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 Introduction 1 Formulation 2 Polyhedral study for MBRT 3 Adaptation 4 Conclusion 5 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 K � � x k Maximize e e k = 1 subject to: x k ( E ( S )) ≤ | S | − 1 , for S ⊆ V and S � = ∅ (1) x k ( δ ( v )) ≤ c v + | K ′ | , for v ∈ V \{ r } , and K ′ ⊆ { 1 , 2 , · · · , K } � (2) k ∈ K ′ K � x k ( δ ( v )) ≤ c v , for v = r (3) k = 1 x k ( δ ( S )) ≥ x k e , for e ∈ E ( S ) , and r ∈ S (4) x k e ≥ 0 , e ∈ E , (5) e ∈ F x k where x ( F ) = � e . Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 9 / 23
Formulation MBRTP Proof Let Pp = { x ∈ R m × K : x satisfies inequalities ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) and ( 5 ) } Xp = { x F ∈ { 0 , 1 } m × K : x F k ∈ { 0 , 1 } m , where ( V ( F k ) ∪ r , F k ) is a Bounded Rooted-Tree (BRT), k = 1 , · · · , K }, where F k := { e ∈ E : x F k = 1 } e Prove the following theorem in two steps. ◮ Prove Xp ⊆ Pp ◮ Prove Pp ∩ Z m × K ⊆ Xp Theorem Pp ∩ Z m × K = Xp. Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 10 / 23
Formulation MBRT � x e Maximize e subject to: x ( E ( S )) ≤ | S | − 1 , for S ⊆ V and S � = ∅ (6) x ( δ ( v )) ≤ c ( v ) (7) x ( δ ( S )) ≥ x e , for e ∈ E ( S ) , and r ∈ S (8) x e ≥ 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 V 1 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 ( V 1 ) ∪ δ ( V 1 ) . 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 x e ≥ 0, it defines a facet iff e is not a bridge ◮ For the constraint x e ≤ 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 ⊆ R n . The separation problem for polyhedron P is to determine for a given x ∗ ∈ R n whether or not x ∗ ∈ P and if not, to produce an inequality α T x ≤ β where α ∈ R n , 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 )) ≥ x e , 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 W v ⊆ V induce a minimum-capacity r − v cut in G with x ( δ ( W v )) . ◮ 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 ) ∩ W v that violate the constraint as x uv > x ( δ ( W v )) . 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 )) ≥ x e , for e ∈ E ( S ) , and r ∈ S Definition � 0 , if E ∗ ( V ( E ∗ ) \ S ) = ∅ x ∗ S = max { x e , e ∈ E ∗ ( V ( E ∗ ) \ S ) } , otherwise Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 18 / 23
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 other models also ◮ Decomposition for K > 1 Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 22 / 23
Thanks for your attention! Jinhua ZHAO (ISIMA and WHU) Third year internship report 11/09/2012 23 / 23
Recommend
More recommend
