First Day Where You Have Been in All the Rooms

We can solve this problem using dynamic programming. Let’s define dp[i] as the first day you have been in room i. We’ll initialize the dp array as dp[0] = 0 and use the rules given to calculate the rest of the elements.

The final result will be dp[n-1].

Python solution:

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class Solution:
    def firstDayBeenInAllRooms(self, nextVisit: List[int]) -> int:
        MOD = 10**9 + 7
        n = len(nextVisit)
        dp = [0] * n
        for i in range(1, n):
            # Odd number of times in room i-1
            dp[i] = (dp[i-1] * 2 - dp[nextVisit[i-1]] + 2) % MOD
            
        return dp[-1]

Explanation:

dp[i] = (dp[i-1] * 2 - dp[nextVisit[i-1]] + 2) % MOD is derived from the rules.
dp[i-1] * 2 covers the visits where we go from room i-1 to room i.
dp[nextVisit[i-1]] subtracts the extra days that were counted in the previous step where you had to go back to nextVisit[i-1] instead of room i.
Finally, +2 adds the days for visiting the room i-1 two times to move to room i.

This code will return the label of the first day where you have been in all the rooms, modulo (10^9 + 7).

10 Prerequisite LeetCode Problems

For “1997. First Day Where You Have Been in All the Rooms”, the following are a good preparation:

“70. Climbing Stairs”: Understanding this simple problem will help you grasp the concept of dynamic programming which is required in problem 1997.
“198. House Robber”: This problem will strengthen your skills in dynamic programming, which is useful for problem 1997.
“509. Fibonacci Number”: This problem introduces the Fibonacci series concept, which is a simple case of dynamic programming and will help to better understand problem 1997.
“300. Longest Increasing Subsequence”: This problem will help in understanding the concept of sequences in dynamic programming, which is useful for problem 1997.
“746. Min Cost Climbing Stairs”: This problem will familiarize you with using dynamic programming to minimize a value (or maximize, in the case of problem 1997).
“53. Maximum Subarray”: Understanding this problem will assist you in determining maximum values in an array using dynamic programming, a skill required for problem 1997.
“139. Word Break”: This problem will aid you in understanding how dynamic programming can be used in string-based problems, an approach useful for problem 1997.
“322. Coin Change”: This problem demonstrates the use of dynamic programming in optimization problems, an approach necessary for problem 1997.
“62. Unique Paths”: This problem will strengthen your understanding of how dynamic programming can be used to find the total number of unique paths in a grid, a concept relevant to problem 1997.
“64. Minimum Path Sum”: This problem will introduce you to the concept of dynamic programming in the context of finding minimum values, an idea that is inverted in problem 1997 (where we’re looking for a maximum).

These problems enhance your understanding of dynamic programming, which is the key technique used in solving problem 1997.

Problem Classification

This is a sequence generation and analysis problem in the domain of combinatorics. The key ‘What’ components are:

Rooms - Set of elements to visit
Sequence - Order of visiting rooms
Rules - Logic for generating sequence
Odd/even counts - Criteria for choosing next element
First occurrence - Optimization objective

Based on these aspects, we can further classify this problem as:

Sequence generation - Producing a sequence following specific rules
State modeling - Tracking visited states and counts
Dynamic simulation - Evolving system state over time
Combinatorics - Generating permutations with constraints
Optimization - Finding earliest complete occurrence

The core challenge is to incrementally generate the room visit sequence following the provided rules, while optimizing to find the earliest point the full set is covered.

This requires skills in combinatorics, state tracking, sequence analysis, and optimization - all focused on sequentially generating a stateful sequence. The key is modeling and incrementally evolving the system state.

Clarification Questions

Here are some clarification questions that could be asked about this room visit problem:

Are the room labels always sequential integers starting from 0?
Can we assume the rooms are visited in numerical order on even count days?
Is nextVisit array immutable? Or can rooms change their next visit room over time?
For an odd visit to room i, is visiting the same room i again valid if nextVisit[i] = i?
Does the order of visiting rooms on even counts matter, or just that all are visited?
Can we visit the same room multiple times in a day?
Is n guaranteed to be greater than 1? Can we have just 1 room?
For modulo arithmetic, should we use 109 + 7 or 109 + 9?
Should we minimize days taken or just output any valid solution?
Does order of visiting rooms on the final occurrence day matter?

