Time and Space Complexity: A Practical Guide with Simple Examples

Time and Space Complexity Explained Simply with Examples
Software rarely fails because one line of code is slow. It fails because the same line runs too many times, stores too much data, or behaves badly when input size grows.

That is what time and space complexity help us understand.

They do not tell us the exact number of seconds a program will take. They tell us how the cost of an algorithm grows as the input becomes larger. This matters in almost every serious system: search, payments, recommendations, analytics, AI inference, fraud detection, logistics, and even a basic dashboard that starts with 1,000 records and later handles 10 million.

A solution that looks fine during testing can become expensive in production simply because the input size changed.

This article explains time and space complexity in a simple, practical way, with examples and enough context to make the concept useful beyond coding interviews.

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What Complexity Really Measures

Complexity measures the relationship between input size and resource usage.

There are two main questions:

1. Time complexity: How does the running time grow as input size increases?
2. Space complexity: How does memory usage grow as input size increases?

The key phrase is as input size increases.

If you search through 10 items, almost any approach works. If you search through 10 crore records, the difference between a good and poor approach becomes costly.

Complexity helps us compare approaches without depending too much on machine speed, programming language, cloud instance type, or temporary system load.

For example, assume two algorithms solve the same problem:

• Algorithm A checks every record one by one.
• Algorithm B uses an index and jumps closer to the answer.

On a small input, both may look fast. At scale, Algorithm B may be the only practical option.

That is the real value of complexity analysis. It helps us reason before production exposes the weakness.

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Big O Notation in Plain English

Big O notation describes the upper growth pattern of an algorithm.

You will usually see expressions like:

• O(1)
• O(log n)
• O(n)
• O(n log n)
• O(n²)
• O(2ⁿ)

Here, n means input size.

If you have a list of 1,000 customers, then n = 1,000.

If you have 1 million transactions, then n = 1,000,000.

Big O ignores minor details and focuses on the dominant growth behaviour. It does not care whether an operation takes 2 milliseconds or 5 milliseconds. It asks: what happens when the input becomes 10 times larger?

Why constants are usually ignored

Consider this code:

for item in items:
    print(item)

for item in items:
    print(item)

The list is looped through twice. Technically, the work is 2n.

In Big O terms, this is still:

O(n)

Why? Because as n grows, the important point is that the work grows linearly with input size. Whether it is 1n, 2n, or 5n matters in real performance tuning, but not in basic complexity classification.

That said, constants are not useless in real systems. A poorly written O(n) implementation can still be slower than a carefully written O(n log n) implementation for some input sizes. Complexity is a guide, not a replacement for measurement.

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O(1): Constant Time

An algorithm is O(1) when the time taken does not depend on input size.

Example:

def get_first_item(items):
    return items[0]

Whether the list has 10 items or 10 lakh items, this function accesses the first element once.

Another example:

user = users_by_id["U123"]

If users_by_id is a hash map or dictionary, looking up a user by ID is generally O(1) on average.

Where this matters

O(1) operations are common in systems that need fast lookups:

• fetching user details by ID
• checking if a session token exists
• reading a cached value
• accessing an array element by index

In AI systems, caches are often used to avoid repeated computation. For example, if a model response, embedding, or metadata lookup is cached against a key, fetching it can be close to O(1). The expensive work has already been done.

The trade-off is usually memory. Fast access often requires storing extra data.

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O(n): Linear Time

An algorithm is O(n) when work grows directly with input size.

Example:

def find_user(users, target_id):
    for user in users:
        if user["id"] == target_id:
            return user
    return None

In the worst case, the target user may be at the end of the list or not present at all. The function may need to check every user.

If there are 100 users, it may check 100.

If there are 10 lakh users, it may check 10 lakh.

That is O(n).

Practical interpretation

O(n) is not bad by itself. Many useful tasks require reading every item:

• calculating total sales
• validating all rows in a file
• scanning logs
• counting word frequency
• processing uploaded records

The issue begins when O(n) operations sit inside frequently used APIs or are repeated unnecessarily.

