Dual Decomposition for Computational
Optimization of Minimum-Power Shared
Broadcast Tree in Wireless Networks
Di Yuan and Dag Haugland
Supplementary Material
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Di Yuan and Dag Haugland, Dual Decomposition for Computational Optimization of
Minimum-Power Shared Broadcast Tree in Wireless Networks, 2012, IEEE Transactions on
Mobile Computing, (11), 12, 2008-2019.
http://dx.doi.org/10.1109/TMC.2011.231
Postprint available at: Linköping University Electronic Press
Appendix: Computing the best 1-edge exchange
Algorithm 11-edge exchange(T )
Require: A spanning tree T
Ensure: Returns leaving and entering edges (k∗, l∗) and (m∗, n∗) and the new power P∗ in the best
1-edge exchange. If no improving move exists, dummy-edges and the current power are returned. p(i, s) ← maxj{pij : (i, j) ∈ Ts} ∀i, s ∈ V
(k∗, l∗, m∗, n∗, P∗) ← 0, 0, 0, 0,P
s∈V
P
i∈V p(i, s) // No improving move is found
// Try all possible edge removals: for all(k, l) ∈ T do
p′(i, s) ← p(i, s) ∀i, s ∈ V
for all s∈ Tkdo p′ (k, s) ← maxj{pkj : (k, j) ∈ Tks} for all s∈ Tldo p′(l, s) ← max j{plj : (l, j) ∈ Tls}
// (i) Power needed for internal forwarding of messages from internal sources: PII ←P i∈Tk P s∈Tkp ′(i, s) +P i∈Tl P s∈Tlp ′(i, s)
// (ii) Power needed for internal forwarding of messages from external sources: for all m∈ Tkdo PmEI ←P i∈Tk|V (Tl)| p ′(i, m) for all n∈ Tldo PnEI ←P i∈Tl|V (Tk)| p ′ (i, n)
// (iii) Power increment needed for external forwarding of messages from internal sources: for U ← Tk, Tldo
for all m∈ U do
Nm ← 0, ¯pm1← 0, ¯pm2← 0 // Correct if |V (U )| = 1
if|V (U )| > 1 then
Find m1 ∈ arg maxi{pmi : (m, i) ∈ U } // Most power-demanding old neighbor
¯
pm1← pm,m1 // The corresponding power ¯
pm2← maxi{pmi: (m, i) ∈ U, i 6= m1} // Second most, if any
Nm← |V (U ) ∩ V (Tm1)| // Counting sources requiring power ¯pm2 for all m∈ Tkdo
for all n∈ Tldo
PIE ← pie(T
k, pmn,p¯m1,p¯m2, Nm) + pie(Tl, pnm,p¯n1,p¯n2, Nn)
// Check quality of the move(k, l, m, n): if PII+ PEI m + PnEI+ PIE < P∗then (k∗ , l∗ , m∗ , n∗ , P∗ ) ← k, l, m, n, PII + PEI m + PnEI+ PIE return (k∗, l∗, m∗, n∗, P∗) 1
Algorithm 2pie(U , pmn, ¯pm1, ¯pm2, Nm)
Require: A tree U , power pmnof the new edge, the two largest power demandsp¯m1andp¯m2of edges
incident to node m, number Nmof sources demanding powerp¯m2at m.
Ensure: Returns power increment necessary for external forwarding of internal messages at node m. PIE ← 0 // No power increment needed for (m, n) so far
if pmn>p¯m2then
PIE ← Nm(pmn− ¯pm2) // Nmsources ask for increment fromp¯m2to pmn
if pmn >p¯m1then
// All but Nmsources in U ask for increment fromp¯m1to pmn
PIE ← PIE+ (|V (U )| − N
m) (pmn− ¯pm1)
return PIE
