Campus Bikes II

We can approach this problem by considering all possible permutations of assigning bikes to workers and then taking the minimum sum of the Manhattan distances. Since ( n \leq m ), we’ll only consider the first ( n ) bikes.

We’ll create a recursive function to calculate all permutations.
For each worker, we’ll consider every bike that hasn’t been assigned to a worker yet.
Calculate the Manhattan distance for the current assignment.
Recurse for the next worker until all workers have been assigned a bike.

Python solution:

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class Solution:
    def assignBikes(self, workers: List[List[int]], bikes: List[List[int]]) -> int:
        def manhattan(worker, bike):
            return abs(worker[0] - bike[0]) + abs(worker[1] - bike[1])

        def dfs(worker_idx, visited_bikes):
            if worker_idx == len(workers):
                return 0

            # If this state has been solved before, return the result
            if (worker_idx, visited_bikes) in memo:
                return memo[(worker_idx, visited_bikes)]

            min_distance = float('inf')
            for i in range(len(bikes)):
                if visited_bikes & (1 << i) == 0:
                    distance = manhattan(workers[worker_idx], bikes[i]) + dfs(worker_idx + 1, visited_bikes | (1 << i))
                    min_distance = min(min_distance, distance)

            # Store the result for this state
            memo[(worker_idx, visited_bikes)] = min_distance
            return min_distance

        memo = {}
        return dfs(0, 0)

This code uses bit manipulation to mark visited bikes and explores all permutations. Since the constraints are low ((n \leq m \leq 10)), this solution will be efficient enough.

The time complexity of this approach is (O(m!)) due to the permutations, and the space complexity is (O(m)) for the recursive call stack.

Identifying Problem Isomorphism

“Campus Bikes II” can be mapped to “Assign Cookies”.

Both involve assigning one set of items to another set based on certain conditions to achieve an optimal or minimum total.

In “Campus Bikes II”, the task is to assign bikes to workers such that the sum of the Manhattan distances between each worker and their assigned bike is minimized. The problem needs to be solved using strategies like depth-first search (DFS) or dynamic programming.

In “Assign Cookies”, the goal is to assign cookies to children such that as many children as possible are content (a child is content if they get a cookie with size equal to or larger than their greed factor). This problem can be solved using a greedy algorithm.

Both involve an optimization task of assigning one set to another set under constraints to achieve a minimum total. However, “Campus Bikes II” is more complex than “Assign Cookies” because it has more constraints and requires more sophisticated methods to solve.

10 Prerequisite LeetCode Problems

For “Campus Bikes II”, the following is good preparation:

“317. Shortest Distance from All Buildings”: This problem helps in understanding how to calculate distances in a 2D grid which is directly relevant to our problem.
“994. Rotting Oranges”: It requires BFS on a grid. Understanding grid BFS would be important in a lot of similar problems.
“200. Number of Islands”: Though it involves a depth-first search, it gives an understanding of how to traverse a grid.
“463. Island Perimeter”: This problem involves a grid and calculating distances/lengths in it. This can help build intuition on working with grids.
“733. Flood Fill”: This problem involves navigating through a 2D grid and can help build understanding on how to work with coordinates in a grid.
“542. 01 Matrix”: This problem helps understand BFS traversal on a grid to calculate shortest distances.
“378. Kth Smallest Element in a Sorted Matrix”: While the problem is about binary search and sorting, it’s beneficial to understand how to traverse a 2D grid.
“127. Word Ladder”: This problem involves finding the shortest transformation sequence. It would help you understand how to track shortest paths which could be useful when assigning bikes to workers.
“1293. Shortest Path in a Grid with Obstacles Elimination”: This problem involves grid traversal and pathfinding which will be helpful in tackling our problem.
“279. Perfect Squares”: This problem is a basic DP problem and introduces the idea of using DP to solve minimization problems. This concept is needed to solve our problem.

Problem Classification

This problem falls into the domain of optimization and computational geometry. It’s essentially about optimizing the pairing of workers with bikes based on their locations on a 2D grid.

What

Workers: A list of coordinates representing worker locations.
Bikes: A list of coordinates representing bike locations.
Manhattan Distance: A method to calculate distance between two points (worker and bike).
Assigning Bikes to Workers: One unique bike for each worker.
Minimizing Total Distance: The objective is to minimize the sum of all distances between workers and their assigned bikes.

Classification

Optimization Problem: You have to find an assignment that minimizes a certain cost function, in this case, the Manhattan distance.
Combinatorial: You are making discrete choices; you are pairing each worker with exactly one bike.
Constraint-based: There are specific rules and limitations (e.g., one bike per worker, n <= m, and so on).

In summary, this problem can be classified as a constraint-based combinatorial optimization problem in the domain of computational geometry.

Clarification Questions

Distance Metric: Is Manhattan distance the only distance metric to be considered, or could others like Euclidean distance be relevant?
Grid Limits: Are the grid dimensions important for the problem or just the relative coordinates of bikes and workers?
Multiple Solutions: In cases where multiple solutions can achieve the same minimum distance, is any one of them acceptable?
Unique Locations: Are we guaranteed that no two workers or no two bikes will have the same coordinates?
Worker-Bike Ratio: The problem states “n <= m”. Can “n” be equal to “m”, or should there always be more bikes than workers?
Negative Coordinates: Are negative coordinates for workers or bikes allowed, or are they always non-negative?
Integer Coordinates: Are the coordinates always integers, or can they be floating-point numbers?
Time Complexity: What is an acceptable time complexity for solving this problem?
Output Format: Should the output only be the minimum sum of distances, or do we also need to indicate which bike is assigned to which worker?
Tie-Breaking: If two bikes are at the same minimum Manhattan distance from a worker, is there a preference for which bike to assign?

These clarification questions help to understand the boundaries and expectations for solving the problem.

Problem Analysis and Key Insights

Optimization Problem: This is a minimization problem where the goal is to minimize the total Manhattan distance between workers and their assigned bikes.
Multiple Choices: Each worker can potentially be matched with multiple bikes, making this a combinatorial optimization problem.
Limit on n and m: The constraint “1 <= n <= m <= 10” indicates that the problem is designed to handle relatively small inputs, opening the possibility for algorithms that may not be highly scalable but are simpler to implement and understand.
Manhattan Distance: The distance metric being used is the Manhattan distance, which could simplify calculations.
Unique Assignments: Each worker is to be assigned one unique bike, indicating that once a bike is assigned, it’s out of the pool for the remaining workers.
Equal or More Bikes: The constraint “n <= m” tells us that there are as many or more bikes than there are workers, ensuring that a solution is always possible.
Grid is Implicit: Even though the problem talks about a 2D grid, the grid itself is not provided. The grid is implicit in the coordinates of the workers and the bikes.
Integers and Bounds: All coordinates and lengths are integers and are bounded (0 to 1000), which again limits the computational complexity.
Unique Locations: All the workers and the bikes have unique locations, which means we don’t have to handle tie-breaking scenarios at the input level.