For example, scanning a list of users once in a background job may be acceptable. Scanning the same list for every request in a high-traffic service is a different problem.

AI context

Suppose you are preparing text chunks before sending them for embedding. If you have n documents and each document must be cleaned once, the preprocessing step is usually O(n), ignoring document length for the moment.

That is reasonable.

But if every new query forces the system to rescan all documents without an index, search quality may be fine during a demo and unacceptable in production.

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O(n²): Quadratic Time

An algorithm is O(n²) when every item is compared with every other item.

Example:

def find_duplicate_pairs(items):
    pairs = []

    for i in range(len(items)):
        for j in range(len(items)):
            if i != j and items[i] == items[j]:
                pairs.append((i, j))

    return pairs

There are two loops. For every item in the outer loop, the inner loop runs through all items again.

If n = 100, the rough number of comparisons is 10,000.

If n = 10,000, it becomes 100,000,000.

This is why nested loops deserve attention. They are not always wrong, but they can become expensive quickly.

Better approach for duplicates

Instead of comparing every item with every other item, we can use a set:

def has_duplicate(items):
    seen = set()

    for item in items:
        if item in seen:
            return True
        seen.add(item)

    return False

This is O(n) time on average because each item is checked once.

But it uses extra memory for the seen set, so space complexity becomes O(n).

This is a common engineering trade-off:

Use more memory to reduce time.

In many production systems, that is a good trade. In memory-constrained environments, it may not be.

AI context

Quadratic behaviour appears often in AI workloads.

One example is comparing each text chunk with every other chunk to find similarity. If you have 1,000 chunks, pairwise comparison means around 1 million comparisons. If you have 1 million chunks, direct pairwise comparison becomes impractical.

This is one reason vector databases, approximate nearest neighbour search, indexing structures, and batching strategies matter. They reduce the need for brute-force comparison.

Another well-known AI example is attention in transformer models. Standard self-attention has a cost that grows roughly with the square of sequence length. This is why very long context windows can become expensive. Longer input does not just mean “a bit more text”. It can mean significantly more compute and memory, depending on model architecture and implementation.

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O(log n): Logarithmic Time

O(log n) means the algorithm reduces the search space at each step, often by half.

The classic example is binary search.

Binary search works only on sorted data.

def binary_search(numbers, target):
    left = 0
    right = len(numbers) - 1

    while left <= right:
        mid = (left + right) // 2

        if numbers[mid] == target:
            return mid
        elif numbers[mid] < target:
            left = mid + 1
        else:
            right = mid - 1

    return -1

If the middle item is not the answer, half the list is eliminated.

For 1,000,000 sorted items, binary search needs around 20 checks, not 1,000,000.

That is the strength of O(log n).

The hidden condition

Binary search is fast because the data is sorted.

If the data is not sorted, you first need to sort it. Sorting has its own cost, usually O(n log n) for efficient comparison-based sorting algorithms.

So the full decision is not:

Binary search is faster than linear search.

The better decision is:

If the data is already sorted, or searched many times after sorting, binary search is useful.

If you sort once and search thousands of times, the cost makes sense.

If you sort once just to search once, a simple linear scan may be better.

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O(n log n): Common in Efficient Sorting

Many efficient sorting algorithms run in O(n log n) time.

Examples include:

• merge sort
• heap sort
• quicksort on average
• built-in sorting functions in many languages, depending on implementation

Example:

numbers = [5, 2, 9, 1, 3]
numbers.sort()

The exact implementation depends on the language, but efficient general-purpose sorting usually sits around O(n log n).

Why sorting matters so much

Sorting is not just a textbook operation. It appears in many real systems:

• ranking search results
• sorting transactions by date
• ordering jobs by priority
• preparing data for reports
• deduplicating records
• grouping and merging datasets

In data and AI pipelines, sorting may appear during preprocessing, batching, ranking, evaluation, and retrieval.

