Longest paths in Planar DAGs in Unambiguous Logspace

Longest paths in Planar D A Gs in Unam biguous Logspace Nutan Lima y e and Meen a Maha jan and Pra jakta Nim bhork ar The Institute of Mathematical Science s, Chennai 600 113, India. { n utan,meena,pra jakta } @imsc.res.in Octob er 23, 2018 Abstract W e sho w via t wo differen t a lgorithms that finding the length of the longest path in plana r directed acyclic graph (D AG) is in unambiguous logspace UL , and also in the complement cla s s co- UL . The result extends to tor o idal D AGs as well. 1 In tro duction Consider the follo wing pr ob lems in graphs: Reach = { ( G, s, t ) | G con tains a path from s to t } Distance = { ( G, s, t, k ) | G con tains a path of lengt h ≤ k f rom s to t } Long-P ath = { ( G, s, t, k ) | G has a simple path of length ≥ k from s to t } These problems ha v e w idely differing complexities: some of the results b elo w are folklore, some are recent adv ances. Reach is NL -complete for general graph s and remains NL -hard ev en if the graphs are acyclic. It is L -complete for un directed graphs [Rei05], and is sandwiched b etw een L and UL ∩ co-UL for planar d irected graph s [BTV07]. Distance is NL -complete for general graphs, and remains NL -hard even if the graph s are acyclic, or if the graphs are undir ected, but it is in UL ∩ co-UL for planar directed graphs [TW07]. Long-P ath is NP -complete f or ge ner al graphs, since it includes Hamiltonian paths as a sp ecial case. It remains NP -hard for planar u ndirected graphs. It is NL -complete for directed acyclic grap h s. Ho w ev er its complexity f or p lanar directed acyclic graphs is, to the b est of our kn owledge, not y et studied. In this note we consider th is com bination of planarit y and acyclici ty for Long-P ath . Our main result is: Theorem 1 PD LP , the Long-P ath pr oblem f or planar dir e cte d acyclic gr aphs, is in UL ∩ co-UL . Th u s Long-P ath shares the c u r ren t b est-kno wn upp er b oun ds f or Reach and Distance for suc h graphs. W e also address th e question of when t h e three problems are indeed equ iv alen t on D AGs, and giv e partial b ound s (Th eorem 8,10). A recen t resu lt in [JT07] shows that for an im p ortan t sub class of planar D A Gs, namely series-parallel graphs , the three pr oblems are indeed equiv alen t and are all L -complete. Theorem 1 is in fact an unobserved corollary of th eir constru ction (see also [J T06]. An analogous result for planar D A Gs equating the thr ee problems w ould b e nice, but is n ot kno wn. F or graphs with em b eddings on the torus, [ADR05] sho ws that reac habilit y is n o h arder than planar reac hability . W e observe that Distance and Long-P ath are also n o h arder than the planar v ersions (Corollary 6). 1 2 Kno wn results, and Prepro cessing W e use the follo w ing results: Lemma 2 ([BTV07]) Reach in planar dir e cte d gr aphs is in UL ∩ co-UL . Lemma 3 ([T W07]) Distance in planar dir e cte d gr aphs is in UL ∩ co-UL . Lemma 4 ([JT 07]) Distance and Long-P ath in series-p ar al lel gr aphs ar e e quivalent. Lemma 5 ([ADR05]) R each (T orus) lo gsp ac e-many-one r e duc es to planar Reach . F or an y sub class C of g rap h s, let Reach ( C ), Distance ( C ), and Long-P ath ( C ) denote the r estric- tion of these problems to instances from C . F or directe d acyclic g raph s, ( G, s, t ) ∈ Reach ⇔ ( G, s, t, | V | ) ∈ Distance ⇔ ( G, s, t, 0) ∈ Long-P ath . S o Distance ( C ) and Long-P ath ( C ) are at least as hard as Reach ( C ) for any sub class C of directed acyclic graphs. Consider any directed acyclic instance ( G, s, t, k ) of D istance or Long-P ath . By (parallel) queries of the form ( G, s, u ) or ( G, u, t ) to Reach , w e can remo v e all vertic es that do not fi gure on some s -to- t path to obtain in L Reach a single-source ( s ) single-sink ( t ) graph G ′ , and all qu eries to Reach in vo lve only th e g raph G . S o no w onw ards w e only consider the case where w e w an t to find a long path b et w een the u nique source and the u nique sink. If the input graph G is not planar but can b e emb ed ded on a torus, then we use the construction of Lemma 5 . This giv es a planar graph G ′ with the fol lowing prop erties: There are l ∈ O ( n ) copies of G cut