Cam's Cyberspace

Recent Notes

Time & Space Complexity

6 min read

Overview

Time and space complexity are tools for predicting how an algorithm’s running time and memory usage grow as input size n increases. Rather than count literal CPU cycles or bytes, analysis adopts a cost model and summarizes behavior with asymptotic notation—$O(\cdot)$, $Ω(\cdot)$, $Θ(\cdot)$. The result is a machine-independent description that guides design, compares alternatives, and surfaces scaling risks early.

Note

Complexity describes growth, not wall-clock time. Two $Θ(n)$ algorithms can differ by large constant factors; profiling still matters in practice.

Motivation

  • Engineering foresight: Choose data structures and algorithms that remain fast and memory-efficient as datasets grow.
  • Communication: Convey guarantees to users (e.g., “lookup is $O(\log n)$”).
  • Comparison: Evaluate trade-offs (e.g., $Θ(n \log n)$ time & $Θ(1)$ extra space vs $Θ(n)$ time & $Θ(n)$ space).
  • Risk control: Detect “looks fine on samples, explodes in production” scenarios.

Definition and Formalism

Let T(x) be the step count (or relevant cost) of an algorithm on input x, with n = |x| its size. Define:

  • Worst-case: $T_{\max}(n) = \max_{|x|=n} T(x)$
  • Best-case: $T_{\min}(n) = \min_{|x|=n} T(x)$
  • Average-case: $E[T(x)\mid |x|=n]$ under a specified input distribution

Asymptotic notation:

  • $O(g(n))$: upper bound up to constant factors
  • $Ω(g(n))$: lower bound
  • $Θ(g(n))$: tight bound (both $O$ and $Ω$)

Space complexity parallels time: let S(n) be the extra memory beyond the input (and sometimes beyond the output, depending on convention). Report in $O(\cdot)$/$Θ(\cdot)$.

Cost models (time)

  • RAM model (unit cost): Arithmetic, comparisons, assignments, and array indexing cost $Θ(1)$. Appropriate for most DS&A with fixed-width machine words.
  • Bit model: Cost depends on operand bit-length b. Adding two b-bit integers costs $Θ(b)$. Needed for big-integer algorithms, cryptography, exact arithmetic.
  • I/O (external memory) model: Dominant cost is block transfers between memory layers (e.g., disk↔RAM). Sorting becomes $Θ\!\left(\frac{n}{B}\log_{M/B}\frac{n}{B}\right)$ in terms of block size B and fast memory M.

Space accounting

  • Input space: memory occupied by the input itself (often excluded from the bound).
  • Auxiliary space: extra scratch structures (stacks, queues, recursion frames, buffers).
  • Output space: storage required to hold results (sometimes excluded if unavoidable).

Tip

State what you count. “$O(1)$ extra space” usually means in-place, excluding input and output.

Example or Illustration

  1. Single loop (RAM model)
sum = 0
for i in 0..n-1:
    sum += A[i]
return sum

Each iteration is constant work → $Θ(n)$ time, $Θ(1)$ extra space.

  1. Nested loops (full square)
for i in 0..n-1:
    for j in 0..n-1:
        work()

Runs n·n bodies → $Θ(n^2)$ time.

  1. Triangular loop
for i in 0..n-1:
    for j in 0..i:
        work()

Sum 1+2+…+n = n(n+1)/2 → $Θ(n^2)$ time.

  1. Binary search halves the interval every iteration → $Θ(\log n)$ time, $Θ(1)$ extra space.

  2. Merge sort solves 2 subproblems of size n/2 and merges in linear time: $T(n)=2T(n/2)+Θ(n)$ → $Θ(n \log n)$ time; auxiliary space $Θ(n)$ (array) or $Θ(\log n)$ (linked lists or bottom-up techniques).

Properties and Relationships

Dominant-term simplification

Drop constants and lower-order terms:

  • 3n^2 + 7n + 10 = Θ(n^2)

  • n \log n + 100n = Θ(n \log n)

Composition rules

  • Sequential parts add: T(n) = T₁(n) + T₂(n); report the dominant term.

  • Conditionals: worst-case takes the max of branch costs; average-case requires branch probabilities.

  • Loops: multiply cost per iteration by the iteration count (often a summation).