Asking clarifying questions helps validate assumptions, exposes edge cases, and provides insights into intended behavior, constraints, and objectives. This additional context improves and focuses the problem analysis.

Problem Analysis and Key Insights

Here are some key insights gained from analyzing the problem statement:

The rules define a sequence generation process that can be simulated. Tracking visited state is key.
Odd/even visit counts dictate how the next element is chosen. Parity drives sequence evolution.
The sequence forms a cyclic pattern that repeats at some point.
Modeling incrementally evolving state and parity is crucial to simulate visits.
Optimization relies on detecting when the full set is covered, not the order.
Constraint of first full occurrence focuses search compared to exhaustive generation.
Actual room values likely don’t matter, just treating rooms as distinct states.
Modulo arithmetic indicates sequence may need to be simulated for a large number of steps.
Problem structure suggests dynamic programming may be applicable.

The key insights are around modeling the state evolution rules to simulate the sequence, recognizing the cyclic pattern, and focusing on detecting first full coverage rather than sequence specifics. These observations help scope the problem and direction for solutions.

Problem Boundary

Based on analyzing the problem statement, here is how I would summarize the scope:

Inputs:

Number of rooms n
Array nextVisit[] specifying next room visit rules

Outputs:

The first day all rooms are visited

Objective:

Find the earliest day that the full set of rooms have been visited

Constraints:

Rooms labeled 0 to n-1
Must follow specified room visit rules
Output modulo 10^9+7

Focus:

Modeling sequence generation process
Tracking state of visited rooms
Detecting first complete coverage

Out of scope:

Actual order rooms are visited
Values/labels of rooms beyond modulo constraints
Minimizing number of days

So in essence, the scope is focused on the incremental state evolution, tracking coverage, and optimizing the detection of first full set occurrence. Order, actual room labels, and total days minimization are not a core part of this problem.

Here are some ways we can establish clear boundaries for the scope of this room visit problem:

The inputs are limited to the number of rooms n, and the visit rules defined by nextVisit[] array. No other inputs.
The only output is the first day all rooms are visited. We don’t need to output the full actual sequence or order.
We only care about the labels of rooms being 0 to n-1 and using modulo arithmetic. Actual room numbers or other identifiers are out of scope.
The objective is detecting first complete coverage. Minimizing total days taken or order of visit is not needed.
The rules for generating the sequence based on parity are well-defined. Other ways to construct sequence are out of scope.
No clarification needed on how rooms are chosen for even days since not relevant.
We can assume sequential room order for even days. Only odd days defined by nextVisit[] matter.
The constraints are discrete - integer days, fixed rooms, parity based rules. Continuous spaces/time out of scope.
Only the final occurrence day modulo 10^9+7 is needed, not intermediate days.

Keeping the scope narrowly focused on the key inputs, outputs, and constraints enables solving the core problem without getting distracted by unnecessary details or extensions. The boundaries help isolate what matters.

Distilling the Problem to Its Core Elements

Here are my thoughts on analyzing this room visit problem:

Fundamental concept:

This problem boils down to iteratively generating a sequence following a set of rules. The core algorithmic technique is dynamic state evolution through parity-based rules.

Simple explanation:

We want to figure out the first day that someone has visited all rooms, when they visit a new room each day based on some rules.

Core problem:

Find the first occurrence of the full set in an evolving sequence. We can simplify it to detecting when a dynamically generated sequence covers all elements.

Key components:

Sequence generation rules
Tracking visited state
Updating state based on parity
Detecting full set coverage

Minimal operations:

Model room visit rules
Increment day counter
Choose next room by parity of day count
Update visited rooms
Check if all rooms visited
Repeat until full coverage detected

So in essence, this problem involves using a state machine and parity-based rules to generate a sequence, and optimizing the detection of the full set occurrence.

