Range Sum Query - Mutable

title: Range Sum Query - Mutable tags: recursion segment tree two-way-comparison

Given an array nums and two types of queries where you should update the value of an index in the array, and retrieve the sum of a range in the array.

Implement the NumArray class:

NumArray(int[] nums) Initializes the object with the integer array nums. void update(int index, int val) Updates the value of nums[index] to be val. int sumRange(int left, int right) Returns the sum of the subarray nums[left, right] which means nums[left] + nums[left + 1], …, nums[right]).

Example 1:
Input
["NumArray", "sumRange", "update", "sumRange"]
[[[1, 3, 5]], [0, 2], [1, 2], [0, 2]]
Output
[null, 9, null, 8]

Explanation

NumArray numArray = new NumArray([1, 3, 5]);
numArray.sumRange(0, 2); // return 9 = sum([1,3,5])
numArray.update(1, 2);   // nums = [1,2,5]
numArray.sumRange(0, 2); // return 8 = sum([1,2,5])

Constraints

1 <= nums.length <= 3 * 104
-100 <= nums[i] <= 100
0 <= index < nums.length
-100 <= val <= 100
0 <= left <= right < nums.length

At most 3 * 104 calls will be made to update and sumRange.

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# input: array
# st: segment tree (array)
# si: index (current index)
# ss: segment range start index
# se: segment range end index
# ue: segment tree update index
def update(st, value, si, ss, se, ui)
  if ss == se
    st[si] = value

    return
  end

   mid = (ss+se)/2
   
   if ui <= mid
     update(st, value, 2*si+1, ss, mid, ui)
   else
     update(st, value, 2*si+2, mid+1, se, ui)  
   end
   
   st[si] = st[2*si+1] + st[2*si+2]
   st
end

def query(st, qs, qe, ss, se, si)
  if qs > se || qe < ss
    return 0
  end
  
  if ss >= qs && se <= qe
    return st[si]
  end
  
  mid = (ss + se)/2
  left = query(st, qs, qe, ss, mid, 2*si+1)
  right = query(st, qs, qe, mid+1, se, 2*si+2)
  
  return left+right
end
# input: array
# st: segment tree (array)
# si: index (current index)
# ss: segment range start index
# se: segment range end index
def construct_tree(st, input, si, ss, se)
  if ss == se    
    st[si] = input[ss]
    
    return
  end

  mid = (ss+se)/2

  construct_tree(st, input, 2*si+1, ss, mid)
  construct_tree(st, input, 2*si+2, mid+1, se)  
  
  st[si] = st[2*si+1] + st[2*si+2] 
  st
end

         # 0 1 2 3 4 5 6 7 8        6 and 15
input = [1,2,3,4,5,6,7,8,9]
st = []
# st = Array.new(4*input.size, 0)

p construct_tree(st, input, 0, 0, input.size-1)
p query(st, 0, 2, 0, input.size-1, 0)
p update(st, 10, 0, 0, input.size-1, 0)
p query(st, 0, 2, 0, input.size-1, 0)

When we update, we need to update the prefix sum from that index to the end. This will be O(n) in time complexity.
Brute Force Approach
- Iterate through the array from index left and right and sum each element.
If we have chunks and prefix for those indices, then we can prevent updating everything in the array from that index. We don’t have overlapping chunks.
How do we know the size of the chunk?
Can we store the prefix sum in a binary tree?
- When we update the index, should we move the node? No
- Updating the rightmost node value, the left side of the tree, the sum will be the same.
Insights
The number of elements is fixed. We update existing elements.
If we have just one chunk (the entire array), this is our immutable array If we have n elements, we can make each element its own chunk (we need to do O(n) work). Both extremes are not ideal and lead to linear time complexity. Can each node in the tree have a range that represents the input. When we fetch the sum range, it can go either left or right and ignore the other half. Better than O(N)

Can we use square root decomposition for non perfect square arrays?

[1, 3, 5]

[1, 4, ]

Construct the segment tree

 0 1 2 3 4 5 6 7 8
[1,2,3,4,5,6,7,8,9]    

[0-8] Root: Sum: 45
middle = 4

Query and update complexity: log m (we reduce the problem by half)

                            [1, 3, 5]

                        [0]

Time and Space Complexity

Time: O(n) - Calculate the sum of one node during each iteration of the for loop. Space: O(n) - Segment tree space used is 2n.

We can use a Fenwick Tree (Binary Indexed Tree) to implement the NumArray class that allows us to update an element and calculate the sum of elements between given indices efficiently. Here’s how to do it:

Initialize NumArray: Initialize the Fenwick Tree with the given nums array.
Update Operation: Update the value of an element at the given index. Since we need to update the Fenwick Tree as well, we’ll adjust the corresponding values in the tree.
Sum Range Operation: Calculate the sum of elements between indices left and right inclusive.

Here’s the code:

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class NumArray:
    def __init__(self, nums: List[int]):
        self.n = len(nums)
        self.nums = nums
        self.bit = [0] * (self.n + 1)
        for i, num in enumerate(nums):
            self._updateBIT(i, num)
            
    def _updateBIT(self, index: int, delta: int) -> None:
        index += 1
        while index <= self.n:
            self.bit[index] += delta
            index += index & -index

    def _queryBIT(self, index: int) -> int:
        index += 1
        result = 0
        while index:
            result += self.bit[index]
            index -= index & -index
        return result

    def update(self, index: int, val: int) -> None:
        delta = val - self.nums[index]
        self.nums[index] = val
        self._updateBIT(index, delta)

    def sumRange(self, left: int, right: int) -> int:
        return self._queryBIT(right) - self._queryBIT(left - 1)

__init__ initializes the Fenwick Tree with the given array.
_updateBIT is a helper function to update the Fenwick Tree.
_queryBIT is a helper function to get the prefix sum up to a given index.
update changes the value at a given index and updates the Fenwick Tree accordingly.
sumRange calculates the sum between given indices by querying the Fenwick Tree.

