Do A* Search and Greedy Best-First Search rank Open List nodes differently?

Yes. A* ranks nodes using $f(n)=g(n)+h(n)$, combining accumulated path cost with estimated remaining cost. Greedy Best-First Search ranks nodes using $h(n)$ alone, so the two algorithms can expand different nodes and return different routes.

Can A* Search and Greedy Best-First Search return the same path?

Yes. Both can return the same route when heuristic ordering leads Greedy toward the path that A* also considers cheapest. Their Open List order may still differ because A* uses $g(n)+h(n)$ while Greedy uses only $h(n)$.

Can Greedy Best-First Search evaluate fewer nodes than A* Search?

Yes. Greedy can follow a short sequence of low-$h(n)$ nodes while A* also evaluates nodes with competitive $g(n)+h(n)$ scores. This can reduce work on some graphs, but Greedy gives up A*'s conditional cheapest-path guarantee.

Does an admissible heuristic make Greedy Best-First Search as reliable as A* Search?

No. Admissibility constrains the heuristic estimate, but A* also includes accumulated path cost when ranking its Open List. Greedy uses only $h(n)$, so admissibility alone cannot guarantee the cheapest path that A* can return under the required conditions.

A* Search vs. Greedy Best-First Search

Last Updated July 20, 2026

A* and Greedy Best-First Search both use heuristic guidance, but they order the Open List differently. A* uses $f(n)=g(n)+h(n)$ , while Greedy Best-First Search uses only $h(n)$ and ignores accumulated path cost when choosing the next node.

Use A* Search when path cost matters and the search must return the cheapest route under valid heuristic and graph-search conditions.
Use Greedy Best-First Search when reaching a reasonable route quickly matters more than proving it is the cheapest.

A* balances route cost with goal direction; Greedy follows goal direction alone.

Head-to-Head Showdown

Open List Priority

A* Search: Lowest $g(n)+h(n)$

Greedy Best First Search: Lowest $h(n)$

The Implication: A* can prefer a node that appears farther from the goal when reaching it has been inexpensive. Greedy selects the smallest heuristic value even when the path to that node has already accumulated a large cost.

Optimality Guarantee

A* Search: Conditional cheapest path

Greedy Best First Search: No cheapest-path guarantee

The Implication: A* can guarantee the cheapest path when its heuristic and graph-search conditions satisfy the required assumptions. Greedy ignores accumulated path cost when ranking nodes, so a goal that appears nearby may be reached through a more expensive route.

Heuristic Influence

A* Search: Balanced against $g(n)$

Greedy Best First Search: Controls node priority

The Implication: A misleading heuristic competes with accumulated path cost inside A*'s priority score, allowing expensive detours to lose priority. In Greedy Best-First Search, that same estimate fully determines which Open List node is selected next.

Selection Criteria

Scenario:Routing an ambulance through streets with varying travel times where the minimum-time route must be guaranteed.

Choose A* Search:A* orders its Open List through $g(n)+h(n)$ , so both travel time already spent and estimated time remaining affect the next node. Under the required correctness conditions, it can guarantee the cheapest route; Greedy Best-First Search cannot.

Scenario:Finding a playable route through a large game map within a strict $5$ ms budget where a slightly longer path is acceptable.

Choose Greedy Best-First Search:Greedy Best-First Search orders its Open List only by $h(n)$ , allowing an informative heuristic to drive the search directly toward the goal. A* may evaluate additional nodes to protect path quality that this latency-bound case does not require.

Scenario:Navigating a maze where straight-line distance repeatedly points toward blocked corridors and expensive detours.

Choose A* Search:A* combines the misleading heuristic with accumulated path cost, so routes that repeatedly become expensive receive larger $f(n)$ values in the Open List. Greedy Best-First Search continues prioritizing whichever node appears closest through $h(n)$ alone.

Side By Side Trace

A directed graph runs from $S$ to $G$ through two competing routes: $S \rightarrow A \rightarrow G$ and $S \rightarrow B \rightarrow C \rightarrow G$ . Edge labels show travel cost, and each node's floating value is $h(n)$ , the estimated remaining cost to $G$ . A* orders its Open List by $f(n)=g(n)+h(n)$ , while Greedy Best-First Search orders its Open List only by $h(n)$ ; no ties occur.

Step 1: Evaluating the Start Node

A* Search

Evaluates $S$ with $g(S)=0$ , $h(S)=6$ , and $f(S)=6$ . Generates $A$ with $f(A)=7+1=8$ and $B$ with $f(B)=2+4=6$ . The Open List becomes $[B(6), A(8)]$ , and $S$ moves to the Closed List.

Greedy Best First Search

Evaluates $S$ with $h(S)=6$ . Generates $A$ with $h(A)=1$ and $B$ with $h(B)=4$ . The Open List becomes $[A(1), B(4)]$ , while edge costs $7$ and $2$ do not affect the ordering, and $S$ moves to the Closed List.

Step 2: Point of Divergence

A* Search

A* selects $B$ because $f(B)=6$ is lower than $f(A)=8$ . Evaluating $B \rightarrow C$ gives $g(C)=2+2=4$ , $h(C)=2$ , and $f(C)=6$ . The Open List becomes $[C(6), A(8)]$ , and $B$ enters the Closed List.

Greedy Best First Search

Greedy Best-First Search selects $A$ because $h(A)=1$ is lower than $h(B)=4$ , even though reaching $A$ already costs $7$ . Evaluating $A \rightarrow G$ adds $G$ with $h(G)=0$ . The Open List becomes $[G(0), B(4)]$ , and $A$ enters the Closed List.

