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ON THE SPACE COMPLEXITY OF DIRECTED GRAPH REACHABILITY

by

Raghunath Tewari

A THESIS

Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Master of Science

Major: Computer Science

Under the Supervision of Professor Vinodchandran Variyam

Lincoln, Nebraska

April, 2007

ON THE SPACE COMPLEXITY OF DIRECTED GRAPH REACHABILITY

Raghunath Tewari, M.S. University of Nebraska, 2007 Advisor: Vinodchandran Variyam Graph reachability problems are fundamental to the study of complexity classes. It is well known that the reachability problem for general directed graphs is known to be complete for non-deterministic logspace (NL). Various restrictions of graph reachability are known to capture low level complexity classes. Recently, in a breakthrough result, it was shown that reachability problem for undirected graphs is complete for deterministic logspace (L). In this thesis we investigate the space complexity of reachability problem for directed planar graphs (denoted by PlanarReach). A graph is planar if it can be drawn on a plane without any crossing edges. Planar graphs are an important and natural subclass of general graphs and are well studied in graph theory. However, the space complexity of reachability problem for directed planar graphs is not yet well understood. It is not known whether PlanarReach is complete for NL. On the other hand researchers did not know any upper bound better than NL for this problem. We make progress on identifying the space complexity of PlanarReach. We show that PlanarReach can be decided in unambigous logspace (UL). UL is the class of problems that are decidable by a nondeterministic logspace machine with atmost one accepting path. As UL is a subclass of NL, our result improves the previously known best upper bound of NL for PlanarReach. We also extend our technique to show that reachability problem for a certain generalization of directed

planar graphs (directed graphs with O(log n) crossing number) can also be decided in unambiguous logarithmic space.

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Acknowledgements First and foremost I thank Dr. Vinodchandran Variyam, my advisor for his support and motivation and without whose guidance this thesis would not have been possible. I would like to thank Dr. Richard Sincovec, Dr. Jitender S. Deogun and Dr. Pavan Aduri for their input and serving on my committee. I would like to thank my colleague, Chris Bourke for his contribution and perpetual help. I would also like to thank the Department of Computer Science and Engineering here at UNL for supporting me during my study at UNL. I would also like to thank Dr. V. Arvind, Institute of Mathematical Sciences, who introduced and motivated me to pursue research in theoretical computer science. Finally, I would like to thank my family and friends for providing me with constant strength and determination.

Contents 1 Introduction

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2 Preliminaries and Earlier Work 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 2.2 Previous results . . . . . . . . . . . . . . . . . . . 2.2.1 Planar Reachability reduces to Grid Graph 2.2.2 Reachability in min-unique graphs is in UL

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3 New upper bounds on Graph Reachability 3.1 Planar reachability is in UL . . . . . . . . . . 3.2 Reachability in non-planar graphs . . . . . . . 3.2.1 Genus 1 graphs . . . . . . . . . . . . . 3.2.2 Graphs with O(log n) crossing number

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4 Conclusion

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Bibliography

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Chapter 1 Introduction Graph reachability problems are fundamental to the study of complexity theory. The general st-connectivity problem for directed graphs perfectly captures the power of nondeterminism in the context of logarithmic space since it is complete for NL. The recent break-through result due to Reingold implies that st-connectivity problem for undirected graphs characterizes deterministic logarithmic space[13]. Various other restricted versions of rechability problem characterize other low-level complexity classes such as L, NC1 and AC0 [11, 7, 8]. A natural and important restriction of the st-connectivity problem is when the graphs involved are planar, which we denote by PlanarReach. A graph is planar if it can be drawn on a plane without any crossing edges. Planar graphs are well studied in graph theory. However, the space complexity of PlanarReach is not yet settled satisfactorily. The best known upper bound in terms of space complexity is NL. Though it is hard for L [11], it is not known whether it is complete for NL. Recently there has been progress in understanding the complexity of PlanarReach. In particular, Allender, Datta and Roy [2] show that PlanarReach logspace reduces to the reachability problem for a strict subclass of planar graphs called grid graphs given a planar combinatorial embedding (i.e. a cyclic ordering of the outgoing edges around each vertex) of the graph. As it is known that given a planar graph we can compute a planar embedding in deterministic logarithmic space [4, 13], we have that reachability in grid graphs and planar graphs are logspace equivalent. We denote the reachability problem for grid graphs as GGR. From this result and the fact that GGR reduces to its complement [8], it follows that PlanarReach reduces to its complement problem of unreachability in planar graphs. In this thesis we make further progress in understanding the space complexity of PlanarReach. Building on earlier work, we show the following theorem. Theorem (Main Theorem). PlanarReach ∈ UL ∩ coUL. Here UL denotes the unambiguous subclass of NL first defined and studied in [10, 6]. A language L is in UL if and only if there exists a nondeterministic logspace machine M accepting L such that, for every instance x, M has at most one accepting

