Number of Music Playlists

10 Prerequisite LeetCode Problems

This involves dynamic programming and combinatorics. Here are some simpler problems to prepare for this:

LeetCode 70. Climbing Stairs
- This problem introduces the concept of dynamic programming in a simple and easy to understand problem.
LeetCode 300. Longest Increasing Subsequence
- This problem is a slightly more advanced dynamic programming problem that involves sequence analysis.
LeetCode 62. Unique Paths
- This problem involves finding the number of unique paths in a grid, which is a fundamental problem in combinatorics and dynamic programming.
LeetCode 63. Unique Paths II
- This problem extends “Unique Paths” by introducing obstacles, adding an extra level of complexity.
LeetCode 139. Word Break
- This problem involves dynamic programming with a twist - instead of working with numbers, you’re working with strings.
LeetCode 377. Combination Sum IV
- This problem is about finding the total number of combinations that sum up to a target. It is a dynamic programming problem that introduces the concept of order sensitivity.
LeetCode 91. Decode Ways
- This problem uses dynamic programming to decode a string of numbers into possible alphabets.
LeetCode 221. Maximal Square
- This problem is about finding the maximal square in a grid using dynamic programming.
LeetCode 198. House Robber
- This problem uses dynamic programming to find the maximum amount of money you can rob from houses.
LeetCode 96. Unique Binary Search Trees
- This problem combines the concept of dynamic programming with combinatorics and binary trees.

These cover dynamic programming and combinatorics, which are key to solving the “920. Number of Music Playlists” problem.

Problem Boundary

The boundary of a problem helps define the scope and limits within which the problem can be solved. Establishing the boundary for this problem involves understanding the given constraints and determining the possible edge cases.

For the given problem, the boundaries are established based on the following factors:

Number of songs (n): The problem states that the number of different songs (n) is within the range [0, 100]. This indicates that we should expect at least one song and at most 100 songs.
Goal (goal): The problem defines that the goal (the total number of songs to be played in the playlist) is within the range [n, 100]. This means the goal cannot be less than the total number of different songs available. The maximum value can be 100, implying that a song may need to be repeated in the playlist.
Repetition constraint (k): The problem defines the condition for song repetition as ‘k’, which is within the range [0, n). This means a song can be repeated immediately (when k=0), or after a certain number of different songs have been played (when k>0). It cannot be greater than or equal to the number of different songs.
Output: The result should be modulo 10^9 + 7, due to potentially large output values. This implies the solution needs to account for large number handling.

The boundary cases for this problem might include scenarios where there’s only one song (n=1), or where the goal equals the number of different songs (n=goal), meaning no repetition is needed, or where a song can be immediately repeated (k=0). Each of these scenarios would require careful consideration in the solution.

Always ensure that your solution can handle these edge cases and fits within the defined boundaries of the problem.

Problem Classification

The problem is about combinatorics, focusing specifically on permutations and arrangements. It’s related to the domain of Discrete Mathematics and falls under the category of combinatorial problems. However, given the context (song playlist creation), it might be seen as a question in the domain of Computer Science too, especially under topics of algorithms and data structures.

Let’s break down the “What” components of the problem:

Songs: The problem involves a music player that contains ’n’ different songs.
Playlist Goal: There is a goal to listen to a certain number of songs (not necessarily different) during a trip.
Playlist Constraints: There are constraints on how the songs can be ordered in the playlist:
- Every song must be played at least once.
- A song can only be played again if ‘k’ other songs have been played.
Playlist Combinations: The problem requires us to calculate the number of possible playlists that satisfy the above conditions.
Output Modulo: The output should be the calculated number of playlists, modulo 10^9 + 7. This is a common technique used in problems involving large numbers to prevent overflow.

The problem can be further classified as a permutation and arrangement problem, with specific constraints. It could be solved using concepts of dynamic programming, as it has overlapping subproblems and possesses an optimal substructure. The dynamic programming solution would involve creating a memoization table and filling it in a bottom-up manner to track the number of possible playlists for each possible subset of songs. The problem is also interesting in that it demonstrates the application of abstract mathematical concepts to practical scenarios, such as creating a song playlist.

Distilling the Problem to Its Core Elements

Fundamental Concept: This problem is primarily based upon the principles of combinatorics and permutation, and is solved using dynamic programming. Combinatorics is a field of mathematics concerned with counting, both as a means and an end in obtaining results, while dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems.
Simplest Description: Imagine you have a music player with a certain number of different songs. You’re going on a trip and want to make a playlist of a certain length. Every song needs to be played at least once, and once a song has been played, you need to play a certain number of different songs before you can replay it. The problem is to figure out how many different playlists you can make given these rules.
Core Problem and Simplification: The core problem is determining the number of possible playlists given the constraints. A simplified statement could be: “You have ’n’ different songs and want to listen to ‘goal’ songs during your trip. You can replay a song only after ‘k’ other songs have been played. How many different ways can you arrange your playlist?”
Key Components:
- ’n’ different songs available.
- A goal of ‘goal’ songs to be played.
- A restriction that a song can only be played again after ‘k’ other songs have been played.
- The task to calculate the number of possible playlists.
Minimal Set of Operations:
- Initialize a two-dimensional array for dynamic programming.
- Iterate over the array, filling each cell based on the rules defined by the problem.
- For each song, calculate the number of ways to add a new song or repeat a previous song (considering the ‘k’ rule).
- Sum up the final row in the dynamic programming table to get the total number of playlists.

