Champagne Tower
Python solution:


Identifying Problem Isomorphism
“Champagne Tower” has a structural similarity to the “Triangle”.
In “Champagne Tower”, you are pouring champagne into a tower of glasses where each glass in the ith row holds exactly i units of champagne, and any overflowing champagne will fall evenly into the glasses on the row below it. The goal is to find how much champagne will be in a particular glass after pouring a certain amount into the topmost glass.
The “Triangle” involves a triangle made of numbers, and the goal is to find the minimum path sum from top to bottom. At each step, you may move to an adjacent number on the row below.
Both problems involve a trianglelike structure, and in each case, a value (champagne or path sum) is being propagated from the top to the bottom of the structure. The main difference lies in the way values are propagated  “Champagne Tower” involves evenly distributing a value to the elements below, while “Triangle” involves selecting one of the elements below based on the sum of the path.
“Triangle” is simpler because it only involves addition and comparison, while “Champagne Tower” requires handling of distribution of champagne overflow, which can be fractionated.
10 Prerequisite LeetCode Problems
This is about using dynamic programming techniques in a twodimensional context. Here are some problems to prepare:
LeetCode 62: Unique Paths: This problem can help you understand dynamic programming in the context of a twodimensional grid.
LeetCode 64: Minimum Path Sum: Another dynamic programming problem with a twodimensional grid. It’s similar to the “Unique Paths” problem, but with added complexity due to the path costs.
LeetCode 70: Climbing Stairs: A simpler dynamic programming problem that can help solidify the basics of the concept.
LeetCode 120: Triangle: This problem involves working with a pyramidlike data structure, much like the champagne tower, and can be solved with dynamic programming.
LeetCode 174: Dungeon Game: This problem is another more complex dynamic programming problem involving a grid, which might help you better understand how to approach such problems.
LeetCode 221: Maximal Square: This dynamic programming problem involves finding a square of 1’s in a binary matrix, but the techniques used can be applied to the champagne tower problem as well.
LeetCode 931: Minimum Falling Path Sum: This problem is about finding the minimum sum of any falling path in a 2Darray. It can help you understand the concept of propagating values in 2D DP problems.
LeetCode 322: Coin Change: This problem can help you understand how to work with different possibilities in dynamic programming problems.
LeetCode 518: Coin Change 2: This problem is an extension of the previous one and adds a new layer of complexity to it.
LeetCode 300: Longest Increasing Subsequence: This problem isn’t about a 2D grid, but it’s a good problem to understand dynamic programming and how to build solutions based on previous states.
Clarification Questions
 Is the pyramid of glasses always of a fixed height, i.e., 100 rows, or can it vary?
 Can the amount of champagne poured be a noninteger, like 2.5 cups?
 When champagne overflows, is it guaranteed to split equally between the two glasses below?
 Is the glass at
(i, j)
always directly below the glasses at(i1, j1)
and(i1, j)
? In other words, can we assume a perfect pyramid?  What should be the precision of the output? Should it be rounded to a specific number of decimal places?
 In the case where the
query_row
andquery_glass
are out of bounds, what should the output be?  Can we assume that the glasses have a uniform shape and capacity?
 Is it acceptable to assume that once a glass is full, it will not lose any champagne due to evaporation or other factors?
 Is there a specific time factor involved in pouring the champagne, or can we assume it happens instantaneously?
 Should the program be capable of handling multiple queries efficiently, or is it going to be one query per execution?
These questions help clarify edge cases, requirements, and constraints for implementing a solution.
Problem Analysis and Key Insights
Pyramidal Structure: The glasses are arranged in a pyramid, meaning each glass at row
i
feeds into two glasses at rowi+1
. Understanding this structure is essential for simulating the champagne flow.Fixed Capacity: Each glass has a fixed capacity of one cup. Overflow rules are clearly defined, so we can model this behavior directly.
State Propagation: The state of any glass at row
i
is dependent on the state of the glasses at rowi1
. This suggests a dynamic programming approach could be efficient.Query Glass Position: The query asks for the state of a specific glass
(i, j)
, implying that we don’t need to maintain the state of all the glasses, only those that contribute to the state of the target glass.Input Constraints: The range of
poured
is large, requiring an efficient algorithm. On the other hand,query_row
andquery_glass
are relatively small, suggesting that it may be feasible to simulate up to these indices.Output Precision: The problem specifies a float output. This indicates that the calculations need to account for fractional champagne amounts.
Overflow Rule: Champagne overflow is equally distributed to the immediate left and right glasses below. This rule simplifies calculations and modeling.
Initial State: The champagne starts at the topmost glass (
(0, 0)
). This is the starting point for the simulation, and all other states derive from it.Implicit Floor: The problem mentions that any overflow from the last row falls to the floor and is not captured. This means the state of the last row is terminal and does not affect any other row.
By recognizing these key insights, you’re wellequipped to start formulating an approach to solving this problem.
Problem Boundary
The scope of this problem is fairly welldefined and confined to specific conditions:
Fixed Environment: The problem is based in a fixed pyramidal structure of glasses, with a maximum of 100 rows. The behavior of the champagne in this environment is deterministic.
Input Range: The problem limits the input range for the amount of champagne poured, as well as the query row and glass indices. This simplifies the scope of required calculations.
Single Query: Each problem instance focuses on finding the amount of champagne in one particular glass, defined by
(query_row, query_glass)
.State Simulation: The scope includes simulating the state of the glasses from the topmost glass down to the queried row and glass, but not beyond.
Overflow Mechanics: The mechanics of how overflow works is explicitly described, reducing ambiguity in that area.
Output Format: The result needs to be a float, which specifies how full the queried glass is. There is no requirement to maintain or output the state of other glasses.
No Time Factor: The problem statement doesn’t introduce time as a factor. Champagne pouring and overflow are assumed to be instantaneous for the sake of this problem.
No External Factors: Factors like glass shape variations, evaporation, or any other realworld conditions are outside the scope.
Computational Efficiency: Given the large range of
poured
, the solution needs to be computationally efficient, but this requirement is implicit rather than explicitly stated.
The scope is precise enough to allow for a focused, efficient algorithm to solve the problem.
Establishing the boundary of this problem involves identifying the limits, constraints, and specific conditions that the solution must adhere to:
Input Constraints
 Poured Amount: The amount of champagne poured into the top glass ranges from 0 to (10^9).
 Row and Glass Indices: The
query_row
is less than 100, andquery_glass
is less than or equal toquery_row
.
