A-star search ($f(s)=g(s)+h(s)$)

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.

Suppose $0\le h(n)\le h^{\ast }(n)\le u(n)$ in a search problem, with $h^{\ast }=\infty$ at dead ends. Every finite $g(n)+u(n)$ bounds some reachable solution cost from above, so their running minimum $B$ is at least the global optimum $C^{\ast }$. Every node of an optimal path has $g+h\le C^{\ast }\le B$, and therefore survives the displayed strict pruning rule. Ordinary A-star search can continue to a selected goal without receiving a path witnessing $u$.

Equality pruning is unsafe without a stored incumbent solution: when $B=C^{\ast }$, it could delete every optimal continuation. A numerical upper bound does not by itself provide a plan to return. This distinguishes an upper-bound certificate from a constructed solution in branch and bound.