CPSC 689: Discrete Algorithms
for Mobile and Wireless Systems
Spring 2009
Prof. Jennifer Welch
Lecture 18
 Topic:
 More location services
 Sources:
 Abraham, Dolev, & Malkhi
 MIT 6.885 Fall 2008 slides
 CPSC 689 slides from Spring 2006 by Sai Siddarth
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Drawbacks of GLS
 Handling of updates and out-of-date information:
 when a node crosses a grid boundary, updating its role
can take a lot of resources
 Lack of scalability:
 two nodes can be very close but the smallest square
containing both of them cain be arbitrarily large
 Expensive in sparse networks:
 routing to location server can be much more expensive
than just directly routing to destination
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Example Drawback of GLS
37
19
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Locality-Aware Location Service (LLS)
 Location service for unit-disk graph model.
 Edge (u,v) iff dist(u,v), Euclidean distance between u and v, is ≤ 1.
 Try to obtain performance for FIND and MOVE that depends on
geographical distances:




Cost of MOVE(s,t), average case: O(d log d), where d = dist(s,t).
Cost of a FIND(t) from s, average case: O(d), where d = dist(s,t).
Averages taken over all pairs (s,t) of locations.
Cost of a FIND(t) from s, worst case: O(d2), where d is the minimum
cost of a path from s to t.
 Costs:
 Cost of an edge = its length (can generalize this measure)
 Cost of a path = sum of costs of edges
 Cost of a MOVE or FIND = total costs of all edges traversed in the
implementation (?)
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Geographic Routing
 Assume availability of geographic routing
algorithm, which can broadcast to a known
location.
 Worst-case cost O(d2), average cost O(d), where
d = dist(source, target).
 (Study such algorithms next.)
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Model
 n mobile nodes in M  M square, coordinates in [0, M], each u
with a unique identifier, u.id.
 Unit disk graph network.
 All nodes know their own locations, and identities and
locations of their neighbors.
 Assume lower bound on dist(u,v); claim this assumption could
be removed using a connected dominating set backbone
network [Kuhn, et al.]
 Virtual nodes:
 For any point p in the square, at any time, define l(p) to be
the closest mobile node to p.
 Assume the geographical routing algorithm, when sending to
p, will reach l(p).
 Essentially assume that each p has a (virtual) node.
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Hierarchical Lattice Structure
 Focus on MOVE/FIND for one mobile
node t.
 Define a hierarchical lattice just for t,
based at a geographical point H(t.id).
 Different lattices for different nodes
are used to balance workload.
 Consists of lattices Lk = Lk(t.id) at
different levels k.
 Lattice Lk is the set of grid points for a
grid of squares of side 2k in both
directions, starting from H = H(t.id).
Discrete Algs for Mobile Wireless Sys
H
2k
8
Origin of Lattices for t
 For each node identifier we build a hierarchy of
lattices associated with the node identifier.
 For node t, we use a double hash function H : V
to [0,M] x [0,M] that maps node identifier to
coordinates inside M x M:
 H(t.id)={h1(t.id), h2(t.id)}
 H(t.id) is the primary virtual home of node t.
 H(t.id) never changes! It serves as the origin for
all the lattices for t.
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Hierarchical Lattice Structure
 For node t and all k = 0, 1, …, log M, define Lk(t.id) to be
the lattice consisting of all corner points of the tiling with
squares of size 2k x 2k having H(t.id) as its origin.
H(t.id)
2k
2k+1
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Hierarchical Lattice Structure
 Now consider any point x in
the overall square (location
where t might move, or from
which someone might try to
find t).
 For each k = 0, 1, …, log M,
define Wk(t.id,x) to be the set
of 4 corner points of the
square of Lk(t.id) that contains
x (break ties somehow).
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H
x
11
Hierarchical Lattice Structure
H(t.id)
- Origin
- Some point x
x
21
20
- W0(t.id,x)
- W1(t.id,x)
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Three algorithms
 Spiral algorithm
 Efficient FIND, average case.
 Expensive MOVE.
 Spiral-Flood algorithm
 Efficient FIND, worst case.
 Expensive MOVE.
 LLS algorithm
 Efficient FIND, worst case.
 Efficient MOVE, average case.
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Spiral Algorithm
 Record t’s location at the 4 corner nodes of the
square that contains t’s current location, in all the
lattices.
 That is, at the points in Wk(x), for all k, where x is
t's current location.
 MOVE(x,y):
 Expensive
 Remove t’s old location information from the
previous Wk(x) nodes.
 Put t’s new location info in all the Wk(y) nodes.
 They depict the strategy, in Figure 2, as
“following a spiral”, which proceeds outward, in
order k = 1, 2,…
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Spiral Algorithm FIND
 FIND(t): Invoked from s, to find the current location x of t.
 This queries corner nodes of the lattice squares that
contain s, that is, the nodes in Wk(s).
 Query proceeds in an outward spiral, in order k = 1, 2,…
 Eventually, the spiral used by this FIND will intersect the
spiral used by the most recent MOVE, that is, it will hit
some point in Wk(x), for some k.
 Guaranteed to happen at the first level of the hierarchy at
which s and x are in (possibly diagonally) adjacent
squares.
 Then the FIND learns the actual location.
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Publish path
Lookup
path
Nodes on
the lookup
spiral
Nodes on
the publish
spiral (or)
Location
servers
Source
Destination
Origin
Complexity Claim
 Theorem: In Spiral, the cost of a FIND(t) from s, averaged over
pairs (s,t), is O(d), where d = dist(s,t).
 Assumes O(d) cost geographical routing.
 Proof sketch:
 FIND gets the info it needs by the time it reaches the lowest level, k,
of the hierarchy where s and t are in adjacent squares.
 It is possible that s and t are arbitrarily close together, and yet this
could be a very high level in the hierarchy.
 They avoid this problem by averaging over all (s,t) pairs, saying:
“On the average”, dist(s,t) is proportional to the length of the side of
the square at level k. (?)
 Node at s must communicate with 4 nodes at each level up to k,
distance proportional to 20 + 21 + …+ 2k, or O(2k), or O(dist(s,t)).
 Geographical routing ensures actual cost of this communication is is
O(dist(s,t)). (?)
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Spiral-Flood