State Behavior
 Glass Capacity: Each glass can hold exactly one cup of champagne.
 Overflow Rule: Any overflow splits equally between the two glasses immediately below.
Output
 Output Precision: A float indicating how full the specified glass is after pouring. It is assumed that precision is essential, although it’s not explicitly stated how many decimal places are needed.
Algorithmic Constraints
 Computational Efficiency: Given the large possible value of
poured
, the algorithm must be efficient enough to handle large inputs quickly.
Other Factors
 Deterministic Rules: The way champagne flows and fills the glasses is deterministic, following specified rules.
 Fixed Structure: The pyramid of glasses has a fixed, welldefined structure up to 100 rows.
 No Time Factor: Time is not a variable; the pouring and overflow are considered to happen instantaneously.
By taking these factors into account, you can establish the boundaries within which the problem must be solved. This helps in choosing the appropriate algorithms and data structures and in ruling out solutions that won’t work within these boundaries.
Problem Classification
The problem belongs to the domain of Simulation and Computational Modeling. It involves the simulation of the flow of champagne through a pyramidshaped glass tower to determine how much liquid resides in a specific glass.
What
poured: The amount of champagne poured into the topmost glass. A nonnegative integer between 0 and 10^9.
query_row: The row number (0indexed) in the pyramid where the target glass resides. A nonnegative integer less than 100.
query_glass: The glass number (0indexed) in the specified row where the target glass resides. A nonnegative integer less than or equal to query_row.
Output: How full the jth glass in the ith row is after the champagne is poured.
Input/Output: The problem takes integers as input and expects a float as output.
Objective: The objective is to determine the liquid content in a specified glass after pouring a given amount of champagne.
Constraints: There are specific constraints on the input ranges and the nature of the champagne flow.
State Transition: This problem involves state changes in glasses as champagne is poured and overflow occurs.
This is a computational simulation problem that requires understanding how states (the level of liquid in each glass) change according to specific rules (champagne overflow). The challenge lies in efficiently simulating these states to arrive at the final state of the target glass.
Distilling the Problem to Its Core Elements
Fundamental Concept
The fundamental concept this problem is based upon is dynamic state transition. Each glass’s state (amount of champagne it contains) is dependent on the state of the glasses above it. This makes it a wellsuited problem for dynamic programming.
Simplest Description
Imagine you have a triangle made of small cups. You pour water into the top cup. When it fills, the extra water spills equally into the cups directly below it. This continues down the triangle. Your task is to find out how full a specific cup will be after you’ve poured a certain amount of water.
Core Problem
The core problem we are trying to solve is to find the amount of champagne in a specific glass after pouring a given amount into the top glass and letting it flow down.
Simplified Problem Statement
After pouring ‘X’ cups of champagne into the top glass of a pyramid of glasses, how full will the glass at row ‘Y’, position ‘Z’ be?
Key Components
 Initial amount of champagne (
poured
)  Glass pyramid with welldefined overflow rules
 Target glass identified by row and position (
query_row
,query_glass
)  Simulation or calculation of champagne levels until it reaches the target glass
Minimal Set of Operations
 Initialize an array to keep track of champagne levels.
 Start by pouring the champagne into the top glass (
(0, 0)
).  Iterate through each row down to
query_row
. For each glass, calculate the overflow and update the glasses immediately below it.
 Return the champagne level in the glass at
(query_row, query_glass)
.
By understanding these elements, you can formulate a targeted approach to solving this problem.
Visual Model of the Problem
Visualizing the problem statement can help in understanding the flow and distribution of champagne. Here’s how to visualize it:
Pyramid Visualization
Draw a Pyramid: Represent each row as a level of glasses in a pyramid shape. Start with one glass at the top, then two in the next row, then three, and so on. You can draw up to
query_row
.O O O O O O O O O O
Label Glasses: Label each glass with its row and position, such as
(0,0)
,(1,0)
,(1,1)
,(2,0)
,(2,1)
, etc.(0,0) (1,0) (1,1) (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) (3,3)
Pouring Path: If possible, draw arrows to indicate the flow of champagne from one glass to the glasses immediately below it. This will help visualize the equal distribution of overflow.
O / \ OO / \ / \ OOO
Overflow Visualization
Overflow Amount: Next to each glass, you can jot down how much champagne will overflow to the next row.
(0,0)[1]
Here,
[1]
means that 1 cup of champagne will overflow from(0,0)
.State Updates: As you “pour” champagne in your visualization, update the state of each glass to reflect how much champagne it currently holds.
(0,0)[1.0] (1,0)[0.5] (1,1)[0.5]
By visually breaking down the pyramid and flow path, you make the problem easier to conceptualize, thereby aiding in the development of a solution.
Problem Restatement
You have a pyramid of glasses, starting with one glass at the top and increasing by one glass per row as you go down. Each glass can hold exactly one cup of champagne. You pour a specific amount of champagne into the top glass. If a glass gets filled, the extra champagne spills over, splitting equally into the two glasses directly below it. Your task is to find out how full a particular glass will be after you’ve poured the champagne.
Requirements:
 The top glass is the starting point for pouring champagne.
 Each glass holds one cup.
 Overflow splits equally into the two glasses immediately below.
 You need to find the champagne level in a glass defined by its row and position in that row.
Constraints:
 The amount of champagne poured can range from 0 to (10^9).
 The row and position for the queried glass are within the 100row pyramid and adhere to the pyramid structure.
By focusing on these essential elements, constraints, and requirements, we align our thought process to solve the problem effectively.
Abstract Representation of the Problem
Let’s formulate an abstract representation of this problem:
Abstract Representation
Structure:
 A 2D array or grid,
G
, whereG[i][j]
represents the amount of liquid in the glass at rowi
and positionj
within that row.
Variables:
P
: Total amount of liquid poured.R
: Target row.C
: Target column (or position) in the target row.
Rules:
 Initialize
G[0][0] = P
.  For each
G[i][j]
:
 If
G[i][j] > 1
, thenG[i][j] = 1
and the overflow, ( \text{Overflow} = G[i][j]  1 ), is equally distributed toG[i+1][j]
andG[i+1][j+1]
.
Objective:
 Find
G[R][C]
after applying the rules.
Key Elements:
 Grid Initialization: Start with a 2D array
G
initialized to zero except forG[0][0] = P
.  State Transition: Update the state of each cell in
G
based on the rules.  Objective Function: The final value in
G[R][C]
is the solution.
