Generating Pascal's Triangle is a classic problem that serves as an excellent introduction to two key concepts: Array manipulation and Dynamic Programming (DP). While the mathematical rules of the triangle are simple, there is a substantial difference in how we implement the array construction in Python to optimize execution and memory. 📐

In this post, we'll dive into how to construct Pascal's Triangle efficiently, comparing an intuitive array-preallocation method with an ultra-clean, memory-optimized dynamic list-building approach. 🚀

🧩 The Problem

Given an integer numRows, return the first numRows of Pascal's triangle.

In Pascal's triangle, each number is the sum of the two numbers directly above it.

Example

Input: numRows = 5

Output:

[
     [1],
    [1,1],
   [1,2,1],
  [1,3,3,1],
 [1,4,6,4,1]
]

🔍 Visualizing the Construction (Dry Run for numRows = 5)

To build row i, we look at row i - 1. The value at index j is computed by taking the values at j - 1 and j from the previous row:

$\text{row}[j] = \text{prev\_row}[j-1] + \text{prev\_row}[j]$

🛠️ Approach 1: Pre-Allocating Rows

The most intuitive way to build the rows is to pre-allocate an array of size i + 1 filled with 1s, and then overwrite the middle indexes with the sum of the elements from the previous row.

class Solution(object):
    def generate(self, numRows):
        """
        :type numRows: int
        :rtype: List[List[int]]
        """
        triangle = []
 
        for i in range(numRows):
            # Pre-allocate row with 1s of correct size
            row = [1] * (i + 1)
 
            # Calculate and overwrite middle values
            for j in range(1, i):
                row[j] = triangle[i - 1][j - 1] + triangle[i - 1][j]
 
            triangle.append(row)
 
        return triangle

⚠️ The Memory Trade-off

While this approach is easy to write and achieves a 100% runtime score (0ms), pre-allocating lists in Python introduces slight memory overhead because it reserves block sizes ahead of time. This gets a ~28% memory beat on LeetCode.

Can we optimize this to get a much higher memory score? 🔄

🚀 Approach 2: Memory-Optimized Dynamic List Building (Beating 94%+)

By starting with a base row of [1], dynamically appending calculated middle sums, and then capping the row with a final .append(1), we avoid the overhead of pre-allocation. This clean builder pattern optimizes Python's internal memory footprint!

class Solution(object):
    def generate(self, numRows):
        """
        :type numRows: int
        :rtype: List[List[int]]
        """
        # Base case: start with the first row
        res = [[1]]
        
        # Loop through each row from 1 to numRows (exclusive)
        for i in range(1, numRows):
            # Start the current row with 1
            row = [1]
            
            # Calculate the middle elements
            for j in range(1, i):
                row.append(res[i - 1][j - 1] + res[i - 1][j])
            
            # Append 1 to close the row boundary
            row.append(1)
            
            res.append(row)
        
        return res

This simple change keeps the 100% execution speed (0ms) while boosting our memory efficiency to 94.36% beats!

⏱️ Complexity Analysis

🧠 How to Recognize This Pattern

Look for these characteristics in dynamic programming/combinatorial problems:

1. Grid Paths / Directed Traversal: Moving in a grid where each state depends on the cells above/left of it.
2. Combination Calculations: Computing $n \choose k$ values recursively (since Pascal's Triangle directly maps to binomial coefficients).
3. Overlapping Subproblems: Solving a larger problem by combining the results of smaller, already-calculated subproblems.

This pattern is also present in:

• Pascal's Triangle II (LeetCode 119)
• Unique Paths (LeetCode 62)
• Climbing Stairs (LeetCode 70)

🎯 Interview Tips

1. Space Optimization: If an interviewer asks you to return only the $k$-th row instead of the entire triangle, show them how to optimize the space complexity down to $O(K)$ by updating a single array in-place!
2. Boundary Checks: Always check for edge cases where numRows = 1 or numRows = 0. Our second approach handles numRows = 1 perfectly by initializing res = [[1]] and executing a for loop that never iterates.
3. Binomial Coefficient Intuition: Connect Pascal's Triangle to mathematics. The value at index $j$ of row $i$ is mathematically equivalent to the combination $\binom{i}{j}$. This mathematical connection is highly impressive to interviewers!