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Analysis of algorithms is the systematic study of how an algorithm behaves as its input grows. It measures resources such as running time, memory, comparisons, disk operations, network messages, energy, or parallel execution depth under a stated computational model.
It is broader than simply assigning a Big O label. A meaningful analysis identifies the problem and algorithm, defines the input size, chooses the resource and cost model, states whether the result is a worst-case, average-case, expected, amortized, or empirical result, and establishes that the algorithm terminates and produces the required answer. As the NIST definition of complexity notes, a resource measure is meaningful only in relation to a computation model.
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What is an algorithm?
An algorithm is a clearly specified computational process or set of rules that produces a prescribed result. It describes the method, not necessarily a particular implementation.
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Several related ideas should be kept separate:
- Algorithm: the abstract procedure for solving a problem.
- Program: a concrete implementation of that procedure in a programming language.
- Specification: what the computation is required to accomplish, including its inputs, outputs, and constraints.
- Data structure: a representation that supports operations such as lookup, insertion, or deletion. Its operation costs become part of the algorithm’s analysis.
- Heuristic: a method that may work well in practice but does not necessarily guarantee an exact answer or an optimal solution.
- Correctness proof: the argument that the algorithm meets its specification.
Analysis asks how much a particular algorithm costs. Computational complexity theory asks how difficult a problem is and what resources the best possible algorithm could require. Performance engineering measures and improves a concrete implementation on particular hardware. These areas overlap, but they answer different questions.
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- INTRODUCTION TO ALGORITHMS, FOURTH EDITION
The five questions every analysis must answer
Before counting loops or writing O(n), specify these five things:
- What is the input? Define the problem and the valid input instances.
- How is input size measured? The symbol
nis a chosen parameter, not a universal definition of size. - What resource is being counted? This might be time, auxiliary memory, comparisons, I/O transfers, communication, energy, work, or parallel span.
- What computational model is assumed? State which operations are treated as constant time and what the machine can access directly.
- Which case or probability model is being analyzed? Say whether the result is worst-case, best-case, average-case, expected, high-probability, amortized, or measured experimentally.
The most important habit is to report the complete claim rather than an isolated label. Instead of saying an algorithm is O(n log n), say something like: this implementation performs Θ(n log n) comparisons in the worst case under the comparison model, uses Θ(n) auxiliary storage, and has an additional expected constant-factor cost from memory allocation.
Input size is not always the number of elements
Input size determines what a growth rate means. A good analysis names every parameter that materially affects the work.
| Input | Useful parameters | Why the distinction matters |
|---|---|---|
| Array of records | n records, or the total number of input bits |
Reading one record is not necessarily the same cost as reading a large record. |
| Graph | V vertices and E edges |
Θ(V + E) describes a sparse graph more accurately than a vague single n. |
| Matrix | m × n dimensions |
A rectangular matrix and a square matrix have different costs. |
| Text search | Text length n and pattern length m |
Searching a short pattern in a long text is a different parameter regime from searching two equally long strings. |
| Large integer | Bit length L, usually Θ(log N) for numerical value N |
Arithmetic on arbitrary-precision integers can take time proportional to the number of bits, not constant time. |
| Output-sensitive problem | Input size plus output size k |
An algorithm may correctly cost O(n + k) because writing the output itself takes time. |
| Parameterized problem | Main size n and parameter k |
A bound such as O(f(k) · poly(n)) may be practical when k is small even if the general problem is difficult. |
For example, an algorithm that scans every edge should normally be described in terms of V and E, not just n. Likewise, treating an arbitrary-precision multiplication as one constant-time operation can produce a misleading result. MIT’s discussions of the Python cost model and word-RAM assumptions illustrate why the representation of data matters.
Cost models and resources
There is no single universal meaning of time or space complexity. The appropriate resource depends on the application.
- Running time: the number of elementary operations or abstract steps.
- Space complexity: memory used while the algorithm runs.
- Auxiliary space: additional memory beyond the input, when that convention is stated.
- Stack space: memory used by recursion and nested function calls.
- I/O complexity: disk, SSD, or external-memory transfers. Moving a block from storage may dominate CPU instructions.
- Communication complexity: messages or bits transmitted between machines, processes, or devices.
- Comparison complexity: the number of comparisons, especially useful for sorting, selection, and searching lower bounds.
- Parallel work and span: total operations versus the length of the critical dependency path.