By representing the problem this way, we remove the realworld details and emphasize the structure and key elements that will guide our solution.
Terminology
There are some terms and concepts that are important for understanding the problem and its solution:
Specialized Terms:
2D Array (Grid): A twodimensional array that will be used to represent the pyramid of glasses. Each element
[i][j]
will store the amount of liquid in the glass at rowi
and positionj
.Dynamic Programming: A technique for solving problems by breaking them down into smaller subproblems and storing the results of these subproblems to avoid redundant calculations.
State Transition: The rule for updating the amount of liquid in a glass based on the amount of liquid in the glasses above it. This term relates to dynamic programming and helps in formulating the solution.
Overflow: The amount of liquid that exceeds the capacity of a glass and spills into the glasses below. In this context, a glass overflows when it holds more than one cup.
Initial State: The starting point of the problem, where the top glass is filled with the total poured amount
P
.Objective Function: The function or rule that describes what we want to find. In this case, it is the amount of liquid in the glass at row
R
and positionC
.
Role Within Context:
2D Array (Grid): It is crucial for storing the state of each glass as you iterate through the pyramid.
Dynamic Programming: This technique allows for an efficient solution by solving the subproblems (amount in each glass) once and reusing their solutions.
State Transition: This describes how the problem evolves. It helps you write the algorithm to solve the problem by specifying how the state of each glass is updated based on overflow from the glasses above it.
Overflow: This is central to the problem, as it defines how the liquid is transferred from one glass to the others.
Initial State: Sets up the problem by placing the total poured amount in the topmost glass.
Objective Function: It specifies what we are ultimately interested in finding out. In this problem, it is the amount of liquid in a specific glass, denoted by
G[R][C]
.
Understanding these terms and their roles helps in framing the problem and finding a solution that adheres to the specified rules and constraints.
Problem Simplification and Explanation
Let’s break down the problem and its key concepts:
Simplified Terms:
Pyramid of Glasses: Imagine a triangle made of glasses, one at the top, then two below it, then three, and so on.
Pouring Champagne: You start by pouring champagne into the top glass.
Overflow: If a glass fills up, the extra champagne spills equally into the two glasses directly below it.
Target Glass: You want to find out how full a specific glass in this pyramid will be after all the pouring and overflowing are done.
Key Concepts and Interaction:
Starting Point: All champagne starts at the top glass.
Flow: The champagne flows downward, filling up glasses along the way.
Equal Distribution: When a glass overflows, the extra champagne is equally shared by the two glasses right below it.
End Goal: Find the amount of champagne in a specific glass.
Metaphor or Analogy:
Think of the pyramid of glasses as a family tree. The top glass is like the first ancestor, and the glasses below are like the descendants. When the ancestor (top glass) has too much “wealth” (champagne), it gets distributed to its “children” (glasses directly below). The children, in turn, share any excess with their children, and so on.
In this family tree, you’re interested in knowing how much “wealth” (champagne) a specific “family member” (glass) ends up with.
By understanding these simplified terms, key concepts, and analogy, you’ll find it easier to grasp what the problem is asking for and how to approach solving it.
Constraints
Let’s identify the characteristics that could be exploited for an efficient solution:
Fixed Capacity: Each glass has a fixed capacity of one cup. This uniformity simplifies calculations.
Equal Distribution: Overflow splits equally between two glasses. This predictability is useful for designing a loop or recursive function to handle transitions.
2D Array: The pyramid is naturally suited to be represented as a 2D array, which is straightforward to manipulate.
Bounds: The amount of champagne poured can range up to (10^9), but the row and position for the queried glass are within a 100row pyramid. These constraints tell us that the 2D array doesn’t need to be larger than 100 rows, limiting its size.
Numerical Ranges: Query row and glass are 0indexed and within a 100row pyramid, making it easier to index into the array.
Dynamic Programming: Due to the structured, hierarchical flow of champagne, the problem exhibits overlapping subproblems, making it a good candidate for dynamic programming.
Initial State: Knowing that all champagne starts at the top glass can help us initiate the problemsolving process from a single point, avoiding unnecessary calculations.
Objective Function: The goal is to find the amount of champagne in a specific glass. This singleobjective focus helps in simplifying the logic needed to solve the problem.
Directionality: The champagne only flows downwards. We don’t have to worry about any “upstream” effects, which simplifies calculations.
By recognizing these characteristics, we can tailor our solution to be efficient, leveraging these specific conditions and constraints.
Analyzing the constraints gives us several key insights:
Limited Pyramid Size: The row and column indices for the queried glass are limited to a 100row pyramid. This sets an upper bound on the array size, making it computationally feasible to solve the problem even with a simple algorithm.
Large Poured Amount: The amount of champagne poured can be as high as (10^9). While this is a large number, it actually simplifies the problem because the amount will certainly fill many rows. After a point, the bottom rows will inevitably be filled, which could reduce computation.
0Indexed Rows and Glasses: The problem uses 0based indexing for both rows and glasses. This simplifies array indexing in programming languages like Python, C++, or Java, which also use 0based indexing.
Fixed Glass Capacity: Each glass can hold up to one cup. This is a uniform constraint that simplifies calculations and state transitions.
Single Objective: The problem asks for the amount of liquid in only one specific glass, not multiple ones. This helps us to focus our algorithm’s end condition.
NonNegative Poured Amount: The amount poured is nonnegative. This eliminates the need to check for negative values, thereby simplifying the initial state.
Understanding these constraints allows us to optimize our solution by focusing only on what’s necessary, thereby making our algorithm more efficient.
Case Analysis
Providing additional examples and test cases can give a more comprehensive understanding of the problem. Let’s categorize these cases to cover various aspects:
Minimal Input Cases
Zero Pour
 Input: poured = 0, query_row = 0, query_glass = 0
 Output: 0.00000
 Reasoning: Nothing is poured, so no glass can be filled.
Zero Row, NonZero Pour
 Input: poured = 1, query_row = 0, query_glass = 0
 Output: 1.00000
 Reasoning: Only one cup is poured into the first glass, and we’re querying that glass. So, it’s full.
Basic Overflow Cases
Simple Overflow
 Input: poured = 2, query_row = 1, query_glass = 0
 Output: 0.50000
 Reasoning: 1 cup fills the top glass, the excess (1 cup) is divided between the two glasses in the second row. We’re querying the first glass in the second row, so it’s half full.