2 computation on input x. Thus, planar reachability can be decided by a nondeterministic machine in logspace with at most one accepting computation. The class UL and related low-space unambiguous classes have been of interest to researchers [10, 6, 12, 3, 14, 2]. Arguably the most interesting result regarding UL, due to Reinhardt and Allender, is that the nonuniform versions of UL contains the whole of NL; that is NL ⊆ UL/poly [14]. Here UL/poly denotes the nonuniform version of UL. Also, Allender, Reinhardt, and Zhou showed that, under the hardness assumption that deterministic linear space has functions that can not be computed by 2n circuits, the constructions given in [14] can be derandomized to show that NL = UL [5]. These results give strong indication that NL equals UL. Our result gives further evidence that this equality might hold. Since PlanarReach reduces to reachability in grid graphs, for our upper bound it suffices to consider grid graphs. Grid graphs are graphs with vertices located on a the planar grid and edges connecting a vertex only with its immediate vertical/horizontal neighbors. Reachability in grid graphs is interesting from a complexitytheoretic point of view. Barrington, Lu, Miltersen, and Skyum showed that stconnectivity on such graphs with constant width captures the complexity of AC0 hierarchy [8]. Recently, complexity of various restrictions of GGR have been studied in [1, 2]. Specifically, in [2] the authors show that the layered grid graph reachability problem (denoted as LGGR) is in UL. A layered grid graph is a restriction to only three cardinal directions. It was open whether this upper bound can be extended to the general grid graph reachabilty. We settle this open problem in this thesis. In Chapter 2 we give formal definitions of the terms that we use. To make the picture complete we also state earlier results and their proofs that we use to show our results. In Chapter 3, we prove our results. In Chapter 4, we conclude by giving an overview of the problems that we solved and related open problems.

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Chapter 2 Preliminaries and Earlier Work 2.1

Definitions

We will mostly be concerned with logarithmic space bounded complexity classes in this thesis. Hence we will be referring to logspace many-one reductions whenever we talk about reductions. All these definitions can be found in any standard textbook on computational complexity theory. Definition 1. NL is the class of languages that can be decided by a non-deterministic logspace bounded Turing machine. Definition 2. UL is the languages that can be decided by an NL machine, such that 1. If the string is in the language, only one computation path accepts. 2. If the string is not in the language, all computation paths reject. It follows from the definition that U L ⊆ N L. Definition 3. A language L1 logspace many one reduces to L2 , denoted by ≤log m , if ∗ there is a logspace bounded machine M , such that for all x ∈ Σ , x ∈ L1 ⇔ M (x) ∈ L2 Definition 4. A language L is said to be logspace hard for a class C if for all languages L0 ∈ C, L0 ≤log m L. Also L is said to be complete for a class C if L is hard for C and L ∈ C. Definition 5. A graph G is said to be a planar graph if G can be embedded on a plane without crossing edges. It is well known that checking for st-connectivity for general directed graphs is NL-complete. We consider the st-connectivity problem for planar graphs and grid graphs.