and stitc hed together, and hence there are l v ertices t 1 , . . . , t l and one sp ecial v ertex, sa y s 1 , suc h that ∃ ρ : s → ∗ G t ∧ | ρ | = l ⇔ ∃ i ∃ ρ i : s 1 → ∗ G ′ t i ∧ | ρ i | = l Hence Corollary 6 Distance ( T orus ) ≤ log m Distance ( Planar ) Long-P ath ( T or oidal DA Gs ) ≤ log m Long-P ath ( Planar DAGs ) 3 Algorithm using distance computation Our first algorithm for Long- Path in planar D A Gs uses a simple extension of Lemma 4. Lemma 7 Distance and Long-P ath in planar dir e cte d acyclic g r aphs ar e e quivalent mo dulo planar r e achability. This actually follo ws from [JT 07] itself; though they claim th eir result only f or ser ies-parallel graphs, it wo rks for single-source single-sink acyclic graph s as w ell, where s and t in the inpu t instance are th e source and sink resp ectiv ely . It d oesn’t even seem to u se planarit y . T o mak e th is clear, we present b elo w in Theorem 8 their pro of simplified by sp ecialising to unw eigh ted graphs, and stated with minim um conditions. Lemma 7 is an ob vious corollary . 2 Theorem 8 L et C b e any sub c lass of dir e cte d acyclic gr aphs. Ther e is a f unction f , c omputable in L with or acle ac c ess to Reach ( C ) , that r e duc es D istance ( C ) to Long-P ath ( C ) and Long-P ath ( C ) to Di stance ( C ) . Pro of: Let G = ( V , E ) b e a directed acyclic graph with a un ique source s an d a unique sink t . Ev ery v ertex of G l ies on some s → ∗ t path. ( G is un we ighted, so all edges hav e w eigh t 1.) Let M b e the n um b er of edges in G . Construct a new graph G ′ = ( V ′ , E ′ ) as follo ws: F or eac h u ∈ V , defi n e P u = { x ∈ V | x → ∗ G u } . S in ce s is the unique sour ce, ∀ u , s ∈ P u . Also define E u = {h x, y i | x ∈ P u , y 6∈ P u } . S ince G is ac yclic, ∀h x, y i ∈ E , h x, y i ∈ E x . Let ρ b e any s → ∗ t path. F or ev ery v ertex u ∈ V , | ρ ∩ E u | = 1. Why? Note that s ∈ P u , t 6∈ P u , and along the path ρ , we tr an s it from b eing in P u to b eing outside P u exactly once. Let this transition o ccur on edge h x, y i . Then h x, y i ∈ E u , and no ot h er edge of ρ ca n b e in E u . T o obtain G ′ , w e replace eac h edge e = h u, v i b y a path o f length l uv determined as follo ws: l uv = 2   X x ∈ V : e ∈ E x out-degree( x )   − 1 = 2   X x ∈ V : u ∈ P x ,v 6∈ P x out-degree( x )   − 1 Since G is acyclic , the v ertex u itself alwa ys qualifies in the ab o v e sum, and so l uv is p ositiv e. No w the crucial claim: eac h s → ∗ t path ρ in G , of length | ρ | in terms of num b er of edges, is transformed by the abov e to a path in G ′ of length exactly 2 | E | − | ρ | . Th is is b ecause the lengt h of the transformed path is X uv ∈ ρ l uv = X uv ∈ ρ   2   X x ∈ V : u ∈ P x ,v 6∈ P x out-degree( x )   − 1   = 2   X uv ∈ ρ X x ∈ V : u ∈ P x ,v 6∈ P x out-degree( x )   − | ρ | = 2 X x ∈ V   out-degree( x ) X e ∈ ρ ∩ E x 1   − | ρ | = 2 X x ∈ V out-degree( x ) · | ρ ∩ E x | ! − | ρ | = 2 X x ∈ V out-degree( x ) − | ρ | = 2 | E | − | ρ | It th us follo ws that t h e longest (shortest) path in G is mapp ed to the shortest (longest, resp ec- tiv ely) path in G ′ . In fact, if the s → ∗ t paths are ord ered m onotonica lly w ith resp ect to length, then the ab o ve transformation precisely rev erses this ordering. Hence the reduction function f maps ( G, s, t, k ) to ( G ′ , s, t, 2 | E | − k ). The next crucial observ ation: G ′ can b e obtained fr om G in logspace w ith oracle access to Reach , wh ere all q u eries inv olv e only th e graph G . This is b ecause obtaining G ′ merely inv olv es finding the sets P u , E u . Theorem 1 follo ws from Theorem 8 and Lemma 3. 3 4 Algorithm using inductiv e coun ting There is another metho d to obtain Theorem 1, bypassing Theorem 8 b ut u s ing Lemmas 2,3 . W e sk etc h it here b ecause it is instructiv e to see ho w doub le inductiv e count ing can b e used, and also b ecause it sa ys so methin g more g eneral as we ll: it places Long-P ath in ( UL ∩ co-UL ) ⊕ Reach (C ) for an y family C of acyclic max-unique graphs (Theorem 10). The initial steps are similar to those used in [TW07] to plac e p lanar Distance in UL ∩ co-UL . 