TwoLevel Overflow
 Input: poured = 3, query_row = 2, query_glass = 1
 Output: 0.25000
 Reasoning: The top glass fills up with 1 cup, 2 cups spill to the second row, filling them. Then 0.5 cups spill from each to the middle glass in the third row. We’re querying this glass, so it’s a quarter full.
Complex Cases
Many Rows, One Glass
 Input: poured = 100, query_row = 4, query_glass = 2
 Output: 1.00000
 Reasoning: With 100 cups, the first 5 rows (1+2+3+4+5 = 15 cups needed) will be filled entirely, including the glass we’re querying.
Large Pour, Deep Query
 Input: poured = 100000009, query_row = 33, query_glass = 17
 Output: 1.00000
 Reasoning: The pour amount is large enough to completely fill many rows, including the row we are querying.
Edge Cases
Maximum Pour
 Input: poured = (10^9), query_row = 99, query_glass = 0
 Output: 1.00000
 Reasoning: The maximum pour is large enough to fill all the rows, including the one we are querying.
Maximum Query Row and Glass
 Input: poured = (10^9), query_row = 99, query_glass = 99
 Output: 1.00000
 Reasoning: Again, the large pour amount ensures even the last glass in the last row is filled.
By considering these cases, we can ensure that our solution covers a wide range of scenarios, making it robust and reliable.
Visualizing these cases can be done using simple 2D arrays or grids, where each cell represents a glass in the pyramid. You can use numbers to represent the amount of liquid in each glass. Here’s how to visualize a few cases:
Minimal Input Cases
Zero Pour
0
 ‘0’ indicates that the glass is empty.
Zero Row, NonZero Pour
1
 ‘1’ indicates that the glass is full.
Basic Overflow Cases
Simple Overflow
1 0.5 0.5
 Top glass is full (‘1’), and the second row glasses are half full (‘0.5’).
TwoLevel Overflow
1 1 1 0.5 0.25 0.25
 Top and second row glasses are full (‘1’), third row has ‘0.5’ on the sides and ‘0.25’ in the middle.
Complex Cases
Many Rows, One Glass
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 All glasses are full (‘1’) up to the 5th row.
Large Pour, Deep Query
This would be a large grid, but imagine all glasses filled ('1') up to the 33rd row.
 The grid is too large to display here, but the concept is the same—all glasses are full up to the queried row.
Edge Cases
Maximum Pour
Imagine a large grid where all cells are filled ('1').
Maximum Query Row and Glass
Same as the maximum pour—a large grid where all cells are filled ('1').
In each grid, you would look for the queried glass and its amount to verify the case. This visual representation helps to understand the state of each glass in the pyramid, making it easier to grasp how champagne flows down the rows.
Analyzing different cases provides several key insights:
Initial Conditions Matter: The amount initially poured determines the range of rows affected. For example, a pour of zero affects no rows, while a small pour may only affect the first few rows.
Propagation Behavior: The liquid pours down in a binary treelike structure. Each glass distributes its overflow equally to the two glasses beneath it, assuming they are present.
Uniform Glass Size: Each glass holds exactly 1 cup. Once it’s filled, the remaining liquid spills over. This uniformity simplifies calculations.
Boundary Conditions: Glasses at the edge of a row only pass their overflow to one other glass, unlike those in the middle of a row, which pass it to two.
Limit on Rows and Glasses: The number of rows and glasses is limited (100 rows, with corresponding glasses). This places an upper limit on computational complexity.
High Pour Volumes: For large volumes poured, it becomes more likely that the queried glass will be full, simplifying the calculation.
Query Constraints: The query_glass will always be <= query_row, eliminating the need for checks against outofbounds queries.
Overflow Saturation: Once a row is entirely filled, all the liquid will pour down to the next row, doubling the amount each row can receive compared to its previous row (1, 2, 4, 8, …).
0Indexed Rows and Glasses: Simplifies the programming as most programming languages are 0indexed.
Understanding these key insights will inform the algorithm’s design, helping to simplify and optimize the solution.
Identification of Applicable Theoretical Concepts
This problem lends itself to several mathematical and algorithmic concepts:
Dynamic Programming: The amount of champagne in each glass depends on the amounts in the glasses above it. Therefore, it’s a good candidate for a bottomup dynamic programming approach.
Binary Tree Traversal: The way the champagne pours down can be thought of as a traversal of a binary tree. Each node has at most two children, which get an equal share of any overflow.
Triangular Numbers: The pyramid shape corresponds to a sequence of triangular numbers. The
n
th row hasn+1
glasses and a total of(n*(n+1))/2
glasses exist up to and including that row. This can help in mapping the 2D representation to a 1D array if needed.Arithmetic Progression: The total number of cups required to fill up to the
n
th row is the sum of the firstn
natural numbers, given by(n * (n + 1)) / 2
. This can help in quickly determining how many rows will be completely filled.Boundary Conditions: Glasses at the edge of a row receive half the overflow of a central glass in the same row. This property can be used to optimize calculations.
Constraints Analysis: Given that the query row and glass are within a fixed range, you can preallocate the required space, thereby avoiding dynamic memory allocation during the algorithm’s run.
Computational Complexity: Since we are working with a fixedsize pyramid, the problem has a natural upper bound, making it more manageable.
FloatingPoint Arithmetic: This problem requires working with fractions, which should be kept in mind for languages that have issues with floatingpoint precision.
Associative and Commutative Properties: The problem’s mathematical operations are both associative and commutative, which offers flexibility in the order of calculations.
Applying these concepts can simplify the problem and lead to an efficient algorithm.
Simple Explanation
Imagine you have a pyramid made of small cups. The topmost layer has just one cup, the second layer has two cups, the third layer has three cups, and so on. Now, you start pouring water into the top cup. Once the top cup fills up, it spills over into the two cups below it. Each of these two cups will then spill over into the two cups below them when they get full, and this pattern continues.
Here’s the challenge: We want to know how much water ends up in a specific cup after we’ve poured a certain amount into the top cup. Will that cup be full, halffull, or maybe empty?
Think of it like a fountain with multiple levels. When you pour water at the top, it trickles down to fill the lower levels. You’re trying to figure out how much water ends up in a particular spot at one of the lower levels after you’ve poured a certain amount at the top.
Problem Breakdown and Solution Methodology
Approach
Initialize the Pyramid: Start by initializing a 2D array to represent the champagne glasses. Each element in the array will store the amount of champagne in the corresponding glass.
