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Improved Sparse Covers for Graphs Excluding a Fixed Minor
Ryan LaFortune (RPI),
Costas Busch (LSU), andSrikanta Tirthapura (ISU)
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Sparse Covers
Distributed systems represented by graphs Fundamental data structure Numerous Applications
– Name-independent compact routing schemes– Directories for mobile objects– Synchronizers
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Definitions
A cover is a collection of connected sets called clusters, where every node belongs to some cluster containing its entire k-neighborhood
Locality parameter Radius and degree (latency and load) Sparse cover
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Extreme Cases
Large radius, small degree– G is one cluster
Small radius, large degree– Every node forms a cluster
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Sparse Cover for Arbitrary Graphs
Impossible to achieve optimality in both metrics [Peleg: 2000]
– Radius O(k) and degree O(1) Best known algorithm [Awerbuch and Peleg: FOCS
1990]
– Radius O(k log n) and degree O(log n) – Based on coarsening
Special graphs?
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Contributions
Improved algorithm for H-minor free graphs– Radius ≤ 4k and degree O(log n)
Improved algorithm for planar graphs– Radius ≤ 24k - 8 and degree ≤ 18
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Outline of Talk
ConsequencesRelated WorkShortest-Path ClusteringP-Path Separable AlgorithmPlanar AlgorithmConclusions
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Name-Independent Compact Routing Schemes
Deliver a message given the ID of the destination node Cannot alter IDs Tradeoff between stretch and memory overhead
General algorithm [Awerbuch and Peleg: FOCS 1990]– Stretch = O(log n) and memory = O(log2 n) bits per node
Planar algorithm– Stretch = O(1) and memory = O(log2 n) bits per node
H-minor free algorithm– Stretch = O(1) and memory = O(log3 n) bits per node
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Directories for Mobile Objects
Given an object’s name, returns its location Wireless sensor networks, cellular phone networks Tradeoff between Stretchfind and Stretchmove
General algorithm [Awerbuch and Peleg: JACM 1995]
– Stretchfind = O(log2 n) and Stretchmove = O(log2 n) Planar algorithm
– Stretchfind = O(1) and Stretchmove = O(log n) H-minor free algorithm
– Stretchfind = O(log n) and Stretchmove = O(log n)
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Synchronizers
Distributed programs that allow the execution of synchronous algorithms in asynchronous systems
Logical rounds simulate time rounds Tradeoff between time steps and average messages per node
ZETA [Shabtay and Segall: WDAG 1994]
– Time steps = O(logz n) and messages = O(z) per node Planar algorithm
– Time steps = O(1) and messages = O(1) per node H-minor free algorithm
– Time steps = O(1) and messages = O(log n) per node
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Related Work
Algorithm for graphs excluding Kr,r
– Diameter = 4(r + 1)2k and degree = O(1) [Abraham et al.: SPAA 2007]
Algorithm for H-minor free graphs– Weak diameter = O(k) and degree = O(1)
[Klein et al.: STOC 1993]
Algorithm for graphs with doubling dimension a– Radius = O(k) and degree = 4a
[Abraham et al.: ICDCS 2006]
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Shortest-Path Clustering
If called with 2k, the path can be removed All nodes are satisfied, radius ≤ 4k, and deg ≤ 3
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H-Minor Free Definitions
The contraction of edge e = (u, v) is the replacement of u and v by a single vertex
A minor of G (subgraph after contractions) H-minor free
– Trees, exclude K3
– Outerplanar graphs, exclude K4 and K2,3
– Series-parallel graphs, exclude K4
– Planar graphs, exclude K5 and K3,3
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P-Path Separable Algorithm
Path Separator (shortest paths, components have at most n/2 nodes)
Every H-minor free graph is P-path separable– P depends on the size of H– [Abraham, Gavoille: PODC 2006]
Recursively cluster path separators
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Initial graph, suppose k=1
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Choose a path separator
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Break the path separator up into sub-paths of length 2k = 2
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Cluster 2k = 2 around the first sub-path
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Radius = O(k) and degree ≤ 3
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≤ n/2 nodes ≤ n/2 nodes
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Continue Recursively(terminates in a logarithmic number of steps)
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…
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Analysis
Only satisfied nodes are removed, thus all nodes are satisfied
Shortest-Path Cluster was always called with 2k, so clearly the radius is O(k)
Degree is O(log n) due to the logarithmic number of steps
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Planar Definitions
The external face of a graph consists of the nodes and edges that surround it
The depth of a node is the minimum distance to an external node
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Planar Algorithm
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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2k4k
Step 1: Take a shortest path (initially a single node)
Step 2: 4k-satisfy it
Step 3: Remove the 2k-neighborhood
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Continue recursively…
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4k-satisfy the path
Remove the 2k-neighborhood
Discard A, and continue
4k
2k 2kA
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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
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Conclusion
We have significantly improved sparse cover construction techniques
– H-minor free graphs– Planar graphs
We can also construct optimal sparse covers for graphs with constant stretch spanners (unit disk graphs)
Name-independent compact routing schemes, directories for mobile objects, and synchronizers