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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

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 (P i )= c. Level 1 Decomposition

Graph Decomposition Length (P i )= c. Level 1 Decomposition

Graph Decomposition Length (P i )= c. Level 1 Decomposition

Graph Decomposition Length (P i )= 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]

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 it Step 3: Remove the 2k-neighborhood 2k 4k

Continue recursively…

4k-satisfy the path Remove the 2k-neighborhood Discard A, and continue 2k 4k 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 S i = G(W i-1 U W i U W i+1 ) at a time Adjust for intra-band overlaps… W i-1 WiWi W i+1 SiSi … …

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