Visual Model of the Problem

One way to visualize this problem is by imagining you’re creating a playlist with your songs. You have ’n’ distinct songs and want to listen to ‘goal’ number of songs.

Let’s take an example: Suppose n=3 (songs are A, B, C), goal=3, and k=1. Here, you want to listen to 3 songs with the condition that you can play a song only if at least 1 different song has been played before it.

A possible visualization would look like this:

Start (no song played)
Play song A (1 song played, 2 more to go)
Options: B or C (because A can’t be replayed until one more unique song has been played)
Play song B (2 songs played, 1 more to go)
Options: A, C, or B (A can be replayed because B has been played after it; B can be replayed because it doesn’t have any restrictions, and C can be played as a new song)
Repeat step 5 until ‘goal’ number of songs have been played.

For each decision point, you have certain options, which create a tree-like structure. You start with one song, then branch out based on the possible songs you can play next. Each branch represents a different playlist. The total number of valid branches (playlists that follow the rules) will be the solution to your problem.

This visualization can help you understand how choices at each step affect the total number of playlists you can create. It also shows why a dynamic programming solution is suitable, as you can store and reuse the number of playlists for smaller goals to build up to your final goal.

Problem Restatement

You have a collection of unique songs, and you’re planning a trip where you want to listen to a certain number of songs. You decide to create a playlist with some rules:

Each song in your collection must be played at least once.
Once a song has been played, you need to play ‘k’ different songs before you can replay it.

Your task is to find out how many different playlists you can create with these rules. Note that the order of songs in the playlist matters, so a playlist with the same songs but in a different order counts as a different playlist.

There are some restrictions on the numbers involved:

The number of unique songs you have (n) is between 0 and 100.
The total number of songs you want to listen to (goal) is between ’n’ and 100.
The number of different songs you need to play before you can replay a song (k) is less than ’n’.

Also, since the number of possible playlists can be quite large, you should return the number modulo 10^9 + 7.

Abstract Representation of the Problem

This problem can be abstracted as a combinatorics problem with certain restrictions. Here’s a formulation:

You have a set of ’n’ distinct elements and a sequence length ‘goal’. You’re required to generate all possible permutations of these elements, with a sequence length of ‘goal’, where:

Each element from the set must appear at least once in each sequence.
An element can only reappear in the sequence after ‘k’ other distinct elements have appeared.

The aim is to determine the total number of such permutations possible. Note that ‘k’ is a limiting factor for element repetition and is less than the total number of distinct elements ’n’.

Due to potentially large output numbers, return the count of permutations modulo 10^9 + 7. The parameters are such that ’n’ lies between 0 and 100, ‘goal’ is between ’n’ and 100, and ‘k’ is less than ’n’.

Terminology

Permutation: This is a term from combinatorics, a branch of mathematics concerned with counting and arranging objects. A permutation refers to the arrangement of items in a specific order. The number of permutations of ’n’ items is given by n! (n factorial), which is the product of all positive integers up to n. In the context of this problem, each playlist can be considered a permutation of songs.
Dynamic Programming: Dynamic programming is an algorithmic paradigm that solves complex problems by breaking them down into simpler subproblems and solving each of those just once, storing their results in case the same subproblem needs to be solved again. In this problem, dynamic programming helps us find the solution efficiently by building up the number of possible playlists of length ‘i’ from the solutions for all lengths less than ‘i’.
Modulo Operation: The modulo operation finds the remainder of division of one number by another. In computing, it is often used to limit the range of numbers to a certain interval. Here, we’re asked to return the answer modulo 10^9 + 7 to prevent overflow of large numbers.
Constraint: In the context of this problem, a constraint is a rule that limits the way we can arrange the songs. The problem has two constraints: each song must be played at least once, and a song can only be replayed after ‘k’ different songs have been played.
Repetition: In this problem, repetition refers to the act of playing the same song more than once in the playlist. The problem places a limit on how soon a song can be replayed, which affects the number of possible playlists.

Problem Simplification and Explanation

Number of Songs (n): This is like having a bag of unique marbles, where each marble represents a different song.
Goal (goal): This is like a target length for a chain of marbles you want to make, where each marble in the chain represents a song in your playlist.
Limit on Repeating Songs (k): This is a rule that you can’t pick the same marble out of the bag until you’ve picked ‘k’ other marbles first.

So, you’re trying to create chains of marbles (songs) where each chain represents a unique playlist, the length of the chain matches your target (goal), every marble is included at least once, and the same marble isn’t picked until ‘k’ other different marbles have been picked.

The challenge is to find out how many different chains (playlists) you can make.

The key concepts involved are permutations and constraints. The permutations refer to all possible ways you can arrange the songs in your playlist, and the constraints limit these arrangements according to the rules about including all songs and not repeating a song until ‘k’ other songs have been played. These concepts interact to form the space of possible playlists.