- Energy and hardware-aware cost: relevant to mobile devices, embedded systems, data centers, accelerators, and heterogeneous computers.
NIST lists comparisons, item moves, disk accesses, messages, memory, and elapsed time as possible cost measures. It also identifies cache-aware, streaming, distributed, energy-aware, heterogeneous, and flash-memory environments in which different algorithms and data structures may be appropriate. See Algorithms and Data Structures for New Models of Computation.
Common computational models
- Unit-cost RAM: basic word-sized operations, array indexing, and simple arithmetic are treated as constant time.
- Word-RAM: operations on machine words are constant time, with explicit assumptions about word size and supported bit operations.
- Comparison model: an algorithm learns ordering through comparisons and cannot exploit the numerical representation of keys.
- Bit-complexity model: the cost of arithmetic depends on operand length and the number of bits manipulated.
- External-memory model: computation distinguishes fast memory from slower storage and counts block transfers.
- Distributed or communication model: messages, latency, bandwidth, and synchronization are first-class costs.
When a textbook says array access or integer arithmetic costs Θ(1), that statement usually belongs to a particular RAM or word-RAM model. It is not a promise that every language, data type, hardware platform, or library operation has that cost.
Asymptotic notation: O, Ω, and Θ
Asymptotic notation describes how a cost function grows for sufficiently large inputs. It intentionally abstracts away constant factors and lower-order terms so that fundamentally different scaling behavior is visible.
Big O: an eventual upper bound
For nonnegative functions f(n) and g(n), f(n) = O(g(n)) if there are constants c > 0 and n₀ such that:
0 ≤ f(n) ≤ c g(n) for every n ≥ n₀.
In plain language, g(n) eventually grows at least fast enough, up to a constant factor, to bound f(n) from above. Big O is a set of upper bounds, not necessarily the tightest one. If an algorithm takes Θ(n) time, it is also O(n²), but O(n²) is a loose description.
Big Omega: an eventual lower bound
f(n) = Ω(g(n)) means that, for suitable positive constants and sufficiently large n, f(n) is at least a constant multiple of g(n). It describes a lower bound on the function being analyzed. It does not automatically mean best-case performance.
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Big Theta: a tight asymptotic bound
f(n) = Θ(g(n)) means both bounds hold:
Θ(g(n)) = O(g(n)) ∩ Ω(g(n)).
Use Θ when the analysis establishes the actual asymptotic order. For example, a loop that always performs exactly n(n + 1)/2 iterations has a Θ(n²) running time, not merely an O(n²) running time.
Little o, little omega, and asymptotic equivalence
f(n) = o(g(n))meansf(n) / g(n) → 0;fgrows strictly more slowly.f(n) = ω(g(n))meansf(n) / g(n) → ∞;fgrows strictly faster.f(n) ~ g(n)meansf(n) / g(n) → 1; the two functions are asymptotically equivalent.
Logarithm bases differ only by a constant factor, so log₂ n, ln n, and log₁₀ n are all Θ(log n) in asymptotic expressions. The formal definitions and notation conventions are summarized in MIT’s algorithm-analysis notes and the Princeton Algorithms cheat sheet.
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Asymptotic notation is not a case label
O, Ω, and Θ describe mathematical bounds. Best case, worst case, average case, expected case, and amortized case describe what function or sequence is being analyzed. Any of these functions can be expressed using asymptotic notation.
Thus, all of the following can be valid, depending on the algorithm:
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Tworst(n) = Θ(n²). - Best-case time:
Tbest(n) = Θ(n). - Expected time:
E[T(n)] = Θ(n log n). - Amortized cost per operation:
Θ(1).
Common orders of growth
| Growth | Typical examples | Important qualification |
|---|---|---|
Θ(1) |
Array access; fixed-size arithmetic | Constant only under an appropriate model and fixed-size representation. |
Θ(log n) |
Binary search; search in a balanced tree | Requires sorted data or a balanced structure and suitable access operations. |
Θ(n) |
Sequential scan; graph traversal with adjacency lists | Often optimal when every relevant input item must be inspected. |
Θ(n log n) |
Mergesort; heapsort; FFT | A common target for comparison sorting, but constants, memory, and assumptions still matter. |
Θ(n²) |
Enumerating all pairs; insertion sort in its worst case | Can be reasonable for small or nearly sorted inputs. |
Θ(n³) |
Floyd–Warshall; naïve matrix multiplication | Matrix dimensions and arithmetic cost must be included. |
Θ(nk) |
Many polynomial-time algorithms | Polynomial does not automatically mean practical; the degree and constants matter. |
Θ(cn) |
Exhaustive subset search; some backtracking algorithms | May be useful for small n or a small fixed parameter. |
Θ(n!) |
Enumerating permutations | Becomes infeasible very quickly as n grows. |
Useful counting identities include:
1 + 2 + ... + n = Θ(n²).1 + 1/2 + ... + 1/n = Θ(log n).- A loop that doubles a counter performs
Θ(log n)iterations.