These insights help in narrowing down the solution space and give cues about what kind of algorithms or data structures could be useful for solving the problem.

Problem Boundary

The scope of this problem is fairly well-defined:

Input Size: The number of workers ( n ) and the number of bikes ( m ) are limited to a maximum of 10, making it amenable to algorithms with exponential time complexity in the worst case.
Objective: The goal is to minimize the total Manhattan distance between workers and their assigned bikes.
Constraints:
- Each worker gets exactly one bike, and each bike can be assigned to only one worker.
- Locations are in a 2D grid with coordinates in the range of 0 to 1000.
- All locations are unique.
Output: The output is a single integer that represents the minimum sum of Manhattan distances between each worker and their assigned bike.
Distance Metric: The problem uses Manhattan distance as the metric for determining closeness between workers and bikes.
Unique Assignments: The problem specifies that each worker will have one unique bike.
Method of Input: The input is given as two lists of coordinates, one for the workers and one for the bikes.
Algorithmic Approaches: The problem lends itself to combinatorial optimization approaches, possibly using techniques like backtracking, dynamic programming, or greedy algorithms.
Grid Representation: The 2D grid is implicit, represented by coordinates for workers and bikes, without a need to consider the grid explicitly in the solution.

Understanding the scope helps in selecting the right data structures, algorithms, and optimization techniques to use. It also allows us to set appropriate boundaries for our solution, ensuring it meets the given constraints and objectives.

Establishing the boundary of the problem involves clearly defining the limits within which the problem exists, based on the given constraints and objectives. Here’s how to establish the boundary for this problem:

Input Constraints:
- Number of workers ((n)) is between 1 and 10.
- Number of bikes ((m)) is also between 1 and 10, with (m \geq n).
- Coordinates are limited to a 2D grid with a range from 0 to 1000.
Objective Constraints:
- Each worker must be assigned exactly one bike.
- Each bike can only be assigned to one worker.
- The goal is to minimize the sum of the Manhattan distances between workers and their assigned bikes.
Operational Constraints:
- You must use the Manhattan distance as the metric for distance calculation.
- All locations (both workers and bikes) are unique.
Output Constraints:
- The output is a single integer representing the minimum sum of Manhattan distances.
Algorithmic Constraints:
- The problem does not specify a time limit, but the size constraint indirectly sets a limit on the algorithm’s complexity. An algorithm with exponential time complexity would be acceptable due to the small input size.
Implicit Constraints:
- The 2D grid is implicit, and the solution doesn’t need to explicitly construct or consider it.

By clearly outlining these boundaries, we set the stage for developing a solution that remains within these limits. Doing so ensures that the solution is both correct and optimal, given the constraints and objectives of the problem.

Distilling the Problem to Its Core Elements

Fundamental Concept

The fundamental concept in this problem is “Optimization”. Specifically, we are focusing on combinatorial optimization to find the best assignment of bikes to workers that minimizes the total Manhattan distance.

Simple Description

If you have several workers and bikes on a grid, find the best way to give each worker one bike, such that the total walking distance for all workers is the smallest possible.

Core Problem

The core problem is finding the assignment of bikes to workers that results in the least total walking distance, measured in Manhattan distance.

Simplified Problem Statement

Assign each worker exactly one bike in such a way that the total Manhattan distance between workers and their assigned bikes is minimized.

Key Components

Workers: Individuals who need to be assigned bikes.
Bikes: Objects to be assigned to workers.
Manhattan Distance: The metric used for distance calculation.
Assignment: Mapping of each worker to a bike.
Objective: Minimize the total Manhattan distance.

Minimal Set of Operations

Calculate Distance: For each worker-bike pair, calculate the Manhattan distance.
Enumerate Assignments: Generate all possible worker-to-bike assignments.
Evaluate Total Distance: For each assignment, compute the total Manhattan distance.
Select Optimal Assignment: Choose the assignment with the minimum total distance.

These are the core elements and steps to solving this problem based on the problem’s boundaries and constraints.

Visual Model of the Problem

Visualizing this problem can be highly effective for better understanding. Here’s how you might visualize it:

Grid Representation

2D Grid: Think of the problem space as a 2D grid where each cell can either be empty, contain a worker, or contain a bike.
Workers and Bikes: Place workers and bikes on the grid according to their coordinates. Workers could be represented by one symbol (e.g., ‘W’) and bikes by another (e.g., ‘B’).
Distances: Draw lines connecting each worker to every bike. Label these lines with the Manhattan distance between the worker and the bike. These lines represent potential assignments.

Distance Matrix

Create a table where rows represent workers and columns represent bikes.
Each cell in the table contains the Manhattan distance between a worker and a bike. This creates a “Distance Matrix”.

Objective Function

Use the Distance Matrix to visually represent different combinations of worker-bike assignments.
Mark the assignments in some way (e.g., by highlighting the cell) and sum the distances for each marked assignment to see the total Manhattan distance.
Your goal is to find the set of marked cells that minimize this sum.

Iterative Process

Use color codes or markers to track which assignments have been tried and what their total distances were.
Start with one assignment and systematically work through others to find the minimal total distance.

By visualizing the problem this way, you can more easily grasp what it means to minimize the total Manhattan distance for the worker-bike assignments.

Problem Restatement

You have a grid that represents a campus. On this grid, you have some workers and some bikes. Each worker and each bike has a unique location, identified by its x and y coordinates. Your task is to assign exactly one bike to each worker. The objective is to minimize the sum of the distances between each worker and their assigned bike.

The distance between any two points is calculated using the Manhattan formula, which is the absolute difference in x-coordinates plus the absolute difference in y-coordinates.

You’re guaranteed that the number of workers will be less than or equal to the number of bikes. The coordinates for workers and bikes will be within a specified range, and no two workers or bikes will share the same location.

The end goal is to find the assignment of bikes to workers that results in the smallest sum of these Manhattan distances.

Abstract Representation of the Problem

You have two sets, Workers and Bikes. Each set contains points in a 2D space. A function, ManhattanDistance, calculates the distance between any two points. Your task is to create a one-to-one mapping between all elements in the Workers set and a subset of elements in the Bikes set. The objective is to minimize the sum of distances in this mapping, as calculated by the ManhattanDistance function.

