Definition
Forv∈V, letWv ={Pv,u|u∈V}.
For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.
Outline of the proof:
First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.
Then we use the lemma to show that there is an edgee∗ that is “quite popular” for “many” nodes, that is,e∗ist-popular fort different nodes, fort= Ω(√
n/∆).
Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine∗, which proves the lower bound.
Proof of the lower bound by Borodin and Hopcroft
Definition
Forv∈V, letWv ={Pv,u|u∈V}.
For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.
Outline of the proof:
First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.
Then we use the lemma to show that there is an edgee∗ that is “quite popular” for “many” nodes, that is,e∗ist-popular fort different nodes, fort= Ω(√
n/∆).
Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine∗, which proves the lower bound.
Proof of the lower bound by Borodin and Hopcroft
Definition
Forv∈V, letWv ={Pv,u|u∈V}.
For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.
Outline of the proof:
First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.
Then we use the lemma to show that there is an edgee∗ that is “quite popular” for “many” nodes, that is,e∗ist-popular fort different nodes, fort= Ω(√
n/∆).
Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine∗, which proves the lower bound.
Proof of the lower bound by Borodin and Hopcroft
Definition
Forv∈V, letWv ={Pv,u|u∈V}.
For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.
Outline of the proof:
First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.
Then we use the lemma to show that there is an edgee∗ that is “quite popular” for “many” nodes, that is,e∗ist-popular fort different nodes, fort= Ω(√
n/∆).
Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine∗, which proves the lower bound.
Proof of the lower bound by Borodin and Hopcroft
Definition
Forv∈V, letWv ={Pv,u|u∈V}.
For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.
Outline of the proof:
First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.
Then we use the lemma to show that there is an edgee∗ that is “quite popular” for “many” nodes, that is,e∗ist-popular fort different nodes, fort= Ω(√
n/∆).
Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine∗, which proves the lower bound.
Proof of the lower bound by Borodin and Hopcroft
Definition
Forv∈V, letWv ={Pv,u|u∈V}.
For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.
Outline of the proof:
First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.
Then we use the lemma to show that there is an edgee∗ that is “quite popular” for “many” nodes, that is,e∗ist-popular fort different nodes, fort= Ω(√
n/∆).
Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine∗, which proves the lower bound.
Proof of the lower bound by Borodin and Hopcroft
Definition
Fort>0, we define a 0-1 matrixA(t):
The matrix hasnrows and|E|columns.
Forv∈V, ande∈E, define Av,e(t) =
1 ife ist-popular forv, and 0 otherwise,
Forv∈V, letAv(t) =P
e∈EAv,e(t)denote the row sum ofv. Fore∈E, letAe(t) =P
v∈VAv,e(t)denote the column sum ofe.
Proof of the lower bound by Borodin and Hopcroft
Definition
Fort>0, we define a 0-1 matrixA(t):
The matrix hasnrows and|E|columns.
Forv∈V, ande∈E, define Av,e(t) =
1 ife ist-popular forv, and 0 otherwise,
Forv∈V, letAv(t) =P
e∈EAv,e(t)denote the row sum ofv. Fore∈E, letAe(t) =P
v∈VAv,e(t)denote the column sum ofe.
Proof of the lower bound by Borodin and Hopcroft
Definition
Fort>0, we define a 0-1 matrixA(t):
The matrix hasnrows and|E|columns.
Forv∈V, ande∈E, define Av,e(t) =
1 ife ist-popular forv, and 0 otherwise,
Forv∈V, letAv(t) =P
e∈EAv,e(t)denote the row sum ofv. Fore∈E, letAe(t) =P
v∈VAv,e(t)denote the column sum ofe.
Proof of the lower bound by Borodin and Hopcroft
Definition
Fort>0, we define a 0-1 matrixA(t):
The matrix hasnrows and|E|columns.
Forv∈V, ande∈E, define Av,e(t) =
1 ife ist-popular forv, and 0 otherwise,
Forv∈V, letAv(t) =P
e∈EAv,e(t)denote the row sum ofv. Fore∈E, letAe(t) =P
v∈VAv,e(t)denote the column sum ofe.
Proof of the lower bound by Borodin and Hopcroft
Definition
Fort>0, we define a 0-1 matrixA(t):
The matrix hasnrows and|E|columns.
Forv∈V, ande∈E, define Av,e(t) =
1 ife ist-popular forv, and 0 otherwise,
Forv∈V, letAv(t) =P
e∈EAv,e(t)denote the row sum ofv. Fore∈E, letAe(t) =P
v∈VAv,e(t)denote the column sum ofe.