(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: peg.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -f peg.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.03.05:1211 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N 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(Theorem)f(1)h(is)h(constructiv)n(e)e(in)i(that)f(it)h(tells)g(us)f(ho) n(w)g(to)g(unhop)g(from)g(a)g(single)g(p)r(eg)h(to)f(an)n(y)0 183 y(feasible)27 b(con\014guration.)f(W)-7 b(e)28 b(simply)g(rev)n (erse)e(this)i(series)e(of)i(mo)n(v)n(es)e(to)h(pla)n(y)g(the)h(game.) 976 b Fg(u)-55 b(t)125 359 y Fu(More)38 b(generally)-7 b(,)38 b(a)h(con\014guration)e(that)j(can)f(b)r(e)h(reduced)f(to)g Fi(k)j Fu(p)r(egs)d(m)n(ust)g(b)r(elong)g(to)h(the)f(regular)f (language)0 459 y(\(0)74 429 y Fj(\003)112 459 y Fi(L)p Fu(0)211 429 y Fj(\003)248 459 y Fu(\))280 429 y Fs(k)321 459 y Fu(,)d(since)g(unhopping)f(cannot)h(in)n(terlea)n(v)n(e)e(the)i (p)r(egs)f(coming)g(from)h(di\013eren)n(t)f(origins)g([7].)g(This)h (leads)f(to)h(the)0 558 y(follo)n(wing)27 b(algorithm:)0 735 y Fl(Theorem)j(2.)41 b Fk(Ther)l(e)36 b(is)f(a)g(line)l(ar-time)g (str)l(ate)l(gy)f(for)i(r)l(e)l(ducing)f(any)g(one-dimensional)h(Pe)l (g)f(Solitair)l(e)h(c)l(on\014gur)l(ation)0 835 y(to)30 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Fi(a;)14 b(i)p Fu(\))25 b(where)h Fi(a)d Fg(2)g Fi(A)j Fu(and)g(0)c Fg(\024)h Fi(i)g Fg(\024)f Fi(n)p Fu(.)k(Dra)n(w)f(an)h(arc)e(from)i (\()p Fi(a;)14 b(i)p Fu(\))26 b(to)f(\()p Fi(b;)14 b(i)h Fu(+)f(1\))26 b(in)g Fi(G)g Fu(whenev)n(er)f Fg(A)0 1410 y Fu(mak)n(es)i(a)h(transition)g(from)g(state)h Fi(a)f Fu(to)g(state)h Fi(b)f Fu(on)g(sym)n(b)r(ol)g Fi(c)1982 1422 y Fs(i)2010 1410 y Fu(.)g(Also,)g(dra)n(w)g(an)g(arc)f(from)i(\()p Fi(t;)14 b(i)p Fu(\))28 b(to)g(\()p Fi(s;)14 b(i)p Fu(\))29 b(for)f(an)n(y)g Fi(t)c Fg(2)h Fi(T)0 1509 y Fu(and)i(an)n(y)g(0)c Fg(\024)f Fi(i)h Fg(\024)g Fi(n)p Fu(.)k(Since)h Fg(jAj)c Fu(=)e Fg(O)r Fu(\(1\),)28 b Fg(j)p Fi(G)p Fg(j)c Fu(=)f Fg(O)r Fu(\()p Fi(n)p Fu(\).)125 1610 y(Then)30 b(an)n(y)g(path)g(from) g(\()p Fi(s;)14 b Fu(0\))30 b(to)h(\()p Fi(s;)14 b(n)p Fu(\))30 b(in)h Fi(G)g Fu(consists)e(of)h Fi(n)h Fu(arcs)e(of)h(the)h (form)f(\()p Fi(a;)14 b(i)p Fu(\))30 b(to)g(\()p Fi(b;)14 b(i)20 b Fu(+)g(1\),)30 b(together)g(with)0 1709 y(some)f(n)n(um)n(b)r (er)g Fi(k)k Fu(of)d(arcs)e(of)i(the)g(form)f(\()p Fi(t;)14 b(i)p Fu(\))30 b(to)f(\()p Fi(s;)14 b(i)p Fu(\).)30 b(Breaking)e(the)i (path)g(in)n(to)f(subpaths)h(b)n(y)f(remo)n(ving)f(all)h(but)i(the)0 1809 y(last)f(arc)f(of)h(this)h(second)e(t)n(yp)r(e)i(corresp)r(onds)d (to)i(partitioning)f(the)i(input)g(string)e(in)n(to)h(substrings)g(of)g (the)g(form)g(0)3703 1779 y Fj(\003)3741 1809 y Fi(L)p Fu(0)3840 1779 y Fj(\003)3877 1809 y Fu(,)0 1908 y(so)25 b(the)g(length)h(of)f(the)h(shortest)e(path)i(from)f(\()p Fi(s;)14 b Fu(0\))25 b(to)g(\()p Fi(s;)14 b(n)p Fu(\))26 b(in)g Fi(G)f Fu(is)g Fi(n)14 b Fu(+)g Fi(k)s Fu(,)25 b(where)g Fi(k)j Fu(is)d(the)h(minim)n(um)g(n)n(um)n(b)r(er)f(of)g(p)r (egs)0 2008 y(to)30 b(whic)n(h)g(the)g(initial)g(con\014guration)e(can) i(b)r(e)g(reduced.)f(Since)i Fi(G)f Fu(is)g(a)f(directed)h(acyclic)f (graph,)g(w)n(e)g(can)h(\014nd)g(shortest)0 2108 y(paths)f(from)g(\()p Fi(s;)14 b Fu(0\))30 b(b)n(y)f(scanning)g(the)g(v)n(ertices)g(\()p Fi(a;)14 b(i)p Fu(\))29 b(in)h(order)e(b)n(y)h Fi(i)p Fu(,)g(resolving)f(ties)i(among)e(v)n(ertices)g(with)i(equal)f Fi(i)g Fu(b)n(y)0 2207 y(scanning)22 b(v)n(ertices)f(\()p Fi(t;)14 b(i)p Fu(\))23 b(\(with)g Fi(t)g Fg(2)h Fi(T)12 b Fu(\))22 b(earlier)f(than)h(v)n(ertex)g(\()p Fi(s;)14 b(i)p Fu(\).)23 b(When)g(w)n(e)f(scan)g(a)g(v)n(ertex,)g(w)n(e)g (compute)h(its)f(distance)0 2307 y(to)28 b(\()p Fi(s;)14 b Fu(0\))28 b(as)f(one)h(plus)g(the)g(minim)n(um)h(distance)f(of)g(an)n (y)f(predecessor)f(of)i(the)g(v)n(ertex.)f(If)i(the)f(v)n(ertex)f(is)h (\()p Fi(s;)14 b Fu(0\))28 b(itself,)h(the)0 2407 y(distance)e(is)h (zero,)e(and)i(all)f(other)g(v)n(ertices)g(\()p Fi(a;)14 b Fu(0\))27 b(ha)n(v)n(e)g(no)g(predecessors)e(and)j(in\014nite)g (distance.)125 2507 y(Th)n(us)e(w)n(e)h(can)g(\014nd)g(the)h(optimal)f (strategy)e(for)i(the)g(initial)h(con\014guration)d(b)n(y)i(forming)f Fi(G)p Fu(,)i(computing)f(its)g(shortest)0 2606 y(path,)d(using)f(the)h (lo)r(cation)e(of)i(the)g(edges)e(from)i(\()p Fi(t;)14 b(i)p Fu(\))23 b(to)g(\()p Fi(s;)14 b(i)p Fu(\))24 b(to)f(partition)g (the)h(con\014guration)e(in)n(to)h(one-p)r(eg)g(sub)r(con\014g-)0 2706 y(urations,)i(and)h(applying)f(Corollary)f(1)h(to)h(eac)n(h)f(sub) r(con\014guration.)g(Since)h Fg(j)p Fi(G)p Fg(j)d Fu(=)g Fg(O)r Fu(\()p Fi(n)p Fu(\),)k(this)f(algorithm)f(runs)h(in)g(linear)0 2806 y(time.)3661 b Fg(u)-55 b(t)125 2982 y Fu(In)22 b(con)n(trast)g(to)h(these)f(results,)h(Uehara)e(and)i(Iw)n(ata)f([11]) g(sho)n(w)n(ed)f(that)i(in)h(t)n(w)n(o)e(or)f(more)h(dimensions)h(P)n (eg)e(Solitaire)h(is)0 3082 y(NP-complete.)f(Ho)n(w)n(ev)n(er,)e(the)j (complexit)n(y)e(of)i(\014nding)f(the)h(minim)n(um)f(n)n(um)n(b)r(er)g (of)h(p)r(egs)f(to)g(whic)n(h)g(a)g Fi(k)8 b Fg(\002)e Fi(n)21 b Fu(con\014guration)0 3181 y(can)27 b(b)r(e)h(reduced,)g(for)f (b)r(ounded)h Fi(k)e(>)c Fu(2,)27 b(remains)g(op)r(en.)0 3450 y Fm(2)112 b(Duotaire)0 3651 y Fu(Ra)n(vikumar)24 b([9])i(has)f(prop)r(osed)f(an)i(impartial)f(t)n(w)n(o-pla)n(y)n(er)e (game,)i(in)h(whic)n(h)f(pla)n(y)n(ers)f(tak)n(e)h(turns)h(making)f(P)n (eg)f(Solitaire)0 3751 y(mo)n(v)n(es,)f(and)h(who)r(ev)n(er)g(is)g (left)h(without)g(a)f(mo)n(v)n(e)g(loses.)f(W)-7 b(e)25 b(call)f(this)h(game)f(\\P)n(eg)e(Duotaire.")i(While)h(he)f(considered) g(the)0 3851 y(v)n(ersion)g(where)h(eac)n(h)g(mo)n(v)n(e)g(consists)g (of)g(a)g(single)h(hop,)f(in)h(the)g(spirit)g(of)f(the)h(game)f(w)n(e)h (will)f(start)h(with)g(the)g(\\m)n(ultihop")0 3950 y(v)n(ersion)g (where)h(a)g(series)g(of)g(hops)h(with)g(a)f(single)g(p)r(eg)g(can)h(b) r(e)g(made)f(in)h(a)f(single)g(mo)n(v)n(e.)125 4050 y(W)-7 b(e)24 b(recall)f(the)h(de\014nition)h(of)f(the)g Fk(Grundy)j(numb)l (er)c Fu(or)g Fk(nim-value)h Fi(G)h Fu(of)f(a)f(p)r(osition)h(in)g(an)g (impartial)f(game,)g(namely)0 4150 y(the)33 b(smallest)f(non-negativ)n (e)f(in)n(teger)h(not)g(app)r(earing)g(among)f(the)i(nim-v)-5 b(alues)33 b(of)f(its)h(options)f([4].)h(The)f Fg(P)7 b Fu(-p)r(ositions,)0 4250 y(in)28 b(whic)n(h)g(the)g(second)g (\(Previous\))e(pla)n(y)n(er)h(can)g(win,)h(are)f(those)h(with)g(nim-v) -5 b(alue)28 b(zero:)f(an)n(y)g(mo)n(v)n(e)g(b)n(y)g(the)i(\014rst)e (\(Next\))0 4349 y(pla)n(y)n(er)d(is)h(to)h(a)f(p)r(osition)g(with)h(a) f(non-zero)f Fi(G)p Fu(,)i(and)g(the)f(second)g(pla)n(y)n(er)f(can)h (then)i(return)e(it)h(to)f(a)g(p)r(osition)g(with)h Fi(G)e Fu(=)e(0.)0 4449 y(This)i(con)n(tin)n(ues)f(un)n(til)h(w)n(e)f(reac)n (h)g(a)g(p)r(osition)h(in)g(whic)n(h)g(there)f(are)g(no)g(mo)n(v)n(es,) g(in)h(whic)n(h)g(case)f Fi(G)g Fu(=)f(0)i(b)n(y)f(de\014nition;)i (then)0 4549 y(Next)i(is)g(stuc)n(k,)f(and)g(Previous)f(wins.)i (Similarly)-7 b(,)26 b(the)h Fg(N)12 b Fu(-p)r(ositions,)27 b(in)f(whic)n(h)h(the)g(\014rst)f(pla)n(y)n(er)f(can)i(win,)g(are)e (those)i(for)0 4648 y(whic)n(h)h Fi(G)23 b Fg(6)p Fu(=)g(0.)125 4748 y(The)28 b(nim-v)-5 b(alue)28 b(of)g(a)g(disjunctiv)n(e)h(sum)f (of)g(games,)g(in)g(whic)n(h)g(eac)n(h)g(mo)n(v)n(e)f(consists)h(of)g (a)g(mo)n(v)n(e)f(in)h(the)h(game)e(of)i(the)0 4848 y(pla)n(y)n(er's)c (c)n(hoice,)i(is)f(the)i Fk(nim-sum)p Fu(,)e(or)g(bit)n(wise)h (exclusiv)n(e)f(or)h(\(binary)f(addition)h(without)h(carrying\))d(of)i (the)g(nim-v)-5 b(alues)0 4948 y(of)31 b(the)f(individual)h(games.)f(W) -7 b(e)31 b(notate)f(this)h Fg(\010)p Fu(,)f(and)g(for)g(instance)h(4) 20 b Fg(\010)g Fu(7)27 b(=)h(5.)i(Lik)n(e)g(man)n(y)g(games,)f(p)r (ositions)i(in)f(P)n(eg)0 5047 y(Duotaire)d(often)h(quic)n(kly)f (reduce)g(to)g(a)h(sum)f(of)h(simple)g(p)r(ositions:)0 5224 y Fl(Lemma)i(1.)40 b Fk(In)29 b(either)g(version)h(of)g(Pe)l(g)f (Duotair)l(e,)g(a)g(p)l(osition)h(of)g(the)f(form)g Fi(x)14 b Fu(0\(01\))2774 5194 y Fj(\003)2812 5224 y Fu(00)g Fi(y)30 b Fk(is)f(e)l(qual)g(to)g(the)g(disjunctive)0 5324 y(sum)g(of)i Fi(x)p Fu(0)e Fk(and)i Fu(0)p Fi(y)s Fk(.)p eop %%Page: 4 4 4 3 bop 0 83 a Fk(Pr)l(o)l(of.)43 b Fu(An)n(y)28 b(attempt)g(to)f (cross)f(this)i(gap)f(only)g(creates)f(a)i(larger)d(gap)i(of)g(the)h (same)f(form;)h(for)f(instance,)g(a)g(hop)g(on)h(the)0 183 y(left)g(end)g(from)f(110\(01\))772 153 y Fs(n)816 183 y Fu(00)f(yields)i(0\(01\))1350 153 y Fs(n)p Ft(+1)1478 183 y Fu(00.)f(Th)n(us)g(the)h(t)n(w)n(o)f(games)g(cannot)g(in)n (teract.)898 b Fg(u)-55 b(t)125 347 y Fu(As)31 b(in)h(the)g(Ha)n(w)n (ai'ian)e(game)h(of)h(Konane)e([12],)h(in)n(teraction)g(across)f(gaps)g (of)i(size)f(2)h(seems)f(to)g(b)r(e)i(rare)d(but)i(b)n(y)f(no)0 447 y(means)24 b(imp)r(ossible.)h(F)-7 b(or)24 b(instance,)h(Previous)e (can)h(win)i(from)e(a)g(p)r(osition)h(of)g(the)g(form)f Fi(w)r Fu(00)p Fi(w)k Fu(b)n(y)c(strategy)g(stealing,)g(i.e.)0 546 y(cop)n(ying)k(eac)n(h)g(of)h(Next's)g(mo)n(v)n(es,)f(unless)h (Next)g(can)g(c)n(hange)f(the)h(parit)n(y)f(b)n(y)h(hopping)g(in)n(to)f (the)i(gap.)e(In)h(the)h(m)n(ultihop)0 646 y(case,)g(ho)n(w)n(ev)n(er,) f(Previous)h(can)h(sometimes)f(reco)n(v)n(er)f(b)n(y)i(hopping)g(in)n (to)g(the)g(gap)f(and)h(o)n(v)n(er)f(the)h(p)r(eg)g(Next)h(has)e (placed)0 746 y(there:)0 910 y Fl(Lemma)g(2.)40 b Fk(In)30 b(multihop)g(Pe)l(g)h(Duotair)l(e,)f(any)g(p)l(alindr)l(ome)h(of)g(the) f(form)800 1083 y Fi(w)16 b Fu(010010)e Fi(w)1202 1049 y Fs(R)1254 1083 y Fi(;)37 b(w)17 b Fu(01100110)d Fi(w)1801 1049 y Fs(R)1852 1083 y Fi(;)43 b Fk(or)30 b Fi(w)17 b Fu(00\(10\))2333 1049 y Fj(\003)2370 1083 y Fu(11100111\(01\))2854 1049 y Fj(\003)2887 