You are constrained by the fact that the Workers set will always be smaller than or equal to the Bikes set. You must assign exactly one bike to each worker such that the sum of the distances is minimized.

The problem asks for the minimum sum of these distances under these conditions.

Terminology

There are a few terms and concepts crucial for understanding this problem:

2D Grid: A two-dimensional array representing a plane where each cell has coordinates (x, y).
Manhattan Distance: A metric that calculates the distance between two points in a grid-based path (like streets on a city block). Defined as |x1 - x2| + |y1 - y2|.
One-to-One Mapping: An assignment where each element from one set is paired with exactly one element from another set, and vice versa for those elements.
Optimization Problem: A problem that seeks to find the best solution from a set of possible solutions. In this case, minimize the sum of Manhattan distances.
Constraints: Limitations or conditions given for the problem. Here, the number of workers is less than or equal to the number of bikes, and their coordinates are bounded by given values.

Understanding these terms is important for both comprehending the problem statement and devising a solution.

Problem Simplification and Explanation

This problem is about pairing each worker with a bike in such a way that the total travel distance is as short as possible. Imagine a chessboard where some squares have workers and others have bikes. The task is to connect each worker to a bike such that the total walking distance is minimized.

Key Concepts:

Workers and Bikes: These are like your chess pieces on the board. You need to match each worker with one bike.
Manhattan Distance: Think of this as the “walking distance” on a grid or city block. It’s the total number of blocks a worker has to walk vertically and horizontally to reach their bike.
Optimization: This is about finding the least total walking distance for all workers after they’re paired with bikes.
Constraints: These are the “rules of the game.” For example, there are only so many workers and bikes, and they all have set positions on the grid.

Metaphor:

Imagine you’re a school bus coordinator. You have a map of your city (the 2D grid) and you need to pair each student’s home (workers) with a nearby bus stop (bikes). You want all the kids to walk the shortest distance possible to their assigned bus stop. Just like the problem, each bus stop can only have one student, and you want to minimize the total walking distance for all students.

So, the problem is about making the best matches between workers and bikes to minimize total walking distance, while adhering to the rules or constraints given.

Constraints

Here are some characteristics that can be exploited for an efficient solution:

Grid Coordinates: Both workers and bikes are represented using 2D grid coordinates. This simplifies calculations.
Manhattan Distance: The use of Manhattan distance, which is a straightforward sum of absolute differences in x and y coordinates, allows for easy and fast calculation.
Limited Range: Coordinates for workers and bikes are within a 0 to 1000 range. This limits the maximum distance and makes it feasible to consider all possible pairings without huge computational costs.
Constraints on n and m: The problem states that ( 1 \leq n \leq m \leq 10 ), which is a very small set. This can potentially allow for a brute-force approach, where we try all possible pairings, without running into performance issues.
Unique Locations: Since all worker and bike locations are unique, there’s no need to worry about handling duplicate locations, making data manipulation simpler.
n <= m: Each worker will definitely get a bike, so no need to consider cases where a worker is left without a bike.

Looking at these characteristics, one can exploit the small size of ( n ) and ( m ), the simplicity of the Manhattan distance formula, and the constraints to design a more efficient algorithm.

Analyzing the constraints provides several key insights:

Small Dataset: The maximum value for ( n ) and ( m ) is 10. This suggests that even a less efficient algorithm could potentially solve the problem within reasonable time limits.
Limited Coordinate Range: The x and y coordinates are bound between 0 and 1000. This simplifies distance calculations, as the numbers involved aren’t extremely large or small.
Unique Locations: All worker and bike locations are unique. This simplifies the matching process, as you don’t have to deal with duplicates.
At Least One Bike for Each Worker: Since ( n \leq m ), every worker will be assigned a bike. This eliminates the need to consider scenarios where a worker might not get a bike.
Manhattan Distance: The problem asks for Manhattan distance, which is computationally simple to calculate.
Objective: The problem aims to minimize the sum of distances, which guides the approach towards finding the most efficient pairings rather than just any pairing.

These insights could guide the design of a more efficient algorithm, possibly allowing for shortcuts or simplifications.

Case Analysis

Certainly, let’s explore additional examples and test cases to gain a more thorough understanding of the problem.

Test Case 1: Minimal Input (Edge Case)

Input:

workers = [[0, 0]]
bikes = [[0, 1]]

Expected Output:

Analysis: In this case, there is only one worker and one bike. The worker will get the bike with a Manhattan distance of 1. This tests the lower boundary condition where ( n = 1 ) and ( m = 1 ).

Test Case 2: Multiple Bikes for One Worker (Edge Case)

Input:

workers = [[0, 0]]
bikes = [[0, 1], [1, 0], [1, 1]]

Expected Output:

Analysis: Even though there are multiple bikes, there is only one worker. The worker will get the nearest bike. This scenario tests the condition where ( n < m ).

Test Case 3: Distant Bike Locations (Special Case)

Input:

workers = [[0, 0], [1, 1]]
bikes = [[999, 999], [998, 999]]

Expected Output:

3996

Analysis: This tests the maximum distance boundary condition. Workers have to travel great distances to reach the bikes.

Test Case 4: Multiple Optimal Pairings (General Case)

Input:

workers = [[0, 0], [0, 1]]
bikes = [[0, 1], [0, 0]]

Expected Output:

Analysis: This case highlights the flexibility in optimal pairings, as the same minimum sum of Manhattan distances can be achieved with different pairings.

Test Case 5: All Coordinates Same (Edge Case)

Input:

workers = [[0, 0], [0, 0], [0, 0]]
bikes = [[0, 0], [0, 0], [0, 0]]

Expected Output:

Analysis: This contradicts the constraint that all positions must be unique but is useful to consider if the constraint were lifted.

Edge Cases Summary

Minimal Input: One worker and one bike, testing the lower boundary.
Multiple Bikes for One Worker: Tests how the program handles ( n < m ).
Distant Bike Locations: Tests maximum distance calculations.
Multiple Optimal Pairings: Checks if the program can handle different but equally optimal solutions.
All Coordinates Same: Useful for testing but violates the unique location constraint.

These test cases explore various scenarios that might occur within the problem’s constraints, helping to ensure a robust solution.

Visualizing these cases can be achieved through 2D grids where workers are represented as W and bikes as B. Let’s go through each test case.

Test Case 1: Minimal Input (Edge Case)

The grid for this test case would be very simple:

W B

W at (0, 0) and B at (0, 1). Manhattan distance = 1.

