Optimal Distribution of Video Stream in Large Under-provisioned - - PowerPoint PPT Presentation

Optimal Distribution of Video Stream in Large Under-provisioned Peer-to-peer Networks

Jinhua Zhao

Supervisor: Dr. Hervé Kerivin

ISIMA School of Electronic Information, Wuhan University

11/09/2012

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 1 / 23

Outline

1

Introduction

2

Formulation

3

Polyhedral study for MBRT

4

Adaptation

5

Conclusion

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 2 / 23

Introduction

Live Stream Delivery (LSD)

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 3 / 23

Introduction

Background

Under-provisioning Upload capacity in P2P networks Dynamic Adaptive Streaming over HTTP (DASH) & Multiple Description Coding (MDC) Rooted-tree based approach

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 4 / 23

Introduction

Problem Description

MBRTP (Maximum Bounded Rooted-Tree Packing) problem

◮ Find a family of K rooted-trees in an undirected graph ◮ Rooted at r ◮ Wrt. the capacity constraint ◮ Maximizing the number of sub-streams the nodes received

MBRT (Maximum Bounded Rooted-Tree) problem

◮ K = 1 Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 5 / 23

Introduction

Problem Description

Complexity

◮ NP-hard ◮ Proof: reduction to 3-SAT problem (by H. Kerivin and G. Simon)

Approximation algorithm

◮ Sub-optimal solution with guarantee ◮ In polynomial time ◮ Upper and lower bounds ◮ Factor k approximation algorithm Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 6 / 23

Introduction

Related Problems

Steiner tree problem Similarity

◮ A required set of vertices to be spanned

Differences

◮ MBRT has only one required vertex ◮ Has degree constraint ◮ No edge-weight function Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 7 / 23

Introduction

Related Problems

Bounded degree spanning tree problem Similarity

◮ Degree bounded

Differences

◮ MBRT does not aim at spanning all vertices but maximizing the

number of vertices spanned

◮ No edge-weight function in MBRT Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 8 / 23

Formulation

MBRTP

Maximize

K

• k=1
• e

xk

e

subject to: xk(E(S)) ≤ |S| − 1, for S ⊆ Vand S = ∅ (1)

• k∈K ′

xk(δ(v)) ≤ cv + |K ′|, for v ∈ V\{r}, and K ′ ⊆ {1, 2, · · · , K} (2)

K

• k=1

xk(δ(v)) ≤ cv, for v = r (3) xk(δ(S)) ≥ xk

e , for e ∈ E(S), and r ∈ S

(4) xk

e ≥ 0, e ∈ E,

(5) where x(F) =

e∈F xk e .

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 9 / 23

Formulation

MBRTP

Proof Let Pp = {x ∈ Rm×K : x satisfies inequalities (1), (2), (3), (4) and (5)} Xp = {xF ∈ {0, 1}m×K : xFk ∈ {0, 1}m, where (V(Fk) ∪ r, Fk) is a Bounded Rooted-Tree (BRT), k = 1, · · · , K}, where Fk := {e ∈ E : xFk

e

= 1} Prove the following theorem in two steps.

◮ Prove Xp ⊆ Pp ◮ Prove Pp ∩ Zm×K ⊆ Xp

Theorem

Pp ∩ Zm×K = Xp.

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 10 / 23

Formulation

MBRT

Maximize

• e

xe subject to: x(E(S)) ≤ |S| − 1, for S ⊆ V and S = ∅ (6) x(δ(v)) ≤ c(v) (7) x(δ(S)) ≥ xe, for e ∈ E(S), and r ∈ S (8) xe ≥ 0, e ∈ E, (9)

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 11 / 23

Polyhedral study for MBRT

Dimension

Definition of F ′

Definition

Define a vertex set V1 as the set of vertices in the connected component containing r of G′ which is G minus all the vertices having capacity 1. Let F ′ be F ′ := E(V1) ∪ δ(V1). Prove the theorem

Theorem

dim(conv(X)) = |F ′|.

