The Complexity of Channel Scheduling in Multi-Radio Multi-

Channel Wireless Networks

Wei Cheng & Xiuzhen Cheng

The George Washington University

Taieb Znati

University of Pittsburgh

Xicheng Lu & Zexin Lu

National University of Defense technology

Outline

Introduction Network Model The Complexity of OWCS/P PTAS for OWCS/P Summary

Introduction – Background

Multi-Radio Multi-Channel (MR-MC) to enhance mesh network throughput• Equipped with multiple radios, nodes can

communicate with multiple neighbors simultaneously over orthogonal channels to improve the network throughput.

• The key problem is the channel scheduling, which aims to maximize the concurrent traffics without interfering each other.

Introduction – Interference Model

P(hysical) interference-free model• if two nodes want to launch bidirectional

communications, any other node whose minimum distance to the two nodes is not larger than the interference range must keep silent.

Hop interference-free model (no position)• … is no larger than H hops must keep silent.

Introduction – Problem Optimal Weighted Channel Scheduling under

the Physical distance constraint (OWCS/P)• Given an edge-weighted graph G(V,E)

representing an MR-MC wireless network, compute an optimal channel scheduling O(G) E, such that ∈O(G) is P interference-free and the weight of O(G) is maximized

Optimal Weighted Channel Scheduling under Hop distance constraint (OWCS/H)

Introduction – Motivation

Both the physical interference-free model and the hop interference-free model are popular but their relations have never been addressed in literature.

Current complexity results for OWCS

Related Research

Channel allocation, routing, and packet scheduling have been jointly considered as a IP problem

Channel Assignment• Common channel

• Default radio for reception

• Code based approach

Related Research

The complexity of scheduling in SR-SC networks• OWCS/H>=1 is NP hard

• OWCS/H>=1 has PTAS

Network Model

Geometric graphs G(V,E), |V | = n a set of C ={c1, c2, · · · , ck} orthogonal

channels ∀ node i V , 1 ≤ i ≤ n, it is equipped ∈

with ri radios and can access a set of Ci C channels, where |C⊆ i| = ki.

Formal Definition

Edge-Physical-Distance Edge-Hop-Distance OWCS/P: Seek an E’ such that any pair

of edges in E’ has an Edge-Physical-Distance >P, and E’ is the maximum

OWCS/H: Seek an E’ such that any pair of edges in E’ has an Edge-Physical-Distance >H, and E’ is the maximum

The Complexity of OWCS/P

Lemma : OWCS/P=1 and OWCS/H=1 are equivalent in SR-SC wireless networks.• Intuition: the interference graphs of G(V,E) for

the cases of P=1 and H=1 are the same

• Proof:• OPT/P=1 is a feasible solution to OWCS/H=1

• We can not add another edge to OPT/P=1 for OWCS/H=1

• Similarly, OPT/H=1 is optimal to OWCS/P=1

The Complexity of OWCS/P

Theorem : OWCS/P>=1 is NP-Hard in SR-SC wireless networks.• OWCS/H=1 is NP-Hard OWCS/P=1 is NP-

Hard

• OWCS/P>1 is polynomial time reducible to OWCS/P=1

The Complexity of OWCS/P

Theorem: OWCS/P>=1 is NP-Hard in MR-MC wireless networks.

Known Known

PTAS for OWCS/P

Polynomial-Time Approximation Scheme (PTAS) for NP-Hard problem.• a polynomial-time approximate solution with a

performance ratio (1 − ε) for an arbitrarily small positive number ε .

• Let Ptas(G) denote the solution given by the PTAS procedure and O(G) the optimal solution for the OWCS/P≥1 problem in a MR-MC network G.• We will prove that W(Ptas(G)) ≥ (1 − ε)W(O(G))

PTAS for OWCS/P-construction

Griding:• Partition network space into small grids with

each having a size of (P + 2) × (P +2).

• Label each grid by (a, b), where a, b = 0, 1, · · · ,N − 1, with N the total number of grids at each row or column. • The id of the grid at the lower-left corner can be

denoted by (0, 0).

• Denote the ith row and the jth column of the grids by Rowi and Colj , respectively.

PTAS for OWCS/P-construction Shifted Dissection:

• Partition vertically the network space • by columns of the grids Colj and rows of the grids Rowi,

where j | (m+1)= k1 , i |(m+1)= k2, k1 k2 = 0, 1, · · · ,m. Remove all the edges whose both end nodes are in Colj or Rowi

• Obtain a number of super-grids with each containing at most m×m grids. Total (m + 1)2 different dissections

• Denote each dissection by Pa,b, where a, b indicate that Pa,b is obtained by shifting Col0 to column b and Row0 to row a.

PTAS for OWCS/P-construction

Computation

• Consider a specific Pa,b

•For each super-grid B in Pa,b,

•compute an maximum weight channel scheduling SB for B.

• Let Sa,b be the union of all SB’s

• Sa,b is a feasible solution for OWCS/P

• Repeat for all Pa,b

PTAS for OWCS/P-algorithm

PTAS for OWCS/P-complexity Computing SB takes polynomial time.

• the area of B is at most (m(P + 2) + 2)2

• For a specific channel

• The number of SB’s edges in each ((P + 2)2) grid is bounded by O(1).

• Then the number of edges in SB is bounded by O(m2)

• Time of computing SB through enumerating is bounded by |EB|O(m2)

• For all K channels

• Time of computing SB through enumerating is bounded by |EB|O(m2)K

PTAS for OWCS/P-performance For all partition Pa,b

• Sa,b is the optimal solution for Ea,b

• Let yields ,

PTAS for OWCS/P-performance

A grid will NOT be included in any super-grid among all (m+ 1)2 partitions for 2m+ 1 times.• An edge will NOT be included in any super-grid

among all (m+ 1)2 partitions for at most 2m+ 1 times.

Summary

Summary

The proposed PTAS for OWCS/P is also a PTAS for OWCS/H in MR-MC wireless networks.• Replace P by H

Summary

OWCS/H=1 is equivalent to OWCS/P>= under the polynomial transformation• OWCS/H=1 is equivalent to OWCS/P=1

• OWCS/P>1 is polynomial time reducible to OWCS/P=1

Physical interference free model is more precise• Need position information

Q&A

Thanks!