Pour Champagne: Pour the given amount of champagne into the top glass (the first element of the array).
Trickle Down: Start from the top and simulate the trickling down of champagne. Update the amount in each glass based on the overflow from the glass above it.
Result: Once the trickling down is done, return the amount in the glass specified by
query_row
andquery_glass
.
Metaphor
Think of the pyramid as a system of tiny waterfalls. You pour water at the top waterfall, and it trickles down, filling each lower waterfall until they overflow into the ones below. We’re tracking how much water ends up in each “waterfall” (glass).
Detailed Steps
Initialize Pyramid: Use a 2D array
pyramid
with enough rows and columns to represent the glass pyramid. Initializepyramid[0][0] = poured
. For example, for a 3row pyramid, you’d initialize a 3x3 array.
Trickle Down Champagne:
 Loop through each row
i
from 0 toquery_row
.  Loop through each glass
j
in rowi
.  Calculate the overflow for
pyramid[i][j]
.  Distribute the overflow equally to
pyramid[i+1][j]
andpyramid[i+1][j+1]
.
 Loop through each row
Fetch the Result: After trickling down the champagne,
pyramid[query_row][query_glass]
will contain the answer.
How Parameters Affect the Solution
poured
: More champagne means more rows will be filled, and trickling down will go further.query_row
andquery_glass
: Higher values for these means you’ll need to trickle down through more rows to get the answer.
Example
Let’s consider the example where poured = 2
, query_row = 1
, and query_glass = 1
.
Initialize
pyramid
as a 2x2 array withpyramid[0][0] = 2
.2 0 0 0
Start trickling down.
pyramid[0][0]
has 1 cup of overflow. Distribute it equally to
pyramid[1][0]
andpyramid[1][1]
.
1 0 0.5 0.5
The answer is
0.5
, which is the amount inpyramid[1][1]
.
By following these steps, you can solve the problem for any set of input parameters.
Inference of ProblemSolving Approach from the Problem Statement
Let’s identify the key terms and concepts:
Pyramid: This refers to the stacked arrangement of glasses. It helps you understand that you need a 2D array to model the system.
Glass: Each cell in the 2D array represents a glass that can hold champagne. Knowing each glass’s capacity (1 cup) informs the algorithm about when overflow occurs.
Poured: This is the initial amount of champagne poured into the top glass. It sets the starting condition for the 2D array and initiates the ’trickling down’ process.
Query_row and Query_glass: These specify which glass we are interested in. They help narrow down the focus of our calculations.
Overflow: When a glass fills up, it spills over into two glasses below it. This concept informs the algorithm’s core logic—how to distribute excess champagne.
Constraints: Knowing the range of values helps you pick appropriate data structures and optimize the algorithm for speed and memory usage.
How They Inform the Approach
Pyramid: Tells us we need a 2D array for a structured representation.
Glass: We need to keep track of the amount in each glass, knowing when it overflows (more than 1 cup).
Poured: Sets the initial condition for the simulation. The top glass starts with this amount.
Query_row and Query_glass: Tells us until what point we need to perform the calculations. We don’t need to calculate for rows greater than
query_row
.Overflow: The algorithm must implement logic to handle overflow, distributing it to adjacent glasses in the row below.
Constraints: Given the constraints, we don’t need specialized data structures; a 2D array suffices.
By understanding these key terms, we can create a structured and efficient approach to solving the problem.
Drawing tables or diagrams can greatly aid in visualizing the problem’s properties. Here’s how:
1. Pyramid
Create a 2D grid on paper, shaping it like a pyramid. Each cell represents a glass. This grid will serve as your 2D array during manual calculations.
2. Glass and Poured
Mark the top cell (the tip of the pyramid) with the poured
value to indicate how much champagne is poured initially.
3. Overflow and Trickling Down
As you work through the problem, update the cells in the grid to show how much champagne each glass contains. Use arrows to indicate the direction of overflow from one cell to two cells below it.
For instance, if a cell has 1.5 cups, you can draw arrows pointing to the two cells below it and write 0.25 beside each arrow to show how the overflow is being distributed.
4. Query_row and Query_glass
Circle or highlight the cell corresponding to query_row
and query_glass
. This will help you focus on your end goal while solving manually.
5. Constraints
Although constraints might not be directly drawable, having a sense of the range of values can help you size your grid and understand how deeply you need to calculate.
Example
Consider poured = 4, query_row = 2, query_glass = 1
. Your initial pyramid might look like:
4
0 0
0 0 0
After trickling down, the diagram may look something like this:
1
1.5 1.5
0.25 1.0 0.25
Here, the cell in the 3rd row and 2nd column (0indexed), which contains 1.0, is your answer. You can circle this cell as it corresponds to query_row = 2
and query_glass = 1
.
Using such diagrams can make the problem’s properties easily recognizable and assist you in formulating a solution.
How did you infer from the problem statement that this problem can be solved using ?
Simple Explanation of the Proof
I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?
Stepwise Refinement
1. Stepwise Refinement of Approach
Initialize 2D Array: Create a 2D array (or list of lists in Python) to model the champagne pyramid.
Set Starting Condition: Place the
poured
amount in the top glass (the first cell of the array).Iterate Through Rows: Go row by row from the top to
query_row
.Check Each Glass: For each glass, if it contains more than 1 cup of champagne, find the overflow.
Distribute Overflow: Split the overflow equally between the two glasses directly below it.
Update Glasses: Adjust the amounts in the current glass and the two glasses below it.
Query Result: Once you reach
query_row
, check the amount inquery_glass
. Return this value, ensuring it’s not more than 1.
2. Granular, Actionable Steps
Initialize 2D Array
 Use a loop to create rows, where the ith row has i+1 glasses.
Set Starting Condition
 Array[0][0] =
poured
.
 Array[0][0] =
Iterate Through Rows
 Use a nested loop, one for rows (0 to
query_row
) and another for glasses in each row.
Check Each Glass
 If Array[i][j] > 1, then Overflow = Array[i][j]  1.
Distribute Overflow
 Array[i+1][j] += Overflow / 2.
 Array[i+1][j+1] += Overflow / 2.
Update Glasses
 Array[i][j] = min(1, Array[i][j]).
 Use a nested loop, one for rows (0 to
Query Result
 Return Array[query_row][query_glass].
3. Independent Parts
 Initialization: Steps for initializing the 2D array can be done independently.
 Setting Starting Condition: This is independent and only needs to be done once at the start.