Step 3: Greedy Reaches the Goal

A* Search

A* selects $C$ because $f(C)=6$ is lower than $f(A)=8$ . Evaluating $C \rightarrow G$ gives $g(G)=4+2=6$ , $h(G)=0$ , and $f(G)=6$ . The Open List becomes $[G(6), A(8)]$ , and $C$ enters the Closed List.

Greedy Best First Search

Greedy Best-First Search selects $G$ because $h(G)=0$ is lower than $h(B)=4$ . The goal is accepted when $G$ is removed from the Open List, returning $S \rightarrow A \rightarrow G$ . Node $B$ remains in the Open List and is never evaluated.

Step 4: A* Reaches the Goal

A* Search

A* selects $G$ because $f(G)=6$ is lower than $f(A)=8$ . The goal is accepted when $G$ is removed from the Open List, returning $S \rightarrow B \rightarrow C \rightarrow G$ . Node $A$ remains in the Open List and is never evaluated.

Greedy Best First Search

Performs no further work. Greedy Best-First Search already ended after selecting $G$ from the Open List with the route $S \rightarrow A \rightarrow G$ .

Final Result

A* Search:A* Search returns $S \rightarrow B \rightarrow C \rightarrow G$ with total cost $6$ after evaluating $4$ nodes: $S$ , $B$ , $C$ , and $G$ . Its Open List ordering kept $A(8)$ behind $B(6)$ , $C(6)$ , and $G(6)$ , so $A$ never entered the Closed List.

Greedy Best First Search:Greedy Best-First Search returns $S \rightarrow A \rightarrow G$ with total cost $13$ after evaluating $3$ nodes: $S$ , $A$ , and $G$ . Its Open List followed the smallest heuristic values, so it performed less work but accepted the more expensive route.

Common Pitfalls & Exam Mistakes

Ordering A*'s Open List using only $h(n)$ .
The Mistake: Students ignore accumulated path cost and order A*'s Open List only by heuristic value, making the trace behave like Greedy Best-First Search.
Why It's Wrong: A* orders the Open List by $f(n)=g(n)+h(n)$ , while Greedy Best-First Search orders it by $h(n)$ alone. Removing $g(n)$ can change the next node selected and the final route.
Assuming Greedy returns A*'s cheapest path.
The Mistake: Students conclude that both algorithms must return the same cheapest route because they use the same heuristic values.
Why It's Wrong: A* combines the heuristic with accumulated path cost and can preserve optimality under the required conditions. Greedy ignores accumulated cost when ordering the Open List, so it can return a more expensive route even with an admissible heuristic.
Stopping when $G$ enters the Open List.
The Mistake: Students accept the goal immediately when it is generated instead of waiting for it to become the next Open List node.
Why It's Wrong: In the standard goal-on-pop trace, the goal is accepted when it is removed from the Open List for evaluation. Another node may still have a lower $f(n)$ in A* or lower $h(n)$ in Greedy Best-First Search.

Comparative Analysis

Attribute	A* Search	Greedy Best First Search
Evaluation Function	$f(n)=g(n)+h(n)$	$f(n)=h(n)$
Accumulated Cost	Included through $g(n)$	Ignored when ranking
Optimality	Conditional guarantee	Not guaranteed
Completeness	Conditional guarantee	Not guaranteed generally
Heuristic Influence	Balanced with path cost	Controls node priority
Typical Trade-Off	Cheapest-path protection	Potentially quicker route

Common Questions & Edge Cases

Do A* Search and Greedy Best-First Search rank Open List nodes differently?
Yes. A* ranks nodes using $f(n)=g(n)+h(n)$ , combining accumulated path cost with estimated remaining cost. Greedy Best-First Search ranks nodes using $h(n)$ alone, so the two algorithms can expand different nodes and return different routes.
Can A* Search and Greedy Best-First Search return the same path?
Yes. Both can return the same route when heuristic ordering leads Greedy toward the path that A* also considers cheapest. Their Open List order may still differ because A* uses $g(n)+h(n)$ while Greedy uses only $h(n)$ .
Can Greedy Best-First Search evaluate fewer nodes than A* Search?
Yes. Greedy can follow a short sequence of low- $h(n)$ nodes while A* also evaluates nodes with competitive $g(n)+h(n)$ scores. This can reduce work on some graphs, but Greedy gives up A*'s conditional cheapest-path guarantee.
Does an admissible heuristic make Greedy Best-First Search as reliable as A* Search?
No. Admissibility constrains the heuristic estimate, but A* also includes accumulated path cost when ranking its Open List. Greedy uses only $h(n)$ , so admissibility alone cannot guarantee the cheapest path that A* can return under the required conditions.

Explore the Algorithms in Action

Open the theory pages or try the interactive solvers for the algorithms compared above.

Try the A* Algorithm Calculator

Balance $g(n)+h(n)$ in the Open List and see how accumulated cost protects the path Greedy's heuristic-only order misses.

A* Algorithm Theory

Review how combining $g(n)$ with $h(n)$ preserves optimality, unlike Greedy's Open List driven by heuristic value alone.

Try the Greedy Best-First Search Calculator

Order the Open List by $h(n)$ alone and test how ignoring accumulated cost can select an expensive route quickly.

Greedy Best-First Search Theory

Clarify how heuristic-only ordering trades cheapest-path guarantees for speed, unlike A*'s cost-aware $g(n)+h(n)$ priority.