Online Packet Admission and Oblivious Routing in Sensor Networks Mohamed Aly Department of Computer Science University of Pittsburgh And John Augustine

Online Packet Admission and Oblivious Routing

in Sensor Networks

Mohamed AlyDepartment of Computer Science

University of Pittsburgh

And

John AugustineDonald Bren School of Information and Computer Sciences

University of California at Irvine

Online Packet Admission and Oblivious Routing in Sensor Networks 2 Mohamed Aly and John Augustine, ISAAC 2006

Agenda

General network model. Previous results for general networks:

Oblivious routing results. Admission control results.

Our Results for sensor networks: Lower Bounds. Logarithmic oblivious routing algorithm for trees. Logarithmic oblivious routing algorithm for 3D grid

networks.


Model of General Networks

We model the network as a graph G=(V,E) of |V| = n nodes and |E| edges.

Edge weights = link bandwidths. For packet-switched networks, a set of demands Dab ≥ 0

for each origin-destination (OD) pair (a , b). A routing algorithm f: determines an s-t path for each

unit demand of each OD pair: fab(i,j) ≥ 0 routing for OD pair (a,b) on edge (i,j)

Flow on edge e=(i,j) when routing D with f: flow(e,f,D)=ab fab (i,j) Dab


Performance Metrics for Routing Algorithms

Congestion on edge e=(i,j) when routing D with f: cong(e,f,D)=flow(e,f,d)/capacity(e)

Congestion of demands D with routing f: cong(f,D)= max cong(e,f,D), for all e E

Optimal Offline Algorithm OPT: has full knowledge of the input set D and can process it optimally.

Competitive Worse-case Analysis: compare the congestion cong(f,D) of any algorithm f with that of the optimal offline algorithm OPT cong(OPT,D) and bound it for all possible inputs D: cong(f,D) ≤ccong(OPT,D) + afor any D c is called the competitive ratio of algorithm f. High probability argument for randomized algorithms


Routing Algorithms under Concern

Online routing algorithms: demands D arrive in an online fashion. For each request, the algorithm determines the path without knowing future requests.

Oblivious routing algorithms: The path chosen for a unit demand is independent of the

current network load. In other words, the routing path of a message (s,t) only depends on s,t and does not depend on all other messages.

We are interested in routing algorithms that are: Online. Distributed. Oblivious Routing. Poly-logarithmic competitive ratio (w.r.t congestion).


Related Problem: Online Call Control

Problem definition: A sequence of requests arrive in an online manner. For each request, the algorithm A has to either accept or

reject the call. When accepting a call, the algorithm has to immediately

select a virtual circuit between the communicating parties, obeying the network constraints.

Arises in the context of circuit-switched networks. Performance Metric: number of accepted calls, i.e. the

throughput of the call control algorithm, P(A,D). Similar bandwidth requirements for all calls.

Competitive analysis: P(A,D) ≤cP(OPT,D) + afor any D


Admission Control Algorithms under Concern

We are interested in routing algorithms that are: Online. Distributed. Oblivious path selection step. Poly-logarithmic competitive ratio (w.r.t throughput).


Oblivious Admission Control & Routing

Online algorithm. Distributed. Call is obliviously admitted/rejected. Path is obliviously determined. Goal of such algorithms:

Maintain poly-logarithmic competitiveness w.r.t either congestion in packet networks or throughput in circuit-switched networks.


Oblivious Routing in General Networks

Räcke, FOCS 2002: Any undirected network has an online distributed oblivious routing with competitive ratio O(log3 n), w.r.t congestion!!

Highlights of the technique: Map the network G=(V,E) into a tree TG=(Vt, Et), the

so-called decomposition tree. Show that: Network decomposition tree. Route the messages on G=(V,E) based on the usage

of TG=(Vt, Et) to determine message paths.


More on Räcke’s Algorithm

Strengths: It is the first routing algorithm that is

Oblivious. Decentralized. Online.

Weaknesses: Decomposition tree construction is not in polynomial time. Not applicable for directed graphs.


Follow-ups of Räcke’s Algorithm

Azar et al., STOC 2003: A polynomial time construction algorithm for Räcke’s decomposition tree (based on the usage of the LP ellipsoid method)

Bienkowski et al., SPAA 2003: A practical algorithm for the decomposition tree construction with a competitive ratio of O(log^3 n).

Harrelson et al., SPAA 2003: drop approximation ratio to O(log^2 n log log n).

Hajiaghayi et al., STOC 2005: oblivious routing for directed graphs which is O(log2 n)-competitive assuming demands chosen at random from a known demand distribution.

Hajiaghayi et al., SODA 2006: lower bounds on oblivious routing in undirected graphs.


Admission Control in General Networks

Awerbuch, Azar, and Plotkin, FOCS 1993 ([AAP] algorithm): First online admission control algorithm. O(log n)-competitive. Admission and routing decisions are based on the

knowledge of all network links. A central node in charge of taking decisions. “Hot spot” in networks with high traffic volumes. “Single point of failure”: operation of the whole network

depend on the availability of the central node. Once a call is admitted, link identities of the links along its

chosen path have to be communicated to the nodes along the path.


