SYMMETRIC CRYPTOSYSTEMS Symmetric Cryptosystems 6/05/2014 | pag. 2.
Symmetric Connectivity With Minimum Power Consumption in Radio Networks
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Transcript of Symmetric Connectivity With Minimum Power Consumption in Radio Networks
Symmetric Connectivity With Minimum Power Consumption
in Radio Networks
G. Calinescu (IL-IT)G. Calinescu (IL-IT)I.I. Mandoiu (UCSD)I.I. Mandoiu (UCSD)A. Zelikovsky (GSU)A. Zelikovsky (GSU)
Ad Hoc Wireless Networks
• Applications in battlefield, disaster relief, etc.• No wired infrastructure• Battery operated power conservation critical• Omni-directional antennas + Uniform power
detection thresholdsTransmission range = disk centered at the node
• Signal power falls inversely proportional to dk
Transmission range radius = kth root of node power
Asymmetric Connectivity
Strongly connected
Nodes transmit messages within a range depending on their battery power, e.g., ab cb,d gf,e,d,a
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Range radii
Message from “a” to “b” has multi-hop acknowledgement route
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Symmetric Connectivity
• Per link acknowledgements symmetric connectivity• Two nodes are symmetrically connected iff they are
within transmission range of each other
Node “a” cannot get acknowledgement directly from “b”
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Asymmetric Connectivity
Increase range of “b” by 1 and decrease “g” by 2
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Symmetric Connectivity
Min-power Symmetric Connectivity Problem• Given: set S of nodes (points in Euclidean plane),
and coefficient k• Find: power levels for each node s.t.
– There exist symmetrically connected paths between any two nodes of S
– Total power is minimized
Power assigned to a node = largest power requirement of incident edges
k=2 total power p(T)=257a
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Power levels for k=2
Distances
Previous Results
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• Max power objective– MST is optimal [Lloyd et al. 02]
• Total power objective– NP-hardness [Clementi,Penna&Silvestri 00] – MST gives factor 2 approximation [Kirousis et al. 00]
Our results
• General graph formulation• Similarity to Steiner tree problem
– t-restricted decompositions• Improved approximation results
– 1+ln2 + 1.69 + – 15/8 for a practical greedy algorithm
• Efficient exact algorithm for Min-Power Symmetric Unicast
• Experimental study
Graph Formulation
Power cost of a node = maximum cost of the incident edge Power cost of a tree = sum of power costs of its nodes
Min-Power Symmetric Connectivity Problem in Graphs: Given: edge-weighted graph G=(V,E,c), where c(e) is the power
required to establish link e
Find: spanning tree with a minimum power cost
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Power costs of nodes arePower costs of nodes are yellow yellowTotal power cost of the tree isTotal power cost of the tree is 68 68
MST Algorithm
Theorem: The power cost of the MST is at most 2 OPTProof
(1) power cost of any tree is at most twice its cost p(T) = u maxv~uc(uv) u v~u c(uv) = 2 c(T)(2) power cost of any tree is at least its cost
(1) (2)
p(MST) 2 c(MST) 2 c(OPT) 2 p(OPT)
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Power cost of MST is n Power cost of OPT is n/2 (1+ ) + n/2 n/2
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Size-restricted Tree Decompositions
• A t-restricted decomposition Q of tree T is a partition into edge-disjoint sub-trees with at most t vertices
• Power-cost of Q = sum of power costs of sub-trees t = supT min {p(Q):Q t-restricted decomposition of T} / p(T)
• E.g., 2 = 2
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p(Q) = 2c(T) = n (1+ )p(T) = n/2 (1+ 2)
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Size-restricted Tree Decompositions
Theorem: For every T and t, there exists a 2t-restricted decomposition Q of T such that p(Q) (1+1/t) p(T)
t 1 + 1 / log k t 1 when t
Theorem: For every T, there exists a 3-restricted decomposition Q of T such that p(Q) 7/4 p(T)
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Gain of a Sub-tree
• t-restricted decompositions are the analogue of t-restricted Steiner trees
• Fork = sub-tree of size 2 = pair of edges sharing an endpoint• The gain of fork F w.r.t. a given tree T = decrease in power cost
obtained by – adding edges in fork F to T– deleting two longest edges in two cycles of T+F
Fork {ac,ab} decreases the power-cost by Fork {ac,ab} decreases the power-cost by gain = 10-3-1-3=3
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Approximation Algorithms
• For a sub-tree H of G=(V,E) the gain w.r.t. spanning tree T is defined by
gain(H) = 2 c(T) – 2 c(T/H) – p(H) where G/H = G with H contracted to a single vertex
• [Camerini, Galbiati & Maffioli 92 / Promel & Steger 00] 3 + 7/4 + approximation
• t-restricted relative greedy algorithm [Zelikovsky 96] 1+ln2 + 1.69 + approximation
• Greedy triple (=fork) contraction algorithm [Zelikovsky 93] (2 + 3) / 2 15/8 approximation
Greedy Fork Contraction Algorithm
Input: Graph G=(V,E,cost) with edge costs
Output: Low power-cost tree spanning VTMST(G)H
Repeat foreverFind fork F with maximum gainIf gain(F) is non-positive, exit loopHH U FTT/F
Output T H
Experimental Study
• Random instances up to 100 points• Compared algorithms
– branch and cut based on novel ILP formulation [Althaus et al. 02]
– Greedy fork-contraction– Incremental power-cost Kruskal– Edge swapping– Delaunay graph versions of the above
Edge Swapping Heuristic
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Remove edge 10 Remove edge 10 power cost decrease = -6power cost decrease = -6
Reconnect components with min increase in power-cost = +5Reconnect components with min increase in power-cost = +5
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For each edge do• Delete an edge• Connect with min increase in power-cost• Undo previous steps if no gain
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Percent Improvement Over MST
Runtime (CPU seconds)
Percent Improvement Over MST
Summary and Ongoing Research
• Graph-based algorithms handle practical constraints– Obstacles, power level upper-bounds
• Improved approximation algorithms based on similarity to Steiner tree problem in graphs
• Ideas extend to Min-Power Symmetric Multicast• Ongoing research
-- Every tree has 3-decomposition with at most 5/3 times larger power-cost 5/3+ approximation using [Camerini et al. 92 / Promel & Steger 00] 11/6 approximation factor for greedy fork-contraction algorithm
Symmetric Connectivity With Minimum Power Consumption
in Radio Networks
G. Calinescu (IL-IT)G. Calinescu (IL-IT)I.I. Mandoiu (UCSD)I.I. Mandoiu (UCSD)A. Zelikovsky (GSU)A. Zelikovsky (GSU)