4 Definition 6. A n × n grid graph is a directed graph whose vertices are {0, . . . , n − 1} × {0, . . . , n − 1} so that if ((i1 , j1 ), (i2 , j2 )) is an edge then |i1 − i2 | + |j1 − j2 | = 1. Grid graphs are a very natural subclass of planar graphs with vertices identified with the n × n grid on the x − y plane oriented at the origin with directed edges connecting only the immediate vertical and horizontal neighbors. It is convenient to view the edges according to the cardinal directions. For a vertex (i, j), the edge (i, j) → (i + 1, j) is an east edge, (i, j) → (i − 1, j) is a west edge, (i, j) → (i, j + 1) is a north edge, and (i, j) → (i, j − 1) is a south edge. The grid graph reachability problem is as follows. Given a grid graph G and vertices s and t, determine if there exists a directed path from s to t in G. The directed planar reachability problem denoted as PlanarReach is the following: Given a planar graph G and vertices s and t, determine if there exists a directed path from s to t in G.

2.2 2.2.1

Previous results Planar Reachability reduces to Grid Graph Reachability

In this section we state the proof from [2]. Theorem 1 ([2]). PlanarReach logspace many-one reduces to GGR. Proof. Let G be the input graph on n vertices, with s and t being two fixed vertices. Without loss of generality we will assume the following and give reasons for our assumptions. 1. A planar embedding of G is given with s and t on the outer face [4, 2, 13]. 2. All edges are unidirectional. Since vertices connected by bidirectional edges can be collapsed by applying the reachability algorithm in undirected graphs [13]. 3. Degree (indegree + outdegree) of all vertices is bounded by 3. To see this, suppose the degree of some vertex v is k > 3. Then replace v with a directed cycle of length k, with each vertex adjacent to one of the k edges that were connected to v. 4. s has indegree 0 and outdegree atmost 2. Since if s has degree more than 2 than an additional vertex can be introduced and renamed s. Compute an undirected spanning tree T of G with s as the root. This can be done as follows. Order the edges of the G in any order such that the edges incident on s are at the beginning. Note that we ignore the directions on the edges. Now the ith edge, ei = (ui , vi ) is in the spanning tree if and only if there is no path from ui to vi

5 in the subgraph induced by the first i − 1 edges in the ordering. Reachability in an undirected graph can be checked using Reingold’s algorithm [13]. It follows from our assumptions on G that T is a binary tree. The planar embedding of G induces an ordering on the children of any vertex. Note that we can compute the height, h(v) of any vertex, v in logspace [4]. If a node in T has only one child, let it be the left child. For notational convenience let h(s) = 0 and if v is a child of u then h(v) = h(u) + 1. Let w(x) denote the number of leaves in T of the subtree rooted at the left child of the parent of x, if x is a right child and let w(x) = 0 if x is a left child. w(s) = 0. Note that w(x) for any vertex can be computed in logspace by traversing the tree T . We now assign vertices to grid points. Assign s to the point (n, n). If u is assigned to the grid point (i, j) then the left child v of u is assigned to the point (i, j + n + 1) and the right child is assigned to the point (i + n.w(v) + 1, j + n + 1). A left edge is represented by a vertical path from (i, j) to (i, j +n+1) and a right edge is represented by a horizontal path from (i, j) to (i + n.w(v) + 1, j) followed by a vertical path to (i + n.w(v) + 1, j + n + 1). For every non-tree edge e = (u, v) in G. Without loss of generality assume u is to the left of v. Let Se = {Set of non-tree edges enclosed by the unique cycle formed by adding e to the tree T } and let g(e) = |Se |. We give a logspace algorithm to calculate g(e). Find the least common ancestor, y of u and v in T . Now for every other non-tree edge e0 = (u0 , v 0 ) (u0 is to the left of v 0 ), let a be the least common ancestor of u and u0 and let b be the least common ancestor of v and v 0 . We divide the problem into several cases. Case 1: If either a does not lie in the path from u to y or b does not lie in the path from v to y then e0 ∈ / Se . Case 2: If a is different from u and u0 , then if u is a left descendent of a then e0 ∈ Se else e0 ∈ / Se . Case 3: If a = u0 , then if e0 is to the right of the path to u from a then e0 ∈ Se else e0 ∈ / Se . Case 4: If a = u, then if e is to the left of the path to u0 from a then e0 ∈ Se else 0 e ∈ / Se . Note that u and v have degree at most 2 in the tree and hence there is no horizontal edge attached to u or v in the tree. Also, since the degree of every vertex in G is bounded by 3, therefore the total number of edges in bounded by d3n/2e. Hence the number of non tree edges in G is strictly bounded by n. Now we need to embed the non-tree edges in the grid so that they do not collide with any other paths in the grid. We divide this into several cases and treat them separately. Let (u, v) be a non-tree edge. 1. Case 1 : (Embedding of (u, v) does not go around any leaf of the tree T ) (a) h(u) > h(v) Draw a horizontal path from u to the column at a distance g(e) from the

