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Best-first search prioritizing the estimated total solution cost $f=g+h$, with $g$ the cost already incurred and $h$ a heuristic function for the remaining cost. With nonnegative estimates, goal estimates zero, finite branching factor and step costs bounded below by a positive constant, the tree search version using an admissible heuristic returns a minimum-cost goal when it removes one for expansion. An optimal solution path always has a frontier node with $f\le C^{\ast }$; a more costly goal cannot be removed before it. The stated finiteness and lower-cost assumptions ensure finitely many earlier nodes when a finite solution exists. Repeated-state handling in graph search needs reopening when a better path is found, unless a consistent heuristic justifies finalizing expanded states.

Best-first search using only a heuristic function estimating remaining goal cost, without adding accumulated path cost. A comparator supplying the same total ordering of estimates suffices to choose expansions even if numeric estimates are unavailable. In general it need not be a complete search algorithm or an optimal search algorithm; particular stronger information or finite-graph policies can supply additional guarantees.

A-star search is best-first search with priorityHere $u$ is a search node, $s(u)$ is its state, $g(u)$ is its accumulated path cost, and $h^{\ast }(s)$ is the minimum remaining cost to any goal state, interpreted as $\infty$ if no goal is reachable. The inequality defines an admissible heuristic. Thus $f(u)$ is a lower bound on the cost of any solution extending that particular action history.

Initialize the search frontier with the root. Repeatedly remove a search node of least $f$, apply the goal test, and, if it is not a goal, insert its applicable-action children with their updated $g$ and $f$. In tree search, different action histories remain distinct search nodes even when they reach the same state; there is no global visited-state set.

For the finite branching factor and unit costs here, an admissible heuristic ensures that any first goal removed is optimal. If a cheaper solution existed, the search frontier would contain an unexpanded prefix of it with $f$ at most that cheaper cost, contradicting selection of the more expensive goal. Nonnegative $h$ also means only finitely many search nodes can have $f$ below any fixed finite cost, so a finite solution will eventually be reached. This does not guarantee termination when no proof exists.

Yes. Greedy best-first search needs only the ordering of remaining-cost estimates. Store generated but unexpanded search nodes in a search frontier. To choose the next search node, compare candidates with $\kappa$ and retain the one it returns; a linear scan suffices, or a priority queue can use the comparison rule. Remove that search node, apply the goal test, and otherwise generate its children.

This implements best-first search ordered by $h^{\ast }$ without ever evaluating $h^{\ast }$. If stable tie handling is wanted, comparing both argument orders distinguishes equality, after which insertion order can break ties. No estimate of accumulated-plus-remaining cost is required for this greedy ordering.

The generated but unexpanded search nodes maintained by a search algorithm. Best-first search selects the next expansion from this frontier using a priority queue or an equivalent ordering operation. A lower-bound argument about a frontier node on an optimal path is central to A-star search correctness.

Best-first search prioritizing accumulated path cost, equivalently A-star search with zero heuristic function. With finite branching factor and positive lower-bounded step costs, tree search is complete for finite solutions and returns a least-cost goal if the goal test is applied on removal from the search frontier. Goal testing at generation can instead return an expensive edge before a cheaper multi-edge route is explored.