These growth classes are a guide to scaling, not a direct conversion from complexity to seconds. A linear algorithm with expensive operations can lose to a quadratic algorithm on small inputs, and two Θ(n log n) implementations can differ substantially because of locality, allocation, or constant factors.
A repeatable workflow for analyzing an algorithm
- State the problem and algorithm. Describe the input, output, preconditions, and the procedure being analyzed.
- Define input parameters. Use parameters such as
n,m,V,E, bit length, output size, or a structural parameter. - Choose the cost model. State whether comparisons, array accesses, arithmetic, pointer traversals, or larger operations count as constant time.
- Identify a basic operation. Choose the operation that dominates the cost, such as a comparison, edge relaxation, or array write.
- Count its executions. Analyze loops, branches, recursive calls, data-structure operations, and repeated allocations.
- Combine costs correctly. Sequential blocks add; independent nested loops often multiply; branches use the relevant case or the more expensive branch; recursion produces a recurrence.
- State the case or probability model. Specify worst, best, average, expected, high-probability, amortized, or another model.
- Simplify to a tight bound. Drop constants and lower-order terms only after obtaining the cost expression.
- Analyze space separately. Include auxiliary arrays, object creation, buffers, and recursion depth according to a declared convention.
- Prove correctness and termination. A fast algorithm that returns the wrong answer is not a solution.
- State assumptions and edge cases. Include sortedness, key ranges, graph representation, nonnegative weights, duplicate handling, overflow, and invalid input behavior.
- Validate practical performance. Use representative benchmarks and profiling when the implementation’s behavior matters.
This workflow follows the approach described in Princeton’s introduction to algorithm analysis: identify basic-operation costs, model the input, count operation frequency, and aggregate the resulting costs.
How to analyze iterative code
A single linear loop
for i = 1 to n:
constant_time_operation()The operation runs exactly n times. Under a constant-time operation model, the cost is Θ(n).
Nested loops with independent ranges
for i = 1 to n:
for j = 1 to n:
constant_time_operation()The inner operation runs n × n times, giving Θ(n²).
A triangular loop
for i = 1 to n:
for j = i to n:
constant_time_operation()The inner loop runs n times when i = 1, then n − 1, and so on:
n + (n − 1) + ... + 1 = n(n + 1)/2 = Θ(n²).
The loops are not being treated as two unrelated full ranges. Their dependency changes the exact sum, even though the final asymptotic order remains quadratic.
Repeated halving or doubling
i = 1
while i < n:
i = 2 * iAfter k iterations, i = 2k. The loop stops when 2k ≥ n, so k = Θ(log n).
A logarithmic inner loop
for i = 1 to n:
j = 1
while j < n:
j = 2 * jThe outer loop runs n times and each inner loop runs Θ(log n) times, so the total is Θ(n log n).
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Sequential blocks
If an algorithm performs an Θ(n) scan followed by an Θ(n log n) sort, the total is:
Θ(n) + Θ(n log n) = Θ(n log n).
The larger asymptotic term dominates the sum. This does not mean the scan is free; it means it does not change the growth class.
Branches, early exits, and multiple parameters
Do not automatically multiply every loop bound. Check whether the inner loop depends on the outer variable, shrinks geometrically, terminates early, or executes differently for different inputs. For a branch, analyze the appropriate branch for best-case or input-specific behavior and the maximum branch cost for a worst-case guarantee.
