We study the minimum backlog problem (MBP). This online problem arises, e.g., in the context of sensor networks. We focus on two main variants of MBP. The discrete MBP is a 2-person game played on a graph $G=(V,E)$. The player is initially located
at a vertex of the graph. In each time step, the adversary pours a total of one unit of water into cups that are located on the vertices of the graph, arbitrarily distributing the water among the cups. The player then moves from her current vertex to an adjacent vertex and empties the cup at that vertex. The players objective is to minimize the backlog, i.e., the maximum amount of water in any cup at any time. The geometric MBP is a continuous-time version of the MBP: the cups are points in the two-dimensional plane, the adversary pours water continuously at a constant rate, and the player moves in the plane with unit speed. Again, the players objective is to minimize the backlog. We show that the competitive ratio of any algorithm for the MBP has a lower bound of $Omega(D)$, where $D$ is the diameter of the graph (for the discrete MBP) or the diameter of the point set (for the geometric MBP). Therefore we focus on determining a strategy for the player that guarantees a uniform upper bound on the absolute value of the backlog. For the absolute value of the backlog there is a trivial lower bound of $Omega(D)$, and the deamortization analysis of Dietz and Sleator gives an upper bound of $O(Dlog N)$ for $N$ cups. Our main result is a tight upper bound for the geometric MBP: we show that there is a strategy for the player that guarantees a backlog of $O(D)$, independently of the number of cups.
This work shows that the following problems are equivalent, both in theory and in practice: - median filtering: given an $n$-element vector, compute the sliding window median with window size $k$, - piecewise sorting: given an $n$-element vector,
divide it in $n/k$ blocks of length $k$ and sort each block. By prior work, median filtering is known to be at least as hard as piecewise sorting: with a single median filter operation we can sort $Theta(n/k)$ blocks of length $Theta(k)$. The present work shows that median filtering is also as easy as piecewise sorting: we can do median filtering with one piecewise sorting operation and linear-time postprocessing. In particular, median filtering can directly benefit from the vast literature on sorting algorithms---for example, adaptive sorting algorithms imply adaptive median filtering algorithms. The reduction is very efficient in practice---for random inputs the performance of the new sorting-based algorithm is on a par with the fastest heap-based algorithms, and for benign data distributions it typically outperforms prior algorithms. The key technical idea is that we can represent the sliding window with a pair of sorted doubly-linked lists: we delete items from one list and add items to the other list. Deletions are easy; additions can be done efficiently if we reverse the time twice: First we construct the full list and delete the items in the reverse order. Then we undo each deletion with Knuths dancing links technique.
Linials seminal result shows that any deterministic distributed algorithm that finds a $3$-colouring of an $n$-cycle requires at least $log^*(n)/2 - 1$ communication rounds. We give a new simpler proof of this theorem.
We show that there is no deterministic local algorithm (constant-time distributed graph algorithm) that finds a $(7-epsilon)$-approximation of a minimum dominating set on planar graphs, for any positive constant $epsilon$. In prior work, the best low
er bound on the approximation ratio has been $5-epsilon$; there is also an upper bound of $52$.
We study the problem of finding large cuts in $d$-regular triangle-free graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a cut of expected size $(1/2 + 0.177/sqrt{d})m$, where $m$ is the number of edges. We give a simpler
algorithm that does much better: it finds a cut of expected size $(1/2 + 0.28125/sqrt{d})m$. As a corollary, this shows that in any $d$-regular triangle-free graph there exists a cut of at least this size. Our algorithm can be interpreted as a very efficient randomised distributed algorithm: each node needs to produce only one random bit, and the algorithm runs in one synchronous communication round. This work is also a case study of applying computational techniques in the design of distributed algorithms: our algorithm was designed by a computer program that searched for optimal algorithms for small values of $d$.
A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured a
nd weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor $(Delta+1)/2$, where $Delta$ is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.
We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose--accept rounds executed by the Gale--Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost s
table matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. This holds even if ties are present in the preference lists. We apply our results to give a distributed $(2+epsilon)$-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
We present a local algorithm (constant-time distributed algorithm) for finding a 3-approximate vertex cover in bounded-degree graphs. The algorithm is deterministic, and no auxiliary information besides port numbering is required.
We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The objective is to maximise $omega$ subject to $Ax le 1$, $Cx ge omega 1$, and $x ge 0$ for nonnegative matrices $A$ and $C$. The approximation ratio o
f our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.
We study the applicability of distributed, local algorithms to 0/1 max-min LPs where the objective is to maximise ${min_k sum_v c_{kv} x_v}$ subject to ${sum_v a_{iv} x_v le 1}$ for each $i$ and ${x_v ge 0}$ for each $v$. Here $c_{kv} in {0,1}$, $a_{
iv} in {0,1}$, and the support sets ${V_i = {v : a_{iv} > 0 }}$ and ${V_k = {v : c_{kv}>0 }}$ have bounded size; in particular, we study the case $|V_k| le 2$. Each agent $v$ is responsible for choosing the value of $x_v$ based on information within its constant-size neighbourhood; the communication network is the hypergraph where the sets $V_k$ and $V_i$ constitute the hyperedges. We present a local approximation algorithm which achieves an approximation ratio arbitrarily close to the theoretical lower bound presented in prior work.
