Array Traversal Patterns - Learning Module

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Traversal with Early Termination

The Power of Knowing When to Stop

In the previous page, we explored linear traversal—visiting every element from start to finish. But many real-world problems don't require examining all elements. Sometimes, finding one specific element is enough. Sometimes, a single counterexample proves a condition false. Sometimes, you've gathered enough information to make a decision.

Early termination is the technique of exiting a traversal before reaching the natural end. When applied correctly, it transforms O(n) worst-case algorithms into O(1) best-case algorithms, dramatically improving average performance in practice.

This isn't just an optimization—it's a fundamental algorithmic pattern that appears in search algorithms, validation routines, short-circuit evaluation, and countless other contexts. Mastering early termination means writing code that's not just correct, but intelligently efficient.

What You Will Learn

By the end of this page, you will understand when and how to apply early termination, recognize the patterns that enable short-circuit behavior, avoid common pitfalls that break early exit logic, and design traversals that gracefully handle both early-exit and full-traversal cases.

The Case for Early Termination

Consider a simple question: Is the number 42 present in this array?

With a full traversal approach, you'd check every element, keeping track of whether you found 42:

found = false
for each element in array:
    if element == 42:
        found = true
return found

This works, but it's inefficient. If the array has 1 million elements and 42 is at index 5, you still check 999,995 more elements for no reason.

With early termination:

for each element in array:
    if element == 42:
        return true  // Stop immediately!
return false

Now, finding 42 at index 5 means only 6 elements are examined. The algorithm is still O(n) in the worst case (42 might be last or absent), but the average case improves dramatically when the target is often found early.

Best, Average, and Worst Case

Early termination doesn't change worst-case complexity—if the target isn't present, you must check all elements. But it dramatically changes best-case (O(1) if target is first) and average-case performance. In practice, average case often determines real-world performance.

When Early Termination Applies:

Early termination is applicable when:

The answer is determined before the end — A single match, mismatch, or threshold crossing decides the outcome
Further processing is unnecessary — Once you know the answer, additional work adds no value
The goal is existence, not enumeration — You need whether something exists, not all instances
Short-circuit logic is valid — Boolean any (true if one matches) or all (false if one fails) semantics apply

When Early Termination Doesn't Apply:

Computing aggregates (sum, average) that require all elements
Collecting all matches, not just the first
Problems where the answer depends on global properties
When you need to process every element regardless of intermediate results

Mechanics of Early Exit

In most programming languages, early termination is achieved through the return statement (exiting the function) or the break statement (exiting only the loop). Understanding when to use each is crucial for correct code.

Return vs. Break:

Mechanism	Scope	Use When
`return`	Exits entire function	The function's purpose is answered
`break`	Exits only the loop	More code must run after the loop

In most cases, if the loop is the main purpose of the function, return is cleaner and more direct. If you need to perform cleanup or processing after finding a result, break is appropriate.

early_termination_mechanics.py
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# Early Termination Mechanics
# ============================
 
# Pattern 1: Return immediately (clean, preferred when possible)
def find_first_negative_return(arr):
    """
    Return the first negative number, or None if none exist.
    Using 'return' exits the function immediately.
    """
    for x in arr:
        if x < 0:
            return x  # Found it, we're done
    return None  # Traversed all, none found
 
 
# Pattern 2: Break and then process (when post-loop work needed)
def find_first_negative_break(arr):
    """
    Same goal, but demonstrating when 'break' is useful.
    """
    result = None
    for x in arr:
        if x < 0:
            result = x
            break  # Exit loop, but stay in function
    
    # Post-loop processing (maybe logging, cleanup, etc.)
    if result is not None:
        print(f"Found negative number: {result}")
    else:
        print("No negative numbers found")
    
    return result
 
 
# Pattern 3: Using a flag variable (avoid if possible)
def find_first_negative_flag(arr):
    """
    Using a flag variable - works but less elegant.
    Shown for completeness; prefer return/break patterns.
    """
    found = False
    result = None
    
    for x in arr:
        if not found and x < 0:  # Only process if not yet found
            found = True
            result = x
        # Loop continues even after finding - inefficient!
    
    return result
 
 
# COMPARISON: Same array, different approaches
test_array = [5, 3, 7, -2, 8, -5, 1]
 
# With return: Examines [5, 3, 7, -2], then exits
# With break: Examines [5, 3, 7, -2], then exits loop
# With flag: Examines ALL elements - wasteful!
 
print("Return:", find_first_negative_return(test_array))
print("Break:", find_first_negative_break(test_array))
print("Flag:", find_first_negative_flag(test_array))

Avoid Flag-Based Pseudo-Termination

A common anti-pattern is setting a flag (like found = True) but continuing the loop. This gives the appearance of early termination but doesn't actually stop iteration. The loop keeps running, checking the flag on each iteration. Use return or break for true early termination.