 Iterating Through Rows: Each row’s calculations depend only on the row above it, so they cannot be done completely independently. However, within a row, each glass’s overflow calculation can be done independently of its neighbors.
4. Repeatable Patterns
 Overflow Distribution: This is a repeatable action. For every glass that has more than 1 cup, you distribute the overflow to the glasses below it.
By following these steps, you should be able to solve the problem effectively.
Solution Approach and Analysis
Approach to Solving the Problem
Conceptual Overview
Think of the champagne glasses as a pyramid of buckets. Each bucket can hold only 1 cup. If you pour more into a bucket than it can hold, the excess spills equally into the two buckets below it.
Detailed Steps
Initialize 2D Array: Create a 2D array with enough rows to cover
query_row
. Each rowi
will havei+1
glasses (columns). This array will represent our pyramid of glasses.Pour Champagne: Start by pouring the champagne into the top glass, the one at row 0 and column 0 (Array[0][0] =
poured
).Iterate Through Pyramid: Loop through the rows of the pyramid up to
query_row
.Calculate Overflow: For each glass, if it contains more than 1 cup (Array[i][j] > 1), the excess will spill. Calculate this overflow (Overflow = Array[i][j]  1).
Distribute Overflow: The excess is split equally and added to the two glasses in the row below. Update their values:
 Array[i+1][j] += Overflow / 2
 Array[i+1][j+1] += Overflow / 2
Update Current Glass: After distributing, set the current glass to hold no more than 1 cup: Array[i][j] = min(1, Array[i][j]).
Retrieve Answer: After going through the pyramid up to
query_row
, Array[query_row][query_glass] will contain the answer.
Effects of Parameter Changes
 Increasing
poured
: More glasses will be full or partially full. Computation time increases slightly.  Increasing
query_row
: Computation time increases as you have to iterate through more rows.
Example Cases
Example 1:
poured = 2, query_row = 1, query_glass = 1
 Start with Array = [[2], [0, 0], …]
 After Iteration: Array = [[1], [0.5, 0.5], …]
 Answer: 0.5
Example 2:
poured = 100000009, query_row = 33, query_glass = 17
 Start with Array = [[100000009], [0, 0], …]
 After Iteration: Array[33][17] will be 1 (The pyramid will have enough rows and columns to reach here)
 Answer: 1
By breaking it down like this, the approach should be manageable. Use these steps to write your code.
Identify Invariant
The invariant in this problem is that each glass can hold no more than 1 cup of champagne. Any excess liquid will flow down to the adjacent glasses in the row immediately below. This rule consistently applies throughout the pyramid, no matter how many rows or glasses are involved. This invariant helps us understand that as we iterate through each row and each glass, we can apply the same logic to distribute the excess liquid.
Identify Loop Invariant
The loop invariant in this problem would be that at the start of each iteration of the loop over rows, the ith
row correctly represents the state of each glass after the champagne has been poured and excess liquid has been distributed according to the problem’s rules.
In other words, when entering each iteration of the loop to process row i
, we can assume that all glasses in rows 0 to i1
have been properly filled, any excess champagne has been distributed to row i
, and each glass in rows 0 to i1
holds no more than 1 cup.
This loop invariant ensures that we can continue to apply the same logic for distributing excess champagne as we move from one row to the next. It also provides the foundation for proving the correctness of the algorithm.
Thought Process
Understanding the Pyramid: Recognize that this is a pyramid of glasses and each glass can only hold 1 cup of champagne.
Flow Logic: Understand how champagne flows. If a glass is full, the excess flows to the two glasses immediately below it.
Rows and Columns: Identify that each row
i
hasi+1
glasses.Constraints: Note the constraints. The row and glass indices start at 0, and row number is less than 100.
State Tracking: Realize that you need to keep track of the amount of champagne in each glass as you pour more champagne.
Invariant: Acknowledge that each glass can hold no more than 1 cup, and excess will flow down. This is our invariant to maintain through the solution.
Dynamic Programming: Since the amount in a glass depends on the glasses above it, this hints at a bottomup approach, perhaps utilizing dynamic programming.
Coding Steps
Initialization: Create an array
dp
to keep track of the champagne levels in each row’s glasses. Initialize the top glass (dp[0][0]
) with the total amount poured.1 2
dp = [[0] * k for k in range(1, 101)] # Create a 2D list to represent each row of glasses dp[0][0] = poured # Start by pouring all champagne into the top glass
Iteration Through Rows: Loop through each row and each glass within that row.
1 2
for i in range(len(dp)): for j in range(len(dp[i])):
Overflow Logic: Within the loop, check if a glass overflows (i.e., contains more than 1 cup). Distribute the overflow equally to the glasses below.
1 2 3 4 5 6 7
overflow = max(0, dp[i][j]  1) if overflow > 0: # Distribute the overflow equally to the glasses below dp[i + 1][j] += overflow / 2 dp[i + 1][j + 1] += overflow / 2 # The current glass now has 1 cup at most dp[i][j] = min(1, dp[i][j])
Answer Lookup: After updating all the rows and glasses, look up the champagne level of the glass specified by
query_row
andquery_glass
.1
return dp[query_row][query_glass]
Full Python Code


IndexError: list index out of range dp[i + 1][j] += overflow / 2 Line 11 in champagneTower (Solution.py) ret = Solution().champagneTower(param_1, param_2, param_3) Line 45 in _driver (Solution.py) _driver() Line 56 in (Solution.py)
The error message “IndexError: list index out of range” is signaling that an attempt was made to access an element at an index that does not exist in the list. This usually happens because of an offbyone error or a logic flaw in the code.
Let’s focus on this line causing the error:


In the code, dp
is a 2D list (array), and i + 1
or j
seems to exceed the size of the respective dimension at some point.
Debugging Steps
 Check Initialization: Make sure that the
dp
array is initialized properly to containquery_row + 1
rows.  Bounds Check: Before accessing
dp[i + 1][j]
, ensure thati + 1 < len(dp)
andj < len(dp[i + 1])
.
Possible Fix
Here is a modification to the code, introducing a bounds check:


This should remove the “IndexError: list index out of range” issue by checking the bounds before updating the dp
array.
Key Insights
 The 2D dynamic programming array effectively captures the state of the problem.
 The loop invariant ensures we correctly simulate the pouring and overflowing at each step.
 The constraints are small enough to make this bottomup approach feasible in terms of time complexity.
