18 research outputs found
Illuminating the x-Axis by ?-Floodlights
Given a set S of regions with piece-wise linear boundary and a positive angle α < 90°, we consider the problem of computing the locations and orientations of the minimum number of α-floodlights positioned at points in S which suffice to illuminate the entire x-axis. We show that the problem can be solved in O(n log n) time and O(n) space, where n is the number of vertices of the set S
A Tight Bound for Shortest Augmenting Paths on Trees
The shortest augmenting path technique is one of the fundamental ideas used
in maximum matching and maximum flow algorithms. Since being introduced by
Edmonds and Karp in 1972, it has been widely applied in many different
settings. Surprisingly, despite this extensive usage, it is still not well
understood even in the simplest case: online bipartite matching problem on
trees. In this problem a bipartite tree is being revealed
online, i.e., in each round one vertex from with its incident edges
arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis,
R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with
augmentations. In INFOCOM 2009] that the total length of all shortest
augmenting paths found is . In this paper, we prove a tight upper bound for the total length of shortest augmenting paths for
trees improving over bound [B. Bosek, D. Leniowski, P.
Sankowski, and A. Zych. Shortest augmenting paths for online matchings on
trees. In WAOA 2015].Comment: 22 pages, 10 figure
Online Service with Delay
In this paper, we introduce the online service with delay problem. In this
problem, there are points in a metric space that issue service requests
over time, and a server that serves these requests. The goal is to minimize the
sum of distance traveled by the server and the total delay in serving the
requests. This problem models the fundamental tradeoff between batching
requests to improve locality and reducing delay to improve response time, that
has many applications in operations management, operating systems, logistics,
supply chain management, and scheduling.
Our main result is to show a poly-logarithmic competitive ratio for the
online service with delay problem. This result is obtained by an algorithm that
we call the preemptive service algorithm. The salient feature of this algorithm
is a process called preemptive service, which uses a novel combination of
(recursive) time forwarding and spatial exploration on a metric space. We hope
this technique will be useful for related problems such as reordering buffer
management, online TSP, vehicle routing, etc. We also generalize our results to
servers.Comment: 30 pages, 11 figures, Appeared in 49th ACM Symposium on Theory of
Computing (STOC), 201
Nearly Optimal Static Las Vegas Succinct Dictionary
Given a set of (distinct) keys from key space , each associated
with a value from , the \emph{static dictionary} problem asks to
preprocess these (key, value) pairs into a data structure, supporting
value-retrieval queries: for any given , must
return the value associated with if , or return if . The special case where is called the \emph{membership}
problem. The "textbook" solution is to use a hash table, which occupies linear
space and answers each query in constant time. On the other hand, the minimum
possible space to encode all (key, value) pairs is only bits, which could be much less.
In this paper, we design a randomized dictionary data structure using
bits of space, and it
has \emph{expected constant} query time, assuming the query algorithm can
access an external lookup table of size . The lookup table depends
only on , and , and not the input. Previously, even for
membership queries and , the best known data structure with
constant query time requires bits of space
(Pagh [Pag01] and P\v{a}tra\c{s}cu [Pat08]); the best-known using
space has query time ; the only known
non-trivial data structure with space has
query time and requires a lookup table of size (!). Our new
data structure answers open questions by P\v{a}tra\c{s}cu and Thorup
[Pat08,Tho13].
We also present a scheme that compresses a sequence to its
zeroth order (empirical) entropy up to extra
bits, supporting decoding each in expected time.Comment: preliminary version appeared in STOC'2
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum