First-difference DAG and min-heap
Infer a Custom Label Alphabet
01. The question
Labels arrive in a claimed alphabetic order. Infer the lexicographically smallest alphabet satisfying all observed first-difference constraints. Return "" for a cycle or a longer word before its prefix.
Before revealing the idea, predict the result and describe what information you would need to track.
02. Worked example
Input["baa","abcd","abca","cab","cad"]
Output"bdac"
- Reject invalid prefix order.
- Build unique precedence edges.
- Choose the smallest currently ready letter.
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: ["baa","abcd","abca","cab","cad"]. Parameter: "". 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 downloads only when you run code, using compressed runtime assets. Code is not saved or uploaded. Enable JavaScript to run it.
Printed output
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
- Reject invalid prefix order.
- Build unique precedence edges.
- Choose the smallest currently ready letter.
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
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
import heapq
def solve(data, parameter=""):
letters = sorted(set("".join(data)))
edges = {letter: set() for letter in letters}
indegree = {letter: 0 for letter in letters}
for first, second in zip(data, data[1:]):
if len(first) > len(second) and first.startswith(second):
return ""
for a, b in zip(first, second):
if a != b:
if b not in edges[a]:
edges[a].add(b)
indegree[b] += 1
break
ready = [letter for letter in letters if indegree[letter] == 0]
heapq.heapify(ready)
order = []
while ready:
letter = heapq.heappop(ready)
order.append(letter)
for neighbor in sorted(edges[letter]):
indegree[neighbor] -= 1
if indegree[neighbor] == 0:
heapq.heappush(ready, neighbor)
return "".join(order) if len(order) == len(letters) else ""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
import heapq
def solve(data, parameter=""):
letters = sorted(set("".join(data)))
edges = {letter: set() for letter in letters}
indegree = {letter: 0 for letter in letters}
for first, second in zip(data, data[1:]):
if len(first) > len(second) and first.startswith(second):
return ""
for a, b in zip(first, second):
if a != b:
if b not in edges[a]:
edges[a].add(b)
indegree[b] += 1
break
ready = [letter for letter in letters if indegree[letter] == 0]
heapq.heapify(ready)
order = []
while ready:
letter = heapq.heappop(ready)
order.append(letter)
for neighbor in sorted(edges[letter]):
indegree[neighbor] -= 1
if indegree[neighbor] == 0:
heapq.heappush(ready, neighbor)
return "".join(order) if len(order) == len(letters) else ""06. Cost & edge cases
- Time
- O(total characters + E + V log V)
- Space
- O(V + E)
- Validate the exact input format and bounds before tracing.
- Equal labels do not collapse vertex identity; missing nodes and invalid pointers are explicit.