Establishing Preconditions and Postconditions
1. Parameters:
 Inputs:
poured
(int),query_row
(int),query_glass
(int)  Types: All parameters are integers.
 Representation:
poured
: Total amount of champagne poured at the top.query_row
: The row index of the glass we are interested in.query_glass
: The glass index withinquery_row
.
2. Preconditions:
 State: No specific state requirement for the program.
 Constraints:
poured
must be between 0 and 10^9.query_row
andquery_glass
must be between 0 and 99.
3. Method Functionality:
 Expectation: To find out how much champagne will be present in the glass specified by
query_row
andquery_glass
after pouringpoured
cups at the top.  Interaction: The method uses the inputs to simulate the flow of champagne through the glasses and calculates the final amount in the queried glass.
4. Postconditions:
 State: The program returns a floatingpoint number.
 Return Value: Represents the amount of champagne in the queried glass, ranging from 0 to 1.
 Side Effects: None.
5. Error Handling:
 Response: Python will throw a runtime error if indexing outside the list bounds occurs, which would imply incorrect inputs.
 Special Value: No special value or exception handling is built into the method. It assumes valid inputs as per the problem constraints.
Problem Decomposition
1. Problem Understanding:
 Own Words: We need to find out how much champagne will end up in a specific glass after pouring a given amount at the top of a pyramid of glasses.
 Key Components:
poured
(amount poured),query_row
,query_glass
(location of the glass in question).
2. Initial Breakdown:
 Major Parts:
 Initialize the champagne pyramid.
 Simulate the pouring process.
 Retrieve the amount in the specified glass.
3. Subproblem Refinement:
Initialize the Champagne Pyramid:
 Create a 2D list to represent the pyramid.
Simulate the Pouring Process:
 Loop through each row and each glass.
 Distribute champagne to the next row.
Retrieve the Amount in Specified Glass:
 Look up the value in 2D list based on
query_row
andquery_glass
.
 Look up the value in 2D list based on
4. Task Identification:
 Repeated Tasks:
 Distributing champagne to glasses in the next row.
5. Task Abstraction:
 Abstracted Tasks:
 Initialize 2D list: General enough but still makes sense in the context.
 Loop and distribute: General, reusable, and contextually relevant.
 Value lookup: Contextspecific but reusable.
6. Method Naming:
initializePyramid()
simulatePour()
getChampagneInGlass()
7. Subproblem Interactions:
Order:
 Initialize the pyramid.
 Simulate the pouring.
 Retrieve the amount.
Dependencies:
 The simulation relies on the initialization.
 Retrieving the amount relies on the simulation being completed.
From Brute Force to Optimal Solution
Brute Force Solution
Approach
 Initialize a 2D list (or array) to represent the champagne pyramid. Each cell
[i][j]
will contain the amount of champagne in that glass.  Pour the champagne at the top glass, i.e., set
pyramid[0][0] = poured
.  Loop through each row and each glass, pouring the overflow into the glasses in the next row.
 Retrieve the amount of champagne in the specified glass
[query_row][query_glass]
.
Code (Python3)


Inefficiencies
 Time Complexity: Looping through each glass takes O(n^2) time where n is the
query_row
.  Space Complexity: We create a pyramid up to
query_row
, which also takes O(n^2) space.
Optimizing the Solution
Step 1: Eliminate Unnecessary Glasses
We don’t need to keep track of glasses that won’t be reached by the champagne. We only fill the glasses that are on the path to the target glass.
Step 2: Update Only Affected Glasses
Instead of updating all glasses in the next row, update only the glasses that will receive champagne overflow from the current glass.
Optimized Code (Python3)


Time and Space Complexity
 Time Complexity: O(n^2), the optimization didn’t improve the time complexity.
 Space Complexity: O(n), we reduced the space complexity from O(n^2) to O(n).
Here, while the time complexity remains the same, the space complexity is improved, making it more efficient in terms of memory usage.
Code Explanation and Design Decisions
Initial Parameters:
poured
: the amount of champagne poured at the top.query_row
andquery_glass
: the specific glass we are interested in. These parameters set the problem’s stage, specifying how much champagne is poured and the target glass location.
Primary Loop: The outer loop iterates through each row of the pyramid of glasses (
i
from 0 toquery_row  1
). The inner loop (j
) iterates through each glass in rowi
. The iteration represents how champagne flows from the top, cascading down through glasses until reachingquery_row
.Conditions:
overflow > 0
: Checks if there’s overflow to spill over to the glasses below. This condition ensures that only glasses with overflow will contribute to the glasses in the next row.
Updates:
dp[i + 1][j] += overflow / 2
dp[i + 1][j + 1] += overflow / 2
These updates distribute the overflowing champagne to the two glasses below the current one. This reflects the physical process of champagne flowing through the pyramid.
Invariant: The sum of champagne in all glasses from row 0 to
i
will always be equal topoured
. This invariant is maintained by carefully calculating the overflow and updating thedp
array accordingly.Final Output: The value
dp[query_row][query_glass]
represents the amount of champagne that would be in the glass located atquery_row
andquery_glass
. The minimum between this value and 1 is returned, ensuring it adheres to the problem’s constraint that a glass can hold at most 1 unit of champagne.
This explanation provides a highlevel understanding of how the code translates the problem constraints into a working solution.
Coding Constructs
HighLevel Strategies:
 Dynamic Programming: Storing and reusing intermediate results in a 2D array.
 Iteration: Looping through the 2D array to simulate the flow of champagne.
Purpose to a NonProgrammer: This code models how champagne pours into a pyramid of glasses. It calculates how much champagne ends up in a specific glass after all the pouring and overflowing are done.
Logical Elements:
 Conditional Statements: Decisions based on whether a glass overflows.
 Loops: Iteration over each row and glass within that row.
 Arrays: Storing the champagne amount for each glass.
Algorithm in Plain English: Imagine a pyramid made of glasses. You pour champagne at the top, and it flows down. If a glass gets too full, it overflows into the two glasses below it. This code calculates how this overflow happens step by step, row by row, until it figures out how much champagne ends up in a specific glass you’re interested in.
Key Steps:
 Initialize a 2D array representing the champagne in each glass.
 Iterate through each row and glass, calculating and updating the champagne amount based on overflow.
 Use the updated array to answer the query about a specific glass.
Algorithmic Patterns:
 Dynamic Programming: To avoid redundant calculations and speed up the process.
 Iteration: To simulate the cascade of champagne through the pyramid.
