Pattern1 Practice5 Interview

Difference Array

GuideCode Labs

Encode range updates at boundaries and reconstruct final values with a prefix accumulation. Learn its decision rule, maintained state, Python cost model, and transfer from a focused drill to a full Interview Problem.

Recognize it when

Consider Difference Array when the prompt's constraints and required operations match this shape: Encode range updates at boundaries and reconstruct final values with a prefix accumulation.

On this page · Store Changes at Boundaries

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Store Changes at Boundaries

A difference array stores how the value changes when crossing each index rather than storing final values directly. Constant regions need only their starting and ending changes.

Apply Range Updates in Constant Time

For an inclusive addition [left, right], add delta at left and subtract it at right plus one when that boundary exists. Each update touches at most two slots.

Trace Difference Boundaries

Reference

def difference_trace(length,updates):
    difference=[0]*(length+1);trace=[]
    for left,right,delta in updates:
        difference[left]+=delta
        if right+1<length:difference[right+1]-=delta
        trace.append(difference[:length])
    return trace

Practice

Implement difference_trace(length,updates). Each update is [left,right,delta] inclusive; return the difference array after each update.

def difference_trace(length,updates):
    pass

Public tests

Add at start and subtract after end

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Reconstruct with One Prefix Pass

The running prefix of differences is the final value at each index. Reconstruction occurs once after all offline updates, giving O(n + updates) total time.

Apply Range Increments

Reference

def apply_increments(length,updates):
    difference=[0]*(length+1)
    for left,right,delta in updates:
        difference[left]+=delta
        if right+1<length:difference[right+1]-=delta
    values=[];current=0
    for index in range(length):current+=difference[index];values.append(current)
    return values

Practice

Implement apply_increments(length,updates). Return final values after inclusive range additions.

def apply_increments(length,updates):
    pass

Public tests

Reconstruct with one prefix pass

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Choose Differences or a Fenwick Tree

Use a difference array for offline range updates followed by one materialization. Choose a Fenwick or segment tree when updates and queries interleave online. Endpoint convention determines whether subtraction occurs at right or right plus one.

Explain It in an Interview

Say: “A range addition begins at left and is canceled immediately after right. One prefix pass carries active deltas into final values.” State offline limitations, boundaries, and O(1) work per update.

Python Version Note

Python 3.11+

The examples use the standard Python syntax and containers supported by the browser Judge; the Difference Array invariant is independent of a minor Python release.

Verify in Python docs

Complexity & Invariants

Interview bounds depend on the stated representation, input model, and real Python operations.

Operation	Average	Worst	Interview note
Difference Array decision loop	`O(n + u)`	`O(n + u)`	Accumulate start and post-end boundary deltas, then prefix-sum the difference array once.

Space

O(n) for the focused Apply Range Additions implementation.

Assumptions

Each update contributes delta beginning at left and removes it immediately after right. Therefore the running prefix at every index equals the sum of exactly the updates covering that index.
The bound includes real Python operations; no repeated slicing, linear list membership, or hidden sorting is treated as free.

Invariants Worth Saying Aloud

State the precise Difference Array invariant before coding and preserve it after every update, traversal step, or recursive return.
The difference entry stores the change from the previous position.
Every inclusive update has one start boundary and at most one stop boundary.
Encode an inclusive update with two boundary changes. Preserve this property after every transition.
Recover final values by prefix accumulation. Preserve this property after every transition.

When To Use Or Avoid Difference Array

Use It When

Use Difference Array when the prompt exposes the ordering, monotonicity, recurrence, or constraint that makes this decision safe.
Use it when the difference entry stores the change from the previous position.

Choose Another Tool When

Avoid Difference Array when its decision rule cannot eliminate work or preserve the required state.
Avoid forcing the label onto a prompt whose simpler direct scan already meets the constraints.

Common Pitfalls

These are the mistakes most likely to survive a happy-path example and fail a boundary case.

Moving state without a proof

Applying the pattern without proving its ordering, monotonicity, or window invariant can silently miss valid candidates.

Prevent it: Write the decision rule beside the loop and verify it against Apply Range Additions before optimizing.

Avoid

def apply_range_additions(length, updates):
    pass

Use instead

def apply_range_additions(length, updates):
    difference = [0] * length
    for left, right, delta in updates:
        difference[left] += delta
        if right + 1 < length:
            difference[right + 1] -= delta
    values = []
    running = 0
    for delta in difference:
        running += delta
        values.append(running)
    return values

Where you will hit this: Apply Range Additions(opens in a new tab)

Breaking the maintained state

Cancels at right and therefore omits the inclusive right endpoint.

Prevent it: Use the public tests and preserve this state: The difference entry stores the change from the previous position.

Avoid

def apply_range_additions(length, updates):
    d=[0]*length
    for l,r,x in updates:
        d[l]+=x
        if r<length: d[r]-=x
    out=[]; run=0
    for x in d: run+=x; out.append(run)
    return out

Use instead

def apply_range_additions(length, updates):
    difference = [0] * length
    for left, right, delta in updates:
        difference[left] += delta
        if right + 1 < length:
            difference[right + 1] -= delta
    values = []
    running = 0
    for delta in difference:
        running += delta
        values.append(running)
    return values

Where you will hit this: Apply Range Additions(opens in a new tab)

Hiding Python work in the hot path

Python slicing, copying, membership in lists, or repeated sorting inside the loop can invalidate the intended complexity.

Prevent it: Count every slice, copy, sort, membership check, and container update before claiming O(n + u).

Avoid

def apply_range_additions(length, updates):
    d=[0]*length
    for l,r,x in updates:
        d[l]+=x
        if r<length: d[r]-=x
    out=[]; run=0
    for x in d: run+=x; out.append(run)
    return out

Use instead

def apply_range_additions(length, updates):
    difference = [0] * length
    for left, right, delta in updates:
        difference[left] += delta
        if right + 1 < length:
            difference[right + 1] -= delta
    values = []
    running = 0
    for delta in difference:
        running += delta
        values.append(running)
    return values

Where you will hit this: Binary Subarrays With Sum(opens in a new tab)

Reviewed References

Official Python documentation supports language behavior. Canonical problem pages provide additional practice context.

Python More on Lists(opens in a new tab)
python-docs · checked 2026-07-27
LeetCode 2090 — K Radius Subarray Averages(opens in a new tab)
leetcode · checked 2026-07-12

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