Search Patterns with Early Exit

Search is the canonical application of early termination. Once you find what you're looking for, there's no reason to continue. Let's explore the full spectrum of search patterns.

search_patterns.py
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# Complete Search Pattern Catalog
# ================================
 
# 1. SIMPLE SEARCH: Find index of target value
def linear_search(arr, target):
    """
    Return index of first occurrence, or -1 if not found.
    
    Best: O(1) - target at index 0
    Worst: O(n) - target at last index or not present
    Average: O(n/2) = O(n) - target at middle on average
    """
    for i in range(len(arr)):
        if arr[i] == target:
            return i  # Early termination on match
    return -1  # Sentinel value indicating "not found"
 
 
# 2. EXISTENCE CHECK: Just need to know if present
def contains(arr, target):
    """
    Return True if target exists, False otherwise.
    Simpler than search when index isn't needed.
    """
    for x in arr:
        if x == target:
            return True  # Found one, that's enough
    return False
 
 
# 3. CONDITIONAL SEARCH: Find first element matching predicate
def find_first(arr, predicate):
    """
    Return first element satisfying predicate, or None.
    
    Example: find_first(arr, lambda x: x > 10) finds first element > 10
    """
    for x in arr:
        if predicate(x):
            return x
    return None
 
 
# 4. CONDITIONAL INDEX SEARCH: Need both element and position
def find_first_index(arr, predicate):
    """
    Return index of first element satisfying predicate, or -1.
    """
    for i in range(len(arr)):
        if predicate(arr[i]):
            return i
    return -1
 
 
# 5. GUARDED SEARCH: Search with boundary conditions
def find_in_range(arr, target, start, end):
    """
    Search only within [start, end) range.
    Useful when you know the target's likely location.
    """
    for i in range(start, min(end, len(arr))):
        if arr[i] == target:
            return i
    return -1
 
 
# 6. MULTI-TARGET SEARCH: Find any of several targets
def find_any_of(arr, targets):
    """
    Return first element that matches ANY target in the set.
    Uses a set for O(1) membership testing.
    """
    target_set = set(targets)  # O(k) preprocessing, k = len(targets)
    for x in arr:
        if x in target_set:  # O(1) lookup
            return x
    return None
 
 
# Demonstration
if __name__ == "__main__":
    data = [4, 8, 15, 16, 23, 42]
    
    print(f"Index of 16: {linear_search(data, 16)}")      # 3
    print(f"Contains 42: {contains(data, 42)}")           # True
    print(f"First > 10: {find_first(data, lambda x: x > 10)}")  # 15
    print(f"Any of [1, 2, 42]: {find_any_of(data, [1, 2, 42])}")  # 42

Choosing the Right Search Pattern

Use contains() when you only need a boolean answer. Use linear_search() when you need the index. Use find_first() with a predicate when the condition is more complex than equality. Match your pattern to your needs—no more, no less.

Boolean Short-Circuit Patterns: Any and All

Two of the most important early termination patterns mirror boolean logic: any (exists) and all (universal). These patterns leverage the mathematical properties of OR and AND operations.

Any (Existential Quantifier):

Returns True if at least one element satisfies a condition
Short-circuits on first True (once one matches, the answer is True)
Only returns False after examining all elements (none matched)

All (Universal Quantifier):

Returns True if every element satisfies a condition
Short-circuits on first False (one counterexample disproves "all")
Only returns True after examining all elements (all matched)

boolean_patterns.py
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# Boolean Short-Circuit Patterns
# ===============================
 
def any_match(arr, predicate):
    """
    Return True if ANY element satisfies the predicate.
    
    Logic: True OR x = True (regardless of x)
    Once we find one True, the result is True.
    
