# Minimum Reverse Operations

## 10 Prerequisite LeetCode Problems

Here are 10 problems to solve before tackling the problem “Minimum Reverse Operations”. These cover Array operations, Dynamic Programming, Queue operations, and Breadth-First Search, which would be essential to solve the given problem.

**189. Rotate Array**: This problem helps understand basic array rotation operation which is crucial in understanding the movement in the given problem.**239. Sliding Window Maximum**: Understanding this problem can help you comprehend sliding window problems. The subarray reversal in the given problem could be treated as a kind of sliding window.**986. Interval List Intersections**: This problem can help in understanding how to deal with ranges and intersections, which could be essential in handling banned positions.**1054. Distant Barcodes**: This problem provides an understanding of how to arrange items in an array based on certain restrictions which is a useful concept for the given problem.**297. Serialize and Deserialize Binary Tree**: This problem is an introduction to the concept of serialization and deserialization which is crucial for any form of array transformation.**862. Shortest Subarray with Sum at Least K**: Understanding this problem helps in dealing with the concept of subarrays, their sums, and the operations performed on them.**621. Task Scheduler**: This problem provides a good understanding of how to arrange tasks (or in the case of the given problem, the operations) based on certain restrictions.**787. Cheapest Flights Within K Stops**: This problem helps understand the concept of achieving a goal (like moving 1 to a certain position) within a certain number of steps or operations (like the minimum number of reverse operations).**649. Dota2 Senate**: This problem introduces the concept of banning certain options until a certain condition is met, similar to how the ‘1’ cannot be placed in the banned positions until certain conditions are met.**752. Open the Lock**: This problem, although appears to be unrelated, introduces the concept of BFS which could be used in figuring out the minimum number of operations to achieve a certain state.

## Problem Classification

This problem falls into the domain of “Array Manipulation” and “Optimization” within the broader category of “Data Structures and Algorithms”. It involves manipulating elements within an array under specific rules and constraints, and the goal is to optimize the number of operations to move a particular element to all other permissible positions.

**‘What’ Components:**

- We are given an integer
`n`

, representing the size of an array. - We are given an integer
`p`

in the range [0, n - 1], which represents the initial position of the ‘1’ in an array that’s otherwise filled with ‘0’s. - We are given an array
`banned`

that contains some positions in the array which must remain ‘0’. - We are given an integer
`k`

, which represents the size of the subarray we can reverse in one operation. - We can perform multiple operations of reversing subarrays of size
`k`

. - The goal is to find the minimum number of operations needed to move the ‘1’ to each position in the array, while abiding by the restrictions imposed by the
`banned`

array.

This problem can be further classified as a “Combinatorial Search” problem. The challenge here is to search through all feasible combinations of reversing subarrays to find an optimal sequence of operations that satisfies the problem’s constraints.

The problem also has elements of “Pathfinding”, as you’re essentially trying to find a valid path for the ‘1’ to reach all permissible positions within the array, avoiding the positions in the `banned`

array.

Since the goal is to minimize the number of operations, this problem also has aspects of “Optimization”.

## Distilling the Problem to Its Core Elements

**Fundamental Concept:**This problem is fundamentally based on the concept of “Combinatorial Search” and “Pathfinding” in arrays. It requires finding the optimal sequence of operations (array reversals) to move a specific element (the ‘1’) from its initial position to other positions in the array, while avoiding certain banned positions.**Simpler Description:**Imagine you have a line of zero tiles, with just one tile flipped to one. You can pick up a group of tiles of a fixed size and flip them all at once. However, certain tiles are forbidden to be flipped to one. Your goal is to find the smallest number of flips needed to move the one tile to each spot on the line.**Core Problem & Simplified Statement:**The core problem is determining the minimum number of subarray reversals needed to move the ‘1’ to every position in the array, avoiding the banned positions. A simplified statement could be: “Given a line of zeros with one ‘1’, and some positions where ‘1’ can never be placed, find the minimum number of flips needed to move the ‘1’ to each valid position.”**Key Components:**- Initial array: An array filled with ‘0’s, except for the ‘1’ at position ‘p’.
- Banned positions: Positions that the ‘1’ should never occupy.
- Flip operation: The operation of reversing a subarray of size ‘k’.
- Goal: For each position in the array, we want to find the minimum number of flip operations needed to bring the ‘1’ to that position.

**Minimal Set of Operations:**- Identify the valid moves for the ‘1’ given its current position, the size of the flips (‘k’), and the banned positions.
- Perform a flip operation that moves the ‘1’ towards its target position.
- Repeat this process until the ‘1’ has reached the target position, or until it’s determined that the target position is unreachable.
- Repeat the entire process for each valid target position.