Constraints

This problem presents a few characteristics and constraints that we can leverage to build an efficient solution:

Dynamic Programming: The problem can be broken down into smaller subproblems, each of which involves finding the number of playlists of a certain length with a given number of unique songs. Once we solve a subproblem, we can store the result to reuse later. This saves time as we don’t need to solve the same subproblem multiple times.
Number Range: The values for ’n’, ‘goal’, and ‘k’ are limited to 0-100. This fixed range allows us to create a dynamic programming table with dimensions that aren’t overly large. This keeps the memory requirements reasonable and the computational time feasible.
Combination of Unique and Repeated Songs: At each step, we can choose to add a new song that hasn’t been played yet, or a song that was played before (as long as it satisfies the ‘k’ condition). This gives us a way to split the problem into two parts: the case where we add a new song and the case where we add a previously played song. We can solve these two parts separately and add the results.
Modulo Operation: The problem asks for the result modulo 10^9 + 7. This operation can be performed at each step of the calculation, preventing number overflow while maintaining the correct result. As modulo operation can be distributed over addition and multiplication, it doesn’t interfere with the overall calculation.

These characteristics suggest that a dynamic programming solution would be an efficient approach to solve this problem.

After analyzing the constraints, we can arrive at several key insights:

Boundaries on Inputs: The constraints set the boundaries for ’n’, ‘goal’, and ‘k’. Knowing these boundaries can help in deciding the most efficient algorithm to solve the problem. For instance, we can confidently use a dynamic programming approach because we know that the maximum possible size of our table (100x100) is feasible in terms of time and space complexity.
Inclusion of All Songs: The constraint that every song should be played at least once directs us to the fact that our playlist length ‘goal’ should always be greater than or equal to the total number of unique songs ’n’. This means every unique song will appear at least once in every valid playlist.
Song Repetition: The constraint that a song can be replayed only after ‘k’ other unique songs have been played helps in structuring our dynamic programming solution. When we consider adding a song to our playlist, there are two scenarios: either we add a new song (one that hasn’t been played yet) or we add a song that has already been played. For the latter case, we know exactly how many such songs are available (’n’ minus ‘k’ most recent songs).
Limit on Repetition: The fact that ‘k’ is less than ’n’ ensures that we can always add a previously played song to our playlist after we have played all unique songs at least once.

These insights provide a deeper understanding of the problem and guide the formulation of our approach to solving it.

Case Analysis

Here are some categorized examples that cover a range of scenarios:

1. Minimal Inputs (Edge Case)

Input: n = 1, goal = 1, k = 0
Output: 1
Explanation: There's only one song and the goal is to play one song, so there's only one possible playlist.

2. Larger Inputs but Small k (Edge Case)

Input: n = 5, goal = 5, k = 0
Output: 120
Explanation: Since k = 0, we can repeat any song immediately. But as the goal is equal to 'n', each song will be played exactly once. So, the output is the number of permutations of 5 songs, which is 5! = 120.

3. Repetition after Playing All Songs (Unique Case)

Input: n = 3, goal = 4, k = 2
Output: 18
Explanation: In this case, k is one less than n. So, after playing all 3 unique songs, we can repeat any song. There are 6 permutations for the first 3 songs, and for the 4th song, we have 3 choices, resulting in 6 * 3 = 18 possible playlists.

4. No Repetition (Boundary Case)

Input: n = 4, goal = 4, k = 3
Output: 24
Explanation: Here, the value of 'k' is such that no song can be repeated (k = n - 1). This reduces the problem to finding the number of permutations of 4 unique songs, which is 4! = 24.

5. High Repetition (Boundary Case)

Input: n = 3, goal = 6, k = 0
Output: 729
Explanation: With k = 0, songs can be repeated immediately, i.e., no need to wait for other songs to be played. So, for each of the 6 positions in the playlist, we have 3 choices (any of the 3 songs). Hence, we get 3^6 = 729 playlists.

These examples cover various aspects of the problem, including cases with minimal inputs, larger inputs but small k, repetition only after playing all unique songs, no repetition, and high repetition. Each of these cases demonstrates how different constraints and conditions affect the total number of possible playlists.

Analyzing different cases helps us to gain several key insights:

Minimal Inputs: When there is only one song and the goal is to play one song, there is just one possible playlist. This highlights the importance of considering edge cases in your solution.
Larger Inputs but Small k: When ‘k’ is zero, any song can be immediately repeated. However, if the goal equals ’n’, each unique song will be played exactly once, and we simply need to calculate the number of permutations of ’n’ songs. This shows the impact of the ‘k’ value on the problem.
Repetition after Playing All Songs: When ‘k’ is one less than ’n’, we have to play all unique songs before we can repeat any song. This case illustrates how ‘k’ limits the number of playlists when ‘goal’ is more than ’n’.
No Repetition: When ‘k’ is equal to ’n - 1’, we can’t repeat any song. This situation reduces the problem to finding the number of permutations of ’n’ unique songs. It highlights the limiting case where repetition is not allowed at all.
High Repetition: When ‘k’ equals zero, songs can be repeated immediately, which results in a significant increase in the number of possible playlists. This illustrates the effect of allowing immediate repetition.

By understanding these insights, you can design your solution to handle a wide range of scenarios and ensure it correctly calculates the number of possible playlists under different conditions.