For example, a recommendation system may generate candidate products and then sort them by score. If the candidate list is large, sorting cost becomes visible. Often, teams do not need to sort everything. They only need the top 10, top 50, or top 100 results. In such cases, using a heap or top-k selection can be more efficient than sorting the entire list.

The practical question is:

Do we need a fully sorted list, or only the best few results?

That one question can reduce unnecessary computation.

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O(2ⁿ): Exponential Time

O(2ⁿ) algorithms grow extremely fast.

A common example is generating all subsets of a set.

def generate_subsets(items):
    result = [[]]

    for item in items:
        new_subsets = []
        for subset in result:
            new_subsets.append(subset + [item])
        result.extend(new_subsets)

    return result

For n items, there are 2ⁿ subsets.

If n = 10, there are 1,024 subsets.

If n = 30, there are over 1 billion subsets.

This type of growth becomes impractical quickly.

Where this appears

Exponential complexity often appears in problems involving:

• all combinations
• all possible paths
• brute-force password attempts
• exhaustive search
• certain optimisation problems
• naive recursive algorithms

In AI and optimisation contexts, many problems are too large for exhaustive search. This is why practical systems use heuristics, pruning, approximation, dynamic programming, sampling, or learned models.

For instance, route optimisation for deliveries cannot usually try every possible route when the number of locations is high. The number of combinations grows too quickly. A good-enough answer found within a practical time limit is often more valuable than a perfect answer that never arrives.

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Time Complexity Versus Actual Running Time

Time complexity and actual running time are related, but they are not the same.

Complexity tells you growth behaviour.

Actual running time depends on:

• hardware
• programming language
• compiler or interpreter
• network calls
• database performance
• disk I/O
• caching
• concurrency
• memory pressure
• data distribution
• implementation quality

A theoretically better algorithm can still perform poorly if implemented badly.

For example, an O(log n) database lookup may be slow if it involves network latency, lock contention, cold storage, or poor indexing. An O(n) scan over a small in-memory list may be faster.

This is why experienced engineers use complexity analysis and profiling together.

Complexity helps you avoid bad design.

Profiling tells you where time is actually going.

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Best Case, Average Case, and Worst Case

The same algorithm may behave differently depending on input.

Consider linear search:

def contains(items, target):
    for item in items:
        if item == target:
            return True
    return False

Best case

The target is the first item.

Only one check is needed.

O(1)

Worst case

The target is the last item or not present.

Every item is checked.

O(n)

Average case

On average, assuming the item is equally likely to appear anywhere, roughly half the list may be checked.

This is still:

O(n)

Big O usually focuses on worst-case or upper-bound behaviour because production systems must handle bad cases as well as normal ones.

In security, payments, AI safety checks, and large-scale APIs, worst-case behaviour matters. Attackers, malformed data, unusual prompts, and edge-case traffic patterns can push systems into expensive paths.

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Space Complexity: Measuring Memory Growth

Space complexity measures how much extra memory an algorithm needs as input size grows.

This includes things like:

• temporary arrays
• hash maps
• sets
• recursion stack
• caches
• copies of data
• intermediate results

Input storage itself may or may not be counted depending on the discussion. In most algorithm analysis, we focus on extra space used by the algorithm.

O(1) space

This function uses a fixed amount of extra memory:

def sum_numbers(numbers):
    total = 0

    for number in numbers:
        total += number

    return total

No matter how large the input list is, the function only keeps one extra variable: total.

So space complexity is:

O(1)

O(n) space

This function creates a new list:

def double_numbers(numbers):
    doubled = []

    for number in numbers:
        doubled.append(number * 2)

    return doubled

If the input has n numbers, the output list also has n numbers.

Space complexity is:

O(n)

Hidden space cost

Some operations look simple but create copies.

For example:

new_items = items[:]

This creates a copy of the list. If the list has n items, extra space is O(n).

Similarly, string operations can create new strings. In data-heavy systems, repeated copying can quietly increase memory usage and garbage collection overhead.