    Best: O(1) - first element matches
    Worst: O(n) - no elements match
    """
    for x in arr:
        if predicate(x):
            return True  # Short-circuit: found one
    return False  # Checked all, none matched
 
 
def all_match(arr, predicate):
    """
    Return True if ALL elements satisfy the predicate.
    
    Logic: False AND x = False (regardless of x)
    Once we find one False, the result is False.
    
    Best: O(1) - first element fails
    Worst: O(n) - all elements pass
    """
    for x in arr:
        if not predicate(x):
            return False  # Short-circuit: found counterexample
    return True  # Checked all, all passed
 
 
def none_match(arr, predicate):
    """
    Return True if NO element satisfies the predicate.
    Logically equivalent to: not any_match(arr, predicate)
    """
    for x in arr:
        if predicate(x):
            return False  # Found one that matches
    return True  # None matched
 
 
# Practical applications
def is_sorted_ascending(arr):
    """Check if array is sorted using all_match pattern."""
    if len(arr) <= 1:
        return True
    
    # All consecutive pairs must satisfy: arr[i] <= arr[i+1]
    for i in range(len(arr) - 1):
        if arr[i] > arr[i + 1]:
            return False  # Early exit on first violation
    return True
 
 
def has_duplicate(arr):
    """Check if array contains any duplicate using any_match logic."""
    seen = set()
    for x in arr:
        if x in seen:
            return True  # Found duplicate, short-circuit
        seen.add(x)
    return False  # All unique
 
 
def all_positive(arr):
    """Practical example: validate all elements are positive."""
    return all_match(arr, lambda x: x > 0)
 
 
def any_negative(arr):
    """Practical example: check for any negative values."""
    return any_match(arr, lambda x: x < 0)
 
 
# Demonstrations
if __name__ == "__main__":
    data1 = [1, 2, 3, 4, 5]
    data2 = [1, -2, 3, 4, 5]
    data3 = [1, 2, 2, 4, 5]
    
    print(f"All positive in {data1}: {all_positive(data1)}")   # True
    print(f"All positive in {data2}: {all_positive(data2)}")   # False (exits at -2)
    
    print(f"Any negative in {data1}: {any_negative(data1)}")   # False
    print(f"Any negative in {data2}: {any_negative(data2)}")   # True (exits at -2)
    
    print(f"{[1,2,3,4,5]} is sorted: {is_sorted_ascending([1,2,3,4,5])}")  # True
    print(f"{[1,3,2,4,5]} is sorted: {is_sorted_ascending([1,3,2,4,5])}")  # False
    
    print(f"{data1} has duplicates: {has_duplicate(data1)}")   # False
    print(f"{data3} has duplicates: {has_duplicate(data3)}")   # True

The Duality of Any and All

Mathematically, not any(predicate) equals all(not predicate), and not all(predicate) equals any(not predicate). This duality helps you choose the cleaner expression. "No element is negative" can be written as none_match(negative) or all_match(non_negative).

Threshold and Limit Patterns

Beyond boolean conditions, early termination applies whenever a computation reaches a decisive threshold. Once you know enough, continuing adds no value—and may waste significant resources.

threshold_patterns.py
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# Threshold and Limit Patterns
# ============================
 
def count_up_to(arr, predicate, limit):
    """
    Count elements matching predicate, but stop at limit.
    
    Use case: "Are there at least 3 errors?" Stop after finding 3.
    """
    count = 0
    for x in arr:
        if predicate(x):
            count += 1
            if count >= limit:
                return count  # Reached limit, no need to continue
    return count  # Didn't reach limit
 
 
def sum_until_threshold(arr, threshold):
    """
    Sum elements until the sum exceeds threshold.
    
    Use case: Stop processing when budget is exhausted.
    """
    total = 0
    for i, x in enumerate(arr):
        total += x
        if total > threshold:
            return total, i + 1  # Return sum and elements used
    return total, len(arr)
 
 
def find_first_n(arr, predicate, n):
    """
    Find the first n elements matching predicate.
    
    Use case: "Get the first 5 matching results"
    """
    results = []
    for x in arr:
        if predicate(x):
            results.append(x)
            if len(results) >= n:
                return results  # Got enough, stop searching
    return results  # Return what we found (may be fewer than n)
 
 
def detect_pattern_early(arr, pattern):
    """
    Check if array starts with a specific pattern.
    