Identification of Applicable Theoretical Concepts

There are a few mathematical and algorithmic concepts that we can use to simplify this problem:

Permutations: The problem of creating a playlist can be seen as finding the permutations of songs with certain restrictions. This allows us to calculate the number of ways songs can be arranged.
Dynamic Programming: This problem can be broken down into subproblems, each of which involves finding the number of playlists of a certain length with a given number of unique songs. Dynamic programming allows us to store and re-use solutions to these subproblems, greatly improving efficiency.
Modulo Arithmetic: The problem asks for the result modulo 10^9 + 7. The properties of modulo arithmetic can be used here to prevent integer overflow and to ensure the result fits within the given constraints. Modulo operation can be distributed over addition and multiplication, which means it won’t interfere with the overall calculation.
Combinatorics: The problem involves counting the number of certain specific arrangements (playlists, in this case), which is a fundamental topic in combinatorics. Combinatorial principles can simplify the counting process.
Recurrence Relations: The problem can be solved using a recurrence relation where the solution to a given state depends on the solutions of some other states. Recurrence relations often appear in dynamic programming problems and can help in defining the state transition.

By applying these concepts, we can develop a solution that efficiently calculates the number of possible playlists. These mathematical and algorithmic concepts not only help in simplifying the problem but also provide a structured approach to tackling the problem.

Simple Explanation

Imagine you have a box of your favorite songs, but you want to create a list (playlist) of these songs to listen to during a long drive. You want to hear all your favorite songs at least once. But you don’t want to listen to the same song again until you’ve heard a certain number of other songs. The question is, how many different playlists can you make following these rules?
Think of this problem as a puzzle game with your favorite music CDs. You want to listen to all your CDs at least once, but after you listen to one CD, you must listen to a certain number of other CDs before you can listen to that first CD again. The challenge is to find out all the different ways you can arrange your CDs to listen to during your game session.
Imagine you’re choosing what dessert to have each night after dinner. You have a few different desserts, and you want to have each one at least once. But, once you’ve had a dessert, you have to try some other desserts before you can have the first one again. The problem is figuring out how many different dessert plans you can make.
You have a deck of cards, and you want to go through all of them at least once, but you don’t want to see the same card until you’ve seen a certain number of other cards. How many different orders can you go through the cards in this way?
Let’s consider a toy shop as an everyday example. Suppose you want to buy several toys, but you want each toy to be different from the previous ones you bought. After buying a toy, you have to buy some other toys before you can buy the same toy again. The task is to find out how many different combinations of toys you can buy following these rules.

Problem Breakdown and Solution Methodology

Step 1: Define the State This problem involves two variables: the number of songs played so far, and the number of unique songs played so far. We can denote these as dp[i][j], where i represents the number of songs played so far and j is the number of unique songs played so far. dp[i][j] represents the number of playlists of length i that have j unique songs.

Step 2: Define the Base Case When no songs have been played (i = 0), there are no unique songs (j = 0). Thus, dp[0][0] = 1.

Step 3: Define the Transition Function Here, for each playlist of length i with j unique songs, there are two scenarios for the next song:

It is a new song. In this case, there are n - j new songs that can be played. So, dp[i + 1][j + 1] += dp[i][j] * (n - j).
It is a song that’s already been played. Here, we have j unique songs we’ve played. However, we can’t play the last k songs again, so we have j - max(0, j - k) choices. So, dp[i + 1][j] += dp[i][j] * max(0, j - k).

We can calculate dp[i][j] for all i and j using these transitions. This way, we will build our solution incrementally, taking into account the previous results.

Step 4: Handle Edge Cases The above transition function doesn’t handle edge cases well. When j = 0 (no unique songs), we can’t play an already-played song because there are none. So, we only need to consider the first scenario for dp[i + 1][j + 1].

Step 5: Find the Final Solution After calculating all states, the final answer will be the sum of all dp[goal][j] for j from n to goal. This represents the total number of playlists of length goal that include all n unique songs, modulo 10^9 + 7.

Now, let’s see how this approach works using an example:

Suppose we have n = 3, goal = 3, k = 1.

Initial dp array will be:

dp[0] = [1, 0, 0, 0]
dp[1] = [0, 3, 0, 0]
dp[2] = [0, 0, 6, 0]
dp[3] = [0, 0, 0, 6]

The final result is dp[3][3] + dp[3][2] + dp[3][1] = 6, which is the expected answer.

Changes in parameters would affect the solution as follows:

As n increases, the number of possible playlists increases as there are more unique songs to choose from.
As goal increases, the number of possible playlists also increases because we have a longer playlist.
As k increases, the number of possible playlists decreases because we have more restrictions on song repetition.

This method ensures that we calculate the number of playlists while considering all unique songs and the restriction on repetition. It’s efficient and handles a wide range of inputs thanks to the power of dynamic programming.

Inference of Problem-Solving Approach from the Problem Statement

The key terms and concepts in this problem are:

Playlist: A playlist refers to an ordered list of songs. This concept introduces the ordering element to the problem. In our context, it represents the sequence of songs, leading us to consider permutations or arrangements of the songs.
Different songs: This implies we have a finite set of unique songs (n). This clues us into the fact that we need to account for all these different items at least once in the final solution, guiding us towards considering combinatorics.
Song can be played again only if k other songs have been played: This constraint imposes an additional rule on the order of the songs, creating a unique condition that complicates how the songs can be arranged. This leads us to consider the principles of permutation with repetition but also with restrictions - a concept that’s crucial in our problem-solving strategy.
Number of possible playlists: This final output, seeking a count of all possible distinct arrangements, firmly establishes this problem within the realms of combinatorics and permutation.
Return it modulo 10^9 + 7: This is a common technique in competitive programming to prevent overflow for very large numbers. It doesn’t influence our method of problem-solving, but it does affect how we manage and calculate the final result.