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Recursion and Space Complexity

Recursive functions use memory on the call stack.

Example:

def factorial(n):
    if n == 0:
        return 1
    return n * factorial(n - 1)

This function makes n recursive calls.

Time complexity:

O(n)

Space complexity:

O(n)

Why? Because each recursive call waits for the next call to finish. The call stack grows with n.

An iterative version uses constant extra space:

def factorial_iterative(n):
    result = 1

    for i in range(1, n + 1):
        result *= i

    return result

Time complexity is still O(n), but space complexity is O(1).

This distinction matters when n is large. Recursion may be elegant, but it can fail due to stack limits or memory pressure.

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Common Time Complexities at a Glance

Complexity	Name	Simple meaning	Example
O(1)	Constant	Same cost regardless of input size	Access array item by index
O(log n)	Logarithmic	Cuts search space repeatedly	Binary search
O(n)	Linear	Checks each item once	Loop through list
O(n log n)	Linearithmic	Common efficient sorting cost	Merge sort, many built-in sorts
O(n²)	Quadratic	Compares many pairs	Nested loop over same list
O(2ⁿ)	Exponential	Tries many combinations	Generate all subsets
O(n!)	Factorial	Tries all permutations	Brute-force travelling salesman

This table is useful, but do not memorise it mechanically. The better habit is to ask:

How many times does the main work happen as input grows?

That question usually reveals the complexity.

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How to Analyse Time Complexity Step by Step

Let us take a practical example.

def print_pairs(items):
    for i in range(len(items)):
        for j in range(len(items)):
            print(items[i], items[j])

Step 1: Identify input size.

n = len(items)

Step 2: Count the main repeated work.

The outer loop runs n times.

The inner loop runs n times for each outer loop.

Total operations:

n × n = n²

Time complexity:

O(n²)

Space complexity:

O(1)

The function does not store all pairs. It only prints them. If it stored them in a list, space complexity would become O(n²).

Now consider:

def print_each_list(a, b):
    for item in a:
        print(item)

    for item in b:
        print(item)

If a has length m and b has length n, complexity is:

O(m + n)

Do not automatically call everything O(n). When there are two independent inputs, name them separately.

Now consider:

def print_combinations(a, b):
    for item_a in a:
        for item_b in b:
            print(item_a, item_b)

If a has length m and b has length n:

O(m × n)

If both lists are roughly the same size, people may simplify this to O(n²). But in real systems, the distinction matters.

For example, if one list has 100 items and the other has 10 million, the behaviour is different from two lists of 10 million each.

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Time and Space Trade-offs

Many algorithm decisions are trade-offs between speed and memory.

Example: checking membership

Suppose you need to check whether a user ID exists.

Using a list:

def exists(user_ids, target):
    return target in user_ids

If user_ids is a list, membership check is O(n).

Using a set:

user_id_set = set(user_ids)

def exists_fast(user_id_set, target):
    return target in user_id_set

Set lookup is O(1) on average.

But the set takes extra memory.

If you check only once, creating the set may not be worth it.

If you check thousands of times, the set is usually worth it.

That is the decision logic:

One-time operation: simple scan may be fine.
Repeated lookup: build an index or set.

This thinking appears everywhere:

• database indexes
• search indexes
• vector indexes
• caches
• precomputed aggregates
• feature stores
• API response caches

You spend memory and storage to reduce repeated computation.

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Complexity in Databases and APIs

Complexity is not limited to code written inside a function.

A database query also has complexity.

This query may look harmless:

SELECT * FROM orders WHERE customer_id = 'C123';

If customer_id is indexed, the database can find matching rows quickly.

If it is not indexed, the database may scan the entire orders table.

For a table with 10,000 rows, this may not matter.

For a table with 100 million rows, it matters a lot.

The same idea applies to APIs.

An endpoint that loops through all records, filters them in application code, and returns the result may work during early development. Later, it becomes slow, expensive, and difficult to scale.

