BRICS BRICS RS-94-31 Nisan & Ta-Shma: Symmetric Logspace is Closed Under Complement Basic Research in Computer Science Symmetric Logspace is Closed Under Complement Noam Nisan Amnon Ta-Shma BRICS Report Series RS-94-31 ISSN 0909-0878 September 1994 Copyright c 1994, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK - 8000 Aarhus C Denmark Telephone:+45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] Symmetric Logspace is Closed Under Complement Noam Nisan Amnon TaShma noamcshujiacil amcshujiacil September Abstract We present a Logspace manyone reduction from the undirected stconnectivity prob lem to its complement This shows that SL co SL Intro duction This pap er deals with the complexity class symmetric Logspace SL dened by Lewis and Papadimitriou in LP This class can b e dened in several equivalentways Languages which can b e recognised by symmetric nondeterministic Turing Machines that run within logarithmic space See LP Languages that can b e accepted by a uniform family of p olynomial size contact schemes also sometimes called switching networks See Raz Languages which can b e reduced in Logspace via a manyone reduction to USTCON the undirected stconnectivity problem A ma jor reason for the interest in this class is that it captures the complexityofUSTCON The input to USTCON is an undirected graph G and twovertices in it s t and the input should b e accepted if s and t are connected via a path in G The similar problem ST CON where the graph G is allowed to b e directed is complete for NL nondeterministic Logspace wn to b e in SL or co SL eg colourabilityis Several combinatorial problems are kno complete in co SL Rei The following facts are known regarding SL relative to other complexity classes in the vicinity L SL RL NL Here L is the class deterministic Logspace and RL is the class of problems that can b e accepted with onesided error by a randomized Logspace machine running in p olynomial This work was supp orted byBSFgrant and byaWolfeson award administered by the Israeli Academy of Sciences The work was revised while visiting BRICS Basic Research in Computer Science Centre of the Danish National ResearchFoundation time The containment SL RL is the only nontrivial one in the line ab ove and follows directly from the randomized Logspace algorithm for USTCON of AKL It is also L known that SL SC Nis SL L KWandSL DS P AC E log n NSW After the surprising pro ofs that NL is closed under complementwere found Imm Sze Boro din et al BCD asked whether the same is true for SL They could prove only the weaker statement namely that SL co RL and left SL co SL as an op en problem In this pap er wesolve the problem in the armativeby exhibiting a Logspace manyone reduction from USTCON to its complement Quite surprisingly the pro of of our theorem do es not use inductive counting as do the pro ofs of NL co NL and is in fact even simpler than them however it uses the AKS sorting networks Theorem SL co SL It should b e noted that the monotone analogues see GS of SL and co SL are known to b e dierent KW SL SL SL As a direct corollary of our theorem we get that L SL SL where L is the class of languages accepted by Log space oracle Turing machines with oracle from SL and SL SL is dened similarly b eing careful with the waywe allow queries see RST SL SL Corollary L SL SL This also shows that the symmetric Logspace hierarchy dened in Rei collapses to SL Pro of of Theorem Overview of pro of We show that we can upp er and lower b ound the numb er of connected comp onents of a graph using connectivity problems We upp er b ound this numb er using a transitiveclosure metho d which can b e easily done since we are allowed to freely use connectivity problems However trying to lowerb ound the numb er of connected comp onents this way requires nega tion The heart of the pro of lies in lowerb ounding the numb er of connected comp onents and weachieve this in a surprisingly easy wayby computing a spanning forest In subsection weshowhowtocombine many connectivity problems to one single con nectivity problem In subsection we showhow to nd a spanning forest using connectivity problems In subsection weshowhow to use this spanning forest to nd the number of connected comp onents of a graph and howwe solvethest nonconnectivity problem with it Pro jections to USTCON In this pap er we will use only the simplest kind of reductions ie Log S pace uniform pro jec tion reductions SV Moreover we will b e interested only in reductions to