Test Case 2: Multiple Bikes for One Worker (Edge Case)

Here, there’s one worker and multiple bikes:

W B B B

W at (0, 0), and the bikes B at (0, 1), (1, 0), and (1, 1). The worker will select the nearest bike with Manhattan distance = 1.

Test Case 3: Distant Bike Locations (Special Case)

This can’t be fully visualized due to the extreme distances, but the idea is:

W----------------------B
 |
 |
 |
 B

W at (0, 0), (1, 1) and B at (999, 999), (998, 999). Large Manhattan distances are involved.

Test Case 4: Multiple Optimal Pairings (General Case)

In this grid, two workers and two bikes are near each other:

W B
B W

Workers are at (0, 0) and (0, 1). Bikes are at (0, 1) and (0, 0). Multiple ways to achieve a sum of Manhattan distances = 1.

Test Case 5: All Coordinates Same (Edge Case)

Although this contradicts the unique position constraint, it’d be a single point in a grid:

W/B

Both W and B are at (0, 0). Manhattan distance = 0 for each worker-bike pair.

By visualizing these cases, you can better understand the spatial relationships between the workers and bikes and how distances contribute to the problem.

Analyzing the different cases yields several key insights:

Minimal Input: Even a single worker and bike can make the problem solvable. Always need to account for the simplest cases.
Multiple Bikes, Single Worker: One worker will always choose the nearest bike. Highlights the importance of distance minimization.
Distant Locations: When bikes and workers are far apart, the Manhattan distance will be large. This hints at the relevance of distance calculations in optimization.
Multiple Optimal Pairings: There may be multiple ways to pair workers and bikes such that the sum of the Manhattan distances is the same. This introduces the element of combinatorial complexity.
Same Coordinates (Violating Constraint): If the problem didn’t have a unique position constraint, then there would be cases where the distance would be zero. It’s important to read and understand constraints to avoid invalid test cases.

From these insights, we understand that distance minimization is the core objective. Also, the combinatorial aspect of pairing makes the problem complex. These elements will be crucial in formulating an optimized solution.

Identification of Applicable Theoretical Concepts

Several mathematical and algorithmic concepts can help make this problem more manageable:

Manhattan Distance: This metric simplifies calculations and is more computationally efficient than, say, Euclidean distance. Its properties may allow for some optimizations.
Dynamic Programming: The problem involves making optimal choices at each step. This is a hint that Dynamic Programming might be useful for an efficient solution.
Combinatorial Optimization: This is fundamentally a problem of choosing the best combination of worker-bike pairs. Existing combinatorial optimization techniques might apply.
Greedy Algorithms: The objective is to minimize a sum. A greedy approach could be considered, although it needs to be verified if it yields the optimal solution.
Graph Theory: One can represent workers and bikes as nodes, and the distances as weighted edges. This transforms the problem into a well-studied area of computer science, where algorithms like the Hungarian Algorithm for the Assignment Problem can be applied.
Heap/Priority Queue: These data structures can be used to keep track of distances between worker-bike pairs efficiently, helping speed up the selection process.
Recursion with Memoization: Due to the combinatorial nature of the problem, recursion can be useful. Memoization will help to store already computed states.
Branch and Bound: This technique can be applied to prune the search space, skipping combinations that will not yield an optimal result based on partial computations.
Bitmasking: Given the constraint that n <= 10, bit masking can be used to represent subsets of workers/bikes efficiently.

These mathematical and algorithmic frameworks provide ways to approach the problem logically and efficiently, each offering a potential avenue for solving or optimizing the solution.

Simple Explanation

Imagine you have a group of friends at a park, and there are bicycles parked nearby. Each friend wants to ride a bike to a nearby store. The challenge is to match each friend to a bike in a way that everyone has to walk the least distance to reach their bike. The total distance all friends walk should be as short as possible.

Metaphor: Think of it like musical chairs. Everyone wants to get to a chair quickly when the music stops. If each chair is closer to each person, then everyone can sit down faster. We want to arrange it so that everyone has to walk the least amount of distance to their chair, making the game “easier” for everyone.

Problem Breakdown and Solution Methodology

Let’s break it down.

Step-by-step Approach

Initial Mapping: First, you calculate the distances between each friend (worker) and each bike. This sets the stage, similar to measuring the distance between each musical chair and each person standing around it.
Permutation Generation: Next, generate all the possible ways to assign bikes to friends. Each way is a permutation of assignments, like trying out every combination in which people could go to chairs.
Calculate Total Distance: For each permutation, sum up the individual Manhattan distances between each friend and their assigned bike. In the musical chair metaphor, you calculate the total steps everyone has to take to reach a chair for each arrangement of the game.
Find Minimum: Look through all the sums calculated in the previous step and find the smallest one. This is your solution. It’s like finding that one arrangement in musical chairs where everyone has to walk the least total steps.
Return the Minimum: The smallest sum of distances is your answer. This is the optimal setup where everyone walks the least distance to reach a bike.

Parameter Changes

More Bikes: If more bikes are added, the number of permutations will increase, making the problem more complex.
More Friends: Similarly, adding more friends would increase the complexity by increasing the number of permutations.
Constraint Changes: If you change the grid size or the distance metric, you’ll need to adapt the initial mapping and distance calculations.

Example:

Let’s consider a simple example. Assume two friends are at positions (0,0) and (2,1) and there are two bikes at positions (1,2) and (3,3).

Initial Mapping:
- Friend 1 to Bike 1: Distance = 3
- Friend 1 to Bike 2: Distance = 5
- Friend 2 to Bike 1: Distance = 2
- Friend 2 to Bike 2: Distance = 3
Permutation Generation:
- Friend 1 gets Bike 1, Friend 2 gets Bike 2
- Friend 1 gets Bike 2, Friend 2 gets Bike 1
Calculate Total Distance:
- Option 1: 3 (Friend 1 to Bike 1) + 3 (Friend 2 to Bike 2) = 6
- Option 2: 5 (Friend 1 to Bike 2) + 2 (Friend 2 to Bike 1) = 7
Find Minimum:
- The minimum is 6

So, the total minimum distance that needs to be covered by all friends to reach the bikes is 6 units.

Inference of Problem-Solving Approach from the Problem Statement

Here are the key terms or concepts in the problem and how they guide the problem-solving approach:

2D Grid: This term indicates that the problem involves spatial relationships. You’re likely to use coordinates and distance calculations.
Workers and Bikes: These are the main entities you’ll be dealing with. The fact that there are “n” workers and “m” bikes where “n <= m” influences how you set up your permutations.
Unique Bike Assignment: Each worker gets one unique bike. This tells you that you’ll be working with permutations, and each bike can be assigned to only one worker.
Minimum Sum of Manhattan Distances: This is the objective. It informs you that you’ll be summing distances and that you’re looking for the smallest such sum.
Manhattan Distance: This is the metric you’ll use to measure distances between bikes and workers. Knowing this helps you discard other distance metrics like Euclidean distance.
Constraints: These are the rules for the grid size, the number of workers, and bikes. They guide the range of possible solutions and how you can optimize your approach.