6 column of u. Then draw a vertical path to the row containing v and then again horizontal to v. (b) h(u) ≤ h(v) Draw a horizontal path from u to the column at a distance g(e) from the column of v. Then draw a vertical path to the row containing v and then again horizontal to v. 2. Case 2 : (Embedding of (u, v) goes around a leaf of the tree T ) Let z be the leaf with maximum height among the leaves that (u, v) goes around in T . Draw a horizontal path from u of length n−g(e) towards the column of v. Then draw a vertical path to a row g(e) above the level of z, followed by a horizontal path to a column at a distance n − g(e) from the column containing v. Then draw a vertical path down to the row containing v and then horizontally to attach it to v. It can be seen that the paths do not collide. Theorem 2 ([8, 2]). PlanarReach logspace many-one reduces to its complement. Because of the above reductions and the fact that UL is closed under logspace many-one reductions, it is enough to show that GGR ∈ UL.

2.2.2

Reachability in min-unique graphs is in UL

In [14] Reinhardt and Allender give a general technique for showing membership in UL which we will make use of. Definition 7. A min-unique graph is a directed graph with positive weights associated with each edge where for every pair of vertices u, v, if there is a path from u to v, then there is a unique minimum weight path from u to v. Here, the weight of a path is the sum of the weights on its edges. Reinhardt and Allender [14] actually define min-uniqueness for unweighted graphs, but these two definitions are essentially same in this context as one can replace an edge with positive weight w, with a path of length w. In this section we will give a shorter version of the proof in [14], thus showing that reachability in min-unique graphs is in UL. Theorem 3. Let G be a min-unique graph with weight function, f : E → N. Then st-connectivity problem for G is in UL. Proof. Construct a graph G0 from G, replacing every edge e in G with a path of length f (e). Easy to see that st-connectivity is preserved from G to G0 . Note that the shortest path between any two vertices in G0 is unique.

7 Let ck and Σk denote the number of vertices which are at a distance at most k from s and the sum of the lengths of the shortest path to each of them, respectively. Let d(v) denote the length of the shortest path from s to v. If no such path exists, then let d(v) = |V | + 1.Therefore we have, X Σk = d(v). {v|d(v)≤k}

Note that the subgraph induced by the vertices of G0 which are at a distance at most k from s is min-unique. We give an unambiguous routine (Algorithm 1) to evaluate the predicate “d(v) ≤ k” given the values of ck and Σk . Input: (G, v, k, ck , Σk ) Output: true if d(v) ≤ k else f alse Initialize count ← 0; sum ← 0; path.to.v ← f alse ; for x ∈ V do Nondeterministically guess if d(x) ≤ k ; if guess is correct then Guess a path of length l ≤ k from s to x ; if guess is correct then Set count ← count + 1; sum ← sum + l ; if x == v then set path.to.v ← true ; end else halt and reject end end end if count == ck & sum = Σk then return path.to.v ; else halt and reject Algorithm 1: Determining if d(v) ≤ k or not To show unambiguity of Algorithm 1 note that 1. If the above algorithm guesses incorrectly for some vertex x that d(x) > k then count < ck and the algorithm rejects. Thus the only paths that run to completion are the ones that correctly guesses the set {x|d(x) ≤ k} exactly. 2. If the algorithm at any point incorrectly guesses the length l of the shortest path to x, then if d(x) > l then no such path would be found, and if d(x) < l then the variable sum would be incremented by a value greater than d(x) and thus sum would be greater than Σk leading to rejection. The subgraph consisting of s alone (d(x) ≤ 0) is min-unique and c0 = 1 and Σ0 = 0. Now ck = ck−1 + |{v|d(v) = k}| and Σk = Σk−1 + k|{v|d(v) = k}|. Also,