    No need to examine the entire array if we only care
    about the beginning.
    """
    if len(pattern) > len(arr):
        return False
    
    for i in range(len(pattern)):
        if arr[i] != pattern[i]:
            return False  # Mismatch, pattern doesn't match
    return True  # Pattern fully matched
 
 
def check_k_consecutive(arr, predicate, k):
    """
    Return True if k consecutive elements satisfy predicate.
    
    Use case: "Were there 3 consecutive failures?"
    """
    consecutive_count = 0
    for x in arr:
        if predicate(x):
            consecutive_count += 1
            if consecutive_count >= k:
                return True  # Found k consecutive
        else:
            consecutive_count = 0  # Reset counter
    return False
 
 
# Demonstrations
if __name__ == "__main__":
    numbers = [3, 7, 2, 9, 1, 8, 4, 6, 5, 10]
    
    # Count at most 3 numbers > 5
    result = count_up_to(numbers, lambda x: x > 5, 3)
    print(f"At least 3 numbers > 5 in {numbers}: {result >= 3}")
    
    # Sum until exceeding 20
    total, used = sum_until_threshold(numbers, 20)
    print(f"Sum reached {total} after {used} elements")
    
    # Find first 2 even numbers
    evens = find_first_n(numbers, lambda x: x % 2 == 0, 2)
    print(f"First 2 even numbers: {evens}")
    
    # Check pattern
    print(f"Starts with [3, 7, 2]: {detect_pattern_early(numbers, [3, 7, 2])}")
    print(f"Starts with [3, 7, 9]: {detect_pattern_early(numbers, [3, 7, 9])}")

Business Logic Thresholds

In real systems, thresholds often come from business requirements: "Validate the first 100 records", "Process orders until daily limit reached", "Find 10 recommendations". Recognizing these as early termination opportunities avoids wasting computation.

Validation with Fail-Fast

Fail-fast is an engineering principle: when something goes wrong, detect and report it as early as possible. In validation contexts, this means stopping at the first error rather than collecting all errors.

When to use fail-fast:

Performance is critical and errors are rare
A single error invalidates the entire input
Quick feedback is more valuable than comprehensive feedback

When to collect all errors:

User feedback (show all form errors at once)
Debugging (find the full scope of problems)
When errors might be related and you need the whole picture

fail_fast_validation.py
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# Fail-Fast Validation Patterns
# ==============================
 
def validate_fail_fast(items, validators):
    """
    Apply multiple validators; stop at first failure.
    
    Returns: (True, None) if all pass, (False, error_message) on first failure
    """
    for item in items:
        for validator in validators:
            is_valid, error = validator(item)
            if not is_valid:
                return False, f"Validation failed for {item}: {error}"
    return True, None
 
 
def validate_collect_all(items, validators):
    """
    Apply all validators; collect all failures.
    
    Returns: list of all error messages (empty if all pass)
    """
    errors = []
    for item in items:
        for validator in validators:
            is_valid, error = validator(item)
            if not is_valid:
                errors.append(f"{item}: {error}")
    return errors
 
 
# Example validators for a list of ages
def is_positive(x):
    if x > 0:
        return True, None
    return False, "must be positive"
 
def is_under_limit(x, limit=150):
    if x < limit:
        return True, None
    return False, f"must be less than {limit}"
 
def is_integer_like(x):
    if isinstance(x, int) or (isinstance(x, float) and x.is_integer()):
        return True, None
    return False, "must be a whole number"
 
 
# Real-world example: API request validation
def validate_request(request):
    """
    Validate an API request - fail fast on first issue.
    """
    # Check required fields
    required_fields = ['user_id', 'action', 'timestamp']
    for field in required_fields:
        if field not in request:
            return False, f"Missing required field: {field}"
    
    # Validate user_id format
    if not isinstance(request['user_id'], str):
        return False, "user_id must be a string"
    if len(request['user_id']) < 5:
        return False, "user_id must be at least 5 characters"
    
    # Validate action
    valid_actions = ['create', 'read', 'update', 'delete']
    if request['action'] not in valid_actions:
        return False, f"Invalid action. Must be one of: {valid_actions}"
    