The update and sumRange operations are efficiently handled in O(log n) time complexity, fulfilling the problem’s constraints.

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class NumArray:
    class SegmentTreeNode:
        def __init__(self, start, end):
            self.start = start
            self.end = end
            self.left = None
            self.right = None
            self.sum = 0

    def __init__(self, nums):
        self.root = self.buildTree(nums, 0, len(nums) - 1)

    def buildTree(self, nums, start, end):
        if start > end:
            return None
        else:
            ret = self.SegmentTreeNode(start, end)
            if start == end:
                ret.sum = nums[start]
            else:
                mid = start + (end - start) // 2
                ret.left = self.buildTree(nums, start, mid)
                ret.right = self.buildTree(nums, mid + 1, end)
                ret.sum = ret.left.sum + ret.right.sum
            return ret

    def update(self, i, val):
        self._update(self.root, i, val)

    def _update(self, root, pos, val):
        if root.start == root.end:
            root.sum = val
        else:
            mid = root.start + (root.end - root.start) // 2
            if pos <= mid:
                self._update(root.left, pos, val)
            else:
                self._update(root.right, pos, val)
            root.sum = root.left.sum + root.right.sum

    def sumRange(self, i, j):
        return self._sumRange(self.root, i, j)

    def _sumRange(self, root, start, end):
        if root.end == end and root.start == start:
            return root.sum
        else:
            mid = root.start + (root.end - root.start) // 2
            if end <= mid:
                return self._sumRange(root.left, start, end)
            elif start >= mid + 1:
                return self._sumRange(root.right, start, end)
            else:
                return self._sumRange(root.right, mid + 1, end) + self._sumRange(root.left, start, mid)

10 Prerequisite LeetCode Problems

For “307. Range Sum Query - Mutable”, the following are a good preparation:

“303. Range Sum Query - Immutable” - This problem is a simpler version of the main problem. It involves only the sumRange operation, without any updates.
“304. Range Sum Query 2D - Immutable” - This problem generalizes the main problem to 2D arrays.
“308. Range Sum Query 2D - Mutable” - Similar to 304, but allows updates. This problem is more complex than the main problem, but solving it may help understand the main problem better.
“198. House Robber” - This problem requires understanding of dynamic programming and can help develop an intuition for range sum queries.
“53. Maximum Subarray” - It introduces the concept of subarrays which is a fundamental part of understanding how to solve range queries.
“70. Climbing Stairs” - It helps understand the principle of problem decomposition, which is needed to implement a segment tree for the main problem.
“121. Best Time to Buy and Sell Stock” - It introduces the concept of tracking minimum and maximum values over an array, similar to tracking sums over a range.
“169. Majority Element” - It provides practice for handling arrays and understanding their properties.
“217. Contains Duplicate” - It provides practice for handling arrays, which are a basic data structure in the main problem.
“283. Move Zeroes” - This problem gives practice for manipulating elements within an array, similar to the update operation in the main problem.

These cover dynamic programming, subarrays, and handling arrays, which are crucial in solving the “Range Sum Query - Mutable” problem.

Problem Classification

Problem Statement:Given an integer array nums, handle multiple queries of the following types:

Update the value of an element in nums. Calculate the sum of the elements of nums between indices left and right inclusive where left <= right. Implement the NumArray class:

NumArray(int[] nums) Initializes the object with the integer array nums. void update(int index, int val) Updates the value of nums[index] to be val. int sumRange(int left, int right) Returns the sum of the elements of nums between indices left and right inclusive (i.e. nums[left] + nums[left + 1] + … + nums[right]).

Example 1:

Input [“NumArray”, “sumRange”, “update”, “sumRange”] [[[1, 3, 5]], [0, 2], [1, 2], [0, 2]] Output [null, 9, null, 8]

Explanation NumArray numArray = new NumArray([1, 3, 5]); numArray.sumRange(0, 2); // return 1 + 3 + 5 = 9 numArray.update(1, 2); // nums = [1, 2, 5] numArray.sumRange(0, 2); // return 1 + 2 + 5 = 8

Constraints:

1 <= nums.length <= 3 * 104 -100 <= nums[i] <= 100 0 <= index < nums.length -100 <= val <= 100 0 <= left <= right < nums.length At most 3 * 104 calls will be made to update and sumRange.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Clarification Questions

What are the clarification questions we can ask about this problem?

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

How to visualize these cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Identify Recursion Invariant

Is there an invariant during recursion in this problem?

Is invariant and invariant during recursion the same for this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the code for the solution of this problem.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Range Sum Query - Mutable

Constraints

10 Prerequisite LeetCode Problems

Problem Classification

Clarification Questions

Problem Analysis and Key Insights

Problem Boundary

Distilling the Problem to Its Core Elements

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Identify Invariant

Identify Loop Invariant

Thought Process

Establishing Preconditions and Postconditions

Problem Decomposition

From Brute Force to Optimal Solution

Code Explanation and Design Decisions

Coding Constructs

Language Agnostic Coding Drills

Targeted Drills in Python

Q&A

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