8 d(v) = k if and only if ∃(x, v) ∈ E such that d(x) ≤ k − 1 and ¬(d(v) ≤ k − 1). Both of these predicates can be computed using Algorithm 1. Input: (G, k, ck−1 , Σk−1 ) Output: ck , Σk Initialize ck ← ck−1 andΣk ← Σk−1 ; for v ∈ V do if ¬(d(v) ≤ k − 1) then for (x, v) ∈ E do if d(x) ≤ k − 1 then Set ck ← ck + 1; Σk ← Σk + k ; end end end end return ck and Σk ; Algorithm 2: Computing ck and Σk Now we give an algorithm that invokes Algorithm 2 to check for st connectivity in a min-unique graph. Note that there exists a path from s to t if and only if d(t) ≤ k. Input: (G) Output: true if there is a path from s to t else f alse Initialize c0 ← 1; Σ0 ← 0; k ← 0; prev ← −1 ; while ck 6= prev do Set prev ← ck ; Set k ← k + 1 ; Invoke Algorithm 2 on (G, k, ck−1 , Σk−1 ) and store the output to ck and Σk ; end Invoke Algorithm 1 on(G, t, k, ck , Σk ) and return the value ; Algorithm 3: Determining if there exists a path from s to t in G Thus we have exhibited the fact that reachability in min-unique graphs can be decided in UL. Therefore to show a that graph reachability is in UL for any class of graphs, it is sufficient to show that the class of graphs is min-unique. In [2], the authors exploited this fact to show that reachability in layered grid graphs is in UL.

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Chapter 3 New upper bounds on Graph Reachability 3.1

Planar reachability is in UL

Theorem 4. PlanarReach ∈ UL ∩ coUL. To prove Theorem 4, in light of Theorems 1, 2, and 3, it suffices to show a logspace computable positive weight function which produces a min-unique graph for grid graphs. Proof of Theorem 4. Let G be a grid graph with the rows and columns of G indexed from 0 to n − 1. We define a weight function w on the edges of G as follows.  n4 if e is an east or west edge  i + n4 if e is a north edge in column i w(e) =  −i + n4 if e is a south edge in column i Clearly, w is log-space computable. Since i < n, the weight on any edge is positive. Since a minimum weight path has to be simple, we will only focus on simple paths. The weight of any path P in G, denoted by w(P ) is of the form a + bn4 . Note that b < n2 , since the total number of edges in G is (n − 1)2 . Also, |a| < n3 , since the weight of an edge is bounded by n − 1. (Actually a careful analysis will show that |a| < n2 but n3 suffices for the purpose of the proof). For a given path P , let a(P ) denote the ‘a’ component and b(P ) denote the ‘b’ component of its weight. Here, a(P ) serves to weight a path’s vertical edges while b(P ) serves to count the total number of edges in the path. Let P1 and P2 be two paths in G having the same weight. Then it is easy to see that a(P1 ) = a(P2 ) and b(P1 ) = b(P2 ). To see this, let w(P1 ) = a1 + b1 n4 and

