Strategy and board game programming

Hashing and Move Ordering

I didn't really finish describing alpha-beta -- my pseudocode included a mysterious "sort list of moves" step that I didn't explain. I'll continue to leave that dangling while I talk about hashing; we'll connect it up in a little while.

The idea of hashing is very simple. Many games allow
*transpositions* of moves, meaning different sequences of moves that
end up leading to the same position. For instance, in chess, the opening
moves 1. d4 Nf6 2. c4 and 1. c4 Nf6 2. d4 both give the same position
(known as an Indian defense). White's two pawn moves could be made in
either order without changing the result. As an example of a more complicated
transposition, the moves 1. e4 c6 2. d4 d5 3. ed Qxd5 4. Nc3 Qd6 (Caro-Kann
defense), 1. e4 d5 2. ed Qxd5 3. Nc3 Qd6 4. d4 c6 (Scandinavian opening),
and 1. e4 Nf6 2. e5 Ng8 3. d4 d6 4. ed Qxd6 5. Nc3 c6 (Alekhine defense)
all lead to the same position, after different numbers of moves.

Because of transpositions, the same positions can show up many places in the alpha-beta search tree. If we store a data structure that remembers what the results of searching each position were, we can look it up rather than searching it again. But...we don't have enough memory to store all the positions we search. And, lookups must be very fast to make it save time over just searching. Fortunately, we have one advantage: it's ok if we sometimes don't find the results from a position we already searched, and search the same position again, as long as it doesn't happen too often.

The answer: hash tables. Make a big array: as large as possible without blowing out your physical memory (you don't want to eat into virtual memory, it will be slow.)

struct { long checksum; // or long long might be even better int depth; enum { exact, lower_bound, upper_bound } entry_type; double eval; } hashtable[HASH_TABLE_SIZE];For each position you search, compute a "hash value" x indexing into the hash table and another "hash value" y for checking whether you've found the right position.

Before searching a position, lookup hashtable[x]. If hashtable[x].checksum == y, hashtable[x].entry_type == exact, and hashtable[x].depth is at least the depth you are currently searching, return the eval stored there.

After searching the position, store y, the current depth, and the eval you just computed, into hashtable[x].

Some further tips for using hashing effectively:

- Don't clean out the array after making a move, it only wastes time and some hashed positions might actually still be useful after the move.
- If the same position occurs at different levels in the tree (as in the second transposition example we listed above) this can actually give you deeper searches than you originally asked for; that's ok.
- Don't hash the positions very near the leaves of the search tree, there are too many of them (they'll take away hash table space from more valuable positions) and you're not saving much time by avoiding searching them.

A large fraction of chess program bugs are related to hashing. Partly, because it interacts in confusing ways with alpha-beta search. But, you can't avoid dealing with the issue, because you need both hashing and alpha-beta to have an efficient searcher. Recall that, when we call alphabeta(depth,alpha,beta) on a position, one of three things can happen: a fail high, in which we know the eval is at least beta but not exactly what it is; a fail low, in which we know the eval is at most alpha but not exactly what it is; and an exact result, alpha < eva < beta. We can only store an exact result in the hash table if we know the exact result. But a fail high or fail low result could still help us prune later. So, along with exact evals, store two other kinds of eval in hash table: a lower bound stating that the eval is at least beta, and an upper bound stating that the eval is at most alpha. We use the entry_type field of the hash table entry to specify what kind of eval is being stored. If the hash lookup comes back with one of these, we need to see whether it's useful enough to prune immediately without searching the node. If so, we return it, and otherwise search the node again. Here's some pseudocode for alpha-beta search with hashing. We maintain the hashtable index x and checksum y in global variables, that are updated as part of the process of making and unmaking moves.

double alphabeta(int depth, double alpha, double beta) { if (depth <= 0 || game is over) return evaluation(); if (hashtable[x].checksum == y && hashtable[x].depth >= depth) switch (hashtable[x].entry_type) { case exact: return hashtable[x].eval; case lower_bound: if (hashtable[x].eval >= beta) return (hashtable[x].eval); else break; case upper_bound: if (hashtable[x].eval <= alpha) return (hashtable[x].eval); else break; } int eval_is_exact = 0; generate and sort list of moves available in the position for (each move m) { make move m; double val = -alphabeta(depth - 1, -beta, -alpha); unmake move m; if (val >= beta) { hashtable[x].checksum = y; hashtable[x].depth = depth; hashtable[x].entry_type = lower_bound; hashtable[x].eval = val; return val; } if (val > alpha) { alpha = val; eval_is_exact = 1; } } hashtable[x].checksum = y; hashtable[x].depth = depth; if (eval_is_exact) hashtable[x].entry_type = exact; else hashtable[x].entry_type = upper_bound; hashtable[x].eval = alpha; return alpha; }

I said we'd return to alpha-beta; here it is. We did an optimistic analysis last time of alpha-beta, showing that it can double your search depth if it prunes whenever it can. The condition that "it prunes whenever it can" can be expressed more simply: good moves are searched before bad ones. The moves don't have to be completely sorted, but the best one should be first or at least one of the best should be one of the first. What happens if not? Then we don't do any pruning and we don't search very deeply.

If we classify nodes into type A (all children get searched) and type B (we prune after finding a good child) then move ordering is important in both cases: in type B, you want to start with a child that will let you prune the rest. In type A, you want to choose a first child that is good enough to let all the other children be type B.

Of course, finding good moves is hard: it's the whole reason we're doing the search in the first place. But we have some clues: (1) we may have hashtable entries from previous iterations of iterated deepening that give approximations to search values (same positions searched less deeply). (2) we may have some game-specific information, e.g. in chess captures are often good moves, try them first. (3) the killer heuristic: if move m was best in a sibling, and is valid here too, try it.

So, before searching children, add a step: sort them by expected quality. Then do the search in the sorted order. (Sometimes you can modify the move generator to output moves in roughly-sorted order e.g. captures first, and save doing an explicit sort.)

One additional trick: if you think you're going to prune, you don't need to sort everything, you just need to output the first few items in sorted order. So you may want to use a sort that you can take items one by one from and stop early, e.g. selection sort or heapsort.

David Eppstein, Dept. Information & Computer Science, UC Irvine, .