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When dimensions are independent, keep them independent. For example, scanning every cell in an Recursive algorithms are usually analyzed by writing a recurrence for the cost on an input of size With substitution, first guess a bound, then prove it by induction. This method is flexible and works when a recurrence does not fit a standard theorem. A recursion tree expands the recurrence level by level and sums the work. It is especially useful for seeing whether the root, internal levels, or leaves dominate the total. The basic Master Theorem handles many recurrences of the form: Do these 3 things before closing this tab: where there are For the standard theorem, the last case requires a condition such as The best-case cost is the minimum cost over all valid inputs of size The worst-case cost is the maximum cost over all valid inputs of size Average-case analysis calculates expected cost under a stated probability distribution over inputs. Saying average case without specifying the distribution is incomplete. A distribution in which data is already nearly sorted can produce a very different result from a uniform random distribution. For a randomized algorithm, expected cost may be taken over the algorithm’s internal random choices for every fixed input. That differs from average-case analysis over randomly distributed inputs. For example, randomized quicksort can have expected A high-probability bound holds except with a specified small probability. A complete statement should identify the source of randomness, the probability, and the quantifiers. For example, it matters whether the claim holds for a fixed input with probability at least Amortized analysis gives a worst-case guarantee over a sequence of operations. It is not ordinary statistical averaging and does not require random inputs. A dynamic array illustrates the distinction. Most insertions cost constant time, but an insertion that fills the array may allocate a larger array and copy Worst-case analysis remains valuable, but it can be too pessimistic when pathological inputs are rare or structurally unrealistic. Other models include: These models are useful when an unconditional worst-case bound does not represent the application’s actual conditions. Background on beyond-worst-case and smoothed analysis is available from Beyond Worst-Case Analysis and Stanford’s smoothed-analysis lecture. Time and space should be analyzed separately. When reporting space, state whether the input and output are included. Recommended Free Tools Actual memory use can differ substantially from an abstract element count because of object headers, references, padding, allocator behavior, machine word size, and language runtime overhead. Princeton’s memory-analysis material explains these implementation-level effects. A complexity bound for an algorithm that does not solve the stated problem is meaningless. A complete analysis should answer three questions: Useful proof techniques include: A practical template is: state the returned result, prove termination, prove correctness, analyze time, analyze space, and document assumptions and failure conditions. MIT’s Mathematics for Computer Science materials cover invariants, induction, well-founded orderings, and recurrences as core algorithmic techniques. Do not confuse these four claims: A comparison sort learns about keys only through pairwise comparisons. Its execution can be represented as a decision tree. There are Free tools Windows power users keep installed One-click scans. No signup required. Therefore every comparison sort has a worst-case lower bound of Likewise, saying that a scan is optimal requires a reason that all relevant input must be read, or that the output itself is large enough to require the claimed work. Not every linear-time algorithm is automatically optimal. These are typical bounds, not unconditional promises. The relevant assumptions include input representation, implementation details, key distribution, stability requirements, preprocessing, and whether the bound is worst-case or expected. Princeton’s algorithm reference tables collect standard sorting, searching, priority-queue, symbol-table, and graph bounds. University algorithms curricula commonly connect asymptotic analysis with divide and conquer, dynamic programming, greedy methods, randomization, amortized analysis, graph algorithms, reductions, approximation, and intractability. Examples include the course materials from Stanford and Princeton. What’s actually slowing this PC down? Pick the symptom - the matching free tool is one click away. Theoretical and empirical analysis answer different questions. Theoretical analysis can compare algorithms independently of one particular computer, but it abstracts away constants, cache effects, compiler optimizations, memory allocation, operating-system behavior, and implementation overhead. A finite benchmark cannot establish a universal asymptotic bound. It supports a claim about the tested implementation, hardware, software environment, input distribution, and size range. OpenStax’s discussion of formal and experimental analysis distinguishes mathematical runtime reasoning from profiling and measurement. Princeton also recommends experiments while warning that an incorrect input model can make a mathematically sound analysis irrelevant to a particular application. Quick wins for a faster PC: Modern systems often make the simple sequential RAM model incomplete: The right algorithm for a disk-backed database, a sensor, a distributed service, and an in-memory desktop program may differ even when they solve the same abstract problem. NIST’s overview of new computational models describes why algorithms must account for these environments. Use this checklist when analyzing code for a class assignment, interview, design review, or production decision: No. Big O is an asymptotic upper-bound