10 w(P2 ) = a2 + b2 n4 . Since, w(P1 ) ⇒ (a1 − a2 ) + (b1 − b2 )n4 ⇒ a1 = a2

= = and

w(P2 ) 0 b1 = b2

The final implication follows since |ai |’s and bi , and hence their respective differences, are bounded by n3 . Now we will argue that, with respect to this weight function for any u and v, the minimum weight path from u to v, if it exists, is unique. First we prove a very nice property of this weight function that the ‘a’ component of the weight of any nontrivial simple cycle in G is non-zero. In fact, we prove the following stronger property of this weight function. For a simple cycle C, let A(C) denote the number of unit squares it encloses. Lemma 5. Let C be a simple directed cycle in G. Then a(C) = +A(C) if C is a counter-clockwise cycle and a(C) = −A(C) if C is a clockwise cycle. Proof. We prove the lemma for a counter-clockwise simple cycle. Let C be a counterclockwise cycle in G. Look at the restriction of the cycle to the set of edges between two consecutive rows, say j and j + 1. We can view this as an ordered set, where an edge e appears before an edge e0 in the ordering if e is to the left of e0 in the graph. Denote this ordered set by Sj . If we order these edges from left to right it alternates between south and north edges with the left most edge being a south edge and the rightmost edge being a north edge. Suppose not, then there are two adjacent north edges in Sj , say e1 = (u1 , v1 ) and e2 = (u2 , v2 ). This implies that C does not have any edge between e1 and e2 in G. Also there is a simple path from v2 to u1 and from v1 to u2 without taking any edge between e1 and e2 in G. This implies that the paths intersect and hence contradicts the fact that C is a simple cycle. The reason why the first edge is a south edge is because it is a counter-clockwise cycle. Now lets look at the set of squares that lie between row j and j + 1. Denote this set by Rj . a(C) restricted to Sj counts the number of squares in Rj that lie between adjacent south and north edges, with the north edge being to the right of the south edge (cf. Figure 1). This is because the weight of the kth south edge plus the weight of the kth north edge is equal to the number of squares between these two edges. A square in Rj is in C iff it is between a south and a north edge. This is because if we look at a partition of the set Rj induced by the edges in Sj , then the partitions alternately fall within and outside of C, with the set of squares between the first south and north edge lying within the cycle. Now we sum our index j from 0 to n − 2 and thus get that the sum of the edge weights of the cycle is equal to the number of squares it encloses. Lemma 6. Let G be a grid graph. With respect to the weight function w, for any two vertices u and v, the minimum weight path from u to v is unique.

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j+1 j

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Figure 3.1: A view of a grid graph between row j and j + 1 of a counter-clockwise cycle. The cycle has an ‘a’ component weight of (2 − 1) + (7 − 4) = 4 with respect to rows j and j + 1, equal to the number of unit squares it encloses. Proof. Suppose there exists two different minimum weight paths P1 and P2 between u and v having the same weight. Let u0 be the vertex at which P1 and P2 diverge for the first time and let v 0 be the vertex where they meet after their first divergence. Denote the subpath of P1 from u0 to v 0 by P10 and the subpath of P2 from u0 to v 0 by P20 (cf. Figure 2). u0 exists since P1 and P2 are not the same, and existence of u0 implies existence of v 0 . If P10 and P20 have different weights then without loss of generality assume w(P10 ) < w(P20 ). This implies that the subpath of P2 from v 0 to v has lesser weight than the subpath of P1 from v 0 to v. Hence taking P1 from u to v 0 and then taking P2 to v gets a path of lesser weight from u to v, hence a contradiction. v v0

P10

P20

u0 u

Figure 3.2: Paths P1 , P2 from u to v. On the other hand, suppose P10 and P20 have the same weight. Then a(P10 ) = a(P20 ). Now consider a simple cycle C from u0 to v 0 following the path P10 and back to u0 following the path −P20 . Here for a path P , −P denotes the path obtained by reversing the edges in P . It is clear that a(−P ) = −a(P ) for any path P . Hence a(C) = a(P10 ) − a(P20 ) = 0. This is a contradiction since C is a nontrivial simple cycle and hence |a(C)| > 0 by Lemma 5. Therefore our weight function makes G a min-unique graph.