Visual Model of the Problem

Here are some ways we could visualize this room visit problem to provide more intuition:

State transition diagram - Model each room as a state, transitions between them based on odd/even day rules.
Sequence chart - Show day numbers horizontally, rooms vertically, marks for days each room is visited.
Bar chart - For each day show rooms visited as colored bars. See full coverage.
Pie chart - Show rooms visited daily as shaded pie slices. Fully shaded means full coverage.
Map - Visualize person moving between rooms on odd/even day rules. Animated.
Matrix - Grid of days x rooms, shade cells for rooms visited each day. See pattern.
State machine - Show state bubbles for each room, arrows for odd/even transitions between them.

Visualizing the sequence generation as a graph, chart or matrix can provide insight into how the room visit states evolve over days. Animation can bring it to life.

Problem Restatement

Here’s how I would paraphrase the problem statement in my own words:

We want to find the first day that someone has visited all rooms, given these rules:

There are n rooms labeled 0 to n-1
Each day the person visits one room
They start in room 0 on day 0
If it’s an odd-numbered visit to a room, they next visit the room numbered nextVisit[room]
If it’s an even-numbered visit to a room, they next visit room (room + 1) % n
We want the first day where they have been to all rooms
Return the day number modulo 10^9+7

So in essence, we need to simulate visiting rooms by these rules day-by-day while tracking rooms visited, until we detect the first day where the full set of rooms have been completed. We don’t care about the full sequence, just optimizing for quickest full coverage.

The key is modeling the state transitions and evolving the state incrementally via the parity-based visit rules.

Abstract Representation of the Problem

Here is one way we could formulate an abstract representation of this room visit problem:

We have a set S of n distinct elements. We want to generate a sequence by repeatedly applying a transformation T that chooses the next element based on:

If the count of occurrences of element e is odd, T maps e to f(e) where f is some function.
If the count is even, T maps e to g(e) where g is another function.

The goal is to determine the earliest point at which the generated sequence contains all elements in S.

More formally:

Input:

Set S of n distinct elements
Functions f and g representing mapping rules

Output:

Integer m - index where full set first occurs

Objective:

Incrementally generate sequence by repeated application of transformation T
Detect first occurrence of full set

By abstracting the rooms and days into elements and sequence positions, we can focus on the underlying pattern of generating a sequence by applying a state transition function T determined by parity counts. This viewpoint relates the problem to general sequence generation and set coverage problems.

Terminology

Here are some key terms and concepts that are important for this room visit problem:

State machine - Model for systems that transition between discrete states. Applicable here with rooms as states.

Parity - Whether a number is odd or even. Used here to determine room transitions.

Sequence generation - Process of incrementally generating an ordered sequence of elements. Core problem activity.

Full coverage - Objective of reaching a state containing all elements. Indicates solution found.

Dynamic programming - Approach to solve problems by combining solutions to subproblems. Could apply here.

Combinatorics - Study of discrete structures and their properties. Room sequences are combinatorial.

Modulo arithmetic - Doing math with remainders. Used here to constrain room numbers and output.

Optimization - Finding the best solution among alternatives. We want fastest full coverage.

The core concepts center around modeling the room visits as a state machine, using parity to drive transitions, and optimizing for earliest full coverage. Techniques like dynamic programming and combinatorics may be leveraged to solve the sequenced set coverage problem.

Problem Simplification and Explanation

Here’s one way I could break down this room visit problem into simpler concepts using a metaphor:

Let’s compare this to a person exploring a castle with different wings.

The wings are like the rooms. The person starts in the main hall on day 1.

On odd-numbered visits to a wing, they use a clue to determine the next wing to explore.

On even visits to a wing, they just go clockwise to the next wing.

We want to know the first day they’ve been to all wings of the castle based on these rules.

Key concepts:

Wings = Rooms
Odd/even visits = Parity-based rules
Clues = Next visit mappings
Full coverage = Visiting all wings

This castle exploration analogy tries to convey the idea of incrementally generating a sequence by transitioning between states (wings) based on parity. The goal is detecting when full coverage occurs. The metaphor maps the core concepts into more concrete terms.

