A* Heuristic Search

Dijkstra's algorithm finds the shortest path by expanding outward from the start node in every direction, always processing whichever unvisited node currently has the smallest known distance — it's guaranteed correct but explores nodes that have nothing to do with where you're actually headed. A* fixes this by adding a heuristic: an estimate $h(n)$ of the remaining distance from any node $n$ to the goal. Instead of ranking nodes purely by distance-so-far $g(n)$, A* ranks them by $f(n) = g(n) + h(n)$ — cost already paid plus estimated cost still to come — so it prioritizes nodes that look like they're on a promising path toward the destination, not just nodes that are cheap to reach so far.

The one condition that keeps A* correct is that the heuristic must be admissible: it must never overestimate the true remaining distance. A classic example is searching a grid where the heuristic is straight-line (Euclidean) distance to the goal — it's guaranteed to underestimate any real path that has to navigate around obstacles, so it's admissible, and A* using it is guaranteed to still find the true shortest path, just typically after visiting far fewer nodes than Dijkstra would. If the heuristic ever overestimates, A* can return a path that looks optimal but isn't.

Quant uses show up less in literal maps and more in state-space search: routing an order through a sequence of venues to minimize total cost, where remaining-cost-to-completion can be estimated from average spreads, or searching a tree of possible portfolio rebalancing trades toward a target allocation.