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3.2 3.2.1

Reachability in non-planar graphs Genus 1 graphs

Allender, Datta and Roy show that, if given an embedding on the torus of a graph of genus 1, the st-connectivity problem is reducible (in deterministic log-space) to the planar case [2]. As a consequence of Theorem 4, we get the following. Corollary 7. The directed st-connectivity problem for graphs of genus 1 is in UL ∩ coUL (when given an embedding).

3.2.2

Graphs with O(log n) crossing number

Definition 8. The crossing number of a directed graph is the minimum number of edge intersections required on any embedding of the graph on a plane. Definition 9. The maximum planar subgraph of G = (V, E) is a graph with vertex set V and edge set E 0 ⊆ E, such that, G0 = (V, E 0 ) is a planar graph and for no other F ⊆ E, such that |F | > |E 0 |, is H = (V, F ) a planar graph. Crossing number is an important property of graphs. In fact reachability in general directed graph can be reduced to reachability in a graph with O(n) crossing number. This fact motivated us to study crossing number in more detail and we came up with the following theorem. Theorem 8. Reachability in a graph G with O(log n) crossing number can be decided in UL. We assume that the maximum planar subgraph of G is given. Proof. The idea behind the proof is that, since there are only O(log n) crossing edges, hence there are only polynomially many subsets of them. Therefore, we are able to assign a logspace computable weight function that assigns a unique weight to each such subset. Let the crossing number of G be k log n for some constant k. Let G0 be the maximum planar subgraph of G. Reduce G0 to a grid graph H 0 . Now add the crossing edges to form the graph H. Reachability of G is preserved in H. Apply the weight function of the previous section to the grid graph H 0 . Let H have k log n crossing edges where k is a constant. Weight of any path on H 0 is bounded by n4 . This follows from the definition of our weight function. Now weigh the crossing edges with distinct weights from the set {n4 2i |1 ≤ i ≤ k log n} It is easy to see that the weights are polynomially bounded. Now consider two minimum weight simple paths P1 and P2 from s to t. This implies that the coefficient of n4 must the same for both the paths. The coefficients are nothing but sums of

13 powers of 2. Hence they are equal if and only if both path consists of the same set of crossing edges. Claim 9. The sequence in which paths P1 and P2 take crossing edges are same. Proof. Suppose the sequence of crossing edges in P1 is not the same as that in P2 . Let (u, v) be ith and jth crossing edge in P1 and P2 respectively, such that i < j. Now construct the path P 0 that is same as P1 from s to u, takes edge (u, v) and then follows path P2 to t. Since both paths had the same set of crossing edges, therefore P 0 has (j − i) crossing edges less than that of P1 or P2 and hence is strictly of lesser weight. This contradicts minimality of P1 and P2 . Now we can break both paths P1 and P2 into k log n + 1 subpaths, each of which lies on H 0 and have the same start and finish vertex. Let S1 be one such subpath of P1 and let S2 be the corresponding subpath of P2 . That is S1 and S2 have the same start and finish vertices and they lie on the grid. Since the weights on grid edges are the same as in Theorem 4 we can apply Lemma 6 on S1 and S2 . Hence S1 and S2 are minimum weight paths from their start to finish vertices if and only if they are the same. This holds for all subpaths of P1 and P2 on the grid. Hence we have the result.

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Chapter 4 Conclusion We have shown that the st-connectivity problem for directed planar graphs can be decided in UL ∩ coUL, improving over the known upper bound of NL. Our results provably are a step further towards bridging the gap between NL and UL, though it is already known that non-uniformly they are the same [14]. Therefore the most obvious question to ask is that, can the general st-connectivity problem can be done unambiguously? In other words, is NL = UL? Also not much is known about the deterministic space complexity of reachability directed graphs other than the wellknown O(log2 n) upper bound given by Savitch’s theorem. For instance, can we show that PlanarReach ∈ L? Or can we even show any deterministic space upper bound for PlanarReach better than O(log2 n).

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