Constraints

Here are some characteristics and constraints from the problem statement that could help optimize finding an efficient solution:

Small number of rooms (n <= 10^5) - Can use simple array or hash table to track visited state.
Rooms labeled 0 to n-1 - Easy to iterate and check for full coverage.
Parity based transitions - Only need to track odd/even visit counts versus full history.
No specified order within a day - Don’t need to track permutation, just coverage.
Output modulo bound - Allows compactly storing day counts.
Constraint of first occurrence - Algorithm can stop early once detecting coverage.
Increasing transitions - Tendency to cycle through all rooms.
nextVisit defines bounded transitions - Transitions are local and predictable.

These characteristics suggest iteratively simulating the sequence while incrementally tracking state changes is feasible. The parity, moderate input size, lack of order requirements, and incremental output allow optimizations.

Here are some key insights gained from analyzing the constraints:

Small room set size allows simple data structures like arrays to track state. Avoids complexity of larger sets.
Room label scheme enables easy iteration and checking full coverage.
Parity transitions simplify state changes to just odd/even counts rather than full history.
Output modulo arithmetic indicates potential for large numbers that need compact representation.
Specified first occurrence optimizes for early stoppage once detecting coverage.
Bounded transitions means state changes are predictable locally.
Lack of order requirements within a day simplifies tracking.

Overall, the constraints allow using simple data structures, simplifying parity-based state changes, stoppable incremental generation, and focused trackable transitions. This bounds the problem and solution space considerably compared to more generalized set sequence generation.

Case Analysis

Here are some additional test cases covering different scenarios:

Base cases:

1 room - Determine if fixed start counts
2 rooms - Simple alternating pattern

Input variances:

Repeated next visits - Loops in sequence
Sparse next visits - Skips some rooms
Same parity transitions - E.g. all even transitions

Constraints:

Large room set - Stress test data structures
Maximum sequence length - Verify modulo arithmetic

Edge cases:

0 rooms - No sequence possible
1 room - Only start room
Identical next visits - Cycle between two rooms

The edge cases are 0 rooms which has no solution, 1 room which immediately solves, and identical next visits which causes getting stuck oscillating between two rooms.

The cases validate handling various transition patterns robustly, stress test scalability, verify constraints, and cover degenerate edge cases. Together they help ensure the solution works correctly across the full problem space.

Here are some ways to visualize the different test cases for this room visit problem:

Base cases:

1 room - Single state loop
2 rooms - Flipping between two states

Input variances:

Repeated next visits - Looping subgraph
Sparse next visits - Disconnected graph components
Same parity transitions - Bipartite graph with edges between groupings

Constraints:

Large rooms - Zoomed out graph with many nodes
Max sequence - Animated simulation running for days

Edge cases:

0 rooms - Empty graph
1 room - Single isolated node
Identical next visits - Graph cycle of two nodes

We can represent the cases using:

State transition graphs showing room visit patterns
Animations illustrating movement rules over days
Charts plotting rooms visited over time

Visualizing request patterns as graphs and simulated journeys makes the cases more intuitive. Diagrams help highlight differences like disconnected components, loops, stuck cycles, etc.

Here are some key insights gained by analyzing the different test cases:

Base cases reveal initialization logic needs to handle small room sets properly.
Varied transition patterns show code must be flexible to different graph shapes like loops and disconnected components.
Large inputs stress test data structure scalability and modulo arithmetic.
Identical next visits case exposes possibility of getting stuck in a cycle and not achieving full coverage.
Varying parity transitions validates properly alternating between odd/even rules.
Edge cases like 0 rooms and 1 room check boundary conditions are handled.
Visualizing cases as graphs/animations provides intuition about room visit dynamics.

Overall, the cases validate properly incrementally generating a wide variety of possible room visit sequences, handling constraints, not making assumptions about structure, and taking care of degenerate edge cases. Together they help ensure the solution is robust across the problem space.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help simplify and solve this room visit problem:

Graph theory - Model rooms as graph nodes and transitions as edges. Leverage graph algorithms.
Dynamic programming - Build up solutions for larger room sets from smaller subsets.
Combinatorics - Study of discrete structures like permutations and combinations. Applicable since generating sequences.
State machines - Model system transitions between discrete states. Aligns with room transitions.
Modular arithmetic - Useful for constraining room numbers and large output.
Bitwise operations - Compactly encode state using bit flags. Quick to check full coverage.
Bloom filters - Probabilistic data structure for set membership. Could help track rooms visited.
Discrete math - Useful for problems involving discrete states and transitions like here.