Spiral has a problem near the boundaries, which prevents worstcase FIND behavior from depending only on geographical
distance.
Spiral-Flood FIND:




Combines (interleaves) Spiral FIND algorithm with an Iterative
Bounded Flooding (IBF) algorithm, which floods FIND requests from s
to successively doubling distances.
Namely, s queries nodes along the same spiral-shaped path followed
by the Spiral FIND algorithm.
But when the Spiral FIND has accumulated sufficient cost (22i), s
floods a FIND query to depth 2i; this policy ensures that the combined
cost is not more than a constant times the cost of Spiral alone.
But also, the cost doesn't exceed the cost of IBF alone, which is O(d2)
in the worst case.


Here, d is min cost of a path from s to t, rather than geographical
distance.
Uses density restriction.
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Spiral-Flood Complexity
 Theorem: In Spiral-Flood:
 The average cost of a FIND(t) from s is O(d),
where d = dist(s,t).
 The worst-case cost of a FIND(t) from s is
O(d2), where d is the minimum cost of a path
from s to t.
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LLS
 Improves Spiral-Flood’s MOVE cost.
 So far, info about t’s location is inserted in, and
deleted from, all the Wk nodes by brute force;
leads to cost O(k), where k is the max level that
changes.
 Could be much larger than distance moved.
 New strategy attempts to update lazily, using
ideas of [Awerbuch, Peleg], to keep cost
proportional to distance moved.
 But, they claim this only on the average (?).
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LLS






For each level k, store t’s location info not only in the corners of the
square containing x (in Wk(x), where x is t’s current location), but
also the corners of the 8 surrounding squares---16 nodes, called
Zk(x).
When t moves within a 9-square area at level k, nothing changes at
level k.
Update location info when t leaves the 9-square area, by which time
the distance traveled by t must be at least 2k, the side of one
square; thus, amortized cost of updating is kept low, a function of
distance traveled.
Info stored at the Zk(x) nodes is not simply t’s location x, but rather,
a forwarding pointer to the Wk-1(x) nodes (similar to [Awerbuch,
Peleg]).
These nodes are lower in the hierarchy, and should contain better
(finer, more recent) information for reaching t.
The Z0(x) nodes contain the actual location x.
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H(t.id)
W10
W11
W9
W8
W7
W13
W14
W6
X
W12
W16
W1
W2
W15 W5
W3
W4
2k
LLS MOVE Operation
 MOVE from x to y:



If W0(x) = W0(y), no update is required (same lowest-level
square).
Otherwise, do a spiral search from the new location y, to find
the lowest level k, all of whose Wk(y) nodes have location info
for t.
If k = 0, update Z0(x) and Z0(y) to delete and add location info,
and return.
(? Not in paper)




Otherwise, k > 0.
This means that t has moved outside the current level-(k-1) 9square region.
Starting from location info in Wk(x), erase all location info for
levels < k.
Then update each Zj(y) node, 1 ≤ j ≤ k, with a forwarding
pointer to Wj-1(y).
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LLS FIND Operation
 FIND, starting from s:
 Start a spiral from s as before, but now the
search ends when a forwarding pointer of the
form Wk-1(y) is found in some Zk(y) node.
 Once such a pointer is found, FIND can just
follow the forwarding pointers to locate t.
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LLS Analysis
 Theorem: In LLS:
 The average cost of a FIND(t) from s is O(d), where d =
dist(s ,t).
 The worst-case cost of a FIND(t) from s is O(d2), where d
is the minimum cost of a path from s to t.
 The average cost of a MOVE(x, y) is O(d), where d =
dist(x, y).
 Analysis: LTTR.
 The key idea is to amortize costs over a series of MOVE
operations, making sure that before info is updated, the
node has moved correspondingly far.
 Justify their average calculations---are some
independence assumptions needed?
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LLS Analysis
 Simulations validate the theoretical averagecase claims.
 Fault-tolerance: They claim some faulttolerance properties, because of:
 Redundancy provided by using multiple nodes.
 Locality of information-processing
 The use of flooding.
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Simulations
Routing:
Connectivity percentage
and cost/distance
Inference:
Cost of routing to
the closest virtual
point is only a
constant times the
distance.
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Simulations
Lookup:
Cost/distance,
Cost/shortest path,
Connectivity and
flood percentages.
Inference:
The average cost of
locating a mobile
node is a constant
factor from the
optimum.
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Simulations
Publish:
Average hop
distance Vs
Average publish
Cost
Inference:
The cumulative cost of
updating location
pointers is
proportional to length
of average hop.
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