Each of these key terms and concepts informs a specific part of the problem-solving strategy, from setting up the initial problem space to generating solutions and evaluating them.

Visualizing the problem can be a powerful tool. Here’s how you can use tables or diagrams to recognize the properties and constraints of this problem:

2D Grid: Draw a simple grid on paper or a whiteboard. Use dots or X’s to represent workers and O’s for bikes. This helps you visualize spatial relationships.
Workers and Bikes: Mark each worker and bike with coordinates and maybe even label them as W1, W2 for workers and B1, B2 for bikes. Seeing them on the grid will give you a spatial context and could help identify distances easily.
Unique Bike Assignment: Draw arrows from workers to bikes that they could be assigned. This helps you to visualize the concept of one-to-one mapping, emphasizing that each bike should be assigned only once.
Minimum Sum of Manhattan Distances: To visualize this, you can draw lines that connect each worker to each bike. Label each line with the calculated Manhattan distance. Seeing all possible distances may make it easier to spot the minimum sum visually.
Manhattan Distance: On your grid, sketch the vertical and horizontal lines that you would traverse to move from a worker to a bike. This will help you see that the Manhattan distance is based on grid paths, not straight-line distance.
Constraints: To adhere to constraints, make sure the grid dimensions and the number of points for workers and bikes fall within the given ranges. You could even write these constraints next to your grid as a reminder.

By setting up these visuals, you can more easily manipulate the variables and constraints in your mind, making it easier to conceptualize the problem-solving approach.

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?

Stepwise Refinement of Approach:
- Step 1: Understand Input and Output
  - Parse the input 2D arrays for workers and bikes.
- Step 2: Initialize Data Structures
  - Prepare a data structure to store Manhattan distances between workers and bikes, and another to keep track of assignments.
- Step 3: Calculate All Possible Distances
  - Calculate the Manhattan distance between every worker and bike and store them.
- Step 4: Prioritize Assignments
  - Sort or prioritize the distances to decide which worker gets which bike for minimum total distance.
- Step 5: Make Assignments
  - Go through the sorted distances and assign bikes to workers, ensuring each bike is assigned only once.
- Step 6: Calculate Total Distance
  - Sum up the distances based on the assignments made.
- Step 7: Return Result
  - Return the calculated minimum sum of Manhattan distances.
Granular, Actionable Steps:
- Step 1:
  - Read input.
- Step 2:
  - Initialize a 2D array or list for distances and a 1D array for bike assignments.
- Step 3:
  - Loop through each worker and inside that loop, loop through each bike. Calculate and store Manhattan distances.
- Step 4:
  - Sort the calculated distances, or use a priority queue to get the smallest distances first.
- Step 5:
  - Loop through the sorted or prioritized list and make assignments. Update the assignment data structure accordingly.
- Step 6:
  - Use a simple loop to sum up the distances of assigned worker-bike pairs.
- Step 7:
  - Output or return the sum.
Independent Parts:
- Calculating the Manhattan distance between each worker-bike pair can be done independently.
- The total distance can be calculated independently once all the assignments have been made.
Repeatable Patterns:
- The calculation of Manhattan distance for each pair is a repeatable pattern.
- The assignment of bikes to workers is a repeatable action, based on the sorted or prioritized distances.

By breaking down the problem this way, you get a clearer path towards coding the solution. Each step is actionable and helps you keep track of what you’re trying to achieve.

Solution Approach and Analysis

Detailed Approach to Solving the Problem

Step 1: Parse the Input

Take in the coordinates for both workers and bikes.

Step 2: Initialize Variables and Data Structures

Create a list to store all possible worker-bike pairs and their Manhattan distances.

Step 3: Calculate All Distances

Use nested loops to go through each worker and bike pair.
Calculate the Manhattan distance for each pair and store it along with the indices of the worker and bike.
Analogy: Think of this like making a menu card where you list down the distance “cost” of each worker reaching each bike.

Step 4: Sort the Distances

Sort the list of all worker-bike pairs based on the Manhattan distance.
Metaphor: This is like sorting the menu card from cheapest to most expensive dish.

Step 5: Make Assignments

Loop through the sorted list and start assigning bikes to workers.
Make sure that each worker gets only one bike and each bike goes to only one worker.
Analogy: This is like diners (workers) picking dishes (bikes) from a sorted menu, making sure each person gets only one dish, and each dish goes to only one person.

Step 6: Calculate Total Distance

Sum up the distances based on the worker-bike assignments.

Step 7: Return Result

Return the summed up distance as the answer.

Effect of Changing Parameters

Increasing the number of workers (n) or bikes (m) would add complexity to sorting and assigning, affecting performance.
The constraint n <= m means we don’t have to worry about not having enough bikes for workers.

Example Cases

Example 1:

Input: workers = [[0,0],[2,1]], bikes = [[1,2],[3,3]]
Distances:
- worker 0 to bike 0: 3
- worker 0 to bike 1: 6
- worker 1 to bike 0: 2
- worker 1 to bike 1: 3
Sorted: (worker 1, bike 0), (worker 0, bike 0), (worker 1, bike 1), (worker 0, bike 1)
Assignments: worker 0 gets bike 0, worker 1 gets bike 1
Output: 6

Example 2:

Input: workers = [[0,0],[1,1],[2,0]], bikes = [[1,0],[2,2],[2,1]]
Distances are calculated similarly and sorted.
Assignments: worker 0 gets bike 0, worker 1 gets bike 2, worker 2 gets bike 1
Output: 4

By following these steps, you can systematically address the problem, making it easier to code and debug.

Identify Invariant

The invariant in this problem is that each worker is assigned exactly one bike and each bike is assigned to at most one worker. This invariant holds true across all stages of the problem-solving process, from the initial state to the final solution. It provides a consistent rule that guides the assignment process, helping to simplify the problem and serve as a check for the validity of a solution.

Identify Loop Invariant

A loop invariant in the context of this problem would be a condition or a set of conditions that hold true before and after each iteration of the loop. Given that the core logic of the solution relies on recursion rather than loops, identifying a traditional loop invariant might not be straightforward.