## Visual Model of the Problem

Visualizing this problem can be done through a straightforward representation of the array. Given the problem’s nature, it is conducive to a one-dimensional visualization.

Let’s take an example from the problem statement to create our visualization. Consider this scenario:

`n = 5, p = 0, banned = [2,4], k = 3`

We can represent the initial array as follows (with ‘P’ representing the initial position of ‘1’, ‘X’ for banned positions, and ‘O’ for other positions):

```
P O X O X
0 1 2 3 4
```

We can use the array’s index for each position. The ‘P’ is our ‘1’, the ‘X’ are the banned positions that ‘1’ cannot occupy, and ‘O’ are positions that ‘1’ can move to.

Now, let’s say we want to move ‘1’ to position 3, and k=3. We need to select a subarray of size 3 for reversal, but no matter which subarray of size 3 we select, ‘1’ will land on a banned position (index 2 or 4), making it impossible.

This visual representation helps to understand the problem constraints, the position of the ‘1’, and the banned positions. We can easily track the ‘1’ as we try different moves (reversals), and quickly identify if a move is not possible due to a banned position.

## Problem Restatement

You have an array of zeroes with length ’n’, and a single ‘1’ placed at a specified position ‘p’. There are also some banned positions in the array, which should always remain as zeroes.

You are allowed to reverse subarrays of a specified length ‘k’, with the aim of moving the ‘1’ around in the array. However, it’s crucial that the ‘1’ does not end up in a banned position after a reversal.

Your task is to determine the minimum number of reversal operations needed to move the ‘1’ to every position in the array, where each position should be considered separately. If it’s impossible to move the ‘1’ to a specific position due to the banned positions, mark that position with a ‘-1’.

The output should be an array where each index ‘i’ shows the minimum number of operations needed to bring the ‘1’ to the position ‘i’ in the original array, or ‘-1’ if it’s impossible.

## Abstract Representation of the Problem

Let’s think about this problem in abstract terms:

You have a data structure (initially an array) of a specific size, with one unique element in a particular position. Certain positions within this data structure are not permissible for the unique element.

Your task involves manipulating the structure (by reversing parts of it) with the aim of moving the unique element around. You can reverse segments of a specific size, and you’re seeking to move the unique element to every possible position, without placing it in the non-permissible positions.

For each permissible position, you aim to determine the minimum number of manipulations required to place the unique element there. If it’s impossible to place the unique element in a particular position due to the non-permissible positions, this should be explicitly indicated.

The result should be a data structure that, for each position, indicates either the minimum number of manipulations needed to bring the unique element there, or signifies that it’s impossible to place the unique element there.

This abstract representation keeps the structure and key elements of the problem, but removes the specifics, allowing for broader application of the problem-solving strategy.

## Terminology

Here are a few key terms and concepts relevant to this problem:

**Array**: An array is a data structure that stores elements of the same type in a contiguous block of memory. In the context of this problem, we’re dealing with an array of integers.**Subarray**: A subarray is a contiguous portion of an array. The problem requires reversing subarrays of a specified size (k).**Reversing a Subarray**: This operation involves reversing the order of elements in a specified segment of the array. If we have an array [1, 2, 3, 4] and we reverse the subarray from indices 1 to 3, the array becomes [1, 4, 3, 2].**Index**: An index is a position in an array. In this problem, the index represents a position in the array where an element can be placed. The indices are 0-based, which means they start counting from 0.**Unique Element**: In this problem, there is a unique element which is different from the other elements in the array. In the given problem, ‘1’ is the unique element.**Banned Positions**: These are specific indices in the array where the unique element is not allowed to be placed. The other positions in the array are either ‘0’ or the unique element ‘1’.**Minimum Number of Operations**: The goal is to find the minimum number of times we need to reverse subarrays to move the unique element to a new valid position in the array.

Understanding these terms and concepts is crucial for grasping the problem statement and working towards a solution.

## Problem Simplification and Explanation

Let’s break this problem down into simpler terms.

Imagine you are a coach and your team has to move a special ball (represented by 1) from its initial position to various other positions on a long, linear field. However, the ball can only be moved through a specific method - flipping sections of the field.

In more detail, you can select a segment of the field (subarray of size k), and reverse or flip that segment. When you do this, all the players and the special ball within that segment reverse their positions.

Now, there are certain positions on the field (the banned positions) where the special ball is not allowed to go. These are like puddles or holes on the field, and the ball will get stuck if it lands there.

