Algorithm Notesworking through it, one step at a time

Prefix sums + frequency map

MediumPractice ↗

Count Balanced Shipment Runs

01. The question

A depot records signed stock changes: arrivals are positive and departures are negative. Count every nonempty contiguous run whose total equals the requested change. Overlapping runs count separately.

Before revealing the idea, predict the result and describe what information you would need to track.

02. Worked example

Inputchanges = [2, -2, 2, 0], target = 2

Output5

  1. Prefix totals, including the empty prefix, are 0, 2, 0, 2, 2.
  2. Each current prefix needs an earlier prefix equal to current - target.
  3. The three prefixes equal to 2 find 1, 2, and 2 earlier zero prefixes. That gives 5 runs.

Try your code

Define solve(data, parameter="") and return your result. Your code runs locally, not on a server.

Edit your test cases

Use 1-10 objects with data and an optional string parameter. Expected outputs are computed from the lesson. An optional expected field must be correct.

Input contract & reference code

Example data: [2,-2,2,0]. Parameter: "2". Numeric parameters arrive as strings; convert them with int(parameter) when needed.

The example and generated cases use the question’s exact variant. This checks behavior on a small test batch—not performance on every possible input.

Python runtime notices

Python downloads only when you run code, using compressed runtime assets. Code is not saved or uploaded. Enable JavaScript to run it.

Five-second execution limit per batch. Standard-library Python only; additional packages and network access are unavailable. Stop ends execution without blocking the lesson.

03. The idea

Try sketching your own approach first. What would you track, and why would it work?

Reveal the idea
  1. Begin with one occurrence of prefix total 0, representing the empty prefix.
  2. Add the current change. Count all earlier prefixes equal to prefix - target.
  3. Only then remember this prefix, so empty runs are never counted.

04. Watch it happen

Use Step to move slowly, or Play to follow the whole trace. Change a preset or load your own input.

Input & settings

Up to 30 integer changes, each -1,000 to 1,000. Target: -10,000 to 10,000. Count nonempty runs.

Red = current item / blue = active window or cells / green = completed work. A value docks at its actual destination; the original input stays unchanged. Positions start at 0.

Working state / what changes

Ready to begin

Enable JavaScript to step through the animation. The worked example and solution below remain readable without it.

Python / why it changes

The line behind the step

def balanced_runs(changes, target):
    frequencies = {0: 1}
    prefix = count = 0
    for change in changes:
        prefix += change
        count += frequencies.get(prefix - target, 0)
        frequencies[prefix] = frequencies.get(prefix, 0) + 1
    return count
How to use this walkthrough

Hover or focus a control for its label. Play forms the input and follows values into the working state. Step shows one change; Back and the slider restore exact snapshots. Reduced motion uses instant updates. Saved snapshots and sorted heap displays are teaching overhead, not algorithm complexity.

05. Python solution

def balanced_runs(changes, target):
    frequencies = {0: 1}
    prefix = count = 0
    for change in changes:
        prefix += change
        count += frequencies.get(prefix - target, 0)
        frequencies[prefix] = frequencies.get(prefix, 0) + 1
    return count

06. Cost & edge cases

Time
O(n) expected
Space
O(n)
  • Empty input returns 0.
  • Zeros and negative changes are allowed; a positive-only shrinking window is not suitable.
  • For [0, 0] and target 0, count 3 nonempty runs, not 2.

Keep going

Browse all questionsBack to Arrays & hash mapsPack a Two-Item Parcel Find a Clean Signal Close a Nested Ledger Scan the Warehouse Shelves