    # Validate timestamp
    if not isinstance(request['timestamp'], (int, float)):
        return False, "timestamp must be a number"
    if request['timestamp'] < 0:
        return False, "timestamp must be non-negative"
    
    return True, None  # All validations passed
 
 
# Guard pattern: validate preconditions before processing
def process_data(data):
    """
    Demonstrates guard clauses - early returns for invalid input.
    """
    # Guard: Check for None
    if data is None:
        raise ValueError("Data cannot be None")
    
    # Guard: Check for empty
    if len(data) == 0:
        return []  # Valid edge case, return early
    
    # Guard: Check type
    if not isinstance(data, list):
        raise TypeError(f"Expected list, got {type(data).__name__}")
    
    # Now safe to process
    return [x * 2 for x in data]
 
 
# Demonstration
if __name__ == "__main__":
    ages = [25, 30, -5, 40, 200, 35]
    validators = [is_positive, lambda x: is_under_limit(x, 150)]
    
    # Fail-fast: stops at first error
    print("Fail-fast validation:")
    success, error = validate_fail_fast(ages, validators)
    print(f"  Success: {success}, Error: {error}")
    
    # Collect all: finds all errors
    print("\nCollect all errors:")
    all_errors = validate_collect_all(ages, validators)
    for e in all_errors:
        print(f"  - {e}")
    
    # API request validation
    print("\nAPI request validation:")
    good_request = {'user_id': 'user123', 'action': 'read', 'timestamp': 1234567890}
    bad_request = {'user_id': 'u1', 'action': 'destroy'}
    
    print(f"Good request: {validate_request(good_request)}")
    print(f"Bad request: {validate_request(bad_request)}")

Guard Clauses

Guard clauses are a specific fail-fast pattern: put validation checks at the beginning of a function and return/throw immediately if preconditions aren't met. This keeps the main logic at a single indentation level, improving readability while ensuring invalid inputs are rejected early.

Common Pitfalls in Early Termination

Early termination, while powerful, introduces opportunities for subtle bugs. Being aware of common pitfalls helps you write correct code the first time.

Pitfalls to Avoid

•Forgetting the fallback return — If the loop exits normally (no early return), what should the function return? Missing fallback returns cause runtime errors or return the wrong default.
•Wrong return value type — Returning -1, None, or False as sentinels can be confused with valid data. Use types or exceptions to disambiguate when possible.
•Premature termination — Exiting too early can miss the correct answer. Ensure your condition truly indicates that further iteration is unnecessary.
•Breaking instead of returning — Using break when return is appropriate leaves you with extra code and potential for bugs after the loop.
•Side effects after early exit — If your loop has cleanup code at the end, early returns skip it. Use try/finally or restructure to ensure cleanup runs.
•Empty array edge case — An empty array means the loop body never executes. Ensure your fallback return handles this correctly.

pitfall_examples.py
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# Common Pitfalls Illustrated
# ===========================
 
# PITFALL 1: Missing fallback return
def find_even_BUGGY(arr):
    for x in arr:
        if x % 2 == 0:
            return x
    # BUG: No return statement if no even numbers!
    # Function implicitly returns None, which might be confused
    # with "found None as an even number"
 
def find_even_FIXED(arr):
    for x in arr:
        if x % 2 == 0:
            return x
    return None  # Explicit: no match found
 
 
# PITFALL 2: Sentinel value ambiguity  
def find_index_AMBIGUOUS(arr, target):
    for i in range(len(arr)):
        if arr[i] == target:
            return i
    return -1  # What if valid indices are negative? (rare but possible)
 
def find_index_BETTER(arr, target):
    for i in range(len(arr)):
        if arr[i] == target:
            return i
    raise ValueError(f"{target} not found in array")
 
# Or use Optional type in typed languages:
# def find_index(arr, target) -> Optional[int]:
 
 
# PITFALL 3: Premature termination
def has_two_evens_BUGGY(arr):
    """Check if array has at least two even numbers."""
    count = 0
    for x in arr:
        if x % 2 == 0:
            count += 1
            return True  # BUG: Returns True on FIRST even!
    return False
 
def has_two_evens_FIXED(arr):
    """Correct: wait until we find TWO evens."""
    count = 0
    for x in arr:
        if x % 2 == 0:
            count += 1
            if count >= 2:  # Correct condition
                return True
    return False
 