1. Given a graph ˆ G , make it single-source s ingle-sink G as describ ed in the p repro cessing step. Reduce the degree of eac h verte x to 3. (T o r educe the degree of no des, in [ADR05] a vertex of degree d is replaced b y a cycle of length d . Sin ce we cannot afford to in tro duce cycles, w e use the trick of [CD06 ]; insert in coming and outgoing tr ees at eac h v ertex.) This constru ction maps edges to paths, a n d we can iden tify a unique new edge as ”resp onsible” for eac h orig inal edge. W e mark su ch edges. Em b ed G in to a grid using the [ADR05] reac habilit y-preserving construction. The output of this step is a grid graph G ′ , with the edges of G (original edges) marked in G ′ and is obtained in logspace. If the original graph G had n v ertices, the new grid graph is of dimensions n 2 × n 2 . 2. T he graph G ′ is then sub ject to a weig hting sc heme building up on t h at of [BTV07], and can b e describ ed as follo ws: ev ery horizon tal edge e gets weigh t n 4 + (mark( e ) × n 8 ), and ev ery v ertical edge e gets w eigh t n 4 + (mark( e ) × n 8 ) + (up(e) × col(e)), where mark( e ) is one if the edge e is marked; zero otherwise, col( e ) equals the column num b er in wh ic h the edge e app ears, and up( e ) is +1 if t h e ed ge e is u p wa rd s, − 1 o therw ise. T h is is t h e graph G ′′ . 3. T he last step in [TW07] is to u se the double counting tec hn iqu e of [RA97] on the min-unique graph G ′′ . Th e idea h ere is to use the indu ctiv e counting coun ter c k that k eeps trac k of num b er of v ertices within distance k , and to u se a cumula tive p aths counte r s k that kee p s trac k of the shortest paths of the no des so coun ted. The first coun ter allo ws c hec king th e complement of reac hability , the second allo ws doing so unambiguously . As men tioned in [TW07], a third coun ter m k trac king cumulativ e marked edges can b e add ed, allo wing distance computation uniquely . Can w e directly use this strategy for long paths as w ell? The argument of [T W07] concerning S tep 2 is restricted to shortest paths; ho wev er, one can observ e something more general ab out the ab o v e weigh ting sc heme. Observ at ion 9 F or any length l , al l the st p aths of length l in G wil l b e map p e d to p aths of weight gr e ater than ( l × n 8 ) and less than (( l + 1) × n 8 ) in G ′′ , and the maximum weight and the minimum weight p ath s in this r ange wil l b e unique. Thus G ′′ is b oth min-un iqu e and max-un ique : f or e ach p air u, v , if ther e is a p ath fr om u to v , then the shortest and the longest p aths ar e unique. Observ ation 9 al r eady guaran tees a max-unique graph. Step 3 ab ov e can not b e us ed as it is. F or computing the shortest path, we can initialise c 0 = 1 and Σ 0 = 0. If the same semantics is to b e used for computing th e longest path, then c 0 should b e the n um b er of v ertices havi n g length of the longe st path from s at least 0, and should be initialised to n . How ev er Σ 0 should then con tain the total lengths of all the longest paths, whic h 4 is an unknown quan tit y . T o han d le this, we redefin e Σ k to b e s u m of lengths of the longest paths for those v ertices wh ose longest p ath to t is of length at m ost k . This allo ws a pro cedure similar to [RA97] to work correctly , bu t no w it is n o longer unambiguous. T o make it unambiguous, we in tro duce more n ondeterminism into the [RA97] p ro cedure. W e guess the sum of lengths of a ll the u → ∗ t longest paths a priori and tally it in the end with the final s k . The detailed pro cedures are giv en b elo w, whic h imply the follo wing: Theorem 10 L et G b e a dir e cte d acyclic gr aph with a unique sour c e s , a unique sink t , such that G is max-uni que. (F or e ach p air u, v , if ther e is a p ath f r om u to v , then the longest uv p ath is unique.) Then the length of the longest st p ath c an b e c ompute d in UL ∩ co-UL . The pro of follo ws from Claims 11, 12, 13 and 1 4 . Notatio n : D ( v ) = Length of the lo n gest path from v to t . S k = { v | D ( v ) ≥ k } , c k = | S k | , Σ k = X v ∈ V \ S k D ( v ) , T = X v ∈ V D ( v ) Algorithm 1 M ain Input: G, s, t Guess nondeterministically M = P v ∈ V D ( v ). n ≤ M ≤ n 2 . c 0 ← n, Σ 0 ← 0 , k ← 0 while c k 6 = 0 do k ← k + 1 Up date ( c k and Σ k ) end while if Σ k 6 = M then Halt and reject else