546 research outputs found
Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs
We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC\u2702, Bernstein STOC\u2713] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation).
Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
How Fast Can We Play Tetris Greedily With Rectangular Pieces?
Consider a variant of Tetris played on a board of width and infinite
height, where the pieces are axis-aligned rectangles of arbitrary integer
dimensions, the pieces can only be moved before letting them drop, and a row
does not disappear once it is full. Suppose we want to follow a greedy
strategy: let each rectangle fall where it will end up the lowest given the
current state of the board. To do so, we want a data structure which can always
suggest a greedy move. In other words, we want a data structure which maintains
a set of rectangles, supports queries which return where to drop the
rectangle, and updates which insert a rectangle dropped at a certain position
and return the height of the highest point in the updated set of rectangles. We
show via a reduction to the Multiphase problem [P\u{a}tra\c{s}cu, 2010] that on
a board of width , if the OMv conjecture [Henzinger et al., 2015]
is true, then both operations cannot be supported in time
simultaneously. The reduction also implies polynomial bounds from the 3-SUM
conjecture and the APSP conjecture. On the other hand, we show that there is a
data structure supporting both operations in time on
boards of width , matching the lower bound up to a factor.Comment: Correction of typos and other minor correction
Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk)
Algorithmic research strives to develop fast algorithms for fundamental problems. Despite its many successes, however, many problems still do not have very efficient algorithms. For years researchers have explained the hardness for key problems by proving NP-hardness, utilizing polynomial time reductions to base the hardness of key problems on the famous conjecture P != NP. For problems that already have polynomial time algorithms, however, it does not seem that one can show any sort of hardness based on P != NP. Nevertheless, we would like to provide evidence that a problem with a running time O(n^k) that has not been improved in decades, also requires n^{k-o(1)} time, thus explaining the lack of progress on the problem. Such unconditional time lower bounds seem very difficult to obtain, unfortunately. Recent work has concentrated on an approach mimicking NP-hardness: (1) select a few key problems that are conjectured to require T(n) time to solve, (2) use special, fine-grained reductions to prove time lower bounds for many diverse problems in P based on the conjectured hardness of the key problems. In this abstract we outline the approach, give some examples of hardness results based on the Strong Exponential Time Hypothesis, and present an overview of some of the recent work on the topic
Tight Bounds for Graph Problems in Insertion Streams
Despite the large amount of work on solving graph problems in the data stream model, there do not exist tight space bounds for almost any of them, even in a stream with only edge insertions. For example, for testing connectivity, the upper bound is O(n * log(n)) bits, while the lower bound is only Omega(n) bits. We remedy this situation by providing the first tight Omega(n * log(n)) space lower bounds for randomized algorithms which succeed with constant probability in a stream of edge insertions for a number of graph problems. Our lower bounds apply to testing bipartiteness, connectivity, cycle-freeness, whether a graph is Eulerian, planarity, H-minor freeness, finding a minimum spanning tree of a connected graph, and testing if the diameter of a sparse graph is constant. We also give the first Omega(n * k * log(n)) space lower bounds for deterministic algorithms for k-edge connectivity and k-vertex connectivity; these are optimal in light of known deterministic upper bounds (for k-vertex connectivity we also need to allow edge duplications, which known upper bounds allow). Finally, we give an Omega(n * log^2(n)) lower bound for randomized algorithms approximating the minimum cut up to a constant factor with constant probability in a graph with integer weights between 1 and n, presented as a stream of insertions and deletions to its edges. This lower bound also holds for cut sparsifiers, and gives the first separation of maintaining a sparsifier in the data stream model versus the offline model
Hardness of Approximate Nearest Neighbor Search
We prove conditional near-quadratic running time lower bounds for approximate
Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance.
Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false,
for every there exists a constant such that computing a
-approximation to the Bichromatic Closest Pair requires
time. In particular, this implies a near-linear query time for
Approximate Nearest Neighbor search with polynomial preprocessing time.
Our reduction uses the Distributed PCP framework of [ARW'17], but obtains
improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG
codes have been constructed in other settings before [BKKMS'16, BCGRS'17], but
our construction is the first to yield new hardness results
On-Line File Caching
In the on-line file-caching problem problem, the input is a sequence of
requests for files, given on-line (one at a time). Each file has a non-negative
size and a non-negative retrieval cost. The problem is to decide which files to
keep in a fixed-size cache so as to minimize the sum of the retrieval costs for
files that are not in the cache when requested. The problem arises in web
caching by browsers and by proxies. This paper describes a natural
generalization of LRU called Landlord and gives an analysis showing that it has
an optimal performance guarantee (among deterministic on-line algorithms).
The paper also gives an analysis of the algorithm in a so-called ``loosely''
competitive model, showing that on a ``typical'' cache size, either the
performance guarantee is O(1) or the total retrieval cost is insignificant.Comment: ACM-SIAM Symposium on Discrete Algorithms (1998
Fast Deterministic Consensus in a Noisy Environment
It is well known that the consensus problem cannot be solved
deterministically in an asynchronous environment, but that randomized solutions
are possible. We propose a new model, called noisy scheduling, in which an
adversarial schedule is perturbed randomly, and show that in this model
randomness in the environment can substitute for randomness in the algorithm.
In particular, we show that a simplified, deterministic version of Chandra's
wait-free shared-memory consensus algorithm (PODC, 1996, pp. 166-175) solves
consensus in time at most logarithmic in the number of active processes. The
proof of termination is based on showing that a race between independent
delayed renewal processes produces a winner quickly. In addition, we show that
the protocol finishes in constant time using quantum and priority-based
scheduling on a uniprocessor, suggesting that it is robust against the choice
of model over a wide range.Comment: Typographical errors fixe
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