Given this scenario, your task is to figure out the minimum number of flips needed to get the special ball to each valid position on the field. If there’s a position where it’s impossible to get the ball to without it landing in a puddle, you should denote that with a -1.

In terms of the key concepts:

- The
**“special ball”**is the unique 1 in the array. This is what we’re trying to move. - The
**“field”**represents our array. Positions on the field are array indices. - The
**“flips”**are the reversal operations we can perform on subarrays. - The
**“puddles”**represent the banned positions in the array where we cannot move the special ball.

Our goal is to find the minimum number of “flips” needed to get the “ball” to each valid position on the “field”.

## Constraints

Based on the problem statement and its constraints, here are a few characteristics we can take advantage of when devising an efficient solution:

**Initial position of the ‘1’:**The initial position of the ‘1’ in the array (i.e., the position ‘p’) can play a significant role in determining the feasibility of reaching certain positions and the number of operations required to do so.**Size of the subarray (k):**The given size of the subarray that can be reversed will greatly affect our strategy. For instance, if k is 1, no changes can occur. If k is equal to n (the array’s length), the entire array can be reversed in one operation. Intermediate values of k will require a more nuanced approach.**Banned positions:**These are the positions we are not allowed to move the ‘1’ to. They act as obstacles in our path, and effectively split the array into separate segments. We will need to consider these banned positions when planning our moves.**Array size (n):**Since the array size can go up to 10^5, any solution with a time complexity more than O(n log n) is likely to exceed the time limit. This rules out brute force approaches, and suggests that we need a more optimized strategy, perhaps involving pre-processing or a clever algorithmic approach.

Remember, our goal is to find an efficient strategy to move the ‘1’ to all possible positions in the array while respecting the constraints, so these factors will be crucial in devising that strategy.

The constraints of this problem offer several insights:

**Banned Positions:**The fact that there are positions in the array where ‘1’ is not allowed to be moved to, effectively divides the array into segments. We need to navigate around these “banned” positions. It is possible that for some configurations, we may not be able to move ‘1’ out of a certain segment due to the size of the subarray we are allowed to reverse (k), making it impossible to get ‘1’ to certain positions.**Array Size (n) and Subarray Size (k):**The constraints of array size (up to 10^5) and the fact that we can select subarrays of length k to reverse provide us a clue towards the possible solution’s complexity. It suggests that we may need to employ an algorithm that operates in linear or near-linear time complexity to be efficient enough. The value of k in relation to n can greatly influence the ease or difficulty of moving ‘1’ around the array.**Initial Position (p):**The initial position of ‘1’ in the array can be anywhere from 0 to n - 1. This indicates that we can’t assume ‘1’ to start from any specific location, and our solution needs to be flexible enough to handle this variability.

These insights are valuable because they help inform our problem-solving strategy and guide us towards a solution that fits within the problem’s constraints. By analyzing these constraints, we can better understand the problem space and devise a more effective approach.

## Case Analysis

Here are some additional examples, with each covering unique aspects of the problem.

**1. Case: Single Element Array**

Input: n = 1, p = 0, banned = [], k = 1 Output: [0] Explanation: This is the simplest possible case, where the array consists of a single element. Since ‘1’ is already at the only available position, no reverse operation is needed.

**2. Case: All Positions Banned Except Initial and Target**

Input: n = 5, p = 0, banned = [1, 2, 3], k = 2 Output: [0, -1, -1, -1, 1] Explanation: In this case, all positions are banned except for the initial and the target (final) position. As long as ‘k’ is equal to or larger than the distance between these two positions, it is possible to move ‘1’ to the final position.

**3. Case: Subarray Size Too Small**

Input: n = 6, p = 0, banned = [2, 4], k = 2 Output: [0, 1, -1, 1, -1, -1] Explanation: Here, the size of the subarray ‘k’ is too small to jump over the banned positions. This means the 1 can only move to positions 1 and 3.

**4. Case: Moving Right to Left**

Input: n = 4, p = 3, banned = [], k = 2 Output: [2, 1, 1, 0] Explanation: This case demonstrates that ‘1’ can be moved from a higher index to lower indices as well. Since no positions are banned and ‘k’ is large enough, ‘1’ can be moved to all positions.

**5. Case: Multiple Possible Routes**

Input: n = 6, p = 0, banned = [2], k = 3 Output: [0, 1, -1, 1, 2, 3] Explanation: This case shows that there may be multiple ways to get to the target position. For instance, to get ‘1’ to position 5, we can either move it directly from position 0 to 5 (3 steps), or we can move it to position 1, then to position 3, and finally to position 5 (1 + 1 + 1 = 3 steps).

These cases illustrate a variety of scenarios and constraints that our solution must be able to handle. By considering these different possibilities, we can better ensure that our solution is robust and comprehensive.

From analyzing the various cases, we can gather some key insights:

**The value of ‘k’ greatly influences the mobility of ‘1’.**If ‘k’ is small, ‘1’ may be limited in how it can move around banned positions. If ‘k’ is large, especially larger or equal to the distance between the initial and target position, it can easily move to any non-banned position.**Banned positions act as barriers that prevent the ‘1’ from moving in certain directions.**However, if ‘k’ is large enough to include a banned position and additional positions in a subarray, it can “jump over” the banned position.**The initial and target positions affect the number of operations required.**If the initial position of ‘1’ is close to the target position and there are no banned positions in between, fewer operations are required. Conversely, if the initial position is far from the target position or there are banned positions in the path, more operations may be required.**The order of operations can influence the result.**As seen in the “Multiple Possible Routes” case, there may be more than one way to get to the target position. The number of operations can differ depending on the path taken.

These insights can guide us in formulating an efficient strategy to solve the problem. It’s crucial to account for these factors while designing the solution.

## Identification of Applicable Theoretical Concepts

Here are a few mathematical and algorithmic concepts that could be helpful for solving this problem:

**Sliding Window**: Since we’re allowed to reverse subarrays of size ‘k’, a sliding window approach could be useful. We could slide a window of size ‘k’ across our array and determine whether we can move ‘1’ within this window, keeping in mind the banned positions.**Breadth-First Search (BFS) or Dijkstra’s Algorithm**: The problem could be visualized as a graph problem where each position in the array is a node, and there is an edge between two nodes if and only if they can be reached by a single operation. ‘1’ can move from one node to another if it’s possible to reverse a subarray of size ‘k’ containing these two nodes. BFS or Dijkstra’s algorithm can then be used to find the shortest path from the starting node to each of the other nodes.**Dynamic Programming**: If there is a certain pattern to the moves or if the minimum moves to a certain position can be computed from the minimum moves to another position, dynamic programming could be a good approach to this problem.**Sorting**: In relation to the BFS approach, sorting the positions based on the number of operations required to reach them could be beneficial. A priority queue (min-heap) can be utilized to ensure we always check the positions that require the least operations first.**Modulo Arithmetic**: This concept could be useful in handling the ‘circular’ nature of the problem when considering array reversal operations.

Remember, it’s not necessary that all of these concepts will be used in the final solution. They’re just possible paths to explore while tackling the problem. The choice would depend on the insights gained from further analysis and exploration of the problem.

## Problem Breakdown and Solution Methodology

Let’s use an analogy of a sliding puzzle game to understand this problem better. You can visualize the array as a puzzle where the ‘1’ is a blank space that can slide to certain positions, and the ‘0’s are puzzle pieces that fill up the remaining space. The banned positions are fixed pieces that can’t be moved or occupied by the ‘1’. Each operation we perform is similar to sliding several pieces of the puzzle at once.

Given this analogy, here’s the approach to solving the problem:

**Initialize a list for the result**: Create a list of length`n`

initialized to`-1`

. This list will eventually hold the minimum number of operations required to bring the ‘1’ to each position in the array.**Set the initial position**: We know that no operations are required to bring the ‘1’ to its initial position`p`

. So, set`result[p]`

to`0`

.**Construct a graph**: Now, we want to find out how many operations it takes to move ‘1’ from one position to another. Since this involves interactions between different positions, we can visualize it as a graph problem. Each position in the array can be a node in the graph, and there is an edge between two nodes if and only if ‘1’ can move from one to the other with a certain number of operations.**Compute the edges**: We need to figure out what the edges in our graph are. Remember that we can reverse a subarray of size ‘k’, which is like sliding a window of size ‘k’ across the array. For each window position, if ‘1’ is in the window and it can be moved to another position within the window that is not banned, add an edge between the current position of ‘1’ and that position.**Find shortest paths**: Once we have our graph, we can apply a shortest path algorithm like Dijkstra’s algorithm or Breadth-First Search (BFS) to find the minimum number of operations required to move ‘1’ from its initial position to every other position. Each edge in the graph represents a possible operation, and the weight of the edge is the number of operations. Our goal is to find the shortest path from node`p`

to each of the other nodes.**Update the result**: For each position`i`

, if a path from`p`

to`i`

exists in the graph, update`result[i]`

to be the number of operations (length of the path). If no path exists,`result[i]`

will remain`-1`

.**Return the result**: After iterating through all positions, return the`result`

list.