notation, not a complete runtime description. A useful claim must also identify the input-size definition, resource, computational model, and case being analyzed. An algorithm can have worst-case For sufficiently large inputs, the linear growth rate is asymptotically better. For real workloads, the answer can depend on constants, input sizes, preprocessing, cache locality, memory, and whether the linear method must repeatedly rescan data. Compare the complete workload rather than only the labels. No. An individual operation can cost Asymptotic notation hides constants and lower-order terms. A theoretically slower algorithm may use contiguous memory, fewer allocations, better cache locality, simpler operations, or a smaller input range. Benchmarking the correct implementations on representative data is necessary. State the problem, input parameters, cost model, dominant operation, operation count or recurrence, case or probability model, tight asymptotic time bound, space convention and bound, correctness argument, termination assumptions, and any important edge cases. Analysis of algorithms is the disciplined process of connecting an algorithm to its input model, resource measure, computational assumptions, and correctness guarantee. Big O is useful, but it is only one part of that process. The most reliable analysis states the parameters explicitly, counts the relevant operations, distinguishes worst-case from expected and amortized behavior, accounts for space and data movement, and proves correctness. A benchmark then answers the complementary question: how does this particular implementation behave on this hardware and workload? Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.m × n matrix costs Θ(mn), not necessarily Θ(n²). A graph algorithm that examines each vertex and edge costs Θ(V + E)Recurrences and recursive algorithms
n. Three standard techniques are substitution, recursion trees, and the Master Theorem.Substitution
Recursion trees
The Master Theorem
T(n) = aT(n/b) + f(n)a subproblems of size n/b and f(n) is the nonrecursive work.
f(n) is polynomially smaller than nlogba, the leaves dominate.f(n) has the same order as nlogba, an additional logarithmic factor appears.f(n) is polynomially larger, the nonrecursive work dominates, subject to the theorem’s regularity condition.a f(n/b) ≤ c f(n) for some constant c < 1. The theorem is powerful but does not cover every recurrence. Unequal subproblem sizes, overlapping subproblems, nonpolynomial toll terms, floors and ceilings, or unusual dependencies may require a recursion tree, substitution, Akra–Bazzi analysis, generating functions, or another method. See MIT’s recurrence and Master Theorem notes.Standard examples
Algorithm or recurrence
Result
Reason
Binary search:
T(n) = T(n/2) + Θ(1)Θ(log n)One subproblem remains, and its size halves.
Mergesort:
T(n) = 2T(n/2) + Θ(n)Θ(n log n)Each level processes
Θ(n)Θ(log n) levels.
Balanced divide-and-conquer traversal:
T(n) = 2T(n/2) + Θ(1)Θ(n)The number of leaves and total nodes is linear.
Naïve recursive Fibonacci:
T(n) = T(n − 1) + T(n − 2) + Θ(1)Exponential
Overlapping subproblems cause a rapidly branching recursion tree.
T(n) = T(n − 1) + Θ(n)Θ(n²)The costs accumulate as
n + (n − 1) + ....Best-case, worst-case, average-case, expected, and amortized analysis
Best case
n. It can explain favorable behavior, such as a sequential search finding its target at the first position, but it is often a weak guarantee because it may describe an unusually convenient input.Worst case
n. It gives a guarantee that applies to every input of that size, which is particularly important for security-sensitive services, real-time systems, and applications exposed to adversarial inputs.Crashes, No Sound, or Screen Glitches?
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Expected analysis of a randomized algorithm
Θ(n log n) comparisons because of its pivot choices, while a poor pivot sequence can still produce Θ(n²)
High-probability analysis
1 − 1/n², or for most inputs drawn from a distribution.Amortized analysis
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Θ(n) elements. Over a long sequence with geometric capacity growth, the total copying cost is linear in the number of insertions, so the insertion cost is Θ(1) amortized even though one individual operation costs Θ(n). MIT introduces this distinction in its amortized-analysis lecture. Common proof methods include aggregate analysis, the accounting method, and the potential method.Beyond worst-case analysis
O(n + k).Space complexity
Algorithm or implementation
Typical auxiliary-space result
Iterative array scan
Θ(1)
Recursive binary search
Θ(log n) stack space
Iterative binary search
Θ(1) auxiliary space
Top-down mergesort
Typically
Θ(n) auxiliary array space
Standard in-place heapsort
Θ(1) extra memory beyond the input
DFS with adjacency lists
Often
Θ(V) for visitation state and the stack, excluding the graph representationCorrectness and termination are part of algorithm analysis
Lower bounds and optimality
Why comparison sorting requires Ω(n log n) comparisons
n! possible orderings of n distinct elements, so the tree must have at least n! leaves. A binary tree of height h has at most 2h leaves, which implies:h ≥ log₂(n!) = Ω(n log n).Ω(n log n) comparisons. Mergesort and heapsort achieve O(n log n) worst-case comparison complexity, making them asymptotically optimal in that model. The qualification matters: the result does not apply to every possible sorting algorithm. Counting sort, radix sort, integer sorting, word-level operations, and bounded key ranges use assumptions outside the pure comparison model. See Stanford’s comparison-sorting lower-bound lecture.Representative algorithm and data-structure comparisons
Task
Method
Typical bound
Conditions and trade-offs
Search an unsorted array
Sequential search
O(n) worst caseRequires no preprocessing; can finish early when the target is near the front.
