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A Note on the Hardness of Tree Isomorphism 1 Introduction A Note on the Hardness of Tree Isomorphism Birgit Jenner Pierre McKenzie Jacob o Toran Tubingen and Ulm Montreal and Tubingen Ulm Decemb er Abstract In this note we prove that the tree isomorphism problem when trees are en 1 1 co ded as strings is NC hard under DLOGTIMEreductions NC completeness 1 thus follows from Busss recent NC upp er b ound By contrast we prove that testing isomorphism of two trees enco ded as p ointer lists is Lcomplete Intro duction GI the graph isomorphism problem is one of the most intensively studied prob lems in Theoretical Computer Science Historically the main source of interest in GI has b een the evidence that GI is probably neither in P nor NPcomplete but other sources of interest include the sophistication of the to ols develop ed to attack the problem for example and the connexions b etween GI and structural complexity see Understandably many GI restrictions have b een considered For example P upp er b ounds are known in the cases of planar graphs or graphs of b ounded valence However none of these GI restrictions are known to b e complete for a natural complexity class and it seemed that the problem lacks the structure needed for a hardness result In the sp ecial case of trees a linear sequential time algorithm for the problem was already known in to Aho Hop croft and Ullman In an NC algorithm was develop ed by Miller and Reif One year later Lindell obtained an L upp er b ound Finally in a subtle algorithm able to test two trees for isomorphism in NC was devised by Buss Buss in asks whether tree isomorphism is NC hard Here we answer this ques tion armatively showing the hardness of the problem under DLOGTIME manyone reducibility Hence tree isomorphism is NC complete Trees thus provide the rst class of graphs for which the isomorphism problem captures a natural complexity Corresp onding author Pierre McKenzie WilhelmSchickardInstitut fur Informatik Univer sitat Tubingen Sand D Tubingen Germany Phone Fax Email mckenzieinformatikunitue binge nde class Moreover so far the problem of evaluating a Bo olean formula and the problem of multiplying p ermutations on ve p oints and some of their varia tions were the only two NC complete problems known Tree isomorphism is a third such problem As noted by Buss cho osing a graph representation is critical when working at the level of NC Buss uses Miller and Reif s string representation for trees on the grounds that when the pointer representation is used the deterministic transitive closure problem can b e reduced to the descendant predicate and the former is known to b e complete for logspace Section We prove here that the tree isomorphism problem is Lhard under manyone NC reducibility when the p ointer representation is used Hence tree isomorphism in the p ointer representation is Lcomplete by Lindell or by Buss Tree isomorphism thus captures in this way another very natural complexity class Preliminaries We assume familiarity with basic notions of complexity theory such as can b e found in the stardard b o oks in the area In particular we simply recall that AC TC NC L where L is the set of languages accepted by deterministic Turing machines using logarithmic space NC is the class of languages recognized by DLOGTIMEuniform families of logarithmic depth p olynomial size Bo olean circuits of b ounded fanin over the basis f g and AC resp TC is the set of languages recognized by DLOGTIMEuniform families of constant depth p olynomial size Bo olean circuits of unb ounded fanin over the basis f g resp over the basis consisting solely of the MAJORITY function For simplicity we will often not distinguish b etween a class of Bo olean func tions like NC and the corresp onding class of functions from f g to the natural numb ers sometimes called FNC Reducibilities We use the following reducibilities D LT DLOGTIME reducibility We use this reducibility for the NC completeness results Following we say that for languages A B a function f manyone reducing A to B is a DLOGTIME reduction if f increases the length of strings only p olynomially jf xj pjxj for a p olynomial p and some DLOGTIME Turing machine can decide given an input of c i x whether the ith symb ol of f x is c We write D LT A B if there exists a DLOGTIME reduction reducing A to B Recall that it is customary in the case of Turing machines op erating in sublinear time to access the input via a sp ecial input index tap e for more details on DLOGTIME Turing machines see for example 1 NC NC manyone reducibility m We use this reducibility for the Lcompleteness and all other results Here the many one reduction f is required to b e computable in FNC