Iterative deepening A*

Iterative-deepening-A* works as follows: at each iteration, perform a depth-first search, cutting off a branch when its total cost f(n) = g(n) + h(n) exceeds a given threshold. This threshold starts at the estimate of the cost at the initial state, and increases for each iteration of the algorithm. At each iteration, the threshold used for the next iteration is the minimum cost of all values that exceeded the current threshold. As in A*, the heuristic has to have particular properties to guarantee optimality (shortest paths). See Properties below. == Pseudocode ==

path current search path (acts like a stack) node current node (last node in current path) g the cost to reach current node f estimated cost of the cheapest path (root..node..goal) h(node) estimated cost of the cheapest path (node..goal) cost(node, succ) step cost function is_goal(node) goal test successors(node) node expanding function, expand nodes ordered by g + h(node) ida_star(root) return either NOT_FOUND or a pair with the best path and its cost procedure ida_star(root) bound := h(root) path := [root] loop t := search(path, 0, bound) if t = FOUND then return (path, bound) if t = ∞ then return NOT_FOUND bound := t end loop end procedure function search(path, g, bound) node := path.last f := g + h(node) if f > bound then return f if is_goal(node) then return FOUND min := ∞ for succ in successors(node) do if succ not in path then path.push(succ) t := search(path, g + cost(node, succ), bound) if t = FOUND then return FOUND if t min then min := t path.pop() end if end for return min end function ==Properties==

Like A*, IDA* is guaranteed to find the shortest path leading from the given start node to any goal node in the problem graph, if the heuristic function is admissible, IDA* is beneficial when the problem is memory constrained. A* search keeps a large queue of unexplored nodes that can quickly fill up memory. By contrast, because IDA* does not remember any node except the ones on the current path, it requires an amount of memory that is only linear in the length of the solution that it constructs. Its time complexity is analyzed by Korf et al. under the assumption that the heuristic cost estimate is consistent, meaning that :h(n) \le \mathrm{cost}(n, n') + h(n') for all nodes and all neighbors of ; they conclude that compared to a brute-force tree search over an exponential-sized problem, IDA* achieves a smaller search depth (by a constant factor), but not a smaller branching factor. Recursive best-first search is another memory-constrained version of A* search that can be faster in practice than IDA*, since it requires less regenerating of nodes. == Time Complexity ==

While IDA* is often praised for its memory efficiency compared to A*, its worst-case time complexity can be significantly worse under certain conditions: IDA* on Trees: Slow Threshold Growth IDA* relies on iterative deepening using an f-cost threshold that increases each iteration. Typically, IDA* performs efficiently when the number of nodes expanded in each iteration grows exponentially. However, Patrick, Almulla, and Newborn (1992) demonstrated that when the f-cost threshold increases by the smallest possible amount each iteration—such as when f-values are unique and strictly increasing—IDA* expands exactly one additional node per iteration. Under these worst-case conditions of uniqueness and monotonicity, IDA* expands N(N+1)/2=\Theta(N^2) nodes, where N is the number of nodes surely-expanded by A*, yielding quadratic complexity compared to A*’s linear O(N) complexity. Mahanti et al. (1992) further showed that this limitation is not exclusive to IDA*: no best-first limited-memory heuristic search algorithm can universally achieve O(N) complexity on trees due to memory constraints. They also specify that IDA* achieves O(N) complexity only if the number of iterations with minimal node growth is bounded, a condition not always satisfied. To avoid the worst-case scenario, Iterative Budgeted Exponential Search (IBEX) has been introduced in 2019. IDA* on Graphs: Impact of Transpositions IDA* explores the search space in a depth-first manner and retains no memory of previously expanded nodes beyond the current path. Consequently, in graphs containing transpositions—where multiple distinct paths lead to the same node—IDA* repeatedly re-expands nodes. Mahanti et al. (1992) illustrated that for directed acyclic graphs, these repeated expansions lead to severe performance degradation, with a worst-case complexity reaching \Omega(2^{2N}), where N is the number of nodes eligible for expansion by A*. Thus, in scenarios involving transpositions or graph structures, IDA* can be significantly less efficient than A*, suggesting that other graph-search algorithms may be more appropriate choices. == Applications ==

Applications of IDA* are found in such problems as planning. Solving the Rubik's Cube is an example of a planning problem that is amenable to solving with IDA*. ==References==