Find the Maximum Number of Elements in Subset

Find the Maximum Number of Elements in Subset

What is this problem about?

The Find the Maximum Number of Elements in Subset coding problem asks you to pick a subset from an array that can be arranged in a specific "pyramid" sequence: $x, x^2, x^4, \dots, x^{2^k}, \dots, x^4, x^2, x$. For example, if $x=2$, a valid subset could be $[2, 4, 16, 4, 2]$. All elements must exist in the original array with the required frequencies.

Why is this asked in interviews?

Amazon and Google use this to test your ability to handle exponential growth and frequency management. It evaluations your proficiency with Hash Tables and Enumeration. Since $x^{2^k}$ grows extremely fast, the number of steps in any sequence is very small (around 5-10 for numbers up to $10^9$), allowing you to use a greedy approach for each possible starting $x$.

Algorithmic pattern used

This problem uses Frequency Counting and Greedy Enumeration.

1. Count the frequency of each number in the array using a Hash Map.
2. Special Case: Handle '1' separately, as any number of 1s can form a sequence $[1, 1, \dots, 1]$. The longest sequence of 1s is the largest odd count $\le freq(1)$.
3. For each other unique number $x > 1$:
   • Try building a sequence starting with $x$.
   • You need at least two $x$'s to continue to $x^2$.
   • If you have two $x$'s, add 2 to the current length and move to $x^2$.
   • If you have only one $x$, it must be the "peak" of your sequence. Add 1 to the length and stop.
   • If you have zero $x$, the sequence must have ended at the previous step. (Backtrack or adjust length).
4. Return the global maximum length.

Example explanation

nums = [2, 4, 8, 2, 4, 3, 9]

1. Counts: {2:2, 4:2, 8:1, 3:1, 9:1}.
2. Start with $x=2$:
   • Have two 2s. Length = 2. Move to $2^2 = 4$.
   • Have two 4s. Length = 4. Move to $4^2 = 16$.
   • Have zero 16s. The sequence must have peaked at 4.
   • Wait, the sequence should be $[2, 4, 2]$ or $[2, 4, 16, 4, 2]$. If we only had one 4, it would be $[2, 4, 2]$.
   • With counts {2:2, 4:2}, the longest is $[2, 4, 2]$? No, $[2, 2, 4, 4]$ isn't a pyramid. Actually, if you have two 2s and one 4, you get $[2, 4, 2]$ (length 3).

Common mistakes candidates make

• Ignoring frequencies: Forgetting that each "wing" of the pyramid needs an instance of the number.
• Infinite loops: Not stopping the $x o x^2$ progression when $x$ exceeds the max value in the array.
• Even length sequences: Forgetting that the pyramid sequence must always have an odd length (one peak and two identical wings).

Interview preparation tip

Always look for the "bottleneck" in sequences. Here, the bottleneck is the frequency of elements. If you don't have at least 2 of a number, it can only be the peak. This simplifies the search significantly.