l t S Low Color Partitions Decomposition of a graph into several - - PowerPoint PPT Presentation

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l t S Low Color Partitions Decomposition of a graph into several - - PowerPoint PPT Presentation

Nov 30, 2023 152 likes •907 views

l t S 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

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l

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S t

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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.

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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).

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Benefit of Low-Color Partitions

Cluster Leader

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Benefit of Low-Color Partitions

Data Sources

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Benefit of Low-Color Partitions

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Benefit of Low-Color Partitions

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Benefit of Low-Color Partitions

Higher Level Leader

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Path Separators

  • A set of shortest paths that partition a graph into two
  • r 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

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Graph Decomposition (Planar Graph)

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Graph Decomposition

Level 1 Cluster

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Graph Decomposition

Length (Pi )= c.

Level 1 Decomposition

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Graph Decomposition

Length (Pi )= c.

Level 1 Decomposition

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Graph Decomposition

Length (Pi )= c.

Level 1 Decomposition

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Graph Decomposition

Length (Pi )= c.

Level 1 Cluster Coloring NOTE: Number of such clusters is small

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Graph Decomposition

Level 2 Components

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Graph Decomposition

Level 2 Decomposition

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Graph Decomposition

Level 2 Clustering

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Graph Decomposition

Level 2 –Cluster Coloring

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Graph Decomposition

Over Coloring of Clusters (upto level 2)

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Level k - 2 Level k - 1 Level k

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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)

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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 ( n n m O 

) log ( n n O ) log D (

* 61 .

n n O 

) log / ( n n D 

) log (

* n

n D O 

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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
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Applicability Contd…

  • Distributed Paging: The constrained file migration problem (Bartal) is the problem
  • f migrating files in a network with limited memory capacity at the processors in
  • rder 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

  • f D times the distance between the two processors. The goal is to minimize the

total cost.

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

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Continue recursively…

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4k-satisfy the path Remove the 2k-neighborhood Discard A, and continue 2k 2k A

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And so on …

…

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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

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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

… …

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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