Minimum Connected Dominating Set in Multihop Wireless Networks: Algorithms and Techniques , Study notes of Computer Science

Download Minimum Connected Dominating Set in Multihop Wireless Networks: Algorithms and Techniques and more Study notes Computer Science in PDF only on Docsity! Minimum Connected Dominating Set in Multihop Wireless Networks Peng-Jun Wan Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 1 / 49 Outline Overview MCDS in Arbitrary Networks MCDS in Symmetric Networks MCDS in Plane Geometric Networks Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 2 / 49 Minimum CDS (MCDS) MCDS: compute a CDS of the smallest size in a network. MCDS in plane geometric networks: NP-hard, but admits PTAS. MCDS in symmetric networks: admits no (1 ε) ln n-approximation for any …xed ε > 0 unless NP  DTIME h nO (log log n) i . MCDS in arbitrary (or asymmetric) networks: at least as hard as MCDS in symmetric networks. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 4 / 49 Minimum CDS (MCDS) MCDS: compute a CDS of the smallest size in a network. MCDS in plane geometric networks: NP-hard, but admits PTAS. MCDS in symmetric networks: admits no (1 ε) ln n-approximation for any …xed ε > 0 unless NP  DTIME h nO (log log n) i . MCDS in arbitrary (or asymmetric) networks: at least as hard as MCDS in symmetric networks. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 4 / 49 Minimum CDS (MCDS) MCDS: compute a CDS of the smallest size in a network. MCDS in plane geometric networks: NP-hard, but admits PTAS. MCDS in symmetric networks: admits no (1 ε) ln n-approximation for any …xed ε > 0 unless NP  DTIME h nO (log log n) i . MCDS in arbitrary (or asymmetric) networks: at least as hard as MCDS in symmetric networks. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 4 / 49 Summary on Algorithms MCDS in arbitrary networks: (4H (n 2) 2)-approximation MCDS in symmetric networks: (2+ ln (∆ 2))-approximation MCDS in plane geometric networks: 6. 075-approximation Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 5 / 49 Roadmap Overview MCDS in Arbitrary Networks MCDS in Symmetric Networks MCDS in Plane Geometric Networks Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 6 / 49 CDS from Arborescences T1: a spanning s-arborescence in D T2: a spanning inward s-arborescence in D Lemma All non-sink nodes of T1 and all non-source nodes of T2 form a CDS of D. (a) (b) (c) (d) sss s Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 7 / 49 From SAFIN to MCDS A: a µ-approximation algorithm for SAFIN s: a node in OPT \ S . DR : reverse of D T1: spanning s-arborescence of D output by A T2: spanning s-arborescence of DR output by A jI (T1) [ I (T2)j  1+ jI (T1) n fsgj+ jI (T2) n fsgj  1+ 2µ (γc 1) = 2µ
γc 2µ+ 1. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 9 / 49 Candidates of Root Fact: for any node u, any CDS must contain at least one node in N in [u] and at least one node in Nout [u]. S : candidates of root u argminv2V min δin (v) , δout (v)
; If δin (u)  δout (u) then S N in [u] ; else S Nout [u] ; Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 10 / 49 Candidates of Root Fact: for any node u, any CDS must contain at least one node in N in [u] and at least one node in Nout [u]. S : candidates of root u argminv2V min δin (v) , δout (v)
; If δin (u)  δout (u) then S N in [u] ; else S Nout [u] ; Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 10 / 49 Greedy Algorithm for SAFIN GBA2: B fsg; while f (B) > 0, …nd a cheapest T 2 T (B) ; B B [ I (T ); output a BFS arborescence of D hBi rooted at s. f (B) = # of orphan components of D hBi = (V ,Sv2B δout (v)). Head of an orphan component: node of smallest ID. price of T = jI (T ) n B j # of heads in T T (B): a set of at most jV j
f (B) candidates Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 12 / 49 Greedy Algorithm for SAFIN GBA2: B fsg; while f (B) > 0, …nd a cheapest T 2 T (B) ; B B [ I (T ); output a BFS arborescence of D hBi rooted at s. f (B) = # of orphan components of D hBi = (V ,Sv2B δout (v)). Head of an orphan component: node of smallest ID. price of T = jI (T ) n B j # of heads in T T (B): a set of at most jV j
f (B) candidates Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 12 / 49 Greedy Algorithm for SAFIN GBA2: B fsg; while f (B) > 0, …nd a cheapest T 2 T (B) ; B B [ I (T ); output a BFS arborescence of D hBi rooted at s. f (B) = # of orphan components of D hBi = (V ,Sv2B δout (v)). Head of an orphan component: node of smallest ID. price of T = jI (T ) n B j # of heads in T T (B): a set of at most jV j