Now, let’s discuss how changes in the problem’s parameters affect the solution:

- If we increase
`n`

, the size of the array (and thus the graph) increases. This could potentially increase the time complexity of the solution as we have more nodes and edges to process. - If we change
`p`

, the initial position of ‘1’, it changes the starting point in our graph, but doesn’t affect the overall approach. - If we add more positions to the
`banned`

list, it increases the number of positions that ‘1’ can’t move to, possibly making the problem harder to solve. It could also reduce the number of edges in our graph. - If we increase
`k`

, the size of the subarray we can reverse, it gives us more flexibility in moving ‘1’ around. This could potentially make the problem easier to solve, but it could also increase the number of edges in our graph.

Let’s demonstrate this approach with an example:

Suppose `n = 4`

, `p = 0`

, `banned = [1,2]`

, `k = 4`

.

- We initialize the result list to
`[-1, -1, -1, -1]`

and set`result[0]`

to`0`

, so now the result is`[0, -1, -1, -1]`

. - We construct the graph with nodes for each position and compute the edges based on the window of size
`k = 4`

. The graph we get is`{0: [3], 3: [0]}`

(there’s an edge from node 0 to node 3 and vice versa). - We find the shortest path from node
`0`

to each of the other nodes. For node`3`

, the shortest path is`0 -> 3`

with a length of 1 operation. - We update
`result[3]`

to`1`

, so now the result is`[0, -1, -1, 1]`

. - Since no paths exist to nodes
`1`

and`2`

,`result[1]`

and`result[2]`

remain`-1`

. - Finally, we return the result list
`[0, -1, -1, 1]`

, which matches the expected output.

In conclusion, by visualizing the problem as a graph and applying shortest path algorithms, we can find the minimum number of operations required to move ‘1’ from its initial position to every other position in the array, while respecting the banned positions.

## Inference of Problem-Solving Approach from the Problem Statement

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

## Stepwise Refinement

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

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.

## Thought Process

Failed to provide working code after several attempts. LC solution:

```
from sortedcontainers import SortedList
class Solution:
def minReverseOperations(self, n, p, banned_vals, K):
remaining = [SortedList(), SortedList()]
banned = set(banned_vals)
for u in range(n):
if u != p and u not in banned:
remaining[u & 1].add(u)
queue = [p]
dist = [-1] * n
dist[p] = 0
for node in queue:
lo = max(node - K + 1, 0)
lo = 2 * lo + K - 1 - node
hi = min(node + K - 1, n - 1) - (K - 1)
hi = 2 * hi + K - 1 - node
for nei in list(remaining[lo % 2].irange(lo, hi)):
queue.append(nei)
dist[nei] = dist[node] + 1
remaining[lo % 2].remove(nei)
return dist
```

This code is a Python solution to a problem involving the manipulation of nodes in a graph. The main task is to find the minimum number of reverse operations to reach each node from a given starting point, with certain nodes being banned. Let’s break down the code:

Import: The code starts by importing the

`SortedList`

class from the`sortedcontainers`

module. This is a Python library for sorted collections, which are faster than built-in collections.Class Definition: The code then defines a class named

`Solution`

with a method named`minReverseOperations`

.The

`minReverseOperations`

method: This method takes five arguments -`n`

(the total number of nodes),`p`

(the starting point),`banned_vals`

(the nodes that are banned), and`K`

(the maximum distance to move in a single operation).It first initializes two sorted lists -

`remaining[0]`

and`remaining[1]`

. These lists will hold the remaining nodes to be visited, separated by whether their index is even or odd.It also creates a set

`banned`

to easily check if a node is in the banned list or not.Then, it adds all the nodes that are not the starting point and not banned to the appropriate

`remaining`

list based on whether they are even or odd.It sets up a queue with the starting node, and a list

`dist`

with size`n`

, initializing all distances to -1, and the distance to the starting node as 0.

Breadth-First Search (BFS): The code then performs a breadth-first search (BFS) from the starting node, updating the

`dist`

list with the minimum number of operations needed to reach each node.It calculates the lower (

`lo`

) and upper (`hi`

) bounds for the nodes that can be reached from the current node (`node`

). It uses the`irange`

method of the`SortedList`

class to find all nodes within these bounds.It then removes each of these nodes from the

`remaining`

list, adds them to the queue, and updates their distance from the starting node.

Return: Finally, the method returns the

`dist`

list, which represents the minimum number of operations needed to reach each node from the starting point.

This solution efficiently uses sorted lists and a BFS approach to solve the problem. It handles the constraints of banned nodes and the maximum operation distance `K`

. The key insight here is the use of the BFS algorithm to ensure the minimum number of operations, and the use of sorted lists to efficiently find the nodes that can be reached from the current node.

