Running time of a ‘for’ loop is __________ the running time of the statements…

By convention in running-time analysis, a figure such as ‘the running time of the statements inside the for loop’ is read as an upper bound on the work done in one iteration — including whatever loop-control overhead (the test and increment) is folded into that per-iteration figure — not as a fixed, exact value. Concept: For a loop that executes N iterations, the loop's total running time is the sum of the running times of its body across all iterations. If every iteration's body takes at most c time units to run — the running time of the statements inside the for loop — then the loop's total running time can be at most N × c; this N × c product is only ever an upper bound on the total cost, not a value the loop is guaranteed to reach.

1. Let c be the running time of the statements inside the for loop (the body), and let N be the number of iterations the loop is set up to perform.

2. If the loop runs through all N iterations and each one performs the full body work, the total running time equals c added to itself N times, i.e. N × c.

3. If instead some iterations skip part of the body through a conditional branch, or the loop exits early via a break or return, less than the full N × c amount of work is actually performed.

4. In every possible execution, the total running time can never exceed N × c — it is bounded above by that product, whether or not the loop happens to do the full work in every pass.

Comparison with the other options:

• ‘at least’ would require the total time to always reach or exceed N × c, but an early exit or lighter iterations make that guarantee false.

• ‘equal to’ would require every iteration to do exactly the same fixed amount of work every time, which is only a special case, not a general law.

• ‘independent of’ is false because the total time is built up directly from N and c — changing either one changes the total.

Hence, the running time of a for loop is at most the running time of the statements inside it times the number of iterations — the relation is an upper bound, matching the option ‘at most’.