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TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: . t. S. Low Color Partitions. Decomposition of a graph into several components (disjoint). Properties of this partition: The components have bounded diameter Coloring: - PowerPoint PPT PresentationTRANSCRIPT
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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