A better design pushes filtering, indexing, pagination, and aggregation closer to the data layer.

Complexity thinking helps you catch these problems before they become incidents.

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Complexity in AI Systems

AI systems make complexity more visible because they often combine large data, heavy computation, and memory-intensive models.

Here are a few places where time and space complexity matter.

1. Token length and inference cost

Large language models process tokens. A longer prompt usually means more computation and memory.

If you send a model a short question, inference is cheaper.

If you send a model a long document, chat history, retrieved context, and detailed instructions, cost increases.

This is why prompt design is not only a quality concern. It is also a performance and cost concern.

In production, teams often need to decide:

• How much chat history should be retained?
• How many retrieved chunks should be added?
• Should old context be summarised?
• Should large documents be processed in parts?
• What is the acceptable latency?

The answer is not always “send more context”. More context can improve quality, but it can also increase latency, cost, and risk of irrelevant information affecting the response.

2. Embedding search

In retrieval-augmented generation systems, documents are converted into embeddings and searched by similarity.

A naive approach compares the query vector with every document vector.

If there are n vectors, that is O(n) per query.

For a small knowledge base, this may be fine.

For millions of vectors, brute-force search can become expensive. Vector databases and approximate nearest neighbour algorithms reduce search time by using specialised indexes.

The trade-off is that approximate search may not always return the mathematically exact nearest vector. In many applications, that trade-off is acceptable because the result is close enough and much faster.

3. Training data pipelines

Before model training or fine-tuning, data often goes through cleaning, deduplication, filtering, tokenisation, validation, and transformation.

Some steps are linear. Others may become expensive.

Deduplication, for example, can be simple if exact matches are enough. Use hashes and sets.

But near-duplicate detection is harder. Comparing every document with every other document can become O(n²), which is not practical for large datasets.

Teams then use hashing techniques, clustering, sampling, or approximate matching to reduce cost.

4. Batch size and memory

In AI training and inference, larger batches can improve hardware utilisation, but they also use more memory.

If the batch is too small, the GPU may be underused.

If the batch is too large, the process may run out of memory.

This is a space complexity problem in practice. The model, activations, gradients, input tensors, cache, and intermediate outputs all occupy memory.

The best setting depends on model size, sequence length, hardware, precision, and workload pattern.

5. Caching AI responses

Caching can reduce repeated model calls.

For example, if many users ask the same standard policy question, the system may serve a cached answer instead of calling the model every time.

This improves time and cost for repeated queries.

But caching has constraints:

• answers may become stale
• user-specific context may make caching unsafe
• sensitive data must not leak across users
• cache keys must be designed carefully
• storage grows over time

Again, complexity thinking is not only academic. It affects architecture.

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Practical Rules for Estimating Complexity

You do not need to be a mathematician to analyse most code. A few habits are enough.

1. Look for loops

One loop over n items is usually O(n).

Two separate loops over the same input are still O(n).

for item in items:
    do_something(item)

for item in items:
    do_something_else(item)

This is O(n), not O(2n) in Big O terms.

2. Look for nested loops

A loop inside another loop often means multiplication.

for a in items:
    for b in items:
        compare(a, b)

This is usually O(n²).

If the loops use different inputs, it may be O(m × n).

3. Look for repeated division

If the algorithm halves the input each time, think O(log n).

Binary search is the standard example.

4. Look for sorting

Sorting is commonly O(n log n).

If your algorithm sorts first and then loops once, the total is:

O(n log n + n)

This simplifies to:

O(n log n)

The sorting dominates for large n.

5. Look for extra data structures

If you create a list, set, map, queue, stack, or cache that grows with input size, space complexity is likely O(n).

6. Watch recursion

Recursive calls add stack space.

If recursion depth grows with n, stack space may also be O(n).

7. Consider hidden work

A single line may hide a loop.

Examples:

sorted(items)
item in list_items
items.copy()
text.split()
",".join(words)

These operations are useful, but they are not free.

In code reviews, it is worth asking what a library call does internally when it appears inside a loop or on a large input.