Together, these concepts guide us to use a combinatorial and dynamic programming approach. We use combinatorics to understand the permutations of song orders while adhering to the constraints, and dynamic programming to efficiently compute the total number of permutations, given the constraints and the unique characteristics of this problem.

The problem statement contains several hints that suggest using dynamic programming as an effective strategy:

Overlapping subproblems: The problem requires us to find the total number of playlists, where each playlist is constructed from smaller ones. This indicates that we can break down the problem into smaller, simpler subproblems, which is a characteristic feature of dynamic programming problems.
Optimal substructure: The problem’s solution depends on the solutions to its subproblems. For instance, the number of playlists of length i that include j unique songs is based on the number of playlists of length i - 1 that include j - 1 unique songs or j unique songs. This property is called optimal substructure and is another hallmark of dynamic programming problems.
Constraints on the variables: The variables n, goal, and k are constrained to be less than or equal to 100. This relatively small input space is suitable for a dynamic programming approach, which often involves filling up a table based on the input parameters.
Counting Problem: The problem asks for the number of ways to do something, not a specific configuration that accomplishes the task. This is a common class of problem that dynamic programming excels at, because dynamic programming is all about enumerating and counting possibilities.

All these aspects together suggested that a dynamic programming approach would be suitable for solving this problem efficiently.

Define State: A state in our dynamic programming solution can be defined as a pair (i, j), where i is the total number of songs played so far and j is the number of unique songs played so far. The reason for defining such a state is that the answer to our problem depends on these two parameters.
Define Base Case: Our base case is when no song has been played, i.e., when i=0 and j=0. The number of such playlists is just 1 (an empty playlist).
Define State Transition: This is where we formulate how to transition from one state to another. We can go from state (i, j) to (i+1, j+1) by picking one of the n-j new songs or go to (i+1, j) by picking one of the previously played songs as long as it isn’t in the last k songs played. Since we can choose from j songs and k of them are restricted, there are j - min(j, k) choices.
Calculate Final Answer: We’ll sum up the number of playlists of length goal that include all n songs.
Optimize Calculation: As the final step, remember that we are to return our answer modulo 10^9 + 7 to prevent overflow. Therefore, every time we perform a calculation, we’ll take the modulo of the result.

Each step is necessary and builds on the previous step. They are intertwined in the sense that the definition of the state directly impacts how the state transitions are formulated. So, they can’t be solved independently but have a sequence to follow.

The pattern here is that each state (i, j) depends on either (i-1, j-1) or (i-1, j), i.e., you always transition from a state with one fewer song. This is the pattern that is repeated for each state until we build up to the final result.

Solution Approach and Analysis

Let’s break down the problem into smaller steps and see how we can construct a solution using a dynamic programming approach. For this, we’ll use an analogy of a DJ arranging a playlist for a concert:

Problem Understanding: Imagine you’re a DJ, and you’ve to arrange a sequence of songs (playlist) for an upcoming concert. The concert requires a particular number of songs (goal) to be played. You’ve a unique collection of songs (n), and the concert rules stipulate that a song can only be repeated after k other songs have been played. The question is: how many different playlists can you create?
State Definition: Consider each song you add to the playlist as a step in your journey of playlist creation. Each state in this journey can be represented as (i, j) where i is the number of songs already on the playlist and j is the number of unique songs used. This is like keeping track of how much of the concert’s requirement you have filled and how many different songs you’ve used so far.
Base Case: When you start, you haven’t added any songs to the playlist, i.e., i = 0 and j = 0. There’s only one way of having an empty playlist - doing nothing.
State Transition: For each state (i, j), you have two choices. Either you can add a song that’s not been used before (from n-j available songs), or you can add a song that you’ve used before, provided it wasn’t in the last k songs played. This is like choosing whether to introduce a new song next, or replay a song from earlier in the concert.
Final Answer: Once you’ve figured out how many playlists are possible for each state, the final answer is the sum of all possible playlists of length goal that include all n songs.
Modulo Operation: Since the number of playlists can be very large, we perform a modulo operation at each step of calculation to keep the numbers manageable.

Let’s walk through an example for better understanding:

Suppose n = 2, goal = 3, k = 1. Here’s how the dynamic programming table gets filled:

Start with dp[0][0] = 1 (base case).
From state (0, 0) -> (1, 1): We can add a new song. There are n = 2 choices. So, dp[1][1] = 2.
From state (1, 1) -> (2, 2): Again, add a new song. There are n - j = 1 choice. So, dp[2][2] = 2.
From state (2, 2) -> (3, 2): We can’t add a new song (no more left), but can repeat one, because j > k. There are j - k = 1 choice. So, dp[3][2] = 2.

In the end, since our goal length is 3, we sum up dp[3][j] for all j. The answer is 2.

Identify Invariant

An invariant, in the context of algorithm design, is a condition that can be relied upon to be true during the execution of an algorithm.