## From Brute Force to Optimal Solution

Given a graph of (n) nodes, we need to find the minimum number of “reverse operations” needed to reach every node starting from a particular node (p). A reverse operation is defined as moving from the current node to any other node within a maximum distance of (K). However, some nodes are “banned”, meaning we cannot visit them.

Now, let’s start with a brute force approach and then gradually optimize it.

### Brute Force Approach:

A simple, brute-force way to approach this problem would be to perform a Depth-First Search (DFS) from the starting node, exploring every possible path.

In each step, we would iterate over all (n) nodes and try to move to that node if it’s within the distance (K) and it’s not a banned node. We would keep track of the minimum number of operations taken to reach each node.

However, this approach has a time complexity of (O(n^2)) because in the worst case, we are visiting all other nodes from each node. Moreover, if there are multiple ways to reach a node, we would be calculating the minimum distance multiple times. This redundancy makes this solution highly inefficient.

### Optimized Approach:

Let’s now consider how we can optimize this brute force approach.

**Use Breadth-First Search (BFS) instead of DFS**: BFS is an ideal choice for this problem because we’re trying to find the minimum number of operations to reach all nodes. BFS starts from a node and visits all its neighbors before moving to the next level. This ensures that the first time we reach a node, it’s through the shortest path. This eliminates the need to calculate the minimum distance multiple times as in the brute force approach.**Use a queue for BFS**: As standard for a BFS approach, we use a queue data structure. We start by adding the starting node (p) to the queue. Then, in each step, we remove a node from the front of the queue, add its reachable nodes to the back of the queue, and update their distances.**Use sorted lists for reachable nodes**: Instead of iterating over all (n) nodes to find the nodes that we can move to, we can keep two sorted lists of remaining nodes, separated by whether their index is even or odd. We use the`irange`

method of the SortedList class to efficiently find all nodes within a maximum distance (K). This reduces the time complexity of each step from (O(n)) to (O(\log n)).**Use a set for banned nodes**: To quickly check whether a node is banned, we can keep the banned nodes in a set. Checking membership in a set takes constant time (O(1)), whereas in a list it would take linear time (O(n)).

After these optimizations, the time complexity of our solution becomes (O(n \log n)), which is a significant improvement over the brute force approach. The space complexity is (O(n)) for storing the distances, the queue, the remaining nodes, and the banned nodes.

These optimizations help us avoid unnecessary computations and make efficient use of data structures to reduce the time complexity while maintaining a reasonable space complexity.

## Code Explanation and Design Decisions

- Initial Parameters:
`n`

: Represents the total number of nodes in the graph.`p`

: Represents the starting node from where we need to perform the operations.`banned_vals`

: Contains the nodes that are banned or off-limits.`K`

: Represents the maximum distance that we can move in a single operation.

These parameters define the structure of the graph, the starting point, the constraints (banned nodes and maximum distance), and the end goal (minimum operations).

Primary Loop: The primary loop is the Breadth-First Search (BFS) over the nodes in the graph. Each iteration of the loop represents a single operation of moving from the current node to a reachable node. The iteration advances the solution by exploring all nodes that can be reached from the current node and updating their distances.

Conditions or Branches: The condition within the loop checks whether a node falls within the range

`[lo, hi]`

and therefore can be reached from the current node. This condition is significant as it ensures that we only consider nodes that can be reached within the maximum distance`K`

.Updates or Modifications: Within the loop, the distance for the reachable node

`nei`

is updated and`nei`

is removed from the`remaining`

list. These changes are necessary to keep track of the minimum number of operations needed to reach each node and to ensure that each node is visited only once.Invariant: The invariant in this code is the SortedLists

`remaining`

. The lists always contain the nodes that have not been visited yet, and they are always sorted. This invariant helps us efficiently find the nodes that can be reached from the current node and ensures that each node is visited exactly once.Final Output: The final output is the list

`dist`

, which represents the minimum number of operations needed to reach each node from the starting node. This output satisfies the problem’s requirements as it provides the solution for the minimum number of operations needed to reach all nodes given the constraints of the problem.

## Coding Constructs

High-Level Problem-Solving Strategies: This code uses Breadth-First Search (BFS) as the primary problem-solving strategy. It also uses efficient data structures like SortedLists and sets for optimizing the search process. The code is built on graph theory concepts and leverages properties of parity (even and odd numbers) for further efficiency.

Explanation for a Non-Programmer: Imagine you are in a city with many intersections (nodes), and you can only move to other intersections within a certain distance. Some intersections are off-limits (banned). Starting from a specific intersection, you want to find out the minimum number of moves required to reach all other intersections. This code is a set of instructions to solve that task in the most efficient way.