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A Worked Example: Improving a Slow Function

Suppose we have two lists:

registered_users = ["A1", "B2", "C3", "D4"]
active_users = ["B2", "D4"]

We want to find registered users who are active.

A simple approach:

def find_active_registered_users(registered_users, active_users):
    result = []

    for user in registered_users:
        if user in active_users:
            result.append(user)

    return result

At first glance, this looks fine.

But user in active_users scans the active users list.

If registered_users has m items and active_users has n items, time complexity is:

O(m × n)

For small lists, no issue.

For large lists, this becomes slow.

A better approach:

def find_active_registered_users(registered_users, active_users):
    active_set = set(active_users)
    result = []

    for user in registered_users:
        if user in active_set:
            result.append(user)

    return result

Now:

• creating the set takes O(n)
• looping over registered users takes O(m)
• lookup in the set is O(1) on average

Total time complexity:

O(m + n)

Space complexity:

O(n)

We improved time by using extra memory.

This is a good trade if the lists are large or the lookup happens often.

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When Complexity Is Not the Biggest Problem

Complexity matters, but it is not the only thing that matters.

In real systems, the slowest part may be:

• a remote API call
• a database lock
• a missing index
• serial processing where parallelism is possible
• large object serialisation
• slow disk reads
• excessive logging
• poor cache behaviour
• oversized payloads
• network latency

An O(n) algorithm that makes a network call inside the loop may be far worse than an O(n²) in-memory operation on a tiny input.

Example:

for user in users:
    profile = call_external_api(user)

If there are 10,000 users, this may make 10,000 network calls.

The complexity may be O(n), but the actual system behaviour can still be unacceptable.

So complexity analysis should be paired with operational judgement.

Ask:

• Is the work CPU-bound, memory-bound, network-bound, or database-bound?
• Does it run once a day or on every request?
• What is the expected input size now and in one year?
• Can the work be indexed, cached, batched, streamed, or moved offline?
• What happens during peak load?

These questions matter more than naming the Big O correctly in isolation.

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Complexity and Clean System Design

Good complexity decisions often show up as simple architecture choices.

Index instead of scan

If a lookup happens repeatedly, build an index.

Examples:

• database index
• dictionary by ID
• search index
• vector index
• feature lookup table

Stream instead of load everything

If data is too large for memory, process it in chunks.

This changes the memory profile even when time complexity remains similar.

For example, reading a 20 GB file fully into memory may fail. Reading it line by line may work.

Precompute when reads are frequent

If a value is expensive to calculate and read often, precompute it.

Examples:

• daily sales totals
• recommendation candidates
• user risk scores
• report aggregates
• embeddings for documents

The trade-off is freshness. Precomputed data may lag behind real-time data.

Avoid repeated work

Repeated work is one of the most common causes of poor performance.

If the same transformation, query, or model call happens again and again, consider caching, memoisation, indexing, or restructuring the flow.

Measure before over-optimising

Do not complicate code only because a faster theoretical option exists.

A simple O(n) solution is often the right choice when:

• n is small
• the code runs rarely
• clarity matters more than speed
• the bottleneck is elsewhere
• the faster solution adds maintenance risk

Good engineering is not about choosing the most complex algorithm. It is about choosing the simplest approach that meets the requirement with enough headroom.

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Final Takeaway

Time complexity tells you how running time grows.

Space complexity tells you how memory usage grows.

Big O notation gives you a shared language to compare approaches, but it does not replace profiling, system knowledge, or production judgement.

For most practical work, the habit is simple:

• check how many times the main operation runs
• look for nested loops and hidden scans
• understand what extra memory is being used
• notice when indexing, caching, batching, or streaming would help
• consider input growth, not just today’s test data

In AI systems, this thinking becomes even more important because cost often grows with tokens, vectors, documents, batches, model size, and repeated inference calls.

A solution is not efficient because it works on a sample. It is efficient when it continues to behave well as the input, traffic, and business usage grow.