http://csiweb.ucd.ie/Staff/acater/comp30260.html Artificial Intelligence for Games and Puzzles

1
Artificially Narrow Alpha Beta

• AlphaBeta cuts off the search of the rest of a
  nodes children if/when it encounters one child
  whose minimaxed value exceeds ? (hope).
• Reduced values of ? result in more child nodes
  being cut off.
• Searching a tree - or any subtree - with a
  narrower window between ? ? will cause more
  cutoffs.
• The search may be more efficient.
• But it may no longer return the right answer!

Increased values of ? become reduced values for ?
at the next deeper level So increasing ? results
in more grandchildren being cut off.
2
Aspiration Search - at the root of a tree

• If you are about to search a tree to some depth
  D,
• And you have a fair idea Vguess what the minimax
  value is likely to be,
• (You can use the result of a search to depth D-1)
• Then you can search with a window Vguess-?,
  Vguess? instead of -?, ?
• If you are right, you will obtain the true value
  Vtrue with less search.
• The smaller ? is, the less search you do, but the
  less likely you are right.
• If you are wrong, and the true value Vtrue
  Vguess-?
• Fail Low Search again using window -?, Vguess
• If you are wrong, and the true value Vtrue
  Vguess?
• Fail High Search again using window Vguess, ?

3
Iterative-Deepening Aspiration Search Algorithm

• int guess 0, depth 1, delta
  Pick_A_Number()
• while Resources_Available()
• int score, alphaguess-delta,
  betaguessdelta
• score??(alpha,beta,depth)
• if score beta / Fail High
  /
• then score ??(score,?,depth)
• else if score alpha / Fail Low
  /
• then score ??(-?,score,depth)
• guessscore
• depthdepth1
• return guess

4
Null windows

• If ? ?1,
• (and if evaluations return integers not reals),
• ?? search will always either fail high or fail
  low.
• Such a search can be thought of as answering the
  boolean question is Vtrue Vguess?
• Such a search is used in NegaScout / PVS
• to confirm cheaply that subsequent move choices
  are in fact inferior to the move believed to be
  best
• but when subsequent moves are not in fact
  inferior, they must be searched again.

5
Negascout / PVS algorithm

• PVS(?, ?, d) / ?-?, ?? at root node /
• if d0
• then return evaluation(?, ?)
• else mk(1) score-PVS(-?,-?,d-1) unmk(1)
• if score lt ?
• then for m from 2 to Move_Count()
• LBmax(?,score) UBLB1
• mk(m) temp-PVS(-UB,-LB,d-1)
• if (tempUB and templt?)
• then temp-PVS(-?,-temp,d-1)
  
• unmk(m)
• scoremax(score,temp)
• if temp ? then break
• return score

6
NegaScout / PVS effect

(?-20, ?30)
-8
-14
-9
-7
-6
Suppose a negamax search of the suppressed
subtrees would give the indicated values for the
child nodes the top node would have value 14
7
NegaScout / PVS effect

(?-20, ?30)
-9
(-30, 20) ? -9
8
NegaScout / PVS effect

(?-20, ?30)
UB9
9
-9
(-30, 20) ? -9
9
NegaScout / PVS effect

(?-20, ?30)
UB9
7
-9
-7
(-30, 20) ? -9
(-10, -9) ? -7
10
NegaScout / PVS effect

(?-20, ?30)
UB9
6
-9
-7
-6
(-30, 20) ? -9
(-10, -9) ? -6
(-10, -9) ? -7
11
NegaScout / PVS effect

(?-20, ?30)
UB9
12
-12
-9
-7
-6
(-30, 20) ? -9
(-10, -9) ? -12
(-10, -9) ? -6
(-10, -9) ? -7
12
NegaScout / PVS effect

(?-20, ?30)
UB9
12
-12
-9
-7
-6
(-30, 20) ? -9
(-10, -9) ? -12
(-10, -9) ? -6
(-10, -9) ? -7
13
NegaScout / PVS effect

(?-20, ?30)
UB9
-9
-7
-6
(-30, 20) ? -9
(-10, -9) ? -12
(-10, -9) ? -6
(-10, -9) ? -7
(-30, 12) ?-14
14
NegaScout / PVS effect

(?-20, ?30)
UB14
14
-14
-9
-7
-6
(-30, 20) ? -9
(-10, -9) ? -12
(-10, -9) ? -6
(-10, -9) ? -7
(-30, 12) ?-14
15
NegaScout / PVS effect

(?-20, ?30)
UB14
8
-8
-12
-9
-7
-6
(-30, 20) ? -9
(-15, -14) ? -8
(-10, -9) ? -12
(-10, -9) ? -6
(-10, -9) ? -7
(-30, 12) ?-14
16
The SCOUT algorithm

• Takes to an extreme the idea that minimal-window
  ?? searches are cheap
• Replaces general ?? with a divide-and-conquer
  approach
• First guess midway in range of possible values -
  say -512,512
• Guess 0, use window 0,1
• A result of say 24 means only that the true
  result exceeds 0
• Try again, with window 256,257
• A result of say 24 means only that the true
  result is no more than 256
• Try again, window 128,129
• and so on, cutting range in half each time,
• eventually finding the N such that value gt N-1
  value N

17
MTD(f) Algorithm

• Scouts divide-and-conquer approach does not take
  advantage of the result of a previous
  iterative-deepening search to make a good 1st
  guess.
• MTD(f) does, then creeps toward correct answer.
• score0 d0
• while resources permit,
• dd1 LB-? UB?
• while LB lt UB
• if score LB
• then windowscore
• else windowscore-1
• score??(window,window1,d)
• if score lt window
• then UBscore
• else LBscore
• return score

18
Reflection on ?? variants

• Aspiration Search, NegaScout, Scout, MTD(f)
• All these variations on ?? may involve repeating
  searches with different choices for ? and ?.
• This can make savings,
• but probably only if a transposition table is
  also used, to allow results of previous partial
  searches to be reused.
• For chess, MTD(f) seems to need on average 3 - 5
  iterations to converge, and saves 5-15 in terms
  of number of nodes generated. ref Schaeffer
• http//www.cs.ualberta.ca/jonathan/Courses/657/N
  otes/7.Windows.pdf
• ?? is fundamentally a depth-first tree search
  algorithm.
• Iterative Deepening gives an effect rather like
  breadth-first search.