Routing and Network Design:Algorithmic Issues

Kamesh Munagala

Duke University

Graph Model for the Links

Model sensor nodes as vertices in a graph

Gateway

d(7,8) = “Length” of link

“Length” of link models communication cost per bit“Length” should be a function of #bits being sent (Why?)

Specialized Nature “Geometric Random” graph

• Nodes on a 2D plane• Each node has a fixed communication radius

Correlation Structures:• Spatial Gaussian models• Simple AR(1) temporal models

Assumptions do not always hold!

Unique Features Distributed algorithms:

• Reconfigure routes around failures

• Learning network topology

• Learning correlation structures

• Query processing

Light-weight implementations:• Low compute power and memory

• Limited communication and battery life

• Noisy sensing and transmission

Goals in this Lecture General algorithmic ideas capturing:

• Simplicity and efficiency• Some performance guarantees• Distributed implementations• Low reliance on specific assumptions

Caveats: • Ideas need to be tailored to context• Specialized algorithms might work better

Topics What constitutes good routing?

• Measures of quality

Algorithm design framework• Basic problem statements

• Spanning, shortest path, and Steiner trees

• Aggregation networks

• Location and subset selection problems

• Solution techniques• Types of guarantees on solution quality

Models of information in a sensor network• Tailoring generic algorithms to specific models

Problem 1:Information Aggregation

Routing Tree Problem Statement:

Route information from nodes to gateway Choose subset of edges to route data Edges “connect” all nodes to gateway

• Tree Property

Minimize: Long-term average “cost” of routing Answer will depend on:

• What constitutes “cost”• Correlations in data being collected

Toy Example

66 6

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Gateway

Each node has 100 bits of information to send to gateway

Value on link (edge) is the cost of transmitting one bit

How should we route the bits?

“Star” network

Depends on Correlations

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Suppose information is perfectly correlatedInformation from all sources together is also 100 bits!

Spanning tree is optimal

Cost = 100 * (6+1+1+1) = 900 units

Ignore cost of compression

Other Extreme: No Correlation

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Suppose information is not correlated at allInformation from all sources together is now 400 bits

Shortest path tree is optimal

Cost = 100 * (6+6+6+6) = 2400 units

Had we used a Spanning Tree

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Suppose information is not correlated at allInformation from all sources together is now 400 bits

Shortest path tree is optimal

Cost = 100 * (6+7+8+9) = 3000 units > 2400 units!

In summary… Moral of the story:

• Choosing good routes is important• Choice depends on correlation structure

Issues to address:• How do we specify correlations

• Simple yet faithful specifications desirable

• Algorithms for finding (near-)optimal routes• Efficient and simple to implement

• Reliability and “backup” routes

There could be nn-2 many spanning trees in general

Exhaustive enumeration is out of question

Minimum Spanning Tree

Cost of MST = 23

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Spanning Tree Algorithm

“Greedy” schemes add edges one at a time in clever fashionNo backtracking

Kruskal's algorithm: Consider edges in ascending order of cost. Insert an edge unless doing so would create a cycle.

Prim's algorithm: Start with gateway and greedily grow a tree from the gateway outward. At each step, add the cheapest edge that has exactly one endpoint in current tree.

Prim’s Algorithm: Execution10

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“Distributed” Algorithm?Nodes connect in arbitrary order

Each node simply connects to “closest” existing neighbor

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Cost = 25

Guarantee on “Online” Scheme

n nodes in graph

Cost of “online” tree is within log n factor of cost of MST

Irrespective of order in which nodes join the system!

Intuition: In “star” network, “online” scheme produces MST!

Natural implementation: Greedy starting from gateway

Such a guarantee is called an “approximation guarantee”

Shortest Paths: OSPF

Key algorithmic idea: Greedy local updates

Each node v maintains “tentative” distance d(v) to gateway

Initially, all these distances are infinity

Each node v does a greedy check:

If for some neighbor u, d(v) > d(u) + Length(u,v), then:

Route v through u

Set d(v) = d(u) + Length(u,v)

Run this till it stabilizes

OSPF Execution10

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∞

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Rate of Convergence

n nodes in graph

1. The protocol converges to the shortest path tree

2. The number of rounds till convergence is roughly n

Intermediate CorrelationsOne tree for all correlation values?

