Layer depth rule
A newly discovered neighbor sits one edge deeper than the node that found it.
Graph layout 0 Start node 0 Target node 7
Concept module
Breadth-First Search and Layered Frontiers
Keep the queue frontier, visited count, and graph layers visible together so breadth-first search reads as a layered process instead of a procedure list.
The simulation shows one labeled graph with the current node, the queue frontier, and the visited nodes colored differently so the BFS layers stay readable. A readout card reports the traversal mode, current node, frontier size, visited count, and target, while a cue panel shows the frontier order and the current neighbor list. Breadth-first search is running on the layered campus graph from A toward H. The start node is waiting on the frontier. The frontier currently holds 1 node, and 0 nodes have already been visited.
Interactive lab
Graph traversal bench
Keep one live graph, the current frontier, and the visited state visible together so breadth-first and depth-first search read like different process choices on the same structure.
Controls
Swap among a few bounded graph scenes without leaving the shared bench.
Choose where the traversal begins.
Mark the node the search is trying to reach.
Compare queue-like breadth-first expansion with stack-like depth-first branching.
More tools
Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.
Show
More tools
Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.
More presets
Presets
Time
0.00 s / 20.3 sLivePause to inspect a specific moment, then step or scrub through it.0.00 s20.3 s
Predict -> manipulate -> observe
Keep the active prompt next to the controls so each change has an immediate visible consequence.
ObservationPrompt 1 of 2
Breadth-first search settles the shallow layer before it lets the graph dive into deeper branches.
Graphs
Switch graph views without breaking the live stage and time link.
Visited nodes versus frontier size
Watch how many nodes have already been expanded while the waiting frontier grows or shrinks.
step: -1.62 to 21.9nodes: -0.64 to 8.64
visited nodesfrontier size
Equation map
See each variable before you move it.
Select a symbol to highlight the matching control and the graph or overlay it most directly changes.
Graph layout
0Switches among the bounded graph scenes on the same traversal bench.
Graph: Visited nodes versus frontier sizeGraph: Current depth versus deepest claimed depthOverlay: Adjacency cueOverlay: Visited cue
Equations in play
Choose an equation to sync the active symbol, control highlight, and related graph mapping.
More tools
Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.
Hide
More tools
Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.
What to notice
Keep the graph and one traversal graph visible together.
ObservationPrompt 1 of 2
Graph: Visited nodes versus frontier size
Breadth-first search settles the shallow layer before it lets the graph dive into deeper branches.
Control: Start nodeControl: Traversal modeGraph: Current depth versus deepest claimed depthGraph: Visited nodes versus frontier sizeOverlay: Frontier cueOverlay: Visited cueEquationEquation
Guided overlays
Focus one overlay at a time to see what it represents and what to notice in the live motion.
3 visible
Overlay focus
Adjacency cue
Keep the current node's neighbor list visible.
What to notice
- The local neighborhood is the input that every BFS layer grows from.
Why it matters
It keeps breadth-first search tied to real graph structure instead of to a slogan about breadth.
Control: GraphControl: Start nodeGraph: Current depth versus deepest claimed depthEquationEquation
Breadth-first search is running on the layered campus graph from A toward H. The start node is waiting on the frontier. The frontier currently holds 1 node, and 0 nodes have already been visited.
Equation detailsDeeper interpretation, notes, and worked variable context.
Queue frontier update
Breadth-first search removes the oldest frontier node, then adds its new neighbors to the back.
Start node 0 Traversal mode 0
Progress
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Stable links
Starter track
Step 4 of 60 / 6 completeAlgorithms and Search Foundations
Earlier steps still set up Breadth-First Search and Layered Frontiers.
1. Sorting and Algorithmic Trade-offs2. Binary Search / Halving the Search Space3. Graph Representation and Adjacency Intuition4. Breadth-First Search and Layered Frontiers+2 more steps
Previous step: Graph Representation and Adjacency Intuition.
Short explanation
What the system is doing
Breadth-first search keeps the frontier organized like a queue. The oldest claimed node gets expanded next, so the search spreads outward in visible layers instead of diving down one branch immediately.
On this bench the graph, the queue frontier, and the visited count stay coupled. That makes the layered shape of BFS honest: nearby nodes get settled first because the queue keeps earlier discoveries ahead of later ones.
Key ideas
01Breadth-first search expands the oldest waiting node first.
02On an unweighted graph, breadth-first layers match the fewest-edge distance from the start.
03The queue frontier grows when a node discovers new neighbors and shrinks when those waiting nodes get expanded.
Worked example
Read the full frozen walkthrough.
Frozen walkthrough
Read the queue frontier directly from the layered-campus bench.
Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.
View plansFrozen valuesUsing frozen parameters
On the layered campus graph, what should BFS settle first from A?
Graph layout
Layered campus
Start node
A
Queue frontier
B then C
1. Read the first layer
Starting from A, the first neighbors B and C become the first queue frontier.
2. Keep the queue order honest
Because BFS uses a queue frontier, B should leave the frontier before the later discovery C.
3. Read what that means
That oldest-first rule makes the search finish the shallow layer before it moves into deeper nodes like D, E, F, and H.
BFS layer read
From A, BFS settles the B/C layer before it pushes deeper.
The queue frontier is what turns a graph traversal into a layered search rather than a branch-first dive.
Common misconception
Breadth-first search just means moving around the page in a wide-looking pattern.
The important rule is not the drawing. The important rule is that the oldest claimed node leaves the frontier next.
That queue rule is what makes BFS settle the graph layer by layer.
Mini challenge
Build a BFS case where the frontier gets wide before the target is reached.
Prediction prompt
Decide whether you need an edge start or a hub start before you switch presets.
Check your reasoning
Use a hub-like start so BFS can claim several same-depth neighbors before it narrows again.
Breadth-first search widens fastest when one node has several direct neighbors at the same shallow depth.
Quick test
Reasoning
Question 1 of 2
Answer from the visible queue and depth pattern, not from memorized acronyms.
Why does BFS discover shallow nodes before deeper ones on this bench?
Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.
Accessible description
The simulation shows one labeled graph with the current node, the queue frontier, and the visited nodes colored differently so the BFS layers stay readable.
A readout card reports the traversal mode, current node, frontier size, visited count, and target, while a cue panel shows the frontier order and the current neighbor list.
Graph summary
One graph tracks visited nodes against frontier size, a second tracks current depth against the deepest claimed depth, and a third compares new discoveries with repeat skips.
Together they show how breadth-first search widens and then settles the graph layer by layer.
Keep the queue frontier and the graph layers in view
Keep moving through the concept map.
These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.
Compare a branch-first frontier on the same graphAlgorithms and Search
Depth-First Search and Backtracking Paths
Keep the stack frontier, current depth, and branch order visible together so depth-first search feels like disciplined backtracking instead of random wandering.
See how cycles and repeat skips change the bookkeepingAlgorithms and Search
Frontier and Visited State on Graphs
Keep repeat skips, waiting frontier nodes, and already-expanded nodes visible together so cycle handling feels like honest bookkeeping on one graph bench.
Next in Graph neighborhoods and adjacencyAlgorithms and Search
Graph Representation and Adjacency Intuition
Keep one live graph, one local neighborhood, and one frontier cue visible together so graph structure feels readable before traversal rules get formal.