In this problem, the invariant is that every song must be played at least once, and a song can be repeated only after k other songs have been played. These rules do not change and hold true irrespective of the state of the system, and thus guide the creation of all possible playlists.

In terms of our dynamic programming approach, the invariant can be thought of as the logic we follow when we fill up our dp[i][j] table. The way we transition from one state to another is always guided by the same principle: at the i-th step, we either play a new song that has not been played in the playlist yet, or repeat a song that’s not among the last k songs played. This principle remains invariant throughout our solution.

Identify Loop Invariant

A loop invariant in computer science is a condition that is initially true and remains true after each iteration of a loop.

In this problem, a loop invariant in our dynamic programming solution could be stated as follows:

For every possible playlist length i and for each number of unique songs j, the computed value of dp[i][j] correctly represents the number of different playlists of length i that include j unique songs, following the rules of the problem. This invariant must hold before we enter the two nested loops used to fill up the dp table and also after each iteration of these loops.

This invariant helps us to make sure that our algorithm correctly counts all valid playlists and ensures the correctness of our dynamic programming approach. The key here is the condition: every song is played at least once, and a song can only be played again only if k other songs have been played, which is maintained throughout the iterations.

Thought Process

The problem involves determining the number of ways we can create a playlist with given constraints. A common way to approach such problems is by using dynamic programming. Here, we divide the problem into smaller sub-problems and build up the solution.

Here are some key insights and cues from the problem statement:

Permutations: Since the order of songs in the playlist matters, it’s a permutation problem, not a combination problem.
Constrained Repetition: A song can be repeated in the playlist, but only if ‘k’ other unique songs have been played in between.
Complete Coverage: Every song must be played at least once. This implies that the minimum length of any valid playlist is ’n’ (the total number of unique songs).

These insights suggest that a dynamic programming approach would be suitable. We can construct a DP table where dp[i][j] represents the number of ways to create a playlist of length ‘i’ with ‘j’ unique songs.

Here are the steps for the solution:

Initialize the DP Table: Create a 2D DP table of size (goal + 1) x (n + 1), initialize dp[0][0] = 1, and all other cells as 0.
Bottom-Up DP Table Filling: For each cell dp[i][j], we have two cases:
- Add a new song to the playlist. There are (n - j + 1) options. So, add dp[i - 1][j - 1] * (n - j + 1) to dp[i][j].
- Repeat a song in the playlist. There are max(j - k, 0) options. So, add dp[i - 1][j] * max(j - k, 0) to dp[i][j].
Modulo Operation: Apply modulo operation (10^9 + 7) to each dp[i][j] after the calculations.
Final Result: The answer is the sum of dp[goal][j] for all j from 1 to n.

Let’s translate this into Python code:

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class Solution:
    def numMusicPlaylists(self, n: int, goal: int, k: int) -> int:
        mod = 10**9 + 7
        dp = [[0]*(n + 1) for _ in range(goal + 1)]
        dp[0][0] = 1

        for i in range(1, goal + 1):
            for j in range(1, n + 1):
                dp[i][j] = (dp[i - 1][j - 1] * (n - j + 1)) % mod   # adding a new song
                dp[i][j] = (dp[i][j] + dp[i - 1][j] * max(j - k, 0)) % mod   # repeating a song

        return dp[goal][n]

This function calculates and returns the number of playlists possible, given the number of unique songs (n), the playlist length (goal), and the constraint on song repetition (k), following the approach we discussed.

Establishing Preconditions and Postconditions

Problem Name: The problem we’re trying to solve can be named as “Music Playlist Generation”.
Method Name: The method we’re using to solve this problem can be named as countPlaylists.
Parameters:
- n (integer): It represents the total number of different songs.
- goal (integer): It represents the total number of songs you want to listen during the trip.
- k (integer): It represents the minimum number of different songs that need to be played after a song before it can be replayed.
Preconditions:
- The parameters n, goal, and k must be integers.
- The constraints are: 0 <= k < n <= goal <= 100.
Method Functionality:
- This method is expected to compute and return the number of possible playlists that can be created considering the constraints.
- The method takes as input the total number of different songs, the total number of songs you want to listen during the trip, and the minimum number of different songs that need to be played after a song before it can be replayed, and then uses dynamic programming to calculate the number of possible playlists.
Postconditions:
- After the method has been called, it returns the total number of possible playlists.
- The return value represents the total number of possible playlists that you can create with the given constraints.
Error Handling:
- If the preconditions are not met, Python will automatically raise an error (like TypeError or ValueError). As these are basic data type requirements, we do not have a special error handling mechanism in the method. It is expected that valid parameters are supplied.