Accept end if Algorithm 2 U p date : Pro cedure for up dating c k and Σ k Input: G, s, t, c k − 1 , Σ k − 1 c k ← c k − 1 , Σ k ← Σ k − 1 for all v ∈ V do if T est ( G, k − 1 , c k − 1 , Σ k − 1 , v )=tru e then if for all out-neigh b our s x of v , T est ( G, k − 1 , c k − 1 , Σ k − 1 , x )=false then c k ← c k − 1 , Σ k ← Σ k + k − 1 end if end if end for Claim 11 If the g uesse d value of M is c orr e ct (i.e. M = T ), then algorithm T est , given th e c orr e ct values of c k and Σ k as input, r ep ort s a de cision on exactly one p ath. 5 Algorithm 3 T est : An unambiguous pro cedure t o determin e if D ( v ) ≥ k Input: G, k , c k , Σ k , v count = n, s um = 0 , path.to.v = tr ue, sum ′ = 0 for all x ∈ V do Guess nondeterministically if D ( x ) ≥ k if Guess is n o t hen Guess a path of le n gth l < k fr om x to t . { If this fails then reject and halt. } count ← count − 1 sum ← sum + l if x = v then path.to.v =false end if else Guess a path of le n gth l ′ ≥ k from x to t . { If this fails then reject and halt. } sum ′ ← sum ′ + l ′ end if end for if count = c k and sum = Σ k and sum ′ + sum = M then return path.to.v else Reject and halt. end if Pro of: The p rocedu re T est , on eac h run R , guesses an x → ∗ t path R x for eac h v ertex x . Dep end ing on its guess for D ( x ) ≥ k , it adds the length of R x to either sum or su m’. Finally these ha ve to add up to M for T est to rep ort a decision. When M = T , M is ind eed the sum of all D ( x ). This can matc h sum +sum’ exactly when all the guessed paths R x are longest. Since G is max-unique, this happ ens o n exactly one run. Claim 12 F or any guesse d value o f M , given the c orr e ct values of c k and Σ k as input, al l p aths of algorithm T est that do not le ad to r eje ction always r eturn the c orr e ct de cision. Pro of: As describ ed in th e preceding pro of, eac h run of T est guesses a path R x for eac h x . It may guess a path of le n gth shorter than D ( x ), but not longer. Sin ce count is d ecremen ted only wh en it guesses that D ( x ) < k , and for other guesses some w itnessing p ath of le n gth a t least k is found, at the end the v alue of count is at most as large as c k . Supp ose on some run T est returns a decision. T hen on this run count = c k . S upp ose further that the decision is wrong. Case 1: D ( v ) < k , but T est rep orts that it is larger. This cannot happ en, sin ce T est has to find a witnessing path of length at least k . Case 2: D ( v ) ≥ k , but T est rep orts t h at it is smaller. Then this run o f T est do es not ac count for v in count . S o at the end of the run, co unt < c k , a con tradiction. Claim 13 If the q ueries ( D ( v ) ≥ k ) ar e answer e d c orr e ctly by T est , then given c k − 1 and Σ k − 1 , the values of c k and Σ k ar e up date d c orr e ctly b y algorithm Up date . 6 Pro of: Up date starts by assuming that S k = S k − 1 and so c k = c k − 1 . Note that S k ⊆ S k − 1 , so Up date only has to detec t when to remo v e vertices from its curren t S k . F or eac h v , Up da te c hec ks whether D ( v ) ≥ k − 1 and D ( u ) < k − 1 for all out-neigh b our s u of v . If this h olds, then the longest p ath from v to t is of length exactly k − 1 and v / ∈ S k . Thus the pro cedure decremen ts c k b y 1 and incremen ts Σ k b y k − 1. So if all the queries are answ ered correctly by T est , then wh at Up date do es is correct. Claim 14 The algorithm Mai n is c orr e ct and unambiguous. Pro of: Main s tarts with the correct v alues of c 0 and Σ 0 . F rom claims 12 and 13, the correctness of Main is immediate. In p articular, the final v alue of Σ k is alw a ys correct. If M = T , then by C laim 11, pro cedure T est alwa ys return s a decision, unambiguously . Thus exactly one p ath of Main (amongst those where M = T wa s guessed) leads to a decision, and this decision is correct. If M > T , then no ru n of T est , at any stage k , can trace paths adding up to M . So T est , and hence Up date , and Main ha v e n o accepting r un. If M < T , consid er the run s on which T est and U p date p ro ceed to finally compute Σ k . S ince Main is correct, w e kno w that Σ k = T . No w the c hec k M = Σ k fails and Main reject s and halts. 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