Admission Control in General Networks

Räcke and Rosén, SODA 2005 ([RR] algorithm): Distributed randomized online call control algorithm. Poly-logarithmically competitive (w.r.t throughput). Handles concurrent requests in the network. Two components:

Path establishment and call admission. Operates in a hop-by-hop manner.

Given a request, a sequence of messages is sent along some path that is defined on a hop-by-hop basis.

Path establishment based on Räcke’s decomposition tree. If a path is established, call routed on that path. Otherwise,

the call is dropped. The network incurs a cost only when path is admitted.


Open Question

Is it possible to get a poly-logarithmic oblivious routing algorithm for sensor networks (or in general, in node-capacitated undirected graphs)??

Previous results: Assuming random demands, a lower bound of Ω(log n/log

log n) on oblivious routing schemes for a family of large graphs (grids) was shown by Hajiaghayi et al., SODA 2006.

Is it possible to use Räcke’s decomposition or the RR algorithm to achieve an upper bound?? No, unfortunately.


Why Not RR Algorithm for Sensor Networks??

Sensor network: Energy-capacitated sensor nodes. Main goal of a routing algorithm: maximize number of

successfully sent messages, i.e. throughput. Sending any message consumes an amount of the

node’s total energy!!! Node deaths, Network partitioning, network lifetime

(longevity). RR’s Path establishment messages are not free of

charge as in general networks. RR Algorithm is not poly-logarithmic (w.r.t. throughput)

for sensor networks. Is there a poly-logarithmic (w.r.t. throughput) admission

control and oblivious routing algorithm??


Our Results

Blend call control with oblivious routing. Lower Bounds:

Any distributed deterministic packet-admission and oblivious routing algorithm cannot be poly-logarithmically competitive against adversarial demands.

An always-send distributed oblivious routing algorithm cannot be poly-logarithmically competitive when:

1. Demands are either adversarial or following a general distribution that is unknown to all sensor nodes.

2. An adversary sets the tree node capacities.


Our Results

Upper Bounds: A O(α log n)-competitive algorithm (w.r.t. throughput)

for tree networks assuming adversarial demands and node capacities [k, αk], where k is Ω(log n).

A O(log n)-competitive algorithm (w.r.t. throughput) for grid networks assuming random demands as well as node capacities are k ≥ log2 n + 6 log n.


Lower Bounds: Adversarial Demands

Lemma 1: There exists at least one set of adversarial demands D that makes any deterministic distributed routing algorithm A at least Ω(n) competitive.

Proof on balanced binary tree T. Assumptions:

Adversarial demands. Adversary knows all tree node capacities at any time. Tree T follows the nesting property. Each leaf has a capacity e, such that e << n.


Lower Bounds: Adversarial Demands• Tree T=(V,E), |V| = n.

• Messages from left subtree to right subtree.

• Bit vector of size n at each of the left subtree leaves. Message to rk is sent if bit k is set to 1.

• Message (s,t)

Tree One path

• Adversary inputs messages till depleting all tree nodes.

log n

l1 l2 l3 l4 r4r1 r2 r3


Lower Bounds: Adversarial Demands• Main idea: make any deterministic algorithm send messages that OPT would have not sent.

• Adversarial strategy operates in n rounds.

• At each round, adversary selects at random one still alive node rk.

• For each left leaf li, input messages (li,rk) till li changes bit entry k to 1.

log n

l1 l2 l3 l4 r4r1 r2 r3


Lower Bounds: Adversarial Demands• For each round, first e messages deplete node rk.

• Rest of messages (at least n) are falsely sent by any deterministic algorithm.

• After e rounds, both algorithms have sent at most e2 messages.

•After e rounds, online algorithm depletes all tree nodes (nesting prop.).

• After n rounds, OPT sends n.e messages. Thus, approximation ratio is n/e.

log n

l1 l2 l3 l4 r4r1 r2 r3


Lower Bounds: Demands Drawn from Unknown Distribution

Lemma 2: Assuming demands D are drawn from a distribution that is oblivious to all sensors, there exists a distribution that makes an always-send algorithm Aas at least Ω(n) competitive.


n is a power of 2. Adversary knows all tree node capacities at any time. Tree T follows the nesting property. Leaf have equal capacity belonging to log n.


Lower Bounds: Adversarial Leaf Node Capacities

Lemma 3: Assuming demands D are drawn from a uniform distribution, there exists at least one adversarial setting for leaf node capacities distribution that makes an always-send algorithm Aas at least Ω(n) competitive.


n is a power of 2. Adversary knows all tree node capacities at any time. Tree T follows the nesting property.

Lemma implies a lower bound on adversarial setting of all tree nodes rather than just leaves.