Logical Elements: The code uses a queue data structure for BFS, sorted lists for maintaining remaining nodes, and a set for quick membership check for banned nodes. It uses loops to iterate through the data and conditional statements to check for certain conditions (like whether a node is banned or not). The core logic is built around graph traversal.

Algorithmic Approach: Starting from a given node, the code uses BFS to explore all reachable nodes within a certain distance. It keeps track of the minimum number of operations needed to reach each node. It cleverly categorizes nodes into even and odd groups to streamline the search process, reducing the number of nodes that need to be considered in each step.

Key Steps or Operations: The code starts by categorizing the nodes based on their parity and putting them in sorted lists. It then performs a BFS, in each step, identifying nodes that can be reached from the current node based on a calculated range. Once a node is visited, it’s removed from the sorted list and its minimum distance is updated. The BFS continues until all nodes have been visited.

Algorithmic Patterns: The primary algorithmic pattern here is BFS, a strategy for traversing or searching tree or graph data structures. It efficiently finds the shortest paths in a graph. The code also uses efficient data structures (SortedList and set) and properties of numbers (parity) to optimize the search process. It leverages the divide and conquer strategy by separating nodes into two groups (even and odd) and handling them separately.

## Language Agnostic Coding Drills

Coding Concepts:

a.

**Variables and Data Types**: Understanding basic data types like integers, lists, and sets.b.

**Loops and Conditional Statements**: Using`for`

loops and`if`

statements to control the flow of the program.c.

**Functions and Methods**: Understanding built-in functions and methods for different data types. For example, list’s`append`

method, set’s`add`

method, and SortedList’s`irange`

and`remove`

methods.d.

**Data Structures**: Understanding more advanced data structures like SortedList, queue, and set.e.

**Bitwise Operations**: Understanding bitwise operations like`&`

(AND). In this code, it’s used to check the parity (even or odd) of a number.f.

**Breadth-First Search (BFS)**: Understanding the BFS algorithm for graph traversal.g.

**Graph Theory**: Understanding concepts of nodes and edges, and how to represent a graph.h.

**Complexity Analysis**: Understanding how to analyze the time and space complexity of the code.Order of Increasing Difficulty:

a.

**Variables and Data Types**: This is the most basic concept that every programmer should know. It forms the foundation for all other concepts.b.

**Loops and Conditional Statements**: These concepts are slightly more complex as they involve controlling the flow of the program based on certain conditions.c.

**Functions and Methods**: While functions and methods are a core part of programming, understanding how different functions and methods work for different data types requires some experience.d.

**Data Structures**: Understanding and using more advanced data structures like SortedList, queue, and set is more complex and requires a good understanding of the previous concepts.e.

**Bitwise Operations**: Bitwise operations are more specialized and may not be used as often as the previous concepts. Understanding them requires a good understanding of binary numbers.f.

**Breadth-First Search (BFS)**: BFS is a specific algorithm used for graph traversal. It’s more complex as it involves using a queue and often requires modifying data during traversal.g.

**Graph Theory**: Graph theory is a specific field in computer science. Understanding it requires a good understanding of data structures and algorithms.h.

**Complexity Analysis**: Complexity analysis is one of the most complex concepts as it requires a deep understanding of algorithms, data structures, and the underlying computer architecture.Problem-Solving Approach:

a.

**Understand the Problem**: The first step is to understand the problem and its constraints. In this case, we’re trying to find the minimum number of operations to reach all nodes from a given node, with certain nodes being banned and a maximum distance for each operation.b.

**Define the Graph**: Next, we define the graph with`n`

nodes. We set the distance to each node as -1, except for the starting node which is 0. We also separate the nodes into two groups based on their parity (even or odd).c.

**Perform BFS**: We then perform a BFS from the starting node. In each step, we calculate the range of nodes that can be reached from the current node and update their distances. We continue the BFS until all nodes have been visited.d.

**Analyze the Complexity**: Finally, we analyze the time and space complexity of our solution to ensure that it’s efficient. In this case, the time complexity is O(n log n) and the space complexity is O(n).

## 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.

## Q&A

Splitting by parity seems to be critical for the performance (TLE if you don’t do this and simply use one sorted list, while checking for appropriate parity in the condition within the loop). But I don’t understand why. It supposed to be (2N)^4/3 = 2.5 N^4/3. Yet it slows it down hugely for some reason.

The division of nodes into two separate lists based on parity (even or odd) is an optimization that is particularly effective in reducing the number of nodes that we need to consider when identifying reachable nodes in each step.