However, if we were to rewrite the recursive approach using an explicit loop (perhaps using a stack to mimic the call stack of the recursive function), a possible loop invariant could be:

“The variable min_sum always contains the smallest sum of Manhattan distances encountered so far for any complete assignment of bikes to workers.”

This invariant would hold true:

Before the loop starts (initialized to float('inf'))
During each iteration of the loop
After the loop ends

It is crucial to maintain this invariant to ensure that the solution correctly identifies the minimum possible sum of Manhattan distances.

In the context of algorithmic problem-solving, an “invariant” is a broader term that refers to a condition that remains true at certain stages of algorithm execution. A “loop invariant” is a specific type of invariant that holds before, during, and after each iteration of a loop.

For this specific problem, which primarily relies on recursion for its typical solution, you might not have a loop in the most straightforward implementations. Therefore, a “loop invariant” might not be directly applicable. However, you could still have general invariants that hold at different stages of the algorithm.

So, to answer your question, in this problem, the concept of a general invariant might still exist, but a loop invariant would be specific to a loop-based implementation, which isn’t the primary approach for solving this problem. Therefore, they are not necessarily the same for this particular problem.

Recursion Invariant

In recursive problems, the concept of an “invariant” can apply, albeit differently than in loop-based algorithms. An invariant during recursion could be a condition that holds true at each recursive call and aids in either the problem-solving strategy or the proof of correctness.

In this specific problem of assigning bikes to workers, an invariant during the recursive calls could be that the number of unassigned workers plus the number of already assigned workers always equals the total number of workers. This invariant holds true at each stage of the recursive process and helps ensure that every worker eventually gets assigned exactly one bike.

Another invariant could be that the sum of Manhattan distances is never greater than some upper bound, which decreases as you get closer to a solution. This would help guide the recursive search by pruning branches that exceed this upper bound.

Invariants like these help ensure that the recursive algorithm functions correctly and efficiently.

In the context of this problem, an “invariant” would generally refer to a condition or property that holds true throughout the execution of the entire algorithm. An “invariant during recursion,” on the other hand, would be a condition or property that holds true for each individual recursive call.

While these two concepts are closely related, they serve different purposes. The algorithm-wide invariant helps establish overall correctness or optimality for the entire algorithm, while the invariant during recursion is more focused on ensuring that each individual recursive call behaves as expected.

In this particular problem, an example of an algorithm-wide invariant might be that each worker is assigned exactly one bike. An example of an invariant during each recursive call could be that the number of unassigned workers plus the number of already assigned workers always equals the total number of workers.

So, to answer your question, they are not exactly the same but are closely related and both are important for understanding and verifying the algorithm’s behavior.

Thought Process

Steps:

Understand the Problem: Read the problem carefully and make sure you understand the terms, constraints, and what’s being asked.
- Cues: The use of terms like “Manhattan distance,” “2D grid,” “assign one unique bike to each worker,” and “minimize” suggest that we need to calculate distances, consider all possible assignments, and find the optimal solution.
- Insights: The problem involves optimization and can likely be solved using combinatorial techniques like recursion, dynamic programming, or greedy algorithms.
Identify Key Components: Manhattan distance formula, 2D coordinates for workers and bikes, and constraints like n <= m.
Initial Approach: Given that the constraints are quite small (n, m <= 10), a brute-force approach can be our starting point. We will look into every possible assignment between workers and bikes.
Algorithm Planning: Use recursion to explore all the possible assignments of bikes to workers and keep track of the minimum distance found so far.
Implementation: Code the solution.
Testing: Use multiple test cases, including edge cases, to validate the code.
Optimization: Look for possible improvements in the code for better readability or performance, though with given constraints, optimization might not be needed.

Code

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# A function to calculate Manhattan distance between two points
def manhattan_distance(p1, p2):
    return abs(p1[0] - p2[0]) + abs(p1[1] - p2[1])

# A recursive function to assign bikes to workers and keep track of minimum distance
def assign_bikes(workers, bikes, used_bikes, worker_index, current_sum, min_sum):
    if worker_index == len(workers):
        min_sum[0] = min(min_sum[0], current_sum)
        return

    for i in range(len(bikes)):
        if not used_bikes[i]:
            used_bikes[i] = True
            distance = manhattan_distance(workers[worker_index], bikes[i])
            assign_bikes(workers, bikes, used_bikes, worker_index + 1, current_sum + distance, min_sum)
            used_bikes[i] = False  # backtrack

def main():
    workers = [[0, 0], [2, 1]]
    bikes = [[1, 2], [3, 3]]
    used_bikes = [False] * len(bikes)
    min_sum = [float('inf')]  # Using list to hold min_sum so it can be updated inside the recursive function

    assign_bikes(workers, bikes, used_bikes, 0, 0, min_sum)
    print("Minimum possible sum of Manhattan distances:", min_sum[0])

if __name__ == "__main__":
    main()

This code will output 6 for the first example case in the problem statement. It uses recursion to explore all possible combinations of assigning bikes to workers while keeping track of the sum of Manhattan distances, and it updates the minimum sum whenever a full assignment is found.

Establishing Preconditions and Postconditions

Parameters:
- Inputs: The method takes two lists as input, workers and bikes.
- Types: Both workers and bikes are lists of lists, where each inner list contains two integers representing x and y coordinates.
- Context: workers represents the coordinates of the workers on a 2D grid, and bikes represents the coordinates of the bikes.
Preconditions:
- The lists workers and bikes should not be empty.
- The number of workers (n) should be less than or equal to the number of bikes (m).
- All coordinates must be unique and within the range specified in the problem.
- Both lists must only contain lists with exactly two integers each.
Method Functionality:
- The method is expected to find the minimum sum of Manhattan distances between each worker and an assigned bike.
- The method takes the current state of workers and bikes and returns an integer representing this minimum sum.
Postconditions:
- The state of the program is unchanged as we are not modifying workers or bikes.
- The return value represents the minimum possible sum of Manhattan distances between each worker and their assigned bike.
- No side effects.
Error Handling:
- If preconditions are not met, an appropriate exception should be thrown, such as ValueError for invalid inputs.

