Saurabh Sawlani

Reversible Pebble Game on Trees

A surprising equivalence between different forms of pebble games on graphs - Dymond-Tompa pebble game (studied in [4]), Raz-McKenzie pebble game (studied in [10]) and reversible pebbling (studied in [1]) - was established recently by Chan[2]. Motivated by this equivalence, we study the reversible pebble game and establish the following results.

Pebbling meets coloring: Reversible pebble game on trees

The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett [1] motivated by applications in designing space efficient reversible algorithms. Recently, Siu Man Chan [2] showed that the reversible pebble game number of any DAG is the same as its Dymond-Tompa pebble number and Raz-Mckenzie pebble number. We show, as our main result, that for any rooted directed tree T , its reversible pebble game number is always just one more than the edge rank coloring number of the underlying undirected tree U of T . The most striking implication of this result is that the reversible pebble game number of a tree does not depend upon the direction of edges, a fact that does not hold in general for DAGs. It is known that given a DAG G as input, determining its reversible pebble game number is PSPACE-hard. Our result implies that the reversible pebble game number of trees can be computed in polynomial time as edge rank coloring number of trees can be computed in linear time ([8]). We also address the question of finding the number of steps required to optimally pebble various families of trees. It is known that trees can be pebbled in nO(log(n)) steps where n is the number of nodes in the tree. Using the equivalence between reversible pebble game and the Dymond-Tompa pebble game [2], we show that complete binary trees can be pebbled in nO(log log(n)) steps, a substantial improvement over the naive upper bound of nO(log(n)). It remains open whether complete binary trees can be pebbled in polynomial number of steps (i.e., nk for some constant k). Towards this end, we show that almost optimal (i.e., within a factor of (1 + ǫ) for any constant ǫ > 0) pebblings of complete binary trees can be done in polynomial number of steps. ∗Sponsored by TCS Research Fellowship

Min/Max-Poly Weighting Schemes and the NL versus UL Problem

For a graph G(V, E) (|V| = n) and a vertex s ∈ V, a weighting scheme (W : E ↦ Z+) is called a min-unique (resp. max-unique) weighting scheme if, for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v, where weight of a path is the sum of the weights assigned to the edges. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by nc for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result in Reinhardt and Allender [2000], in which a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique involves triple inductive counting and generalizes the techniques of Immerman [1988], Szelepcsényi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al. [1984] (also used in Garvin et al. [2014]). We combine this with a complementary unambiguous verification method to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph. Using our techniques, we generalize the double inductive counting method in Limaye et al. [2009], in which the UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for layered DAGs.

Polynomial Min/Max-weighted Reachability is in Unambiguous Log-space

For a graph G(V,E) and a vertex s in V, a weighting scheme (w : E -> N) is called a min-unique (resp. max-unique) weighting scheme, if for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by n^c for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this paper, we propose an unambiguous non-deterministic log-space (UL) algorithm for the problem of testing reachability in layered directed acyclic graphs (DAGs) augmented with a min-poly weighting scheme. This improves the result due to Reinhardt and Allender [Reinhardt/Allender, SIAM J. Comp., 2000] where a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique is a triple inductive counting, which generalizes the techniques of [Immermann, Siam J. Comp.,1988; Szelepcsenyi, Acta Inf.,1988] and [Reinhardt/Allender, SIAM J. Comp., 2000], combined with a hashing technique due to [Fredman et al.,J. ACM, 1984] (also used in [Garvin et al., Comp. Compl.,2014]). We combine this with a complementary unambiguous verification method, to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-poly property of the graph. Using our techniques, we generalize the double inductive counting method in [Limaye et al., CATS, 2009] where UL algorithms were given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for DAGs.

On Directed Tree Realizations of Degree Sets

Given a degree set D = {a 1 < a 2 < … < a n } of non-negative integers, the minimum number of vertices in any tree realizing the set D is known [11]. In this paper, we study the number of vertices and multiplicity of distinct degrees as parameters of tree realizations of degree sets. We explore this in the context of both directed and undirected trees and asymmetric directed graphs. We show a tight lower bound on the maximum multiplicity needed for any tree realization of a degree set. For the directed trees, we study two natural notions of realizability by directed graphs and show tight lower bounds on the number of vertices needed to realize any degree set. For asymmetric graphs, if μ A (D) denotes the minimum number of vertices needed to realize any degree set, we show that a 1 + a n + 1 ≤ μ A (D) ≤ a n − 1 + a n + 1. We also derive sufficiency conditions on a i ’s under which the lower bound is achieved.