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MinimumLatency Broadcast Scheduling in Wireless Ad Hoc Networks

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Basic Broadcast Scheduling (BBS): Overview. Transmitters: ... Differs from BBS only in connectors' selection and transmission scheduling. For 1 i R-1, ... – PowerPoint PPT presentation

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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'
  • i-i' 2 far separation
  • 0 transmissions.
  • ii' and jj' Sij ends before Si'j' starts.

24
Correctness
  • Every v?Vij is informed before tij
  • Let u be the parent of v and j' rank(u)
  • j' j v is informed in the slot ti-1,j
  • j' j v is informed in Si-1,j', which ends
    before Sij starts

25
Bound on Latency
  • All nodes in V are informed before tR0 ?
    latency ? tR0 R3(L1)r ? R3(L1) logn
  • Replace L by 16 ? latency ? R51?logn

26
Pipelining Broadcast Scheduling (PBS)
  • Select an MIS U
  • Construct the SPT T from s to U V(T) O(R3)
  • Broadcast in GV(T) from s ? R153logR O(1)
    slots
  • Transmissions by U ? 12 slots
  • ? latency RO(153logR)
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