The Complexity of Channel Scheduling in Multi-Radio Multi-Channel Wireless Networks
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Transcript of The Complexity of Channel Scheduling in Multi-Radio Multi-Channel Wireless Networks
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!