l

S

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Low Color Partitions• Decomposition of a graph into several components (disjoint).

• Properties of this partition:– The components have bounded diameter

Coloring:– Components that are “close” to each other cannot have

the same color. Parameter

– Color the partition (at each level) with minimal # of colors.

Why Low-Color Partitions?• Clusters of same color are far away from each other.

• Leaders of these clusters are mutually far off.

• The real data sources that feed those leaders will also be mutually far away.

• The number of such real data sources that are mutually far away are significant (compared to those that are closeby).

Benefit of Low-Color Partitions

Cluster Leader

Benefit of Low-Color Partitions

Data Sources

Benefit of Low-Color Partitions

Benefit of Low-Color Partitions

Benefit of Low-Color Partitions

Higher Level Leader

Path Separators

• A set of shortest paths that partition a graph into two or more components of size atmost n/2 (n is total size of the graph).

• Path Separators can be computed in polynomial time– Planar Graphs are 3-path separable– H-Minor Free Graphs are k-path separable

Graph Decomposition (Planar Graph)

Graph Decomposition

Level 1 Cluster

Graph Decomposition

Length (Pi )= c.

Level 1 Decomposition

Graph Decomposition

Length (Pi )= c.

Level 1 Decomposition

Graph Decomposition

Length (Pi )= c.

Level 1 Decomposition

Graph Decomposition

Length (Pi )= c.

Level 1 Cluster Coloring

NOTE: Number of such clusters is small

Graph Decomposition

Level 2 Components

Graph Decomposition

Level 2 Decomposition

Graph Decomposition

Level 2 Clustering

Graph Decomposition

Level 2 –Cluster Coloring

Graph Decomposition

Over Coloring of Clusters (upto level 2)

Level k - 2

Level k - 1

Level k

61

RSMT Problem

• Rectilinear Steiner minimal tree (RSMT) problem:– Given pin positions, find a rectilinear Steiner tree with minimum WL– NP-complete

• Optimal algorithms:– Hwang, Richards, Winter [ADM 92]– Warme, Winter, Zachariasen [AST 00] GeoSteiner package

• Near-optimal algorithms:– Griffith et al. [TCAD 94] Batched 1-Steiner heuristic (BI1S)– Mandoiu, Vazirani, Ganley [ICCAD-99]

• Low-complexity algorithms:– Borah, Owens, Irwin [TCAD 94] Edge-based heuristic, O(n log n)– Zhou [ISPD 03] Spanning graph based, O(n log n)

• Algorithms targeting low-degree nets (VLSI applications):– Soukup [Proc. IEEE 81] Single Trunk Steiner Tree (STST)– Chen et al. [SLIP 02] Refined Single Trunk Tree (RST-T)

Minimum Spanning Trees• The basic algorithm [Gallagher-Humblet-Spira 83]

– messages and time

• Improved time and/or message complexity [Chin-Ting 85, Gafni 86, Awerbuch 87]

• First sub-linear time algorithm [Garay-Kutten-Peleg 93]:

• Improved to

• Taxonomy and experimental analysis [Faloutsos-Molle 96]• lower bound [Rabinovich-Peleg 00]

)log( nnmO )log( nnO

)logD( *61.0 nnO

)log/( nnD

)log( * nnDO

Steiner Tree Approximations

• Gabriel Robins and Alexander Zelikovsky: [J. Discrete Mathematics, 2005]– 1.55 approximation polynomial-time heuristic.– 1.28 approximation for quasi-bipartite graphs.

• Hougardy and Prommel : [SODA 1999] – 1.59 approximation

• Unless P = NP, the Steiner Tree Problem for general graphs cannot be approximated within a factor of 1 + ε for sufficiently small ε > 0.

• Rajagopalan and Vazirani [SODA 1999] : Approximation > 1.5– Primal-Dual Algorithm

• Zelikovsky [Algorithmica1993)]: 11/6 approximation

Applicability Contd…• Distributed Paging: The constrained file migration problem (Bartal) is the problem

of migrating files in a network with limited memory capacity at the processors in order to minimize the file access and migration costs. This is a natural generalization of uniprocessor paging problem and a special case of distributed paging problem.

In a network G = (V,E,w), a set of files resides in different nodes in the network. Processor v can accommodate in its local memory upto k_v files. The cost of an access to file F initiated by processor v is the distance from v to the processor holding the file F. A file may be migrated from one processor to another at a cost of D times the distance between the two processors. The goal is to minimize the total cost.

Planar Algorithm [Busch, LaFortune, Tirthapura: PODC 2007]

• If depth(G) ≤ k, we only need to 2k-satisfy the external nodes to satisfy all of G

• Suppose that this is the case

Step 1: Take a shortest path (initially a single node)Step 2: 4k-satisfy itStep 3: Remove the 2k-neighborhood

2k4k

Continue recursively…

4k-satisfy the pathRemove the 2k-neighborhoodDiscard A, and continue

2k2k4k

A

And so on …

…

Analysis

• All nodes are satisfied because all external nodes are 2k-satisfied

• Shortest-Path Cluster was always called with 4k, so clearly the radius is O(k)

• Nodes are removed upon first or second clustering, so degree ≤ 6

If depth(G) > k

• Satisfy one zone Si = G(Wi-1 U Wi U Wi+1) at a time

• Adjust for intra-band overlaps…Wi-1

Wi

Wi+1

Si

… …

Final Analysis

• We can now cluster an entire planar graph• Radius increased due to the depth of the

zones, but is still O(k)• Overlaps between bands increase the degree

by a factor of 3, degree ≤ 18