USTCONIn this subsection we dene this kind of reduction and weshow some of its basic prop erties n Notation Given f f g f g denote by f f g f g the restriction n n of f to inputs of length n Denote by f the k th bit function of f ieiff f g nk n n k n f g then f f f n n nk n n Notation We represent an nnode undirectedgraph G using variables x fx g ij ij n st x is i i j E GIff x operates on graphswewillwrite f G meaning that ij n the input to f is a binary vector of length representing G Definition We say that f f g f g reduces to USTCONm m mn if there is a uniform family of S pacelog n functions f g st for al l n and k nk is a projection ie is a mapping from fi j g to f x x g nk nk ij m i i in Given x dene G to bethegraph G fmgE where x x g E fi j j i j or i j x and x or i j x and x nk nk i i nk i i It should hold that f x thereisapath from to m in G nk x If is restricted to the set f x g we say that f monotonically reduces to USTCONm i in Lemma If f has uniform monotone formulae of size sn then f is monotonical ly re ducible to USTCONO sn Pro of Given a formula recursively build G s t as follows If x then build a graph with twovertices s and t and one edge b et ween them i lab elled with x i If and G s t the graphs for i then identify s with t and i i i i dene s s t t If and G s t the graphs for i then identify s with t and i i i i s with t and dene s s t and t s t Using the AK S sorting networks AKS which b elong to NC we get Corollary Sort f g f g which given a binary vector sorts it is monotoni cal ly reducible to USTCONpol y Lemma If f monotonical ly reduces to USTCONm and g reduces to USTCONm then f g reduces to USTCONm m where is the standard function composition operator Pro of f monotonically reduces to a graph with m vertices where each edge is lab elled with one of f x g In the comp osition function f g each x is replaced by x g y i i i i which can b e reduced to a connectivity problem of size m Replace each edge lab elled x i with its corresp onding connectivity problem Finding a spanning forest In this section we showhow to build a spanning forest using USTCON This construction was also noticed by Reif and indep endentl y by Co ok Rei Given a graph G index the edges from to mWe can view the indices as weights to the edges and as no two edges have the same weight weknow that there is a unique minimal spanning forest F In our case where the edges are indexed this minimal forest is the lexicographically rst spanning forest It is well known that the greedy algorithm nds a minimal spanning forest Let us recall how the greedy algorithm works in our case The algorithm builds a spanning forest F which is at the b eginning empty F Then the algorithm checks the edges one by one according to their order for each edge e if e do es not close a cycle in F then e is added to the forest ie F F feg At rst glance the algorithm lo oks sequential however claim shows that the greedy algorithm is actually highly parallel Moreover all we need to check that an edge do es not participate in the forest is one st connectivity problem over a simple to get graph Definition For an undirectedgraph G denote by LF F G the lexicographical ly rst span ning forest of GLet n SF G f g be i j LF F G SF G ij otherwise Lemma SF reduces to USTCONpol y Pro of Let F b e the lexicographically rst spanning forest of GFor e E dene G to e b e the subgraph of G containing only the edges fe E j indexe indexeg e E i is not connected to j in G Claim e i j F e Pro of Let e i j E Denote by F the forest which the greedy algorithm built at the e time it was checking eSoe F e do es not close a cycle in F e e F and therefore e do es not close a cycle in F but then e do es not close a cycle e in the transitive closure of F and in particular e do es not close a cycle in G e e e do es not close a cycle in G therefore e do es not close a cycle in F and e F e e Therefore SF G x i is connected to j in G ij ij ij Since x can b e viewed as the connectivity problem over the graph with twovertices ij and one edge lab elled x it follows from lemmas that SF reduces to USTCON ij Notice however that the reduction is not monotone Putting it together First wewant to build a function that takes one representative from each connected com p onent We dene LI Gtobe ithevertex i has the largest index in its connected i comp onent n Definition LI G f g i has the largest index in its connectedcomponent LI G i otherwise Lemma LI reduces to USTCONpol y Pro of W n LI G i is connected