array ranges.

Structure

  • Binary Tree Representation: The Segment Tree is a binary tree where each node contains a range of the array. The root node represents the whole array, and the leaf nodes represent individual array elements.
  • Array Representation: The Segment Tree can be represented as a 1-dimensional array.

Operations

  • Sum Query: Computing the sum of elements over a specific range.
  • Update: Modifying a specific element in the array and updating the relevant nodes in the Segment Tree.

Step-by-Step Example

1. Build the Segment Tree Assume we have an array arr = [1, 2, 3, 4, 5].

def build_segment_tree(arr, node, l, r, seg_tree):
    if l == r:  # Leaf node
        seg_tree[node] = arr[l]
    else:
        mid = (l + r) // 2
        left_child = 2 * node + 1
        right_child = 2 * node + 2
        build_segment_tree(arr, left_child, l, mid, seg_tree)
        build_segment_tree(arr, right_child, mid + 1, r, seg_tree)
        seg_tree[node] = seg_tree[left_child] + seg_tree[right_child]

def initialize_segment_tree(n):
    size = 1
    while size < n:
        size *= 2
    return [0] * (2 * size - 1)

arr = [1, 2, 3, 4, 5]
n = len(arr)
seg_tree = initialize_segment_tree(n)
build_segment_tree(arr, 0, 0, n - 1, seg_tree)
print(seg_tree)  # Output: [15, 6, 9, 3, 3, 4, 5, 0, 0]

2. Query the Segment Tree To get the sum of elements in a specific range [L, R], we traverse the Segment Tree.

def query_segment_tree(node, l, r, ql, qr, seg_tree):
    if ql > r or qr < l:  # No overlap
        return 0
    if ql <= l and r <= qr:  # Total overlap
        return seg_tree[node]
    mid = (l + r) // 2
    left_child = 2 * node + 1
    right_child = 2 * node + 2
    return (query_segment_tree(left_child, l, mid, ql, qr, seg_tree) +
            query_segment_tree(right_child, mid + 1, r, ql, qr, seg_tree))

print(query_segment_tree(0, 0, n - 1, 1, 3, seg_tree))  # Output: 9 (Sum of arr[1] to arr[3])

3. Update the Segment Tree To modify an element and update the Segment Tree, we traverse the tree and update the nodes.

Top 10 Interview Questions & Answers on Algorithm Fenwick Tree and Segment Tree Intro

Top 10 Questions and Answers on Fenwick Tree and Segment Tree Introduction

  • Answer: A Fenwick Tree, also known as a Binary Indexed Tree (BIT), is a data structure that provides efficient methods for cumulative frequency tables. It handles two primary operations: point updates and prefix sum queries in (O(\log n)) time, where (n) is the number of elements. Fenwick Trees are typically used for scenarios involving dynamic arrays where we need to perform frequent prefix sum queries and individual element updates.

2. What is a Segment Tree?

  • Answer: A Segment Tree is a binary tree data structure used for storing and querying ranges of array elements. Each node represents an interval or segment of the array and contains information about that segment. Segment Trees are powerful for handling range queries (like finding the sum or minimum over a segment) and can be updated to reflect changes in the array. Both construction and range queries can be performed in (O(\log n)) time.

3. How does a Fenwick Tree differ from a Segment Tree?

  • Answer: While both data structures are used for range queries and updates, they differ in their structure and use cases. Fenwick Trees are more space-efficient, requiring only an array of size (n). They are simpler to implement and primarily used for prefix sum queries. Segment Trees are more versatile and can handle a wider variety of range operations (e.g., range maximum, minimum, and sum) and can be extended to 2D, 3D, or higher dimensions. However, they require (\Theta(n)) additional space and are more complex to implement.

4. How do you construct a Fenwick Tree?

  • Answer: The construction of a Fenwick Tree involves initializing it with zeros and then updating it with the values of the original array. The tree is built by iterating through each element of the array and performing an update operation similar to what would be done in a regular point update. The update operation involves adding the value to the current index and then moving to the next index obtained by adding the least significant bit of the current index to itself.

5. How do you construct a Segment Tree?

  • Answer: A Segment Tree is typically built recursively. The process starts by defining the root node to represent the entire array. Each leaf node in the tree will represent a single element in the array. Each internal node will represent a subarray, with the left and right children representing the left and right halves of the parent node’s subarray, respectively. During construction, each node stores the computed result for its segment, such as the sum or minimum of its subarray. The time complexity for this construction is (O(n)).

6. What are common use cases of Fenwick Trees?

  • Answer: Fenwick Trees are commonly used in scenarios that require frequent prefix sums or rank/percentile queries, such as in order statistics, frequency counting, and in problems involving cumulative sums or differences. They are particularly useful when the array is large and dynamic, involving frequent updates.

7. What are common use cases of Segment Trees?

  • Answer: Segment Trees are used in scenarios where range queries, such as finding sums or minimum values over intervals, are required frequently, and the array can be large and dynamic. They are also used in problems that require updates over intervals, such as setting all elements in a range to a given value, incrementing/decrementing all elements in a range, etc. Segment Trees extend beyond just point updates and support more advanced types of queries compared to Fenwick Trees.

8. How do you perform a range sum query on a Fenwick Tree?

  • Answer: A range sum query on a Fenwick Tree can be performed efficiently in (O(\log n)) time by leveraging the property that the tree stores prefix sums. For a single point prefix sum query, the query starts at a given index and accumulates the values by moving to the parent nodes obtained by subtracting the least significant bit of the current index from itself. To convert this into a range sum query between indices (l) and (r), perform two prefix sum queries: one for (r) and the other for (l-1), and then subtract the result of the latter from the former.

9. How do you perform a range sum query on a Segment Tree?

  • Answer: A range sum query on a Segment Tree is performed by traversing the tree and combining information from relevant nodes that correspond to the query range. The query starts at the root and recursively examines the left and right children depending on whether their segments intersect with the query range. If a segment is fully contained within the query range, its stored value is used directly. The query process ensures that only a logarithmic number of nodes are accessed and combined, resulting in a time complexity of (O(\log n)).

10. Are there variations of the Fenwick Tree and Segment Tree?

  • Answer: Yes, there are various variations of both Fenwick Trees and Segment Trees to optimize them for specific types of queries and operations. For Fenwick Trees, there are 2D and 3D BITs that extend the concept to handle multi-dimensional prefix sum queries. For Segment Trees, there are Range Minimum Query (RMQ) Segment Trees, Lazy Propagation Segment Trees (which can handle range updates), Persistent Segment Trees (which can preserve all versions of the segment tree after updates), and Range Maximum Query (RMQ) Segment Trees. These variations offer enhanced capabilities and flexibility to handle diverse problems.

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