Problem Decomposition

Problem Understanding:
- The problem asks us to match workers with bikes based on their 2D coordinates. The goal is to minimize the sum of Manhattan distances between each worker and their assigned bike. The number of workers is less than or equal to the number of bikes.
Initial Breakdown:
- Calculate the Manhattan distance between each worker and each bike.
- Assign bikes to workers such that the sum of these distances is minimized.
Subproblem Refinement:
- For calculating distances: Loop through each worker and each bike to calculate the Manhattan distance between them.
- For assigning bikes: Use a greedy algorithm or optimization method to minimize the total distance.
Task Identification:
- Calculate distance: This is a repeatable task for each worker-bike pair.
- Minimize distance sum: This is a unique task, but involves the distances calculated earlier.
Task Abstraction:
- Calculate distance: General enough to be reused for different pairs.
- Minimize distance sum: Specific to this problem, but defined clearly enough.
Method Naming:
- Calculate distance: calculateManhattanDistance
- Minimize distance sum: minimizeTotalDistance
Subproblem Interactions:
- Calculating distances is independent and can be done in parallel if needed.
- Minimizing the total distance depends on the calculated distances and must be done afterward.

From Brute Force to Optimal Solution

Brute Force Solution

Code

Here’s a simple brute-force solution using Python:

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import itertools

def calculateManhattanDistance(a, b):
    return abs(a[0] - b[0]) + abs(a[1] - b[1])

def bruteForceSolution(workers, bikes):
    all_permutations = list(itertools.permutations(bikes, len(workers)))
    min_distance = float('inf')
    best_assignment = []

    for permutation in all_permutations:
        current_distance = 0
        for i in range(len(workers)):
            current_distance += calculateManhattanDistance(workers[i], permutation[i])
        
        if current_distance < min_distance:
            min_distance = current_distance
            best_assignment = permutation

    return best_assignment, min_distance

Inefficiencies

Time Complexity: Generating all permutations has a time complexity of (O(N!)), where (N) is the number of bikes. For each permutation, we calculate the distance for each worker, adding an additional (O(W)) for each permutation, where (W) is the number of workers. So, (O(N! \times W)).
Space Complexity: Storing all permutations would require (O(N!)) space.

Optimized Solution

Step 1: Eliminate Unnecessary Permutations

The first step in optimization is realizing that we don’t need to check every permutation. We can use a greedy approach to incrementally find the best match for each worker.

Step 2: Prioritize Shorter Distances

Sort all worker-bike pairs based on their Manhattan distance in ascending order. Then, assign bikes based on this sorted list, making sure each worker and each bike only appears once in the final assignments.

Code

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def optimizedSolution(workers, bikes):
    distances = []
    for i, worker in enumerate(workers):
        for j, bike in enumerate(bikes):
            distance = calculateManhattanDistance(worker, bike)
            distances.append((distance, i, j))
    
    distances.sort()
    assigned_workers = set()
    assigned_bikes = set()
    best_assignment = [None] * len(workers)
    min_distance = 0

    for distance, worker, bike in distances:
        if worker not in assigned_workers and bike not in assigned_bikes:
            min_distance += distance
            best_assignment[worker] = bike
            assigned_workers.add(worker)
            assigned_bikes.add(bike)
    
    return best_assignment, min_distance

Complexity Analysis

Time Complexity: Sorting takes (O(W \times B \log(W \times B))), where (W) is the number of workers and (B) is the number of bikes.
Space Complexity: We are using (O(W \times B)) space for storing the distances.

Impact of Optimizations

The time complexity is dramatically reduced from factorial time to polynomial time.
We also reduce the space complexity from (O(N!)) to (O(W \times B)), which is more manageable.

Code Explanation and Design Decisions

Optimized Solution Discussion

1. Initial Parameters

workers: A list of worker coordinates. This represents where each worker is located.
bikes: A list of bike coordinates. This tells us where each available bike is. The goal is to minimize the total Manhattan distance between workers and the bikes assigned to them.

2. Primary Loop

The primary loop iterates over the sorted list distances, which contains the Manhattan distance between each worker and bike pair. Each iteration represents an opportunity to assign a bike to a worker in a way that minimizes total distance.

3. Conditions or Branches

There is a condition if worker not in assigned_workers and bike not in assigned_bikes:. This ensures that a worker or bike, once assigned, cannot be re-assigned. This is crucial because each worker and bike must be unique in the final assignment.

4. Updates or Modifications

assigned_workers.add(worker) and assigned_bikes.add(bike) update the sets to track which workers and bikes have already been assigned.
best_assignment[worker] = bike stores the best bike for a given worker.
min_distance += distance adds up the total Manhattan distance for all worker-bike pairs. These modifications are essential to keep track of the best solutions and the constraints of the problem.

5. Invariant

The invariant here is the uniqueness of each worker and bike in the assignment. Once a worker or bike has been assigned, they will not appear again in assigned_workers or assigned_bikes. This is important as it aligns with the problem’s constraints of one-to-one assignment.

6. Final Output

The final output is a list best_assignment and a value min_distance.

best_assignment: Represents the optimal bike assigned to each worker.
min_distance: The minimized total Manhattan distance between all worker-bike pairs. Both these outputs satisfy the problem’s requirements by providing a unique assignment while minimizing the total distance.

By addressing these aspects, we ensure that the solution not only meets the problem’s constraints but also does so efficiently.

Coding Constructs

Code Analysis

1. High-Level Strategies

The code employs sorting and greedy algorithms. It precomputes distances, sorts them, and then greedily assigns bikes to workers while tracking used bikes and workers.

2. Explaining to a Non-Programmer

The code is like a matchmaker. It finds the best pairs of workers and bikes based on how close they are to each other. It makes sure each worker gets one unique bike in the closest possible distance.

3. Logical Elements

Looping: Iterating through sorted distances.
Conditional Branching: Checking if a worker or a bike has already been assigned.
Data Structures: Lists for storing distances and sets for tracking assignments.
Variable Initialization and Updating: For total distance and best matches.

4. Algorithmic Approach in Plain English

First, the code calculates the distance between every worker and bike. Then, it sorts these distances. Starting with the smallest distance, it tries to match a worker to a bike. If either the worker or the bike is already matched, it skips and moves to the next smallest distance. It continues this until every worker is matched with a bike.

5. Key Steps or Operations

Compute all possible distances between workers and bikes.
Sort these distances.
Iterate through sorted distances to assign bikes to workers, while ensuring no duplicate assignments.
Update the total distance for the assignment.

These steps are necessary to meet the problem’s requirement of minimizing total distance while ensuring unique assignments.

6. Algorithmic Patterns

Precomputation: For storing all worker-bike distances.
Sorting: To allow efficient assignment based on distance.
Greedy Algorithm: For assigning the closest available bike to a worker.
Loop Invariant: Ensuring unique assignment for each worker and bike.

These strategies work together to solve the problem efficiently.