1083 y Fu(00)d Fi(w)3046 1049 y Fs(R)0 1256 y Fk(is)30 b(a)g Fg(P)7 b Fk(-p)l(osition.)0 1421 y(Pr)l(o)l(of.)43 b Fu(Previous)31 b(steals)h(Next's)h(strategy)e(un)n (til)i(Next)h(hops)e(in)n(to)g(the)h(gap.)f(Previous)f(then)i(hops)g (in)n(to)f(the)h(gap)f(and)0 1520 y(o)n(v)n(er)h(Next's)j(p)r(eg,)f (lea)n(ving)f(a)g(p)r(osition)h(of)g(the)h(form)f(in)g(Lemma)g(1.)g (The)g(games)f(then)h(separate)f(and)h(Previous)e(can)0 1620 y(con)n(tin)n(ue)27 b(stealing)g(Next's)h(strategy)-7 b(,)26 b(so)h(the)h(nim-v)-5 b(alue)28 b(is)f Fi(G)p Fu(\()p Fi(w)r Fu(0\))20 b Fg(\010)e Fi(G)p Fu(\(0)p Fi(w)2477 1590 y Fs(R)2532 1620 y Fu(\))23 b(=)g(0.)125 1719 y(T)-7 b(o)27 b(sho)n(w)f(that)i(this)g(remains)f(true)g(ev)n(en)h (if)g(Next)g(tries)f(to)g(hop)h(from)f Fi(w)f Fu(=)c Fi(v)s Fu(11,)27 b(consider)g(the)h(follo)n(wing)e(game:)1015 1887 y Fi(v)s Fu(11)14 b(001110011100)f(11)o Fi(v)1799 1857 y Fs(R)1015 1987 y Fi(v)s Fu(00)h(101110011100)f(11)o Fi(v)1799 1956 y Fs(R)1894 1987 y Fu(Next)28 b(hops)f(from)g(the)h (left)1015 2086 y Fi(v)s Fu(00)14 b(101110011101)f(00)o Fi(v)1799 2056 y Fs(R)1894 2086 y Fu(Previous)26 b(steals)h(his)g (strategy)1015 2186 y Fi(v)s Fu(00)14 b(101001011101)f(00)o Fi(v)1799 2156 y Fs(R)1894 2186 y Fu(Next)28 b(hops)f(in)n(to)h(the)g (breac)n(h)1015 2286 y Fi(v)s Fu(00)14 b(101010000101)f(00)o Fi(v)1799 2256 y Fs(R)1894 2286 y Fu(Previous)26 b(hops)h(t)n(wice)0 2467 y(No)n(w)c Fi(v)28 b Fu(and)23 b Fi(v)453 2437 y Fs(R)532 2467 y Fu(are)g(separated)f(b)n(y)i(t)n(w)n(o)f(gaps)g(of)h 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Fq(Lecture)g(Notes)f(in)h(Computer)f (Science)h Fo(1350)p Fq(,)h(Springer)e(\(1997\))i(233{242.)0 4875 y(10.)43 b(J.E.)20 b(Hop)r(croft)g(and)f(J.D.)h(Ullman,)f Fa(Intr)l(o)l(duction)k(to)f(A)n(utomata)h(The)l(ory,)g(L)l(anguages,)g (and)f(Computation.)e Fq(Addison-W)-6 b(esley)g(,)140 4967 y(1979.)0 5058 y(11.)43 b(R.)25 b(Uehara)h(and)f(S.)h(Iw)n(ata,)g (\\Generalized)i(Hi-Q)d(is)h(NP-complete.")f Fa(T)-6 b(r)l(ans)29 b(IEICE)d(73)p Fq(.)0 5149 y(12.)43 b(W.S.)25 b(Sizer,)i(\\Mathematical)g(notions)f(in)f(preliterate)i(so)r (cieties.")h Fa(Mathematic)l(al)h(Intel)t(ligenc)l(er)d Fo(13)g Fq(\(1991\))h(53{59.)0 5240 y(13.)43 b(R.J.)26 b(P)n(arikh,)g(\\On)f(con)n(text-free)h(languages.")i Fa(Journal)g(of)f(the)i(A)n(CM)c Fo(13)h Fq(\(1966\))h(570{581.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF