TY - UNPB

T1 - Linear-Size Hopsets with Small Hopbound, and Distributed Routing with Low Memory

AU - Elkin, Michael

AU - Neiman, Ofer

PY - 2017

Y1 - 2017

N2 - For a positive parameter β, the β-bounded distance between a pair of vertices u, v in a weighted
undirected graph G = (V, E, ω) is the length of the shortest u − v path in G with at most β edges, aka
hops. For β as above and ǫ > 0, a (β, ǫ)-hopset of G = (V, E, ω) is a graph G′ = (V, H, ωH) on the
same vertex set, such that all distances in G are (1 +ǫ)-approximated by β-bounded distances in G∪G′
.
Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used
for distance-related problems in a variety of computational settings. Currently existing constructions of
hopsets produce hopsets either with Ω(n log n) edges, or with a hopbound n
Ω(1). In this paper we devise
a construction of linear-size hopsets with hopbound (ignoring the dependence on ǫ) (log n)
log(3) n+O(1)
.
This improves the previous bound almost exponentially.
We also devise efficient implementations of our construction in PRAM and distributed settings. The
only existing PRAM algorithm [EN16a] for computing hopsets with a constant (i.e., independent of
n) hopbound requires n
Ω(1) time. We devise a PRAM algorithm with polylogarithmic running time
for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than
the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from
[EN16a].
We use our hopsets to devise a distributed routing scheme that exhibits near-optimal tradeoff between
individual memory requirement O˜(n
1/k) of vertices throughout preprocessing and routing phases of the
algorithm, and stretch O(k), along with a near-optimal construction time ≈ D + n
1/2+1/k, where D
is the hop-diameter of the input graph. Previous distributed routing algorithms either suffered from a
prohibitively large memory req

AB - For a positive parameter β, the β-bounded distance between a pair of vertices u, v in a weighted
undirected graph G = (V, E, ω) is the length of the shortest u − v path in G with at most β edges, aka
hops. For β as above and ǫ > 0, a (β, ǫ)-hopset of G = (V, E, ω) is a graph G′ = (V, H, ωH) on the
same vertex set, such that all distances in G are (1 +ǫ)-approximated by β-bounded distances in G∪G′
.
Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used
for distance-related problems in a variety of computational settings. Currently existing constructions of
hopsets produce hopsets either with Ω(n log n) edges, or with a hopbound n
Ω(1). In this paper we devise
a construction of linear-size hopsets with hopbound (ignoring the dependence on ǫ) (log n)
log(3) n+O(1)
.
This improves the previous bound almost exponentially.
We also devise efficient implementations of our construction in PRAM and distributed settings. The
only existing PRAM algorithm [EN16a] for computing hopsets with a constant (i.e., independent of
n) hopbound requires n
Ω(1) time. We devise a PRAM algorithm with polylogarithmic running time
for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than
the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from
[EN16a].
We use our hopsets to devise a distributed routing scheme that exhibits near-optimal tradeoff between
individual memory requirement O˜(n
1/k) of vertices throughout preprocessing and routing phases of the
algorithm, and stretch O(k), along with a near-optimal construction time ≈ D + n
1/2+1/k, where D
is the hop-diameter of the input graph. Previous distributed routing algorithms either suffered from a
prohibitively large memory req

KW - cs.DS

M3 - ???researchoutput.researchoutputtypes.workingpaper.preprint???

T3 - Arxiv preprint

BT - Linear-Size Hopsets with Small Hopbound, and Distributed Routing with Low Memory

ER -