Problem Decomposition

Problem Name: The complex problem at hand is “Generating Possible Playlists with Constraints”.
Problem Understanding: The problem involves generating a list of all possible music playlists given three parameters: n - the total number of unique songs, goal - the total number of songs to be in the playlist, and k - a constraint which specifies that a song can be replayed only after k other songs have been played. The challenge is to calculate the total number of such possible playlists.
Initial Breakdown: This problem can be broken down into a few broad subproblems:
- Initialize a structure to track possible playlist counts.
- For each song length from 1 to goal, calculate how it can be composed using the available songs and considering the k constraint.
Subproblem Refinement: For each subproblem:
- Initialization: Prepare a 2D structure to track possible playlist counts, considering n and goal.
- Playlist calculation: For each song length and for each song, we calculate the playlist count considering two cases: the song has been played before, and the song is new.
Task Identification: The repeated task here is the calculation of the playlist count for each song length and for each song, considering the k constraint.
Task Abstraction: The task abstracted here would be: given a song length and a song, calculate the count of playlists that can be made.
Method Naming: We could call the abstracted task calculatePlaylistCount.
Subproblem Interactions: The initialization subproblem is the first thing we need to do. The task calculatePlaylistCount is performed iteratively for each song length and for each song. It relies on the previously calculated results, hence these tasks are not independent but depend on the calculations of previous states.

From Brute Force to Optimal Solution

A brute force approach to this problem would involve generating all possible permutations of songs and then eliminating those that do not meet the requirements, i.e., a song being played again only after k other songs have been played.

This brute force approach would entail the following steps:

Generate all possible playlists: This can be done using a recursive function that adds every song to the playlist until the playlist length reaches the goal.
Validate the playlists: For each generated playlist, check if it meets the k distance constraint. If not, discard it.

However, this brute force method is very inefficient. The number of permutations of n songs taken goal at a time is n^goal, which becomes exceedingly large even for moderate values of n and goal. Thus, the time complexity of generating all permutations is O(n^goal), and then we also need to validate each of these. Space complexity is also quite high because we need to store all these permutations.

Now let’s consider optimizing this approach.

Given the constraints of the problem, it’s clear that we’re dealing with overlapping subproblems: the problem of generating a playlist of length l with k constraint depends on the problem of generating a playlist of length l-1, l-2, …, l-k. This is a classic indication that dynamic programming could be an efficient strategy.

Here’s a dynamic programming approach:

Create a 2D DP table with n+1 rows and goal+1 columns, where dp[i][j] represents the number of possible playlists of length j using i unique songs.
Initialize dp[0][0] to 1 and all other dp[i][0] values to 0 as there’s only one way to make a playlist of length 0 - no songs.
For each i from 1 to n and each j from 1 to goal, calculate dp[i][j] as the sum of two scenarios:
- The song is new: There are n - i + 1 ways to pick a new song, so dp[i][j] = dp[i-1][j-1] * (n - i + 1).
- The song is not new: There are i songs that could be replayed, but we can’t replay the last k songs, so there are max(0, i - k) choices. dp[i][j] += dp[i][j-1] * max(0, i - k).
The total number of valid playlists is stored in dp[n][goal].

The time complexity of this dynamic programming solution is O(ngoal), and the space complexity is also O(ngoal). Both are significant improvements over the brute force method. This optimized approach provides an efficient and practical way to solve the problem.

Code Explanation and Design Decisions

Initial Parameters: There are three main parameters for this problem: n (the number of unique songs), goal (the total length of the playlist), and k (the number of songs that must be played before a song can be repeated). These parameters dictate the constraints and objective of our problem.
Primary Loop: The outer loop of the DP algorithm iterates over the number of unique songs (i from 1 to n). Each iteration of this loop signifies considering an additional unique song for our playlist. The inner loop iterates over the playlist length (j from 1 to goal). Each iteration of this loop signifies the creation of a playlist of an increasing length.
Conditions/Branches: There are no explicit branches in our loops, but the DP update inside the inner loop essentially signifies two scenarios or conditions: a new song is added or an old song (not among the last k played) is repeated. This follows from the problem’s constraint that a song can only be repeated after k other songs have been played.
Parameter Updates: The DP table is updated in every iteration of the inner loop. Each cell dp[i][j] stores the number of valid playlists of length j using i unique songs, calculated based on the two scenarios: adding a new song and repeating an old song. These updates gradually build up the solution, moving from smaller subproblems to larger ones, reflecting the typical bottom-up approach of dynamic programming.
Invariant: The invariant here is the rule that every song in the DP table, from the first to the last, respects the problem’s constraints. Specifically, in any playlist considered, a song is only repeated after k other songs have been played. This invariant is crucial as it guarantees the validity of our generated playlists and ensures we’re working within the problem’s requirements at all stages of our algorithm.
Final Output: The final output of our algorithm is dp[n][goal], which represents the total number of valid playlists of length goal using all n unique songs. This output is significant as it directly answers our problem statement, satisfying the requirements of determining the number of such playlists while respecting the constraint of k.

