Range Sum Query - Immutable

The problem asks for an implementation that can efficiently answer multiple queries for the sum of elements in a given range of an integer array nums.

We can solve this problem using a concept called “Prefix Sum”. A prefix sum array contains the sum of all elements up to a given index, which allows us to quickly calculate the sum of elements in any given range.

Approach:

1. Create Prefix Sum Array: In the constructor, create a prefix sum array by iterating through the given nums array and calculating the sum of elements up to each index.

2. Calculate Sum of Range: In the sumRange method, use the prefix sum array to calculate the sum of elements between indices left and right inclusive. The sum is given by the difference between the prefix sums at indices right and left - 1.

Here’s the code:

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class NumArray:

    def __init__(self, nums: List[int]):
        self.prefix_sum = [0] * (len(nums) + 1)
        for i in range(len(nums)):
            self.prefix_sum[i + 1] = self.prefix_sum[i] + nums[i]

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

This implementation ensures that the sumRange query is handled in (O(1)) time, and the space complexity is (O(n)), where (n) is the length of the nums array. The constructor takes (O(n)) time to create the prefix sum array.

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private int[] sum;

public NumArray(int[] nums) {
    sum = new int[nums.length + 1];
    for (int i = 0; i < nums.length; i++) {
        sum[i + 1] = sum[i] + nums[i];
    }
}

public int sumRange(int i, int j) {
    return sum[j + 1] - sum[i];
}

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    def __init__(self, nums: List[int]):
        self.sum = []
        sum_till = 0
        for i in nums:
            sum_till += i
            self.sum.append(sum_till)

    def sumRange(self, i: int, j: int) -> int:
        if i > 0 and j > 0:
            return self.sum[j] - self.sum[i - 1]
        else:
            return self.sum[i or j]

Problem Classification

This problem falls into the following categories:

1. Data Structure: The problem is asking to implement a class which is a fundamental concept in Object-Oriented Programming. In addition, this class is required to encapsulate an integer array, which is a basic data structure.

2. Array Processing: The problem involves performing computations on an array which includes calculating sums between given indices.

3. Prefix Sum: The nature of the query — the sum of elements between two indices — suggests the use of a prefix sum array. Prefix sum is a common technique used in array processing problems to speed up the calculation of sums in a range of values in an array.

4. Query Handling: The problem involves handling multiple queries, which implies that the solution must efficiently support repeated operations or lookups.

5. Range Query: Specifically, this problem is a kind of “range query” problem, a common problem type where you’re asked to find or calculate something over a specific range in a data structure.

Language Agnostic Coding Drills

1. Understanding the Problem Statement: The problem is asking for a class implementation with a method to calculate the sum of elements between two indices in a given array. The class should be initialized with an array, and then it should be able to answer multiple queries regarding the sum of ranges.

2. Class and Method Implementation: Understanding the basic principles of object-oriented programming, specifically class creation and method definition.

3. Array Manipulation: Understanding how to handle and manipulate arrays is crucial to solving this problem.

4. Prefix Sum Concept: Learning about the prefix sum technique which involves creating an extra array that stores the sum from the start to the current index.

5. Prefix Sum Implementation: Coding a prefix sum array from the given input array.

6. Creating Class Constructor: Writing a constructor for the class that initializes an instance of the class with the given array and calculates its prefix sum array.

7. Method Definition: Implementing the sumRange method, which calculates the sum of the array between two given indices using the prefix sum array.

8. Subtraction Operation: Understanding that the sum of a range in the prefix sum array can be calculated by subtracting the sum at the start index-1 from the sum at the end index.

Here’s the step by step approach:

• Start by understanding how to define a class and its methods in Python. This is a basic concept of Object-Oriented Programming (OOP).

• Understand what the problem asks. You are given an array of integers and a set of queries. Each query asks for the sum of all numbers between two indices in the array.

• To solve this problem efficiently, we need to understand the concept of Prefix Sum. A Prefix Sum array is a data structure that helps to answer multiple queries efficiently. It is an array where each element is the sum of all elements before it in the original array, including the element at the current index.

• The next step is implementing the Prefix Sum array. Given an array, create a Prefix Sum array. The element at index i of the Prefix Sum array is the sum of all elements from the 0th index to the ith index (inclusive) in the original array.

• Now, we’ll define our class constructor. This is where we’ll create our Prefix Sum array from the given input array.

• The next step is defining the sumRange function. This function will take in two parameters i and j, which represent the start and end indices of the range respectively.

• Inside the sumRange function, we calculate the sum of the elements between the ith index and jth index using the Prefix Sum array. If i is greater than 0, subtract the prefix sum at index i-1 from the prefix sum at index j to get the sum of the elements in the range i to j inclusive. If i is 0, return the prefix sum at index j.

• By implementing this method, we can efficiently answer multiple queries in constant time, O(1), after an initial preprocessing time of O(n), where n is the size of the given array.

Targeted Drills in Python

1. Understanding the Problem Statement

   No specific coding drill for this step. This is more about comprehension and analysis of the problem.

2. Class and Method Implementation

   Drill: Create a simple class with an instance variable and a method.

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   class MyClass: def __init__(self, x): self.x = x def print_x(self): print(self.x)

3. Array Manipulation

   Drill: Given an array, write a function to calculate the sum of its elements.

   1 2	def sum_array(arr): return sum(arr)

4. Prefix Sum Concept

   No specific coding drill for this step. This is more about understanding the concept of prefix sums.

5. Prefix Sum Implementation

   Drill: Given an array, write a function to create its prefix sum array.

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   def prefix_sum(arr): prefix = [0] * len(arr) prefix[0] = arr[0] for i in range(1, len(arr)): prefix[i] = prefix[i - 1] + arr[i] return prefix

6. Creating Class Constructor

   Drill: Implement a class with a constructor that takes an array as input and creates a prefix sum array.

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   class PrefixArray: def __init__(self, arr): self.prefix = [0] * len(arr) self.prefix[0] = arr[0] for i in range(1, len(arr)): self.prefix[i] = self.prefix[i - 1] + arr[i]

7. Method Definition

   Drill: Add a method to the class that returns the sum of elements between two given indices.

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   class PrefixArray: # constructor code here ... def range_sum(self, i, j): if i == 0: return self.prefix[j] else: return self.prefix[j] - self.prefix[i - 1]

8. Subtraction Operation

   Drill: Given two numbers, return their difference.

   1 2	def subtract(a, b): return a - b

You can practice these drills separately and finally integrate them to solve the problem.