Main Lower Bound Theorem

Theorem: Given a balanced binary tree T(V,E) and a set of demands D, an always-send oblivious routing algorithm Aas cannot maintain poly-logarithmic competitiveness in either of the following cases: D is a set of adversarial demands, or follows a general

distribution that is unknown to all sensor nodes. An adversary sets the tree node capacities (internal nodes

or leaf nodes). Results extends to general trees and general undirected

graphs. Packet-admission protocol is mandatory poly-

logarithmic competitiveness.


Upper Bound for Sensor Trees

Tree T=(V,E), |V| = n Balanced with height O(log n)

Each v V has energy capacity (k+1)·log n Each message is a pair (s,t)

It originates at s Its intended destination is t T is a tree Unique path for each message


Routing in Trees

s

t

• Tree T=(V,E), |V| = n

• Message (s,t)

Tree One path

• Capacity of each node

is at least (k+1) log n

log n

k (log n) for messages(log n) for control info.


Routing in trees

• Input instance I A tree T A sequence of messages released online

Message i will not be released until (i-1) is A. Rejected, orB. Accepted and routed

• OPT(I) = # of messages routed by an opt. algorithm Without violating capacity constraints

A. Let A be an algorithm that admits A(I) messagesB. A is c-competitive if

OPT(I) c · A(I)


Pivot Vertices

p

s

t

• Every message (s,t)pivots at some p • Pivot of a message = Least Common Ancestor (LCA) of its source s and destination t.


Our Algorithm A

Each message is classified based on its LCA. Each node uses at most k units of energy (1 unit = send 1

message). Each node partitions its energy into log n shares with

energy k/log n each. A node admits messages with LCA at level i as long as its

ith share is not totally consumed. When a message is admitted at any node, the node

decrements its share i by 1.


Our Algorithm A

Each vertex v is pivot for up to k messages

After that quota is met Broadcast a message to all descendants Inform them to reject messages pivoting at v

Theorem: A is (log n)-competitive

k k lognlogn Vertex capacity

Height of Tree


Consider one pivot v

For some input instance I Let OPT(v) = # of messages OPT pivots at v A(v) = # of messages OPT pivots at v

Recall: A routes first k messages OPT can route up to k • log n messages

OPT(v)

A(v)logn


Consider Entire Tree

A(I) A(v)vV

OPT(I) OPT(v)vV

OPT(I)

A(I)

OPT(v)vV

A(v)

vV

(logn)

OPT(v)

(logn)vV

A(v)

vV

(logn)A(v)

vV

A(v)

vV

logn

Recall:

OPT(v)

lognA(v)


Capacity Constraint

When a pivot vertex reaches the limit Broadcasts to descendants: reject further messages. Each vertex can receive at most log n such control

messages from its ancestors Each vertex can participate in ≤ k•(log n) messages

# pivoting at one ancestor

# of ancestors (at most)


Properties of A

Distributed: No central node for call control

Oblivious Decisions can be made without detailed knowledge of the

rest of the tree.

We account for every message hop


Extending Algorithm for General Trees

a partition vertex such that# of vertices in the bigger partition ≤(2/3)n

Recursively partition the tree and generate subtrees. Each tree gets an equal quota. Message (s,t) uses quota from smallest subtree it fits. Shown to be O(log n)-competitive.


Call Control in Grids

Assumption: message generated between nodes chosen uniformly at random.

Techniques: Quadtree style partitioning of the mesh.

Partition each dimension into levels. Assign each level an energy quota. Route message if quotas of its 3 levels are still non-

consumed. Balls in Bins (take advantage of random I/P)

Result: O(log n)-competitive distributed oblivious algorithm for 3D grids.


Conclusion & Future Work

Oblivious routing and admission control in general networks.

Our results: Lower Bounds. Logarithmic oblivious routing algorithm for trees. Logarithmic oblivious routing algorithm for 3D grid

networks. Future work:

Upper bound for oblivious routing on general graphs. Use our theoretical results to present a new robust routing

algorithm for large-scale sensor networks.


Acknowledgment

Our advisors: Kirk Pruhs and Sandy Irani. Harald Räcke (TTI) and MohammadTaghi Hajiaghayi

(CMU). NSF grants:

ANI-0325353 CCF-0448196 CCF-0514058 IIS-0534531 CCF-0514082


Thank You

Questions ?


More on Räcke’s Decomposition Tree

c

f

a b

d

e

hi

g

j

f

ba t

31 1

g

h j

3

42

3

e

12

5

3

3 4

dc1. Cluster nodes (based on bulding a laminar system).

2. Assigning capacities to tree edges.

3. Define bandwidth ratio and weight ratio for each tree level.

4. Initialize network by solving a CMCF problem.

5. For every demand, randomly select one of the paths resulting from the CMCF solution.

Documents

Online Packet Admission and Oblivious Routing in Sensor Networks Mohamed Aly Department of Computer Science University of Pittsburgh And John Augustine