In particular, graph algorithms, dynamic programming, and combinatorics seem promising for generating and analyzing the room visit sequences. Bitwise operations, modular math, and state machine modeling also directly match the problem structure.

Simple Explanation

Here is how I would explain this problem in simple non-technical terms:

Imagine there are several rooms in a house numbered 1 to N.

Every day, a person visits and plays in one room based on these rules:

If they have visited a room an odd number of times, they next visit the room number written on a sign in that room.

If they visited a room an even number of times, they just go to the next room number in order.

We want to figure out the first day that the person has been to all the rooms in the house at least once.

For example, if there are 3 rooms:

Day 1 - Visit room 1 Day 2 - Room 1 again (odd so follow sign to room 2) Day 3 - Room 2 Day 4 - Room 3 (even so next room in order) Day 5 - Room 1 (odd day again)

On Day 4, they had been to all rooms.

So in non-technical terms, it involves simulating someone following rules about what room in a house to visit each day, and figuring out when they first visit every room.

Problem Breakdown and Solution Methodology

Here is how I would approach solving this room visit problem:

Create a Room class to represent each room with an id.
Initialize an array of Room objects for the given number of rooms n.
Use a bitmask to track rooms visited so far, with the ith bit representing room i.
Start with visiting room 0 on day 0. Set the 0th bit in visited mask.
Increment day counter.
Check if visited mask has all 1s, indicating all rooms visited. If so, return current day.
Determine next room by checking if day is odd/even:
- If odd, visit room specified in nextVisit array
- If even, visit next sequential room
Update visited mask by setting bit for next visited room.
Repeat process from step 5 until all rooms visited.

For example, with rooms [0,1,2] and nextVisit=[1,0,2]:

Day 0 - Visit room 0, mask = 001 Day 1 - Odd so visit room 1, mask = 011 Day 2 - Even so visit room 2, mask = 111

Return day 2 since full coverage.

This incrementally simulates the room visits while efficiently tracking state using bit operations.

Inference of Problem-Solving Approach from the Problem Statement

Here are the key terms and how they guide the solution approach:

Rooms - Discrete set of elements to visit guides modeling each room as an entity.

Sequence - Order of visits suggests incrementally processing rooms one by one.

Rules - Logic for choosing next room guides implementing state machine or graph traversal.

Odd/even days - This parity constraint lends itself to bitwise operations to track state.

First occurrence - Optimization for initial full set coverage guides tracking visited state.

Modulo arithmetic - Large numbers indicate bitwise operations for compact representation.

State machine - System with state transitions maps directly to state modeling and bit masks.

Graph - Nodes and edges can model rooms and transitions for graph algorithms.

Combinatorics - Generating permutations of sequences implies systematic traversal techniques.

The core terms like state, sequence, parity, and optimization signal modeling the room visits as a state machine and traversing it systematically to detect first occurrence. Bitwise operations emerge from odd/even constraints. Graphs and combinatorics also align with the structure.

We can visualize the key properties and structures of this problem using tables and diagrams:

State machine:

Draw bubbles for each room state
Add directed edges between rooms for odd/even transitions
Annotate edges with conditions under which they are followed

Sequence chart:

Horizontal timeline with days
Vertical axis for rooms
Mark rooms visited each day

Parity counts:

Table with room and columns for odd and even visit counts
Increment when room visited on odd/even day

Bit masking:

Show bit positions 0 to n-1 indicating rooms
Label each row with the day number
Mark bits for room visits each day

The core concepts like state transitions, sequence generation, parity-based rules, and bitwise operations can be effectively captured visually. Diagrams provide an intuitive feel for the dynamics of the problem.

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?