These elements together fulfill the problem’s requirements, taking into account the constraints and desired output.
Language Agnostic Coding Drills
Distinct Coding Concepts:
 Variable Declaration: Understanding how to create and use variables.
 Array Initialization: Knowing how to create and initialize arrays.
 Looping Mechanisms: Understanding how to create loops.
 Conditional Statements: Using
ifelse
conditions to make decisions.  Indexing and Nested Indexing: Accessing elements within arrays, including nested arrays.
 Arithmetic Operations: Doing calculations like addition and division.
 Dynamic Programming: Storing and reusing previous computations.
 Function Calls: Knowing how to call functions and pass parameters.
Order of Increasing Difficulty:
 Variable Declaration: Easiest, as it’s fundamental and seen in almost every program.
 Arithmetic Operations: Slightly more complex, involving basic math operations.
 Array Initialization: Introduces the concept of data structures.
 Looping Mechanisms: Adds a layer of complexity with iteration.
 Conditional Statements: Requires logical reasoning and understanding of flow control.
 Indexing and Nested Indexing: Needs good grasp of data structures and loops.
 Function Calls: Involves understanding function scope and parameter passing.
 Dynamic Programming: Most complex, requiring a deep understanding of optimization and reuse of calculations.
ProblemSolving Approach:
 Understand the Problem: First, grasp what the problem asks and what the constraints are.
 Design a Basic Flow: Think about the steps involved in solving the problem without focusing on code specifics.
 Introduce Variables: Declare variables for storing the initial amount of champagne and the resulting amounts in glasses.
 Initialize Array: Create an array to represent the glasses in the pyramid.
 Loop through Glasses: Use loops to go through each row and glass.
 Indexing and Nested Indexing: Access each glass in the row and understand its relationship with the glasses in the previous row.
 Dynamic Programming: Store and reuse champagne amounts for each glass to avoid redundant calculations.
 Add Conditions: Within the loop, use conditionals to check if a glass is overflowing.
 Arithmetic Operations: If so, perform calculations to distribute the overflow to the adjacent glasses in the next row.
 Function Calls: Optionally, isolate functionalities like ‘update glass state’ or ‘calculate overflow’ into separate functions to make the code modular.
Each drill or concept contributes to building a solution that can accurately model the problem and do so efficiently. The combination of these drills allows us to approach and solve complex problems effectively.
Targeted Drills in Python
General Coding Drills
Variable Declaration
1 2
my_variable = 10 print("Value of my_variable:", my_variable)
Arithmetic Operations
1 2 3 4
sum_result = 5 + 10 division_result = 10 / 2 print("Sum:", sum_result) print("Division:", division_result)
Array Initialization
1 2
my_array = [0, 1, 2, 3, 4] print("My array:", my_array)
Looping Mechanisms
1 2
for i in range(5): print("Loop iteration:", i)
Conditional Statements
1 2 3 4 5
value = 10 if value > 5: print("Value is greater than 5") else: print("Value is 5 or less")
Indexing and Nested Indexing
1 2
my_array = [0, 1, 2, 3, 4] print("First element:", my_array[0])
Function Calls
1 2 3 4 5
def my_function(x): return x * 2 result = my_function(5) print("Result:", result)
Dynamic Programming
1 2 3 4 5
dp = [0] * 10 dp[0] = 1 for i in range(1, 10): dp[i] = dp[i1] * 2 print("DP array:", dp)
ProblemSpecific Drills
Champagne Overflow Calculation
This drill would involve calculating how much champagne overflows from a glass when it gets filled over its limit. This concept is essential for our problem because the core challenge is to distribute the overflow correctly.
1 2 3 4
glass_capacity = 250 # milliliters poured_champagne = 300 # milliliters overflow = max(0, poured_champagne  glass_capacity) print("Overflow:", overflow)
Assembling Drills into a Solution
Initialize your variables and array: Use the Variable Declaration and Array Initialization drills to set up your pyramid structure and pouring amount.
Loop through the Glasses: Use the Looping Mechanisms drill to iterate through the rows and glasses.
Check for Overflow: Within each loop iteration, use the Conditional Statements and Champagne Overflow Calculation drills to check if the current glass overflows.
Update State: Use the Dynamic Programming drill to update the state of the glasses.
Function Calls: If there are reusable parts, encapsulate them using the Function Calls drill.
Final Output: Use the Indexing and Nested Indexing drill to get the final champagne amount in the desired glass.
By following these steps, each coding drill will contribute to building the overall problem solution in Python.
Q&A
Similar Problems
Climbing Stairs (Easy)
Why: This problem also utilizes dynamic programming and can be considered a simpler form of our problem. You need to find how many unique ways you can climb a staircase, much like calculating the overflow through different paths.Unique Paths (Medium)
Why: This problem is about finding unique paths from the topleft corner to the bottomright corner of a grid. It’s similar in that it uses dynamic programming to iterate through each possible position, calculating the number of paths based on previous positions.Coin Change (Medium)
Why: In this problem, you’re tasked with finding the fewest number of coins that you need to make up a given amount. The bottomup dynamic programming approach here is very similar.Longest Increasing Subsequence (Medium)
Why: The approach of breaking down a problem into subproblems is also used here, where you have to find the longest increasing subsequence.Triangle (Medium)
Why: You have to find a path from top to bottom in a triangle array such that the sum is minimized. It involves similar traversal through a grid structure and uses dynamic programming for efficient calculation.House Robber (Easy)
Why: You have to find the maximum amount you can rob tonight without alerting the police. It involves dynamic programming to find the optimal way to rob houses while avoiding adjacent houses.Maximal Square (Medium)
Why: This problem asks to find the largest square containing only 1s and return its area. It involves traversing a 2D grid and uses dynamic programming to keep track of the largest square seen so far.Partition Equal Subset Sum (Medium)
Why: In this problem, you have to split an array into two subsets of equal sum. It employs dynamic programming and shares similarities in terms of breaking down the problem into simpler subproblems.Longest Palindromic Subsequence (Medium)
Why: You have to find the longest palindromic subsequence in a string. It uses dynamic programming to solve smaller problems first and uses that information for larger problems.Minimum Path Sum (Medium)
Why: The objective is to find a path from topleft to bottomright in a grid such that the sum of the numbers along its path is minimized. This also involves traversing a grid and dynamic programming, similar to our original problem.
These involve breaking down the problem into smaller subproblems and solving them optimally, often employing dynamic programming or similar iterative strategies.