When we’re at a node, we know that we can only reach nodes within the range `[node - K + 1, node + K - 1]`

. Further, the parity of a reachable node will be the same as the parity of the sum of the current node and the distance to that node. This is because adding two even numbers or two odd numbers always results in an even number, while adding one even and one odd number results in an odd number. In other words, a reachable node and the current node will have the same parity if and only if the distance to that node is even.

Therefore, when we’re looking for reachable nodes, we only need to consider nodes in the `remaining`

list with the same parity as `lo % 2`

, where `lo`

is the lower bound of the range of reachable nodes. This effectively halves the number of nodes that we need to consider in each step.

If we were to use a single sorted list for all remaining nodes, we would need to check the parity of each node in the range `[lo, hi]`

to see if it’s a reachable node. This would result in more nodes being examined and removed from the list, which would slow down the algorithm.

Although theoretically, the time complexity of the two approaches may seem to be close ((2N)^(4/3) for the single list approach vs. N^(4/3) for the two list approach), in practice, the constant factors and lower order terms can have a significant impact on performance. The two list approach can be much faster than the single list approach when the number of nodes is large, which is why it is preferred in this case.

Could you please explain why nei is chosen from some interval [lo, lo+2, …, hi]. rather than [lo, lo+1, …, hi] ?

The interval for `nei`

is defined as `[lo, lo+2, ..., hi]`

instead of `[lo, lo+1, ..., hi]`

because of the parity-based sorting of the nodes. The nodes are divided into two separate lists based on their parity (even or odd).

This parity-based separation is based on the observation that a reachable node and the current node will have the same parity if and only if the distance to that node is even.

Given that `lo`

is the lower bound of the range of reachable nodes, the `lo % 2`

operation is used to identify the sorted list (either for even or odd nodes) we should be looking in.

If `lo`

is even, `lo % 2`

will be `0`

and we will be looking at the sorted list of even nodes, thus we want to increment by `2`

to maintain the same parity (even). The same goes for when `lo`

is odd. We want to look at the sorted list of odd nodes, and increment by `2`

to maintain the same parity (odd).

By incrementing `lo`

by `2`

at each step, we ensure that we are only considering nodes of the same parity, thus reducing the number of nodes that we need to consider in each step.

This is why the interval for `nei`

is `[lo, lo+2, ..., hi]`

, because it ensures we only consider nodes within the reachable distance `K`

and of the same parity, which speeds up the algorithm.

## Similar Problems

Here are 10 problems that involve similar concepts or problem-solving strategies as the problem we’ve just solved. The key concepts involved in our problem were breadth-first search (BFS), graph traversal, and the use of optimized data structures like SortedList and sets:

LeetCode 279. Perfect Squares: This problem requires finding the least number of perfect square numbers which sum up to a given number

`n`

. It involves a BFS approach to find the minimum number of steps, similar to our problem.LeetCode 127. Word Ladder: The problem is about finding the shortest transformation sequence from a start word to an end word, with certain transformations allowed. This is similar to our problem because it involves finding the shortest path in a graph using BFS.

LeetCode 207. Course Schedule: This problem requires checking if it’s possible for a student to finish all courses given some conditions. It involves graph traversal and can be solved using BFS.

LeetCode 773. Sliding Puzzle: The problem is to find the least number of moves to solve a sliding puzzle. It requires BFS to find the minimum number of moves.

LeetCode 752. Open the Lock: This problem requires finding the minimum total number of turns to open a lock, which is similar to our problem of finding the minimum number of operations. It can be solved using BFS.

LeetCode 934. Shortest Bridge: The problem is to find the shortest bridge between two islands in a 2D grid. It requires BFS to find the minimum distance.

LeetCode 126. Word Ladder II: The problem is to find all the shortest transformation sequences from a start word to an end word. This is an extension of the Word Ladder problem and involves BFS for finding shortest paths.

LeetCode 1293. Shortest Path in a Grid with Obstacles Elimination: This problem involves finding the shortest path in a grid from the top-left corner to the bottom-right corner with some conditions. BFS is required to find the minimum path.

LeetCode 743. Network Delay Time: The problem is to find how long it will take for all nodes to receive a signal. It involves graph traversal and can be solved using BFS.

LeetCode 787. Cheapest Flights Within K Stops: The problem is to find the cheapest price from a source city to a destination city within a certain number of stops. It involves BFS to find the minimum cost path with a constraint on the number of steps, similar to our problem.

These problems involve similar strategies such as BFS, graph traversal, and optimized data structures. They require the same kind of thinking and approach as our original problem.