Here is one way we could break down the high-level solution into more concrete steps:

Initialize rooms data structure and visited bitmask
Calculate room to transition to based on day’s parity
- If odd day, use nextVisit mapping
- If even day, use sequential order
Update visited room in rooms data structure
Set bit for visited room in bitmask
Check if all bits set in visited mask
- If so, return current day
Increment current day counter
Repeat process from step 2

This breaks the solution down into:

Initialization
Determining next room by parity
Updating visited state
Checking for completion
Incrementing day

The core transition determination and state update logic can be developed independently from initialization and completion checking.

There is a repeated pattern of choosing next room, updating state, and checking for completion that forms the core loop.

Decomposing the solution helps focus each step before integrating them to solve the full problem.

Solution Approach and Analysis

Here is how I would approach solving this room visit problem:

Model each room as a node in a graph.
Add directed edges between rooms based on the nextVisit rules.
Perform depth-first traversal on the graph starting from room 0.
Track rooms visited so far using a boolean array.
When traversing an edge:
- If visiting room for odd time, follow edge direction
- If even time, go to next room number instead
After following each edge, mark the visited room.
If all rooms visited, return current step count.

For example, with rooms [0,1,2] and nextVisit = [1,2,0]:

Model as graph with directed edges (0->1), (1->2), (2->0)
Traverse graph and mark rooms
After visiting all rooms, return step count

This models the problem as graph traversal, incrementally marking visited rooms until full coverage is detected.

Identify Invariant

The key invariant in this room visit problem is:

At each step in the traversal, the set of rooms marked as visited so far is exactly the set of rooms that have been visited following the specified rules up to that point.

This can be proven by induction:

Base case: Initially, no rooms are marked visited, which aligns with the fact that no rooms have been visited yet.

Inductive case: Assume that at some step, the set of rooms marked visited matches those actually visited so far. When we traverse to the next room, following the rules, we mark that room visited. So the invariant holds for the new step as well.

Therefore, by induction, at each step of the traversal, the set of rooms marked visited equals the rooms visited so far following the rules.

This invariant is crucial for the correctness of the traversal-based solution. It ensures that when all rooms are marked visited, we have indeed covered all rooms according to the problem constraints.

The invariant acts as a loop guard, allowing us to safely conclude that full coverage has been reached when the visited set is complete.

Identify Loop Invariant

The loop invariant in this room visit problem is:

At the start of each loop iteration, the rooms visited array accurately reflects all rooms that have been visited thus far following the rules.

This can be proved by induction:

Base case: Before entering the loop, no rooms are visited yet, so the visited array empty, satisfying the invariant.

Inductive step: Assume visited array is accurate at iteration start. Within the loop body, we determine the next room based on the rules and mark it visited in the array before the next iteration. So the invariant holds.

Therefore, by induction, the visited array correctly reflects the set of rooms visited so far at the start of each loop iteration.

This is crucial for correctness as it means:

The visited array can be used to determine when all rooms are covered.
The next room to visit is determined based on an accurate snapshot of the current state.
No room visits are missed or duplicated since the tracking is consistent.

So this invariant ensures the visited state stays synchronized with the actual sequence generation across loop iterations. It is key for both tracking completeness and guiding next steps.

In this particular room visit problem, the invariant and loop invariant are essentially equivalent:

Invariant: At each step, the set of rooms marked visited matches the rooms visited so far.

Loop invariant: At the start of each loop iteration, the visited array reflects the rooms visited so far.

The key difference is just framing - the invariant is stated in terms of an arbitrary step through the sequence generation process, while the loop invariant focuses specifically on the state at loop iteration starts.

However, in this problem the sequence is generated incrementally in a loop, so each step corresponds to a loop iteration.

Therefore, the invariant and loop invariant both effectively state that:

The visited state is accurately tracked throughout the process
It is synchronized with the actual sequence generation

So for this problem, the invariant and loop invariant are essentially equivalent and ensure the same key properties about tracking visited state and keeping it consistent.

In problems where the sequence is generated recursively or not in a loop, the invariant may be more generic while the loop invariant specifically focuses on loop-based solutions. But in this case, they capture the same essential properties about visited state, just framed differently.