Coding Constructs

High-Level Problem-Solving Strategies: This code utilizes a dynamic programming approach to solve the problem. Dynamic programming is a method for solving complex problems by breaking them down into simpler, overlapping subproblems that can be solved independently. The results of these subproblems are then stored and reused, which avoids redundant computation and leads to a more efficient solution.
Purpose of the Code to a Non-Programmer: This code is like a playlist generator, but with a few rules. It’s creating a long playlist from a given set of unique songs, but with a constraint: once a song is played, we need to play a certain number of different songs before we can play that song again. The code works out all possible ways we can create such a playlist.
Logical Elements or Constructs: Irrespective of the programming language, the code uses a nested loop construct where one loop is nested inside another, representing a decision at each point in the playlist. It uses an array to store and reuse previously computed results (memoization), and arithmetic operations to update the array elements.
Algorithmic Approach in Plain English: This code starts with an empty playlist and then gradually builds it up, song by song. At each step, it considers two options: adding a new song that’s not been played yet, or repeating an old song that was played long enough ago. It keeps track of all the possible playlists it can create at each step and builds upon those possibilities. It does this until it’s considered all songs and all positions in the playlist.
Key Steps or Operations on Input Data: The code begins by initializing a 2D array to track the possible playlists. It then iterates over each song and each position in the playlist. For each position and song, it computes the possible playlists if a new song is played or an old song is repeated. This computation is done by accessing previously computed results and updating them according to the problem’s rules. The ultimate goal is to fill up the 2D array with all the possible playlists for every combination of songs and positions.
Algorithmic Patterns or Strategies: The code applies the strategy of Dynamic Programming, specifically a bottom-up approach. It breaks down the problem into smaller, simpler subproblems (number of playlists for each song and each position), solves these subproblems, and stores their results. These stored results are then used to construct solutions for bigger problems. This process repeats until the solution to the original, larger problem is found. It’s a systematic way of filling up a “table” of solutions, based on simpler, previously solved entries.

Language Agnostic Coding Drills

Identifying Coding Concepts

Variables and Basic Arithmetic: These are fundamental to programming, regardless of the language. They’re used to hold values and perform basic operations.
Arrays and Indexing: They are used for storing multiple values and accessing them via indices. It’s a basic way of organizing data in code.
Loops: The process of repeating a block of code. It’s fundamental to perform repetitive tasks.
Conditions (If-Else statements): These are decision-making constructs used to perform different actions based on different conditions.
2D Arrays: This is an extension of an array concept where data is stored in rows and columns.
Dynamic Programming: It’s a method for solving complex problems by breaking them down into smaller subproblems. The results of these subproblems are stored and reused.

Listing Concepts in Order of Increasing Difficulty

Variables and Basic Arithmetic: Easy. These are fundamental concepts that serve as the building blocks of code.
Arrays and Indexing: Easy. This is slightly more complex as it involves organizing multiple values and manipulating them.
Conditions (If-Else statements): Easy. Decision-making is a key aspect of coding, but it’s also a fairly basic concept.
Loops: Medium. Understanding how to correctly set up loops and manage iterations can be tricky for beginners.
2D Arrays: Medium. This introduces another level of complexity in organizing and accessing data.
Dynamic Programming: Hard. This requires a good understanding of problem-solving techniques and an ability to think about problems in a recursive way. It also requires a good command over all the previous concepts.

Problem-Solving Approach

Start with understanding the problem. Identify what is being asked, what are the inputs, and what are the constraints.
Identify that the problem can be broken down into subproblems. This means if we know the solution to smaller problems, we can use those to build up to the solution of larger problems.
Identify the states of the dynamic programming solution. In this case, it’s the number of songs left to be included in the playlist and the last song that was played.
Formulate a plan to fill up the 2D table, which will hold the answers to our subproblems. We fill up this table in a bottom-up manner, starting from the simplest subproblem and gradually building up.
Identify the transitions between states. In other words, how do we go from one state to another and how does that affect the solution?
Finally, take the solution of the subproblem that represents the original problem. In this case, that would be the value stored in the cell representing all songs being included in the playlist.

Targeted Drills in Python

Python Coding Drills for Each Identified Concept

Variables and Basic Arithmetic:

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a = 5
b = 10
sum = a + b
print("Sum is", sum)

Arrays and Indexing:

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arr = [1, 2, 3, 4, 5]
print("Element at index 2 is", arr[2])

Conditions (If-Else statements):

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a = 5
if a > 0:
    print("a is positive")
else:
    print("a is not positive")

Loops:

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for i in range(5):
    print("i is", i)

2D Arrays:

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matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print("Element at (2, 3) is", matrix[1][2])

Dynamic Programming:

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def fibonacci(n):
    dp = [0, 1] + [0] * (n - 1)
    for i in range(2, n + 1):
        dp[i] = dp[i - 1] + dp[i - 2]
    return dp[n]

print("Fibonacci of 6 is", fibonacci(6))

Problem-specific Concepts

The problem-specific concept in this case is primarily the Dynamic Programming approach. In our context, the state of the Dynamic Programming is represented by the number of songs left to be included in the playlist and the last song that was played. The drill to handle this scenario is essentially to build an understanding of how to create and manipulate 2D arrays, which is covered in the drills above.

Integrating Drills

The concept of variables and basic arithmetic forms the basis of setting up the problem and handling simple calculations.
Arrays and indexing are fundamental to setting up the 2D table for dynamic programming, and accessing the data stored in the table.
If-else statements are used for making decisions, such as handling boundary conditions.
Loops are used for iteratively filling up the 2D table.
2D Arrays are used for setting up the dynamic programming table and storing the state of our solution.
Dynamic Programming concept is the main technique used for solving this problem. Once the table is set up, it’s all about identifying the right subproblems, how they relate to each other, and using that relation to fill up the table.

Once all these drills are well understood, they can be assembled to form the solution for the problem. Start with setting up the 2D table, then fill it up iteratively using the dynamic programming concept, while using loops and conditionals to manage the flow of your code.

Q&A

What are the reasons for making these mistakes in the given code?