The Dynamic Array

The Dynamic Array

Data structures are critical for solving complex problems, such as efficiently managing and querying a collection of sequences.

The "Dynamic Array" problem demonstrates how to solve such challenges using Python.

While exploring this problem, we will focus on the practical application of arrays, bitwise operations, and modular arithmetic.

Understanding the Problem

The dynamic array problem revolves around managing a collection of sequences and performing two types of operations:

Appending values to a specific sequence: Determine the sequence index using a combination of inputs and store the value in that sequence.
Querying a value from a sequence: Retrieve a specific value based on the index calculated from inputs. Finally, we update a global variable using the value we have retrieved.

However, the key challenge lies in efficiently calculating the sequence index using bitwise XOR and modulo operations to guarantee quick access and updates.

Problem Statement

Input

1. n(Integer)- refers to the number of sequences to initialize.
2. A list of queries, where each query is either of type 1 or type 2:
   1. Type 1: Append an integer y to a specific sequence.
   2. Type 2: Retrieve a value from a sequence and update a global variable.

Output

A list of integers, each representing the result of a type 2 query.

What are the Constraints

1. The number of queries can be large, making efficient computation possible.
2. Calculations involve bitwise XOR (^) and modulo (%) operations. They determine the sequence indices.

High-Level Thought Process

• Initialize the Data Structure: Create a list of empty lists to represent n sequences.
• Efficient Query Execution:
  • Use bitwise XOR and modulo to calculate the sequence index.
  • For type 1 queries, append the given value to the sequence.
  • For type 2 queries, retrieve and store the result of the global variable update.
• Output the Results: Return the list of results from type 2 queries.

Implementation in Python

Below is the Python implementation of the dynamic array problem:

#!/bin/python3

import os

def dynamicArray(n, queries):
    arr = [[] for _ in range(n)]  # Create a list of n empty lists
    last_answer = 0
    results = []

    for query in queries:
        t, x, y = query  # Unpack the query directly

        # Calculate the index using XOR and modulo
        idx = (x ^ last_answer) % n

        if t == 1:
            arr[idx].append(y)  # Append y to the correct sequence
        elif t == 2:
            # Retrieve the element and update last_answer
            last_answer = arr[idx][y % len(arr[idx])]
            results.append(last_answer)  # Add to results

    return results

if __name__ == '__main__':
    fptr = open(os.environ['OUTPUT_PATH'], 'w')

    first_multiple_input = input().rstrip().split()
    n = int(first_multiple_input[0])
    q = int(first_multiple_input[1])

    queries = []
    for _ in range(q):
        queries.append(list(map(int, input().rstrip().split())))

    result = dynamicArray(n, queries)

    fptr.write('\n'.join(map(str, result)))
    fptr.write('\n')

    fptr.close()

Detailed Breakdown

Initialization

arr = [[] for _ in range(n)]

This creates a list of n empty lists. Each list represents a sequence where values can be appended.

Processing Queries

Each query is processed in a loop. For example:~

Query Type 1: Append

if t == 1:
    arr[idx].append(y)

• Calculate the sequence index using:

  idx = (x ^ last_answer) % n

  Here, x ^ last_answer applies a bitwise XOR operation, and % n ensures the index stays within bounds.

• Append the value y to the sequence at index idx.

Query Type 2: Retrieve and Update

elif t == 2:
    last_answer = arr[idx][y % len(arr[idx])]
    results.append(last_answer)

• Calculate the sequence index as before.

• Use y % len(arr[idx]) to determine the specific element within the sequence.

• Update last_answer and store it in the results list.

Output Results

return results

The function returns the results of all type 2 queries in the order they are encountered.

Example Walkthrough

Input:

n = 2
queries = [
    [1, 0, 5],
    [1, 1, 7],
    [1, 0, 3],
    [2, 1, 0],
    [2, 1, 1]
]

Execution:

1. Initialization:

   arr = [[], []]
   last_answer = 0
   results = []

2. Processing Queries:

   • Query [1, 0, 5]: Append 5 to arr[(0 ^ 0) % 2] (i.e., arr[0]).

     arr = [[5], []]

   • Query [1, 1, 7]: Append 7 to arr[(1 ^ 0) % 2] (i.e., arr[1]).

     arr = [[5], [7]]

   • Query [1, 0, 3]: Append 3 to arr[(0 ^ 0) % 2] (i.e., arr[0]).

     arr = [[5, 3], [7]]

   • Query [2, 1, 0]: Retrieve arr[(1 ^ 0) % 2][0 % len(arr[1])] (i.e., arr[1][0] = 7).

     last_answer = 7
     results = [7]

   • Query [2, 1, 1]: Retrieve arr[(1 ^ 7) % 2][1 % len(arr[1])] (i.e., arr[1][1] = 3).

     last_answer = 3
     results = [7, 3]

Output:

[7, 3]

Benefits of using this approach

Efficiency: Use XOR and modulo to enforce quick index calculation.
Scalability: The dynamic structure adapts to any number of sequences and queries.
Modular Design: Each query type is handled independently, simplifying code maintenance.

What's Next?

Understanding the Dynamic Array problem helps you solve similar challenges.

More so when dealing with dynamic data structures.

To deepen your expertise, apply these concepts to other issues, such as caching mechanisms or custom indexing systems.