 
# PITFALL 4: Break when return was intended
def get_first_positive_AWKWARD(arr):
    result = None
    for x in arr:
        if x > 0:
            result = x
            break
    return result
 
def get_first_positive_CLEAN(arr):
    for x in arr:
        if x > 0:
            return x
    return None
 
 
# PITFALL 5: Side effects skipped by early return
import logging
 
def process_with_cleanup_BUGGY(items, log):
    for item in items:
        if item == "ERROR":
            return None  # BUG: Logging and cleanup skipped!
        process(item)
    
    log.info("Finished processing all items")  # Skipped on early return
    cleanup()
 
def process_with_cleanup_FIXED(items, log):
    result = None
    try:
        for item in items:
            if item == "ERROR":
                result = None
                return result
            process(item)
        result = "success"
        return result
    finally:
        # Always runs, even on early return
        log.info(f"Processing ended with result: {result}")
        cleanup()
 
def process(item):
    pass  # placeholder
 
def cleanup():
    pass  # placeholder

Test Edge Cases

Always test early termination code with: empty arrays, single-element arrays, target at first position, target at last position, target not present, and all elements matching. These edge cases expose most early termination bugs.

Performance Implications

Early termination can dramatically improve performance, but the improvement depends on where the answer is typically found. Understanding the probability distribution of outcomes helps predict actual performance.

Complexity Analysis with Early Termination:

Let p = probability that any given element satisfies the exit condition.

For a search with random target locations:

Best case: O(1) — target at first position
Worst case: O(n) — target at last position or absent
Average case: O(n/2) = O(n) — target equally likely at any position

Expected Iterations by Target Position Distribution
Scenario	Distribution	Expected Iterations	Practical Impact
Uniform random	Equal probability anywhere	n/2	Half the elements examined on average
Target usually first	Front-weighted	O(1) to O(logn)	Very fast in practice
Target usually absent	Rare matches	n	Full traversal usually needed
Multiple matches	k matches in n elements	n/k on average	Faster as k increases

Optimize for the Common Case

If you know that matches are more likely in certain positions, consider reordering the array or using a secondary data structure. For example, if you frequently search for recently added items, keeping them at the front gives O(1) expected time for recent additions.

Micro-Optimization Considerations:

Early termination isn't just about reducing iterations—it also affects:

Branch prediction: CPUs predict that loops continue. Early exits are "mispredictions" with small penalties, but still cheaper than useless iterations.
Cache behavior: Fewer iterations mean less cache pollution from unneeded data.
Compiler optimization: Some compilers may optimize return paths better than complex break logic.

In most cases, these micro-optimizations are negligible compared to the algorithmic benefit of reducing from O(n) to O(k) iterations. Focus on the big picture first.

Summary: Mastering Early Termination

Early termination transforms how we think about traversal. Instead of mechanically visiting every element, we ask: When do we know enough to stop? This question unlocks significant performance improvements and leads to cleaner, more expressive code.

Key Takeaways

•Early termination stops iteration when the answer is known — No need to examine remaining elements once the result is determined.
•Use return for clean exits — It's clearer and more direct than break with post-loop logic in most cases.
•Search patterns (existence, find-first) are canonical early termination examples—stop on first match.
•Boolean patterns (any, all) short-circuit based on boolean logic: any stops on first True, all stops on first False.
•Threshold patterns stop when a count, sum, or other accumulated value reaches a limit.
•Fail-fast validation detects errors early, improving performance and providing quick feedback.
•Avoid pitfalls: Always provide fallback returns, test edge cases, and ensure cleanup code runs when needed.
•Performance gains are real — From O(n) worst-case to O(1) best-case, with O(n/2) average case for uniform distributions.

What's Next:

So far, we've visited elements and made decisions. But often, we need to produce new values during traversal—computing derived values, filtering elements, or transforming data. The next page explores collecting and transforming during traversal: how to build new data structures while iterating through existing ones.

Page Complete

You now understand early termination—when to stop traversal, how to implement clean exits, and how to recognize patterns that enable short-circuit behavior. This powerful technique will serve you in countless algorithms. Next, we'll explore how to collect and transform data during traversal.