Algorithm Notesworking through it, one step at a time

Bit-prefix dynamic programming

EasyPractice ↗

Build a Small Switch Lookup Table

Original & variantCounting Bits ↗

Original examples and wording. The question below states the exact input format and output contract.

01. The question

Return the number of enabled bits for every counter value from 0 through the inclusive maximum. Original lookup-table scenario; the counting-bits contract is unchanged. The educational maximum is 31.

Before revealing the idea, predict the result and describe what information you would need to track.

02. Worked example

Input7

Output[0,1,1,2,1,2,2,3]

  1. The count for 0 is 0.
  2. Dropping the low bit of 6 leaves 3, so count[6] = count[3] + 0 = 2.
  3. For 7, the low bit is 1, so count[7] = count[3] + 1 = 3.

Try your code

Define solve(data, parameter="") and return your result. Your code runs locally, not on a server.

Edit your test cases

Use 1-10 objects with data and an optional string parameter. Expected outputs are computed from the lesson. An optional expected field must be correct.

Input contract & reference code

Example data: 7. Parameter: "". Numeric parameters arrive as strings; convert them with int(parameter) when needed.

The example and generated cases use the question’s exact variant. This checks behavior on a small test batch—not performance on every possible input.

Python runtime notices

Python downloads only when you run code, using compressed runtime assets. Code is not saved or uploaded. Enable JavaScript to run it.

Five-second execution limit per batch. Standard-library Python only; additional packages and network access are unavailable. Stop ends execution without blocking the lesson.

03. The idea

Try sketching your own approach first. What would you track, and why would it work?

Reveal the idea
  1. Initialize the zero entry.
  2. Process values in increasing order.
  3. Reuse the count for value >> 1 and add the low bit.

04. Watch it happen

Use Step to move slowly, or Play to follow the whole trace. Change a preset or load your own input.

Input & settings

Small validated inputs: arrays at most 16 entries, text at most 16 characters, tables at most 20 states. Bit questions use unsigned bytes (0-255). See the task for its exact format.

Red = current item / blue = active window or cells / green = completed work. A value docks at its actual destination; the original input stays unchanged. Positions start at 0.

Working state / what changes

Ready to begin

Enable JavaScript to step through the animation. The worked example and solution below remain readable without it.

Python / why it changes

The line behind the step

def solve(data, parameter=''):
    counts = [0] * (data + 1)
    for value in range(1, data + 1):
        counts[value] = counts[value >> 1] + (value & 1)
    return counts
How to use this walkthrough

Hover or focus a control for its label. Play forms the input and follows values into the working state. Step shows one change; Back and the slider restore exact snapshots. Reduced motion uses instant updates. Saved snapshots and sorted heap displays are teaching overhead, not algorithm complexity.

05. Python solution

def solve(data, parameter=''):
    counts = [0] * (data + 1)
    for value in range(1, data + 1):
        counts[value] = counts[value >> 1] + (value & 1)
    return counts

06. Cost & edge cases

Time
O(n)
Space
O(n) output
  • Maximum 0 returns [0].
  • Powers of two have one enabled bit.
  • Require a JSON integer from 0 to 31.

Keep going

Browse all questionsBack to Bits & binaryAdd Two Byte Counters Count Enabled Byte Switches Find the Unscanned Ticket Mirror an Eight-Switch Panel