Generate Binary Numbers from 1 to N Using a Queue

Given a positive integer N, generate the binary representations of every integer from 1 to N, in increasing order, using a queue-based construction. Start with a queue containing only the string "1". Repeatedly remove the string at the front of the queue, record it as the next binary number, and then push front+"0" and front+"1" (in that order) back onto the queue. Stop once N binary numbers have been recorded and print them in order.

Input: A single line containing the integer N. Output: A single line containing the N binary representations (for integers 1 through N), separated by single spaces.

1 <= N <= 100000

This is a clever, non-obvious use of a queue for GENERATION (not just processing existing data) - recognizing that a queue's FIFO order can drive a breadth-first-style construction process, not just traverse an existing structure, broadens how you think about where queues apply.

Generate the binary representations of every number from 1 to N, in order, using a queue-based construction that builds each new binary string from an earlier one. Brute force: Convert every integer from 1 to N directly into binary using a built-in conversion function or manual division-by-2 - this works but doesn't illustrate or require the queue-based generation technique this problem specifically asks for. Optimised: Since a queue processes elements in the exact order they were ADDED (FIFO), and every new string is derived by appending to an EARLIER string, generating children in a fixed order (0 before 1) and enqueuing them guarantees that dequeuing always produces binary numbers in increasing numeric order - no numeric comparison or sorting ever needed.

1. Start a queue containing just the string '1' (binary numbers never start with a leading zero). 2. Repeatedly dequeue the FRONT string, record it as the next binary number, then enqueue front+'0' followed by front+'1'. 3. Stop once N binary numbers have been recorded.

Time: O(N) - each of the N numbers is generated and recorded exactly once, with O(1) work (string append) per generation. Space: O(N), for the queue holding the generated strings (bounded proportionally to N).

N = 1 should simply print "1"; for N near the upper bound, the queue must be implemented so each operation is O(1) amortized to avoid timing out.

Be ready to explain WHY this queue-based generation naturally produces increasing order without ever comparing numeric values directly - the FIFO property combined with 'always append 0 before 1' is the entire mechanism.