We haveW=O(lognn), w.h.p.
In order to improve the load balancing, we usek virtual nodes for each device. LetV0denote the set ofkn“virtual” nodes.
Each of these nodes gets an address from [0,1) chosen i.u.r.
For addressA∈[0,1), re-define succ(A) =
argmin{a(i)>A|i ∈V0} if∃i ∈V0:a(i)∈[A,1), argmin{a(i)>0|i ∈V0} otherwise.
Objectx ∈Uis mapped to nodesucc(h(x))and stored on the device to which this node belongs.
LetWi denote the weight of devicei, i.e., the sum of the lengths of the intervals corresponding toi’s nodes, andW =maxi∈[n]Wi.
Improved quality of the load balancing
We haveW=O(lognn), w.h.p.
In order to improve the load balancing, we usek virtual nodes for each device. LetV0denote the set ofkn“virtual” nodes.
Each of these nodes gets an address from [0,1) chosen i.u.r.
For addressA∈[0,1), re-define succ(A) =
argmin{a(i)>A|i ∈V0} if∃i ∈V0:a(i)∈[A,1), argmin{a(i)>0|i ∈V0} otherwise.
Objectx ∈Uis mapped to nodesucc(h(x))and stored on the device to which this node belongs.
LetWi denote the weight of devicei, i.e., the sum of the lengths of the intervals corresponding toi’s nodes, andW =maxi∈[n]Wi.
Improved quality of the load balancing
We haveW=O(lognn), w.h.p.
In order to improve the load balancing, we usek virtual nodes for each device. LetV0denote the set ofkn“virtual” nodes.
Each of these nodes gets an address from [0,1) chosen i.u.r.
For addressA∈[0,1), re-define succ(A) =
argmin{a(i)>A|i ∈V0} if∃i ∈V0:a(i)∈[A,1), argmin{a(i)>0|i ∈V0} otherwise.
Objectx ∈Uis mapped to nodesucc(h(x))and stored on the device to which this node belongs.
LetWi denote the weight of devicei, i.e., the sum of the lengths of the intervals corresponding toi’s nodes, andW =maxi∈[n]Wi.
Improved quality of the load balancing
We haveW=O(lognn), w.h.p.
In order to improve the load balancing, we usek virtual nodes for each device. LetV0denote the set ofkn“virtual” nodes.
Each of these nodes gets an address from [0,1) chosen i.u.r.
For addressA∈[0,1), re-define succ(A) =
argmin{a(i)>A|i ∈V0} if∃i ∈V0:a(i)∈[A,1), argmin{a(i)>0|i ∈V0} otherwise.
Objectx ∈Uis mapped to nodesucc(h(x))and stored on the device to which this node belongs.
LetWi denote the weight of devicei, i.e., the sum of the lengths of the intervals corresponding toi’s nodes, andW =maxi∈[n]Wi.
Improved quality of the load balancing
We haveW=O(lognn), w.h.p.
In order to improve the load balancing, we usek virtual nodes for each device. LetV0denote the set ofkn“virtual” nodes.
Each of these nodes gets an address from [0,1) chosen i.u.r.
For addressA∈[0,1), re-define succ(A) =
argmin{a(i)>A|i ∈V0} if∃i ∈V0:a(i)∈[A,1), argmin{a(i)>0|i ∈V0} otherwise.
Objectx ∈Uis mapped to nodesucc(h(x))and stored on the device to which this node belongs.
LetWi denote the weight of devicei, i.e., the sum of the lengths of the intervals corresponding toi’s nodes, andW =maxi∈[n]Wi.
Improved quality of the load balancing
Theorem
For anyk>1,W = 1n·O(1+logkn), w.h.p.
Corollary
Ifk>lognthenW =O(1n), w.h.p.
Proof of the Theorem:
Consider devicej and suppose the address of theknodes of this device are fixed arbitrarily.
For anyt∈[0,1), we want to upper-boundPr[Wj>t].
Improved quality of the load balancing
Theorem
For anyk>1,W = 1n·O(1+logkn), w.h.p.
Corollary
Ifk>lognthenW =O(1n), w.h.p.
Proof of the Theorem:
Consider devicej and suppose the address of theknodes of this device are fixed arbitrarily.
For anyt∈[0,1), we want to upper-boundPr[Wj>t].
Improved quality of the load balancing
Theorem
For anyk>1,W = 1n·O(1+logkn), w.h.p.
Corollary
Ifk>lognthenW =O(1n), w.h.p.
Proof of the Theorem:
Consider devicej and suppose the address of theknodes of this device are fixed arbitrarily.
For anyt∈[0,1), we want to upper-boundPr[Wj>t].
Improved quality of the load balancing
Theorem
For anyk>1,W = 1n·O(1+logkn), w.h.p.
Corollary
Ifk>lognthenW =O(1n), w.h.p.
Proof of the Theorem:
Consider devicej and suppose the address of theknodes of this device are fixed arbitrarily.
For anyt∈[0,1), we want to upper-boundPr[Wj>t].
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
In order to be able to enumerate all possibilities for choosing thesek intervals, we look at a slightly strongernecessary conditionfor the event Wj>t.
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
In order to be able to enumerate all possibilities for choosing thesek intervals, we look at a slightly strongernecessary conditionfor the event Wj>t.
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
In order to be able to enumerate all possibilities for choosing thesek intervals, we look at a slightly strongernecessary conditionfor the event Wj>t.
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
In order to be able to enumerate all possibilities for choosing thesek intervals, we look at a slightly strongernecessary conditionfor the event Wj>t.
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
Necessary condition:
If the eventWj>thappens then there arekintervals left of thekaddresses of j’s nodes so that
the length of each of these intervals is a multiple of kn1
these intervals have a total length oft0wheret0 is the largest multiple of
1
kn such thatt06t−1n, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
Necessary condition:
If the eventWj>thappens then there arekintervals left of thekaddresses of j’s nodes so that
the length of each of these intervals is a multiple of kn1
these intervals have a total length oft0wheret0 is the largest multiple of
1
kn such thatt06t−1n, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
Improved quality of the load balancing
Exact condition:
The eventWj>thappensif and only ifthere arek intervals left of thek addresses ofj’s nodes so that
these intervals have a total length oft, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
Necessary condition:
If the eventWj>thappens then there arekintervals left of thekaddresses of j’s nodes so that
the length of each of these intervals is a multiple of kn1
these intervals have a total length oft0wheret0 is the largest multiple of
1
kn such thatt06t−1n, and
none of the otherk(n−1)nodes have an address that falls into these intervals.
Improved quality of the load balancing
The number of possibilities to choose these intervals corresponds to the number of possibilities to choosekintegersq1, . . . ,qksuch that Pk
i=1qi =q, forq=t0kn.
Theqi’s can be encoded bijectively into binary strings withk−1 ones andqzeros.
Thus, the number of possibilities to choose theqi’s and, hence, the intervals is at most
q+k−1
Improved quality of the load balancing
The number of possibilities to choose these intervals corresponds to the number of possibilities to choosekintegersq1, . . . ,qksuch that Pk
i=1qi =q, forq=t0kn.
Theqi’s can be encoded bijectively into binary strings withk−1 ones andqzeros.
Thus, the number of possibilities to choose theqi’s and, hence, the intervals is at most
q+k−1
Improved quality of the load balancing
The number of possibilities to choose these intervals corresponds to the number of possibilities to choosekintegersq1, . . . ,qksuch that Pk
i=1qi =q, forq=t0kn.
Theqi’s can be encoded bijectively into binary strings withk−1 ones andqzeros.
Thus, the number of possibilities to choose theqi’s and, hence, the intervals is at most
q+k−1
Improved quality of the load balancing
The number of possibilities to choose these intervals corresponds to the number of possibilities to choosekintegersq1, . . . ,qksuch that Pk
i=1qi =q, forq=t0kn.
Theqi’s can be encoded bijectively into binary strings withk−1 ones andqzeros.
Thus, the number of possibilities to choose theqi’s and, hence, the intervals is at most
q+k−1
Improved quality of the load balancing
Once the intervals are fixed, the probability that these intervals with a total length oft0>t−2n are not hit by one otherk(n−1)addresses is at most
e−t0k(n−1)≤e−(t−2n)k(n−1) which follows analogously to the lemma on slide 17.
This gives assumingn>2.
Improved quality of the load balancing
Once the intervals are fixed, the probability that these intervals with a total length oft0>t−2n are not hit by one otherk(n−1)addresses is at most
e−t0k(n−1)≤e−(t−2n)k(n−1) which follows analogously to the lemma on slide 17.
This gives assumingn>2.
Improved quality of the load balancing
Once the intervals are fixed, the probability that these intervals with a total length oft0>t−2n are not hit by one otherk(n−1)addresses is at most
e−t0k(n−1)≤e−(t−2n)k(n−1) which follows analogously to the lemma on slide 17.
This gives assumingn>2.
Improved quality of the load balancing
Once the intervals are fixed, the probability that these intervals with a total length oft0>t−2n are not hit by one otherk(n−1)addresses is at most
e−t0k(n−1)≤e−(t−2n)k(n−1) which follows analogously to the lemma on slide 17.
This gives assumingn>2.
Improved quality of the load balancing
Once the intervals are fixed, the probability that these intervals with a total length oft0>t−2n are not hit by one otherk(n−1)addresses is at most
e−t0k(n−1)≤e−(t−2n)k(n−1) which follows analogously to the lemma on slide 17.
This gives assumingn>2.
Improved quality of the load balancing
decreases exponentially inβ since e12 > 43. Forβ>25, this term is less than 1. Consequently,
Improved quality of the load balancing
decreases exponentially inβ since e12 > 43. Forβ>25, this term is less than 1. Consequently,
Improved quality of the load balancing
decreases exponentially inβ since e12 > 43. Forβ>25, this term is less than 1. Consequently,
Introduction Con.Hash. Ch.NW. Rand.Obl.Routing Path Selection Hypercube General NW.
6:26 Introduction 1/4 Walter Unger 30.1.2017 11:54 WS2016/17 Z