Movement of Robots

This is a variation of a very classic problem: There are some ants on a plank at some positions. They are moving right or left, and change directions when they bump into each other. When does the last ant fall off the plank? At first glance, this may seem difficult to compute, as many collisions may be happening and it is hard to keep track of them all. But there is a very important key observation here: When two ants bump into each other, we can just act like they “phased” through each other! When one ant is going left, and another is going right, and they bump into each other, the left ant is now going right and the right ant is going left. There would be no difference if we had just swapped the ants, or let them pass through each other.

We can now apply this idea to the robots. Let us ignore any collisions, and just imagine all robots simply passed through each other, as if they were ghosts. Then now we can simply just subtract the distance from them if they were going left or add the distance to them if they were going right. Now, we simply need to compute the distance between each pair of robots. Note that (i, j) and (j,i) should not both be counted, so we will only use a prefix sum. If (j,i) was meant to be counted, we would use a prefix and postfix sum. We can compute this easily by sorting the array, then multiplying the number times the index, minus the prefix sum. This gives us the distance between all pairs of robots without double counting.

With these insights in mind, we’ll follow this strategy:

  1. Ignore the collisions, letting robots “phase” through each other.
  2. Adjust the positions of the robots based on the direction they’re headed and the given time d.
  3. To find the sum of distances between all pairs of robots, sort the array and use a prefix sum to calculate the distances.

Here’s how to implement this:

  1. First, determine the final position of each robot if there were no collisions.
  2. Sort the robots based on their final positions.
  3. Compute the sum of distances between robots using a prefix sum.
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class Solution:
    def sumDistance(self, nums: List[int], s: str, d: int) -> int:
        MOD = 10**9 + 7
        
        # Calculate the positions of robots after d seconds.
        for i in range(len(nums)):
            if s[i] == 'R':
                nums[i] += d
            else:
                nums[i] -= d

        # Sort the robots based on their new positions
        nums.sort()
        
        # Calculate the sum of distances between all pairs of robots
        sum_dist = 0
        prefix_sum = 0
        for i, pos in enumerate(nums):
            sum_dist += i * pos - prefix_sum
            prefix_sum += pos
            sum_dist %= MOD
            
        return sum_dist

This solution now calculates the final position of each robot after a given number of seconds and then calculates the sum of distances between them using the prefix sum.

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class Solution:
    def sumDistance(self, nums: List[int], s: str, d: int) -> int:
        n, m = len(nums), int(1e9 + 7)

        for i in range(n):
            if s[i] == 'L':
                nums[i] -= d
            else: 
                nums[i] += d

        nums.sort()
        pre = nums.copy()

        for i in range(1, n):
            pre[i] += pre[i - 1]
            pre[i] %= m

        ans = 0
        for i in range(1, n):
            ans += i * nums[i] - pre[i - 1]
            ans %= m
        return ans

Problem Classification

This falls within the domain of Algorithmic and Mathematical Programming. It involves mathematical computations, the management and manipulation of data structures (arrays), and conceptual understanding of space and movement, as well as the analysis of potential collisions.

The ‘What’ components of the problem statement are:

  1. Robots: The entities that will be moving along an infinite number line. The initial position of each robot is given by a 0-indexed integer array ’nums’.

  2. Movement Command: A string ’s’ is provided which will dictate the direction each robot will move towards. ‘L’ indicates left or negative direction, and ‘R’ indicates right or positive direction.

  3. Collisions: If two robots collide, they will start moving in opposite directions.

  4. Time: The quantity ’d’ denotes the number of seconds each robot will be moving for.

  5. Output: The sum of distances between all pairs of robots after ’d’ seconds. This sum could be large, so the result should be returned modulo 10^9 + 7 to avoid overflow issues.

This problem can be further classified as a dynamic problem that involves state transitions (robots changing directions) and it requires a mathematical understanding to efficiently calculate the sum of distances between all robots pairs. This is essentially a task of handling state changes and computing geometric or spatial information.

Language Agnostic Coding Drills

  1. Concepts contained in the code:
  • Data Types and Variables: Understanding of integer data types and List data types. Creation and manipulation of variables.
  • Condition Checking: Using if statements to check conditions (Here, checking direction of movement for each robot).
  • Loops: Using for loops to iterate over lists. (Here, for updating the position of robots and calculating the sum of distances)
  • List Operations: Knowledge of various list operations like copying and sorting.
  • Arithmetic Operations: Performing arithmetic operations, here modulo operation to avoid overflow and addition/subtraction to update positions.
  • Function Definition: Defining a function with parameters and returning values from the function.
  1. Difficulty Level and Description:
  • Data Types and Variables: Basic - Understanding and using basic data types and variables is one of the first concepts in programming.
  • Condition Checking: Basic - Checking conditions and making decisions based on them is a basic programming concept.
  • Arithmetic Operations: Basic - Arithmetic operations like addition, subtraction, multiplication, and modulo are basic operations in programming.
  • Loops: Intermediate - Loops are an intermediate concept as it requires understanding of control flow and iteration.
  • List Operations: Intermediate - Knowing how to manipulate lists, like sorting and copying, is an important intermediate skill.
  • Function Definition: Intermediate - Defining a function and using parameters correctly requires understanding of the concept of functions and scope.
  1. Problem-solving approach:

The approach towards the solution for this problem involves multiple steps.

First, the positions of the robots after ’d’ seconds are calculated by iterating over the nums array and depending on the direction of movement, either adding or subtracting ’d’ from the position.

Next, the nums array is sorted to find the relative positions of the robots after ’d’ seconds.

A prefix sum array pre is created and populated with the sum of positions up to the current index in the sorted array nums.

Finally, the total sum of distances is calculated by iterating over the sorted list nums and for each robot, subtracting the sum of distances to all robots on its left (which can be quickly found using the prefix sum array) from the sum of distances to all robots on its right.

Each of the identified coding concepts contributes to these steps. Understanding of loops and condition checking helps in updating the positions of robots and calculating the sum of distances, list operations are used to sort the array and copy it, arithmetic operations are used for calculating new positions and the sum of distances, and function definition encapsulates the whole process and provides the final output.

Targeted Drills in Python

  1. Python drills for each concept:
  • Data Types and Variables:
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# Declare an integer variable
x = 10

# Declare a list
nums = [1, 2, 3, 4, 5]
  • Condition Checking:
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# Condition Checking using if statement
if x > 5:
    print("x is greater than 5")
else:
    print("x is less than or equal to 5")
  • Arithmetic Operations:
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# Basic Arithmetic Operations
sum = x + 5  # Addition
diff = x - 5  # Subtraction
product = x * 5  # Multiplication
quotient = x // 5  # Division
mod = x % 5  # Modulo
  • Loops:
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# Using for loop to iterate over a list
for num in nums:
    print(num)
  • List Operations:
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# List operations
nums_copy = nums.copy()  # Copying a list
nums.sort()  # Sorting a list
  • Function Definition:
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# Function definition
def add(x, y):
    return x + y
  1. Drills for specific needs of the problem:
  • Update list items based on condition:
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# Update list items based on condition
directions = ['L', 'R', 'L', 'R', 'L']
for i in range(len(nums)):
    if directions[i] == 'L':
        nums[i] -= 1
    else:
        nums[i] += 1
  • Create and update prefix sum array:
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# Create and update prefix sum array
pre = nums.copy()
for i in range(1, len(nums)):
    pre[i] += pre[i - 1]
  • Calculate sum of distances:
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# Calculate sum of distances
distance_sum = 0
for i in range(1, len(nums)):
    distance_sum += i * nums[i] - pre[i - 1]
  1. Merge these drills for the final solution:

Start with defining the function and initializing necessary variables. Then, update the list of robot positions based on their respective directions. Next, sort the updated list and create a prefix sum array. Finally, calculate the sum of distances between the robots using the prefix sum array and return the result modulo 109 + 7 to avoid overflow. Remember to include all necessary Python concepts learned from the drills like condition checking, loops, arithmetic operations, and list operations within the function.