FOCS 2011
TechTalks from event: FOCS 2011
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10B

Efficient Distributed Medium AccessConsider a wireless network of n nodes represented by a graph G=(V, E) where an edge (i,j) models the fact that transmissions of i and j interfere with each other, i.e. simultaneous transmissions of i and j become unsuccessful. Hence it is required that at each time instance a set of noninterfering nodes (corresponding to an independent set in G) access the wireless medium. To utilize wireless resources efficiently, it is required to arbitrate the access of medium among interfering nodes properly. Moreover, to be of practical use, such a mechanism is required to be totally distributed as well as simple.As the main result of this paper, we provide such a medium access algorithm. It is randomized, totally distributed and simple: each node attempts to access medium at each time with probability that is a function of its local information. We establish efficiency of the algorithm by showing that the corresponding network Markov chain is positive recurrent as long as the demand imposed on the network can be supported by the wireless network (using any algorithm). In that sense, the proposed algorithm is optimal in terms of utilizing wireless resources. The algorithm is oblivious to the network graph structure, in contrast with the socalled `polynomial backoff' algorithm by HastadLeightonRogoff (STOC '87, SICOMP '96) that is established to be optimal for the complete graph and bipartite graphs (by GoldbergMacKenzie (SODA '96, JCSS '99)).

Local Distributed DecisionA central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard $\cal{LOCAL}$ model of computation and define $LD(t)$ (for local decision) as the class of decision problems that can be solved in $t$ communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class $BPLD(t,p,q)$, containing all languages for which there exists a randomized algorithm that runs in $t$ rounds, accepts correct instances with probability at least $p$ and rejects incorrect ones with probability at least $q$. We show that $p^2+q = 1$ is a threshold for the containment of $LD(t)$ in $BPLD(t,p,q)$. More precisely, we show that there exists a language that does not belong to $LD(t)$ for any $t=o(n)$ but does belong to $BPLD(0,p,q)$ for any $p,q\in (0,1]$ such that $p^2+q\leq 1$. On the other hand, we show that, restricted to hereditary languages, $BPLD(t,p,q)=LD(O(t))$, for any function $t$ and any $p,q\in (0,1]$ such that $p^2+q> 1$. In addition, we investigate the impact of nondeterminism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that nondeterminism does help, but that this help is limited, as there exist languages that cannot be decided nondeterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with nondeterminism that enables to decide all languages in constant time. Finally, we introduce the notion of local reduction, and establish some completeness results.

The Complexity of RenamingWe study the complexity of renaming, a fundamental problem in distributed computing in which a set of processes need to pick distinct names from a given namespace. We prove a local lower bound of \Omega(k) process steps for deterministic renaming into any namespace of size subexponential in k, where k is the number of participants. This bound is tight: it draws an exponential separation between deterministic and randomized solutions, and implies tight bounds for deterministic fetchandincrement registers, queues and stacks. The proof of the bound is interesting in its own right, for it relies on the first reduction from renaming to another fundamental problem in distributed computing: mutual exclusion. We complement our local bound with a global lower bound of \Omega(k log(k/c)) on the total step complexity of renaming into a namespace of size ck, for any c \geq 1. This applies to randomized algorithms against a strong adversary, and helps derive new global lower bounds for randomized approximate counter and fetchandincrement implementations, all tight within logarithmic factors.

Mutual Exclusion with O(log2 log n) Amortized WorkThis paper gives a new algorithm for mutual exclusion in which each passage through the critical section costs amortized O(log2 log n) RMRs with high probability. The algorithm operates in a standard asynchronous, local spinning, sharedmemory model. The algorithm works against an oblivious adversary and guarantees that every process enters the critical section with high probability. The algorithm achieves its efficient performance by exploiting a connection between mutual exclusion and approximate counting. A central aspect of the work presented here is the development and application of efficient approximatecounting data structures. Our mutualexclusion algorithm represents a departure from previous algorithms in terms of techniques, adversary model, and performance. Most previous mutual exclusion algorithms are based on tournamenttree constructions. The most efficient prior algorithms require O(log n/ log log n) RMRs and work against an adaptive adversary. In this paper, we focus on an oblivious model, and the algorithm in this paper is the first (for any adversary model) that can beat the O(log n/ log log n) RMR bound.

Algorithms for the Generalized Sorting ProblemWe study the generalized sorting problem where we are given a set of $n$ elements to be sorted but only a subset of all possible pairwise element comparisons is allowed. The goal is to determine the sorted order using the smallest possible number of allowed comparisons. The generalized sorting problem may be equivalently viewed as follows. Given an undirected graph $G(V,E)$ where $V$ is the set of elements to be sorted and $E$ defines the set of allowed comparisons, adaptively find the smallest subset $E' \subseteq E$ of edges to probe such that the directed graph induced by $E'$ contains a Hamiltonian path. When $G$ is a complete graph, we get the standard sorting problem, and it is wellknown that $\Theta(n \log n)$ comparisons are necessary and sufficient. An extensively studied special case of the generalized sorting problem is the nuts and bolts problem where the allowed comparison graph is a complete bipartite graph between two equalsize sets. It is known that for this special case also, there is a deterministic algorithm that sorts using $\Theta(n \log n)$ comparisons. However, when the allowed comparison graph is arbitrary, to our knowledge, no bound better than the trivial $O(n^2)$ bound is known. Our main result is a randomized algorithm that sorts any allowed comparison graph using $\wt{O}(n^{3/2})$ comparisons with high probability (provided the input is sortable). We also study the sorting problem in randomly generated allowed comparison graphs, and show that when the edge probability is $p$, $\wt{O}(\min\{\frac{n}{p^2},n^{3/2} \sqrt{p}\})$ comparisons suffice on average to sort.