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 12 / 23

Polyhedral study for MBRT

Facet-defining constraints

Examples

◮ For the constraint xe ≥ 0, it defines a facet iff e is not a bridge ◮ For the constraint xe ≤ 1, it defines a facet iff r and e are two-edge

connected

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 13 / 23

Polyhedral study for MBRT

Separation

Separation problem

Definition

Consider an optimization problem: max{cx : x ∈ P} where P is a polyhedron and P ⊆ Rn. The separation problem for polyhedron P is to determine for a given x∗ ∈ Rn whether or not x∗ ∈ P and if not, to produce an inequality αTx ≤ β where α ∈ Rn, β ∈ R, so that this inequality is satisfied for all x ∈ P but violated by x∗.

Theorem

The optimization problem is polynomially solvable if and only if the separation problem is polynomially solvable (by M. Grötschel, L. Lovász, and A. Schrijver).

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 14 / 23

Polyhedral study for MBRT

Separation

For acyclicity constraint x(E(S)) ≤ |S| − 1, for S ⊆ V and S = ∅ Minimizing the function g(S) = |S| − x(E(S)) Prove g(S) is a submodular function Minimizing a submodular function is polynomially solvable (see Canningham’s paper and Schrijver’s paper)

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 15 / 23

Polyhedral study for MBRT

Separation

For connectivity constraint x(δ(S)) ≥ xe, for e ∈ E(S), and r ∈ S

Definition

A minimum-capacity r − v cut of G = (V, E) is defined to be min{x(δ(S)) : r ∈ S ⊆ V and v ∈ S} Can be solved by solving at most n − 2 minimum r − v cut problem.

◮ Let Wv ⊆ V induce a minimum-capacity r − v cut in G with

x(δ(Wv)).

◮ If there exists a set U ⊆ V and r ∈ U that violates the constraint

(8), there must exists an edge uv where u ∈ N(v) ∩ Wv that violate the constraint as xuv > x(δ(Wv)).

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 16 / 23

Adaptation

Goemans’ Algorithm

Goemans’ algorithm for MBDST (Maximum Bounded Degree Spanning Tree) problem His process

◮ Obtain an extreme point x∗ and its support E∗ of the linear

programming relaxation

◮ Orient E∗ into a directed graph A∗ with maximum indegree at most

2

◮ Find a spanning tree T of minimum cost such that |T ∩ δ+

A∗(v)| ≤ k

for all v ∈ V.

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 17 / 23

Adaptation

Goemans’ Algorithm

Laminar property proof x(E(S)) ≤ |S| − 1, for S ⊆ V and S = ∅ x(δ(S)) ≥ xe, for e ∈ E(S), and r ∈ S

Definition

x∗

S =

, if E∗(V(E∗)\S) = ∅ max{xe, e ∈ E∗(V(E∗)\S)} , otherwise

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Adaptation

Goemans’ Algorithm

Tight sets for case 1, 2 and 3 Tight sets for case 4

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 19 / 23

Adaptation

Goemans’ Algorithm

Tight sets for case 5 Tight sets for case 6 and 7 Result: Not possible to apply Goemans’ algorithm straightforwardly.

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 20 / 23

Adaptation

Jain’s Algorithm

Half-integral property

Definition

Half-integral property is that a problem always has a optimal fractional solution with half-integral values, which are normally values among 0, 0.5 and 1.

◮ A counter example of half-integral property

Submodular property

Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 21 / 23

Conclusion

Objectives achieved

◮ Learning in a new field ◮ Build foundation ◮ Formulation proof ◮ Started polyhedral study ◮ Attempt on adaptation

Future work in my PhD

◮ Study and adaptation on other algorithms ◮ Polyhedral and computational study for K = 1 and K > 1, and for

• ther models also

◮ Decomposition for K > 1 Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 22 / 23

Thanks for your attention!

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