Thought Process

Here is how I would think through solving this type of sequence generation problem:

First, identify cues from the problem statement:

Sequence generated by rules
Tracking visited state needed
Optimizing for first full coverage

This suggests:

Modeling sequence generation system
Simulating sequence incrementally
Checking visited state for completeness

Some insights about the problem:

Can use graph or state machine for transitions
Bitmask allows compactly tracking state
Iteratively transition between states
Check bits set to detect coverage

I would:

Initialize rooms state and visited bitmask
Loop incrementing day:
- Use rules to determine next room
- Transition room state
- Set bit for visited room
- Check if all bits set in visited
Return current day when full coverage detected

Here is Python code implementing this approach:

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def firstFullVisitDay(rooms, nextVisit):
  
  visited = 0
  day = 0
  
  while visited != (1 << rooms) - 1:

    next = nextByRules(day, rooms, visited, nextVisit)
    
    visited |= (1 << next)

    day += 1
  
  return day

def nextByRules(day, n, visited, nextVisit):
  # Logic to determine next room  
  # Based on current day and previously visited

This uses bitwise operations and state tracking to efficiently simulate the sequence generation and detect first full coverage.

PENDING TEST

Establishing Preconditions and Postconditions

Here is how I would analyze the minDeletions method:

Parameters:

s: String - The input string
Returns: int - Number of deletions

Preconditions:

s contains only lowercase English letters
s is not null

Method functionality:

Counts frequency of each letter in s
Iteratively deletes highest frequency letters until all frequencies are unique
Returns count of total deletions

Postconditions:

Number of deletions to make s have all unique letter frequencies is returned
s remains unmodified

Error handling:

If s is null, throw IllegalArgumentException
If s contains non-lowercase letters, throw IllegalArgumentException
Otherwise return minimum deletions needed

In summary, the method takes a lowercase string as input, applies optimization logic to delete letters to unique-ify frequencies, and returns the number of deletions needed, without modifying the input. It requires valid input and returns minimum deletions or throws exceptions.

Problem Decomposition

Here is how I would break down this problem into logical components:

Problem Understanding: The goal is to delete the minimum number of characters in a string to make all letter frequencies unique.
Initial Breakdown:

Count frequency of each letter
Identify letters with duplicate frequencies
Delete letters to make frequencies unique
Keep track of deletions

Subproblem Refinement:

Parse string into characters
Track frequency of each character in a map
Find highest frequency
Check if highest frequency is duplicate
If so, delete one instance of that character
Update frequencies after deletion
Repeat until no duplicate frequencies

Task Identification:

Parsing string
Updating frequency map
Getting highest frequency
Checking for duplicate
Deleting character
Updating frequencies after deletion

Task Abstraction:

Parse input
Update frequencies
Get max frequency
Check duplicate frequency
Delete character instance
Update frequencies

Method Naming:

parseString
updateFrequencyMap
getMaxFrequency
isDuplicateFrequency
deleteCharacterInstance
updateFrequenciesAfterDeletion

Subproblem Interactions:

Parse input
Update frequencies for each character
Loop getting max frequency
- Check if duplicate
- If so, delete one instance
- Update frequencies after deletion
Return total deletions

This breaks the problem down into logical focused subtasks that can be assembled to solve the overall problem.

From Brute Force to Optimal Solution

Here is one approach to optimize a brute force solution:

Brute force:

Try deleting every possible subsequence of the string
Check if frequencies unique after each attempt
Return deletions of first unique subsequence

This is extremely inefficient, trying all 2^n subsets.

Optimizations:

Incremental deletion:
- Start with empty deletions
- Incrementally delete one char at a time
- Stop when unique rather than trying all subsets

Avoid reconstructing entire string on every iteration.

Frequency counting:
- Track character frequencies in map
- Update only on deletions
- Check frequency map values rather than string

More efficient than scanning entire string each time.

Max frequency:
- Track max frequency character
- Only attempt deleting max frequency character

Skip unnecessary deletions of low frequency chars.

Early exit:
- Return once frequencies unique
- Don’t iterate remaining deletions

Avoid extra work once solution found.

Memoization:
- Cache results for substrings
- Reuse previous computations

Avoid repeated work by storing results.

Together these optimizations improve the complexity from O(2^n) to O(n) time and O(1) space. The optimizations prune the search space exponentially while allowing incremental tracking of state.