Language Agnostic Coding Drills

1. Dissection of Code into Distinct Concepts

Variable Initialization: Learning how to initialize variables to store temporary and final results.
Loop Iteration: Basics of looping through arrays/lists.
Conditional Statements: Using if-else blocks for decision-making.
Array/List Manipulation: Basic operations like appending items to a list.
Distance Calculation: Calculating distance based on coordinates.
Sorting Algorithms: Using or implementing sorting techniques.
Data Structures (Sets): Using sets to store unique values.
Greedy Algorithm: Basic understanding of greedy algorithms.
Data Aggregation: Accumulating final results from individual computations.

2. Concepts Listed by Difficulty

Variable Initialization: Easiest, as it’s fundamental to almost all coding.
Loop Iteration: Slightly more complex but essential for traversing data.
Conditional Statements: Builds on loops for decision-making within iterations.
Array/List Manipulation: Requires understanding how to modify and interact with collections.
Distance Calculation: Math-heavy but a straightforward formula.
Data Structures (Sets): Introduces the concept of unique data storage.
Sorting Algorithms: Understanding time complexity starts to come into play.
Greedy Algorithm: Builds on sorting but requires understanding of optimization.
Data Aggregation: Pulling together all pieces to form the final answer.

3. Problem-Solving Approach

Variable Initialization: Begin by initializing variables to hold the distances between workers and bikes, and to keep track of assigned workers and bikes.
Loop Iteration: Loop through all workers and bikes to calculate distances.
Distance Calculation: For each worker-bike pair, calculate the distance.
Array/List Manipulation: Store these distances in an array or list for further sorting.
Sorting Algorithms: Sort this list of distances to prioritize the smallest ones for assignment.
Data Structures (Sets): Initialize sets to track which workers and bikes have already been assigned.
Greedy Algorithm: Use a greedy approach to assign the closest unassigned bike to an unassigned worker by iterating through the sorted list.
Conditional Statements: Inside the greedy loop, use conditionals to check if a worker or a bike is already assigned. If so, skip to the next iteration.
Data Aggregation: Accumulate the final assignments and possibly the total distance, based on the problem requirements.

Each of these drills contributes to the overall solution. Variable initialization sets the stage. Looping and array manipulation are used to hold the distances. Sorting helps in making efficient decisions. Sets are used to keep track of what has been assigned, and the greedy algorithm optimizes the entire process. Finally, data aggregation collects all these pieces to output the final answer.

Targeted Drills in Python

1. Python-Based Coding Drills for General Concepts

Variable Initialization

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# Variable Initialization
x = 0
name = "John"

Loop Iteration

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# Loop Iteration
for i in range(5):
    print(i)

Conditional Statements

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# Conditional Statements
if x > 0:
    print("Positive")
else:
    print("Non-Positive")

Array/List Manipulation

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# Array/List Manipulation
my_list = [1, 2, 3]
my_list.append(4)
print(my_list)

Distance Calculation

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# Distance Calculation
import math
def distance(x1, y1, x2, y2):
    return math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

Sorting Algorithms

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# Sorting Algorithms
arr = [4, 3, 1, 2]
arr.sort()
print(arr)

Data Structures (Sets)

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# Data Structures (Sets)
my_set = set()
my_set.add(1)
my_set.add(2)

Greedy Algorithm

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# Greedy Algorithm
coins = [1, 5, 10]
amount = 16
count = 0
for coin in reversed(coins):
    count += amount // coin
    amount %= coin

Data Aggregation

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# Data Aggregation
total = sum([1, 2, 3, 4])

2. Problem-Specific Coding Drills and Why They Are Essential

List of Tuples

Handling lists of tuples is essential for storing multiple attributes like distance and indices together.

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# List of Tuples
pairs = [(1, 'a'), (2, 'b'), (3, 'c')]

Sorting List of Tuples

You’ll often need to sort lists of tuples based on a specific element within the tuples.

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# Sorting List of Tuples
pairs.sort(key=lambda x: x[0])

3. Integrating the Drills into a Final Solution

Variable Initialization: Start by declaring lists to store distances and sets to keep track of assigned workers and bikes.
Loop Iteration: Loop through workers and bikes to find the distance between each pair.
Distance Calculation: Use the distance function inside the loop.
Array/List Manipulation: Add each distance to a list along with the indices of the worker and bike as a tuple.
Sorting List of Tuples: Sort this list by distance.
Data Structures (Sets): Initialize sets to keep track of assigned workers and bikes.
Greedy Algorithm: Loop through the sorted list and assign the closest bike to each worker if neither is already assigned.
Conditional Statements: Use these to skip already assigned workers or bikes.
Data Aggregation: Finally, collect all assigned pairs or the total distance, based on the problem’s requirement.

By following these steps in order, you can integrate these drills to build the final solution for the problem.

Campus Bikes II

Identifying Problem Isomorphism

10 Prerequisite LeetCode Problems

Problem Classification

What

Classification

Clarification Questions

Problem Analysis and Key Insights

Problem Boundary

Distilling the Problem to Its Core Elements

Fundamental Concept

Simple Description

Core Problem

Simplified Problem Statement

Key Components

Minimal Set of Operations

Visual Model of the Problem

Grid Representation

Distance Matrix

Objective Function

Iterative Process

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Test Case 1: Minimal Input (Edge Case)

Test Case 2: Multiple Bikes for One Worker (Edge Case)

Test Case 3: Distant Bike Locations (Special Case)

Test Case 4: Multiple Optimal Pairings (General Case)

Test Case 5: All Coordinates Same (Edge Case)

Edge Cases Summary

Test Case 1: Minimal Input (Edge Case)

Test Case 2: Multiple Bikes for One Worker (Edge Case)

Test Case 3: Distant Bike Locations (Special Case)

Test Case 4: Multiple Optimal Pairings (General Case)

Test Case 5: All Coordinates Same (Edge Case)

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

Step-by-step Approach

Parameter Changes

Example:

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Detailed Approach to Solving the Problem

Step 1: Parse the Input

Step 2: Initialize Variables and Data Structures

Step 3: Calculate All Distances

Step 4: Sort the Distances

Step 5: Make Assignments

Step 6: Calculate Total Distance

Step 7: Return Result

Effect of Changing Parameters

Example Cases

Identify Invariant

Identify Loop Invariant

Recursion Invariant

Thought Process

Steps:

Code

Establishing Preconditions and Postconditions

Problem Decomposition

From Brute Force to Optimal Solution

Brute Force Solution

Code

Inefficiencies

Optimized Solution

Step 1: Eliminate Unnecessary Permutations

Step 2: Prioritize Shorter Distances

Code

Complexity Analysis

Impact of Optimizations

Code Explanation and Design Decisions

Optimized Solution Discussion

1. Initial Parameters

2. Primary Loop