Visited and frontier update
Expanding one node moves it into the visited set while new neighbors join the waiting frontier.
Start node 0 Target node 7 Traversal mode 0
Concept module
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.
The simulation shows one labeled graph with the current node, the frontier nodes, and the visited nodes colored differently so waiting work and finished work stay separate. 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 bridge cycle 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 / 18.7 sLivePause to inspect a specific moment, then step or scrub through it.0.00 s18.7 s
Predict -> manipulate -> observe
Keep the active prompt next to the controls so each change has an immediate visible consequence.
ObservationPrompt 1 of 2
The frontier is a waiting line of claimed nodes, not a list of finished nodes.
Graphs
Switch graph views without breaking the live stage and time link.
New discoveries versus repeat skips
Compare fresh discoveries with the repeat edges that visited state prevents from reopening.
step: -1.5 to 20.22checks: -0.72 to 9.72
new discoveriesrepeat skips
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
1Switches among the bounded graph scenes on the same traversal bench.
Graph: New discoveries versus repeat skipsGraph: Visited nodes versus frontier sizeOverlay: Visited cueOverlay: Frontier 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
The frontier is a waiting line of claimed nodes, not a list of finished nodes.
Control: Traversal modeGraph: Visited nodes versus frontier sizeOverlay: Frontier cueEquation
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
Frontier cue
Show which claimed nodes are waiting next.
What to notice
- Frontier nodes are not finished yet, even though the search has already claimed them.
Why it matters
It keeps waiting work distinct from finished work on graphs with cycles.
Control: Traversal modeControl: Start nodeGraph: Visited nodes versus frontier sizeEquationEquation
Breadth-first search is running on the bridge cycle 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.
Repeat-skip count
Repeat work appears when checked neighbors are not new discoveries.
Graph layout 1 Traversal mode 0
Progress
Not startedMastery: NewLocal-firstStart exploring and Open Model Lab will keep this concept's progress on this browser first. No finished quick test, solved challenge, or completion mark is saved yet.
Let the live model runChange one real controlOpen What to notice
Try this setup
Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.
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Current bench
Bridge cycle BFS preset
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Stable links
Starter track
Step 6 of 60 / 6 completeAlgorithms and Search Foundations
Earlier steps still set up Frontier and Visited State on Graphs.
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: Depth-First Search and Backtracking Paths.
Short explanation
What the system is doing
On a graph with cycles, the frontier and the visited state are not the same thing. The frontier holds nodes that have already been claimed but are still waiting to be expanded. The visited state marks nodes whose neighborhoods have already been used.
That separation is what stops graph traversal from wasting time on the same loop over and over. This bench keeps repeat skips, frontier size, and visited nodes visible together so cycle handling feels like visible bookkeeping rather than hidden magic.
Key ideas
01Frontier nodes are waiting work, not finished work.
02Visited state means a node's neighborhood has already been expanded, so repeat edges can be skipped honestly.
03Cycles become manageable when the search keeps claimed, waiting, and finished nodes distinct.
Worked example
Read the full frozen walkthrough.
Frozen walkthrough
Read the frontier and visited sets directly from the bridge-cycle 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 bridge-cycle graph, why are frontier and visited state both needed?
Graph layout
Bridge cycle
Frontier role
claimed but waiting
Visited role
already expanded
1. Read the frontier honestly
Nodes on the frontier have already been discovered, but they are still waiting for their neighborhoods to be expanded.
2. Read the visited role honestly
Visited nodes are the ones whose edges have already been used, so repeat edges back into them can be skipped.
3. Read the cycle consequence
That separation keeps the bridge-cycle graph from reopening the same diamond loops every time one edge points backward.
Cycle-control read
Frontier means waiting; visited means already expanded.
Keeping those roles separate is what turns cycles into manageable bookkeeping instead of repeated work.
Common misconception
Frontier and visited state are basically the same set with two different names.
A frontier node has been claimed but not expanded yet. A visited node has already had its neighborhood used.
That difference is exactly what makes repeat skips and clean backtracking possible on looped graphs.
Mini challenge
Build a graph run where repeat skips become visible before the frontier is empty.
Prediction prompt
Decide whether you need a looped graph or a tree-like graph before you change the preset.
Check your reasoning
Use the bridge-cycle graph so backward edges appear while there is still frontier work waiting.
Repeat skips become visible when the graph contains cycles and the traversal has enough time to encounter them.
Quick test
Reasoning
Question 1 of 2
Answer from the visible cycle bookkeeping, not from memorized vocabulary.
What is the clearest difference between the frontier and the visited state?
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 frontier nodes, and the visited nodes colored differently so waiting work and finished work stay separate.
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 cycle-handling depends on keeping frontier and visited state distinct.
Keep frontier and visited state separate when cycles appear
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 graph-traversal bookkeeping with visible list workAlgorithms and Search
Sorting and Algorithmic Trade-offs
Watch sorting as visible work on a live list so input order, comparisons, and writes stay concrete instead of collapsing into one final answer.
Return to the ordered-search case after the graph branchAlgorithms and Search
Binary Search / Halving the Search Space
Keep an ordered list, the low-mid-high markers, and the shrinking interval visible together so binary search feels visual instead of procedural.
Next in Branch-first searchAlgorithms 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.