#303

Easy

LeetCode Problem Workspace

Range Sum Query - Immutable

The Range Sum Query - Immutable problem involves implementing a data structure to handle range sum queries efficiently.

Problem Statement

You are given an integer array nums. You need to implement the NumArray class which supports the sumRange(left, right) method that returns the sum of elements in the range [left, right] inclusive. The NumArray class should be initialized with the array nums and the sumRange method should be optimized for multiple queries.

The challenge is to provide an efficient way to compute the range sum for multiple queries. A naive approach might involve summing the elements directly for each query, but this would be too slow for large arrays. You must design the class to handle a high volume of queries efficiently, focusing on reducing repeated work.

Examples

Example 1

Input: See original problem statement.

Output: See original problem statement.

Input ["NumArray", "sumRange", "sumRange", "sumRange"] [[[-2, 0, 3, -5, 2, -1]], [0, 2], [2, 5], [0, 5]] Output [null, 1, -1, -3]

Explanation NumArray numArray = new NumArray([-2, 0, 3, -5, 2, -1]); numArray.sumRange(0, 2); // return (-2) + 0 + 3 = 1 numArray.sumRange(2, 5); // return 3 + (-5) + 2 + (-1) = -1 numArray.sumRange(0, 5); // return (-2) + 0 + 3 + (-5) + 2 + (-1) = -3

Constraints

1 <= nums.length <= 104
-105 <= nums[i] <= 105
0 <= left <= right < nums.length
At most 104 calls will be made to sumRange.

Solution Approach

Prefix Sum Array

Use a prefix sum array where each element at index i stores the sum of all elements in nums up to index i. With this, any sum range query can be computed in constant time by using the formula: sumRange(left, right) = prefix[right] - prefix[left - 1].

Optimized Initialization

To minimize overhead, the prefix sum array should be built during the initialization phase. This allows each query to be answered in constant time, making the solution efficient for large arrays and multiple queries.

Space-Time Trade-off

The space complexity increases due to the storage of the prefix sum array, but this is justified by the performance gain in time complexity. The solution offers O(1) time per query after O(n) preprocessing for the prefix sum.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity for preprocessing the array into the prefix sum array is O(n), where n is the length of the input array nums. Each range sum query can then be answered in O(1) time. The space complexity is O(n) due to the storage of the prefix sum array, which requires extra space proportional to the size of nums.

What Interviewers Usually Probe

Understanding the trade-off between preprocessing and query time is crucial.
Candidates should demonstrate an ability to optimize for multiple queries.
Pay attention to how the solution scales with large inputs.

Common Pitfalls or Variants

Common pitfalls

Not optimizing for multiple queries by recalculating sums from scratch.
Incorrectly calculating the range sum due to off-by-one errors with indices.
Failing to properly initialize the prefix sum array, leading to incorrect query results.

Follow-up variants

Query sum over a segment of the array after each update.
Handle range queries with modifications to the array in between.
Designing a solution for a multi-dimensional range sum query.

FAQ

What is the key pattern for solving Range Sum Query - Immutable?

The key pattern is using a prefix sum array to optimize range sum queries. This allows each query to be answered in constant time after an O(n) preprocessing step.

How can I improve my solution to Range Sum Query - Immutable?

Focus on the initialization phase and make sure to build the prefix sum array efficiently. Also, ensure that you handle queries in constant time after the preprocessing.

What are the main challenges in solving Range Sum Query - Immutable?

The main challenge is designing the solution to handle multiple queries efficiently, balancing the space used by the prefix sum array with the time required for queries.

Is Range Sum Query - Immutable a common pattern in coding interviews?

Yes, this problem is commonly used to assess understanding of arrays and optimization techniques such as prefix sum for efficient querying.

How can GhostInterview assist with Range Sum Query - Immutable?

GhostInterview provides insights into the prefix sum technique and helps you avoid common pitfalls, guiding you to optimize for both space and time complexity.

terminal

Solution

Solution 1: Prefix Sum

We create a prefix sum array $s$ of length $n + 1$, where $s[i]$ represents the prefix sum of the first $i$ elements, that is, $s[i] = \sum_{j=0}^{i-1} nums[j]$. Therefore, the sum of the elements between the indices $[left, right]$ can be expressed as $s[right + 1] - s[left]$.

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class NumArray:
    def __init__(self, nums: List[int]):
        self.s = list(accumulate(nums, initial=0))

    def sumRange(self, left: int, right: int) -> int:
        return self.s[right + 1] - self.s[left]


# Your NumArray object will be instantiated and called as such:
# obj = NumArray(nums)
# param_1 = obj.sumRange(left,right)

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