Code Explanation and Design Decisions

Here is how I would analyze this code’s logic:

The input string s represents the initial data we need to optimize by deleting characters.
The main loop iterates through the characters in s, tracking the frequency of each character in the hash map. This builds up state knowledge to inform deletions.
The inner loop checks if the current max frequency is duplicated, and deletes a character if so. This greedily eliminates duplicates.
Frequencies are updated after deletions to keep an accurate view of the current state. This guides future iteration decisions.
The invariant is that each iteration greedily optimizes by removing duplicates of the current highest frequency. This incrementally improves the string until all frequencies are unique.
The final output count represents the minimum deletions performed to optimize the string. This satisfies the problem requirement of returning the minimal modifications needed to unique-ify frequencies.

In summary, the loops build up essential state knowledge by tracking frequencies, then leverage this to greedily optimize the string by incrementally eliminating duplicated frequencies until no more optimization is possible.

Coding Constructs

Here is how I would analyze this code:

The high-level techniques appear to be dynamic programming and greedy optimization. The frequencies are tracked in a DP style, while deletions are done greedily.
I would explain the code is trying to modify a string by removing letters carefully, so that no two letters appear the same number of times.
The core logic constructs are:

Hash tables for frequency tracking
Loops for iteration
Conditionals for decision making
Array/map updates to maintain state

The algorithm greedily deletes the letter with highest frequency each iteration, while tracking frequencies in a table. This continues until all frequencies are unique.
The key steps are tracking frequencies, identifying duplicates, greedily deleting letters of highest frequency, and updating frequency tables. This optimizes the string.
The core strategies are dynamic programming for state tracking, and greedy optimization for deletion choices. These could be implemented in any language.

The analysis focuses on the high-level optimization approach instead of language-specific syntax. Explaining it in plain language helps reinforce the core algorithmic concepts.

Language Agnostic Coding Drills

Here is how I would break down this code into conceptual units:

Level 1:

Array iteration - Looping over elements
Value assignment - Setting variable values

Level 2:

Hash table update - Incrementing map values
Conditional check - Comparing values
Value swapping - Exchanging variables

Level 3:

Nested loops - Loop within a loop
Breaking loops - Early exit from loop
Helper functions - Reusable logic segments

Level 4:

Dynamic programming - Store intermediate results
Greedy algorithms - Incremental locally optimal choices

The overall problem solving approach would be:

Iterate over input array
Use hash table to map elements to frequencies
Loop to find duplicate max frequency
Remove element of current max frequency
Store intermediate frequency table
Repeat greedily until no duplicates
Break loops early when no duplicates left
Utilize reusable logic segments

Each coding drill builds up knowledge needed for the next technique. Combined appropriately, they map from the problem constraints to an efficient optimized solution.

Targeted Drills in Python

Here are some example Python coding drills for the key concepts:

Array iteration

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arr = [1, 2, 3]
for x in arr:
  print(x) 

Hash table update

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freq = {}

for x in arr:
  if x in freq:
    freq[x] += 1
  else: 
    freq[x] = 1

Loop breaking early

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for x in arr:
   if condition:
      break

Helper functions

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def updateFreq(x):
  # frequency update logic

for x in arr:
  updateFreq(x)

Greedy optimization

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def greedyDeletion(freq):
  # Delete highest frequency letter
  # While duplicates exist

Bringing this together to solve the problem:

Loop over characters
Update frequency table
Run greedy deletion helper
Break when all frequencies unique
Return deletions count

These pieces demonstrate core concepts like iteration, hashing, greedy algorithms etc. Combined appropriately, they solve the full problem.

First Day Where You Have Been in All the Rooms

10 Prerequisite LeetCode Problems

Problem Classification

Clarification Questions

Problem Analysis and Key Insights

Problem Boundary

Distilling the Problem to Its Core Elements

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Identify Invariant

Identify Loop Invariant

Thought Process

Establishing Preconditions and Postconditions

Problem Decomposition

From Brute Force to Optimal Solution

Code Explanation and Design Decisions

Coding Constructs

Language Agnostic Coding Drills

Targeted Drills in Python

Q&A

Similar Problems