Search a sorted random-access array
Binary search
O(log n) worst caseRequires sorted data and efficient random access; sorting cost must be included if it is performed for one query.
Sort data
Insertion sort
Θ(n²) worst case; Θ(n) best case for suitable nearly sorted inputSimple and often excellent for small or nearly sorted arrays.
Sort data
Mergesort
Θ(n log n)Stable; commonly requires
Θ(n) auxiliary storage.
Sort data
Heapsort
Θ(n log n) worst caseIn-place in a standard implementation; not stable.
Sort data
Quicksort
Expected
Θ(n log n); worst case Θ(n²) for basic variantsOften good constants and locality; pivot strategy and defenses against adversarial input matter.
Priority queue
Binary heap
O(log n) insert and delete-min; O(1) minimum accessGood general-purpose priority queue; does not provide fully ordered traversal.
Ordered dictionary
Red-black or AVL tree
O(log n) operationsSupports ordered operations; structural and pointer overhead can be significant.
Dictionary lookup
Hash table
Expected
O(1) operations under suitable hashing assumptionsWorst-case operations may be
O(n); performance depends on load factor, collisions, and hash quality.
Graph traversal
BFS or DFS with adjacency lists
Θ(V + E)Assumes standard adjacency-list processing and includes visiting vertices and edges.
All-pairs shortest paths
Floyd–Warshall
Θ(V³)Dynamic programming approach often suited to dense graphs or compact matrix representations.
Nonnegative shortest paths
Dijkstra with a suitable priority queue
Commonly
O(E log V) with a binary heap implementationRequires nonnegative edge weights; exact expression depends on graph representation and priority queue.
How design paradigms change the analysis
Theory versus real-world performance
Best Value
What theoretical analysis tells you
What empirical analysis tells you
How to benchmark responsibly
Advanced models that change the answer
Common mistakes and how to correct them
Correction: Big O is an upper-bound notation; specify which case is being bounded.
Correction: Omega is a lower-bound notation, not a case label.O(n²) when a tight quadratic count was established.
Correction: Prefer Θ(n²) when both bounds are known.n for two independent dimensions.
Correction: Use m and n, or V and E, as appropriate.
Correction: Analyze arbitrary-precision arithmetic in a bit-complexity model when necessary.
Correction: Examine dependencies, geometric changes, early exits, and input-sensitive behavior.
Correction: Amortized analysis is a worst-case bound over an operation sequence and does not require random inputs.
Correction: State whether randomness comes from the input, the algorithm, or both.
Correction: State the expected bound, worst-case protection, constants, and implementation costs.
Correction: It applies to comparison sorting under the comparison model.
Correction: Consider degree, constants, input limits, memory, communication, and data movement.
Correction: Small inputs, small parameters, pruning, and specialized hardware can make some exponential methods useful.
Correction: Repeat controlled benchmarks across relevant input sizes and distributions.
Correction: Prove that the procedure solves the specified problem before relying on its resource bound.O(1) space without a convention.
Correction: State whether input, output, recursion stack, temporary buffers, and library allocations count.
Correction: Analyze the chosen implementation, language runtime, data type, and library guarantees.A practical checklist
Glossary
Frequently Asked Questions
Is Big O the same as an algorithm’s runtime?
O(n²) time and best-case Θ(n) time, for example.Which is better: an O(n) algorithm or an O(n log n) algorithm?
Does amortized O(1) mean every operation takes constant time?
Θ(n), such as a dynamic-array resize. Amortized O(1) means the total cost over a specified sequence is linear in the number of operations.Why can a theoretically slower algorithm be faster in practice?
What should I write in an algorithm-analysis answer?
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