Trees and representations All trees discussed in this pap er are ro oted hence implicitly directed and unordered ie the ordering of the descendants of a no de do es not matter In some of our constructions we use colored trees A tree with n no des is said to b e colored if each no de in the tree is lab elled with a p ositive integer no greater than n An isomorphism b etween two colored trees T and T is a bijection b etween their no de sets which maps the ro ot of T to the ro ot of T preserves the colors and preserves the edges Two colored trees are isomorphic denoted T T i an isomorphism exists b etween them These notions apply mutatis mutandis to noncolored trees The main computational problem of interest in this pap er is Colored Tree Isomorphism CTI Given Two colored trees T and T Problem Determine whether T T We consider two dierent representations for enco ding trees the string repre sentation and the p ointer representation As we will see in the small complexity classes we are dealing with the representation used might change the complexity of the problem In the string representation trees are represented over an alphab et contain ing op ening and closing parentheses A string representation of a tree T is dened recursively in the following way The tree with a single no de is represented by the string and if T is a tree consisting of a ro ot and subtrees T T in k any order with string representations a representation of T is given k by Observe that a tree might have dierent string representations k dep ending on the order of the descendants of any of its no des Colored trees can b e enco ded in the same way using colored parentheses Let C b e the set of colors An op ening parenthesis in a colored tree is represented by followed by log jC j bits enco ding the color Note that we only need to color the op ening parentheses The pointer representation is a more standard way to enco de graphs In this case we consider the trees given by a list of pairs of no des representing directed edges As in the previous case if we deal with colored trees with the representation of a no de we include log jC j bits to enco de its color For many tree problems in NC and L completeness results seem to dep end on the representation used For example the reachability problem on forests which is Lcomplete in the p ointer representation can b e solved in NC and even TC in the string representation Analogously the Bo olean formula value problem which is complete for NC in the string representation b ecomes Lcomplete when describ ed using trees given in the p ointer representation In fact changing from p ointer to string representation is FLcomplete Another imp ortant observation is that it is p ossible to test within the classes NC and L whether a given input is a correct enco ding of a tree in the string representation and resp ectively in the p ointer form Tree Isomorphism for Trees Given by Strings We show rst that deciding whether two trees given in string representation are isomorphic is NC complete We rst consider colored trees for which the hardness pro of is simpler D LT Lemma In the string representation CTI is NC hard under Pro of Following Barrington Immerman and Straubing Lemma it suf ces to reduce the problem of evaluating a balanced DLOGTIMEuniform family of Bo olean expressions made up of alternating layers of ANDs and ORs to CTI The core of the reduction is the simple construction from describ ed in page for the purp ose of simulating ANDs and ORs using graph isomorphism questions We adapt this construction as follows Consider four colored trees G G H and H Then the colored trees G G H H Tree H Tree G ^ ^ have the prop erty that G H i G G H H and the colored trees ^ ^ H H G G H H G G Tree H Tree G _ _ have the prop erty that G H i G G H H Observe that the _ _ ORconstruction doubles the size of the initial trees for G G and H H that is from trees each having the same size s the ORconstruction would pro duce trees each having size s Now pick two singleno de trees T and T and assign them dierent colors Start ing from the CTI instance T T to represent a TRUE input and the CTI instance T T to represent a FALSE input it is trivial to construct from a depthd Bo olean d expression f with no negation gates two colored trees G and H having O no des this is a rough upp er b ound and verifying the prop erty that G H i f evaluates to TRUE To see that this yields a DLOGTIME reduction note it is p ossible to add a few dummy no des in order to mo dify the constructs ab ove in such a way that if G G H and G are full binary colored trees then so are G H G and H ^ ^ _ _ and moreover the color o ccurrences in these resp ective constructs are the same
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