Both spanning and shortest path trees at once?

Do-able if we settle for “nearly” optimal trees

In other words, there exists a tree with:

Cost at most thrice cost of MST

Distances to gateway at most twice S.P. distances

Example: MST

nn n

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n2 nodes

Cost of MST = n2+n

Path length = n2+n

Example: Shortest Path Tree

nn n

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n2 nodes

Cost = n3

Path Length = n2

Example: Balanced Tree

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n nodes

Cost = 2n2

Path Length = 2n

Walk on a Tree

Gateway

Balancing Algorithm

GatewayWalk along Spanning Tree

Add shortcuts to gateway

At node v:

Suppose previous shortcut at u

If SP(u) + Walk(u,v) > 2 SP(v)

Add “shortcut” from v

Walk too long!

Shortcut

Example Revisited

nn n

n

1 1 1

Gateway

11 1

n nodes Walk length = 2n

Proof Idea Final Path Lengths < 2 S.P. Lengths

• Follows from description

Final Cost < 3 MST Cost• Final Cost = MST + Shortest Paths Added• Suppose paths are added at …,u,v… on walk

• SP(u) + Walk(u,v) > 2 SP(v)

• Add these up:• Total Walk Length > Total Length of added Paths

• But, Total Walk Length = 2 MST Cost

Problem 2:Sensor Location

“Most Informative” Placement

Close by locations are not very “informative”

Abstraction Parameters:

• Each node v has communication cost to gateway = cv

• Depends on location

• Subset S of nodes has “information” f(S)• Information is a property of a set of nodes

• Depends on whether “close by” nodes also in set

Problem Statement:• Choose set S so that:

• Sum of costs of nodes in S is at most C

• Maximize Information = f(S)

Algorithmic Issue

Number of subsets of n locations = 2n

• Inefficient to enumerate over them

Given subset S, how do we compute f(S)• Needs a correlation model among locations

Communication costs are not additive• Also depend on location of nodes!

Information Functions

f(S) = Entropy of S

Correlations are multidimensional Gaussian: = Covariance matrix between locationsEntropy log det()

Covariance(j,k) exp(-dist(j,k)2 / h2)

Properties of f(S)

A B

v

Location v is more informative w.r.t A than w.r.t B

Property 2: f(A+v) - f(A) ≥ f(B+v) - f(B)

Property 1: f(A+v) ≥ f(A)

Greedy Algorithm

Start with S =

Repeat till cost of S exceeds C:

• Choose v such that:• ( f(S+v) - f(S) ) / cv is maximized

• “Information gain per unit cost”

• Add v to S

Analysis

Suppose:

All costs cv = 1

O = Best Information set of size at most C

At any stage, if adding v is best greedy decision

Adding entire O cannot give more information per unit cost!

f(S + v) - f(S) ≥ ( f(S + O) - f(S) )/C ≥ ( f(O) - f(S) )/C

Let d(S) = f(O) - f(S) = Deficit w.r.t. Optimal Solution

Implies: d(S) - d(S+v) ≥ d(S) / C

Analysis

d(S+v) ≤ d(S) (1 - 1/C)

d(Initial) = f(O)

d(Final) = f(O) - f(Final)

f(O) - f(Final) = d(Final) ≤ d(Initial) ( 1 - 1/C )C ≤ f(O) / 2

Implies: f(Final) ≥ f(O) / 2

Greedy set has information at least 1/2 information in optimal set

Two-Level Routing

Aggregation Hub

Clustering

Optimal placement of cluster-heads

Minimize routing cost

K-Means Algorithm Start with k arbitrary leaders

Repeat Steps 1 and 2 till convergence: Step 1:

• Assign each node to closest leader• Yields k “clusters” of nodes

Step 2:• For each cluster, choose “best” leader• Minimizes total routing cost within cluster

Analysis Convergence is guaranteed:

Each step reduces total distance• Step 1: Each node travels smaller distance• Step 2: Each cluster’s routing cost reduces

Rate of convergence:• Fast in practice

Quality of solution:• “Local” optimum depending on initial k nodes• Need not be best possible solution• Works very well in practice