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MinimumLatency Broadcast Scheduling in Wireless Ad Hoc Networks

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Title: MinimumLatency Broadcast Scheduling in Wireless Ad Hoc Networks

1
Minimum-Latency Broadcast Scheduling in Wireless
Ad Hoc Networks
2
Network Model

Planar distribution unit transmission range ?
communication topology G (V, E) is a unit-disk
graph
Synchronous transmissions in equal-length
time-slots
Nodes informed by a set U of (simultaneously
transmitting) nodes Inf(U) ? v ?V\U N(v) ?
U1
N(v) the set of neighbors of v in G

3
Broadcast Schedule

A broadcast schedule of latency l from a source
node s?V is a sequence ?U1, U2, ?, Ul? s.t.
U1 s,
Ui ? Inf(U1) ? Inf(U2) ?? ? Inf(Ui-1), ? 2 ? i ?
l
V\s ? Inf(U1) ? Inf(U2) ?? ? Inf(Ul).
Minimum Latency Broadcast Scheduling (MLBS)
seek a broadcast schedule of the smallest
latency.

4
Radius of G w.r.t. s

R maxv ?V dist(v,s) height of any BFS tree
rooted at s
Layer i Vi v ?V dist(v,s)i for each 0 ?
i ? R
Latency ? R

5
MLBS in UDGs

NP hard Gandhi, Parthasarathy and Mishra 2003
Claimed but never appeared
Known approximations
Dessmark and Pelc 2001 ? 1200R
Gandhi, Parthasarathy and Mishra 2003 ? 648R
New approximations
A broadcast schedule with latency ? 16R-15
A broadcast schedule with latency

6
Independence

A set of nodes are indep. if their pairwise
distances 1.
Independent set (IS)
Maximal independent set (MIS)
MIS induced by a node ordering v1, v2, ?, vn
U? ?
for i1 to n, if vi is not adjacent to U, then U?
U vi.

7
Distance-2 Coloring

Distance-2 coloring of U ? V proper coloring of
G2U
Application for each color class Ui, Inf(Ui)
N(Ui).
Any IS is distance-2 12-colorable
Tile Coloring simple but requiring knowledge of
nodes positions
First-Fit Coloring requiring no knowledge of
nodes positions but is less simpler

8
Tile Coloring

Each cell contains at most one dominator.
If u and v lie in two cells with the same label,
then uv 2.

0.5
9
FIRST-FIT Coloring
10
Basic Broadcast Scheduling (BBS) Overview

Transmitters
Dominators MIS induced by BFS ordering
Connectors parents of dominators in BFS tree
Sequential transmission scheduling
Layer by layer in the top-down manner
In each layer, dominators transmit before
connectors

11
BBS Preprocessing

T ? a BFS tree
U ? MIS induced by BFS ordering dominators
For 0 ? i ? R, Ui? dominators in layer i
U0 s, U1 ?
For 2 ? i ? R, Uij 1 ? j ? ci? a dist-2
coloring of Ui with ?12 colors
For 1 ? i ? R-1 and 1 ? j ? ci1, Wij? parents of
nodes in Ui1,j .

12
BBS Scheduling

Layer 0 s only
Layer 1 ?W1j 1 ? j ? c2?
Layer i with 2 ? i ? R-1 ?Uij 1 ? j ? ci?,
?Wij 1 ? j ? ci1?
Layer R ?URj 1 ? j ? cR?
? Latency 12(c2c3?cR) ? 124(R-1)24R-23

13
Improvement on BBS

Fewer connectors minimal cover
Shorter schedule of transmissions by connectors
? 4 slots
No redundancy each connector transmits once

14
Minimal Cover

A subset X ? V covers another subset Y ? V if Y\X
? N(X).
X is a minimal cover (MC) of Y if X is a cover of
Y but no proper subset of X is a cover of Y.
Suppose that X is a cover of Y. Any ordering x1,
x2, ?, xm of X induces a MC W ? X of Y by the
following method
W? X
for i1 to m, if W-xi is a cover of Y, then W?
W -xi.

15
Iterative Minimal Covering (IMC)

Input A bipartite graph H(X,Y E) in which no
node in Y is isolated.
Output disjoint subsets W1, W2, ?, Wl of X
satisfying that
W1 ? W2 ? ? ?Wl is a MC of Y.
Y ? Inf(W1) ? Inf(W2) ?? ? Inf(Wl),
l ? the maximum degree of nodes in Y.
Algorithm IMC
l ? 0, X0 ? X
While Y ? ?,
l ? l 1, Xl ? a MC in Xl-1 of Y
Wl-1 ? Xl-1 - Xl , Y ? Y - Inf(Xl)
Wl ? Xl

16
Enhanced Broadcast Scheduling (EBS)

Differs from BBS only in connectors selection
and transmission scheduling
For 1 ? i ? R-1,
Xi? parents of nodes in Ui1 .
Wij 1 ? j ? li ? output by IMC applied to
GXi, Ui1
li ? 4 since each node in Xi is adjacent to ?4
nodes in Ui1
? Latency 1 (l1l2?lR-1) (c2c3?cR) ?
116(R-1)16R-15

17
Further Improvement on EBS

Idea Pipeline transmissions in more than one
layers
Interleave transmissions to avoid interference
Transmissions from layer i are restricted to the
slots t t ? i mod 3

18
Inter-Layer Broadcast Scheduling (ILBS)

Input A graph H and a partition of the vertex
set into X and Y satisfying that X is a cover of
Y
Output a sequence ?U1, U2, ?, Ul? s.t.
U1 ? X,
Ui ? X ? Inf(U1) ? Inf(U2) ?? ? Inf(Ui-1), ? 2 ?
i ? l
Y ? Inf(U1) ? Inf(U2) ?? ? Inf(Ul).

19
An algorithm for ILBS

U ? an MIS in HY
Wi 1 ? i? k ? output by IMC applied to HX,U
Ui 1 ? i ? c? a dist-2 coloring of U with
?12 colors
Schedule ?Wi 1 ? i? k?, ?Ui 1 ? i ? c?
Latency k c
If H is a UDG, then latency ? 51217
If X and Y lie in two consecutive layers
respectively of a UDG G and HGX?Y, then ?
41216

20
Canonical BFS Tree Node Ranking

0 ? rank(v) ? logn, ?v ?V
u ?Vi is the parent of v ?
rank(u) ? rank(v)
u has the largest rank among all neighbors of v
in layer i.
u ?Vi is the parent of v, rank(u) rank(v) ?
all other children of u have smaller rank
if w?Vi1 is a neighbor of u with the same rank,
then its parent has larger rank

21
Notations

r rank(s) maximum rank
Vij nodes in layer i with rank j
Uij children of the nodes in Vij
Gij GVij?Uij
A an algorithm for ILBS
LL(n) a bound on the latency of the schedule
output by A

22
Scheduling in Gij

tij ? i3(1L)(r-j)
W0 ? parents of the nodes in with rank j
?W1, W2, ?, Wl?? schedule for broadcasting from
Vij to Uij - Inf(W0) output by A
Schedule Sij Wk transmit in slot tij 3k for 0
? k ? l

23
No Interference between Sij and Si'j'