Recurrences and patterns

For recursive algorithms describe T(n) by:

  • Halving + constant work: $T(n)=T(n/2)+Θ(1) ⇒ Θ(\log n)$ (binary search).

  • Split & merge: $T(n)=aT(n/b)+Θ(n^c)$ → use Master Theorem. See Recurrence Relations and Recurrences — Master Theorem.

  • Degenerate partition: $T(n)=T(n-1)+Θ(n) ⇒ Θ(n^2)$ (worst-case quicksort).

Parameterized bounds

Express in the right variables:

  • Graphs: $Θ(n + m)$ for BFS/DFS where n=|V|, m=|E|.

  • Strings: search costs in both text length n and pattern length m.

  • Hash tables: expected $Θ(1)$ at low load factor; worst-case $Θ(n)$.

Note

The parameterization frequently matters more than the big-O class alone; $Θ(n + m)$ conveys structure that $Θ(n^2)$ would obscure.

Implementation or Practical Context

Time: constants that bite

Asymptotics hide constant factors, but implementations reveal:

  • Cache locality: Contiguous arrays beat pointer-chasing structures at the same big-O due to fewer cache misses.

  • Branching: Predictable branches (or branchless code) reduce misprediction penalties.

  • Vectorization: Turns many $Θ(n)$ scans into much faster $Θ(n)$ with smaller constants.

  • Parallelism: Analyze work (T₁) and span (T_\infty) for parallel algorithms; wall-clock time lower-bounded by span and divided by available cores.

Space: what actually fills memory

  • Recursion depth: contributes $Θ(\text{depth})$ stack frames (e.g., DFS).

  • Auxiliary structures: queues, heaps, hash tables; report peak occupancy.

  • Representation choices: 64-bit vs 32-bit indices, dense vs sparse matrices, struct padding, alignment.

Tip

When reporting space, separate categories: “$Θ(n)$ input, $Θ(n)$ output, $Θ(\log n)$ auxiliary (recursion stack).” This is far more actionable.

Model selection guidance

  • Use RAM for fixed-width integers and mainstream DS&A problems.

  • Switch to the bit model when values grow with n (big integers, exact arithmetic).

  • Use the I/O model for disks/SSDs or out-of-core analytics; the right algorithm (e.g., external mergesort) can change feasibility entirely.

Amortized analysis vs average-case

  • Amortized: worst-case over any operation sequence is small on average (dynamic arrays push_back $O(1)$ amortized; union–find $Θ(m α(n))$ for m ops). Distribution-free.

  • Average-case: expectation under a distribution of inputs (e.g., quicksort $Θ(n \log n)$ with random pivots). State assumptions.

See Dynamic Arrays and Disjoint Set Union — Union–Find.

Common Misunderstandings

Warning

“$O(\cdot)$ is an equality.” $O(n)$ includes all smaller classes; only $Θ(n)$ communicates tight growth.

Warning

Counting every statement. Avoid brittle micro-counts; model the dominant operations and simplify.

Warning

Ignoring input representation. If numbers are b bits, an “$O(1)$ add” in RAM becomes $Θ(b)$ in the bit model.

Warning

Conflating amortized and average-case. Amortized requires no input distribution; average-case does.

Warning

Not stating the case. Best/worst/average can differ drastically; say which you report.

Warning

Hiding parameters. Writing $Θ(n)$ when the real bound is $Θ(n + m)$ drops critical information.

Broader Implications

  • Scalability budgeting: If SLA requires 100 ms at n=10^6, $Θ(n^2)$ is off the table; $Θ(n \log n)$ or better is necessary.

  • Algorithm selection: The same task may need different algorithms by regime (e.g., insertion sort for tiny n, mergesort/quicksort for large n).

  • Data layout and hardware: Once asymptotics are acceptable, layout (AoS vs SoA), memory hierarchy, and parallel decomposition often provide larger real-world wins than further asymptotic improvements.

Summary

Time and space complexity abstract away machine idiosyncrasies to expose growth behavior. Pick a cost model (RAM/bit/I/O), express bounds with asymptotics, and account for multiple parameters when relevant. Use recurrences for divide-and-conquer, and amortized analysis for operation sequences. In practice, pair the theoretical bound with attention to constants, memory layout, and parallelism to achieve algorithms that scale on paper and in production.

See also

Graph View