A General Backtrack Algorithm That Eliminates Most Redundant Tests

We def ine a faster algorithm functionally equivalent to the classical backtrack algorithm for assignment p rob lems , of wh ich the Eight Queens puzzle is an e l emen ta r y example (Fi l lmore & Williamson 1974, Knuth 1975 ] . Exper imental measurements (f igure 1) reveal r educ t i on by a factor of 2.5 for the 8-queens puzzle ( fac to r of 8.7 for 16 queens) in T, the number of pairtests p e r f o r m e d be fo re f inding a solution (i.e., f irst solution). A p a i r t e s t in this case determines whether a queen on square ( i j , j j ) attacks a queen on square (12, j2^ i C P U " seconds, net speedup is by a factor of 2.0 and 6.0 for 8and 16-queens , respect ive ly . 16-queens was solved in 0.14 seconds on a PDP KL/10. The speedup can be at t r ibuted to the e l iminat ion of almost all redundant tests otherwise r e c o m p u t e d in many parts of the search tree, as indicated in f i gu re 2, wh ich shows the mean number of times, D, an a r b i t r a r y pa i r t es t is executed. If D = 1 then all tests are d is t inc t (no recomputatiOn). Note that each data point in the f i gu res represen ts the mean over 30 or 70 problem instances that d i f fer as fol lows: instead of instantiating queen 3, say, on square (3,1), then on (3,2),..., then (3,8), these 8 squares are o rdered randomly. A problem instance is de f i ned by choosing a "legal squares order ing" for each queen . Random order ing general ly gives a smaller value of T, on the average, than the "natura l " 1,2,3,...,N ordering (for 2 0 q u e e n s , a factor of 5 00 smaller!). The a lgor i thm exploi ts an advantageous time-space t radeo f f and is def ined below in general form by recursive SAIL p rocedu re BKMARK [Swinehart & Sproull 1971]. The classical backt rack algori thm is defined the same, minus the unde r l i ned por t ions (except that "NEW[VAR]" in line 7 is rep laced by "1") . The algorithm applies to any problem hav ing NVARS var iables (8, for 8-queens), each variable Xj hav ing NVALSH] a p r io r i possible values (8 squares for each queen (one row of board), except 4 for queen 1 for s y m m e t r y reasons). An assignment vector ASSIGN[1:NVARS] of values to variables is a solution iff PAIRTEST0, ASSIGN[i] , j , ASSIGN[j]) is true for all 0 < i < j < NVARS (iff no queen can take any other queen). Be low, ASSIGN contains indices to the actual values. Top leve l invocat ion for 8-queens takes the form tmp <BKMARKO, 8, A, B, C, D) wi th array dimensions D[1:NVARS] and C[1:NVARS, l :k ] , where k is the maximum of the B [ i ] values (-8 for 8-queens). Initial values of A are i r r e l evan t ; C and D values are initially 1. BKMARK returns 1, w i t h solut ion in ASSIGN, or returns 0 if no solution ex is ts . Define PAIRTEST for 8-queens and trace the execu t ion (new vs. old versions) to see how it works. (Suggest ion : def ine an array VALUES wi th same dimensions as MARK, so that an element of VALUES encodes a board locat ion.) For b rev i t y , the symbol st.ands for "comment".