f (B) candidates Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 12 / 49 Candidate Arborescences Supplied by u BFS (u): a BFS u-arborescence in D. Sort all the heads in the increasing order of depth in BFS (u). Th (B, u) for 1  h  f (B): the minimal arborescence in BFS (u) spanning u and the …rst h heads. If u = s, then, T (B, u) = fTh (B, u) : 1  h  f (B)g ; otherwise, T (B, u) = fTh (B, u) : 2  h  f (B)g . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 13 / 49 Candidate Arborescences Supplied by u BFS (u): a BFS u-arborescence in D. Sort all the heads in the increasing order of depth in BFS (u). Th (B, u) for 1  h  f (B): the minimal arborescence in BFS (u) spanning u and the …rst h heads. If u = s, then, T (B, u) = fTh (B, u) : 1  h  f (B)g ; otherwise, T (B, u) = fTh (B, u) : 2  h  f (B)g . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 13 / 49 Candidate Arborescences Supplied by u BFS (u): a BFS u-arborescence in D. Sort all the heads in the increasing order of depth in BFS (u). Th (B, u) for 1  h  f (B): the minimal arborescence in BFS (u) spanning u and the …rst h heads. If u = s, then, T (B, u) = fTh (B, u) : 1  h  f (B)g ; otherwise, T (B, u) = fTh (B, u) : 2  h  f (B)g . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 13 / 49 Roadmap Overview MCDS in Arbitrary Networks MCDS in Symmetric Networks MCDS in Plane Geometric Networks Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 15 / 49 MCDS in Graphs with Max. Degree at Most 2 If ∆  2, G is either a path or a cycle. When G is a path, the MCDS consists of all internal vertices. When G is a cycle, a MCDS can be obtained by deleting two adjacent vertices. (b)(a) So we assume that ∆  3 from now on. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 16 / 49 MCDS in Graphs with Max. Degree at Most 2 If ∆  2, G is either a path or a cycle. When G is a path, the MCDS consists of all internal vertices. When G is a cycle, a MCDS can be obtained by deleting two adjacent vertices. (b)(a) So we assume that ∆  3 from now on. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 16 / 49 MCDS in Graphs with Max. Degree at Most 2 If ∆  2, G is either a path or a cycle. When G is a path, the MCDS consists of all internal vertices. When G is a cycle, a MCDS can be obtained by deleting two adjacent vertices. (b)(a) So we assume that ∆  3 from now on. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 16 / 49 A Two-Phased Algorithm 1 Phase 1: apply the greedy algorithm for minimum submodular cover to produce a dominating set U with jU j  H (∆) γc . 2 Phase 2: select a set W of “connectors” to interconnect U. start with an empty W iteratively reduce the # of components of G [U [W ] by adding at most two connectors to W until G [U [W ] is connected. (b)(a) U2 11 v UU U2 v jW j  2 (jU j 1)) jU [W j  3 jU j 2  3H (∆) γc 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 17 / 49 A Two-Phased Algorithm 1 Phase 1: apply the greedy algorithm for minimum submodular cover to produce a dominating set U with jU j  H (∆) γc . 2 Phase 2: select a set W of “connectors” to interconnect U. start with an empty W iteratively reduce the # of components of G [U [W ] by adding at most two connectors to W until G [U [W ] is connected. (b)(a) U2 11 v UU U2 v jW j  2 (jU j 1)) jU [W j  3 jU j 2  3H (∆) γc 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 17 / 49 A Two-Phased Algorithm 1 Phase 1: apply the greedy algorithm for minimum submodular cover to produce a dominating set U with jU j  H (∆) γc . 2 Phase 2: select a set W of “connectors” to interconnect U. start with an empty W iteratively reduce the # of components of G [U [W ] by adding at most two connectors to W until G [U [W ] is connected. (b)(a) U2 11 v UU U2 v jW j  2 (jU j 1)) jU [W j  3 jU j 2  3H (∆) γc 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 17 / 49 A Two-Phased Algorithm 1 Phase 1: apply the greedy algorithm for minimum submodular cover to produce a dominating set U with jU j  H (∆) γc . 2 Phase 2: select a set W of “connectors” to interconnect U. start with an empty W iteratively reduce the # of components of G [U [W ] by adding at most two connectors to W until G [U [W ] is connected. (b)(a) U2 11 v UU U2 v jW j  2 (jU j 1)) jU [W j  3 jU j 2  3H (∆) γc 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 17 / 49 A Single-Phased Greedy Algorithm Single phase with proper potential function Better approximation bound: 2+ ln (∆ 2) Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 18 / 49 Potential Function 8U  V , f1 (U) = # of components in G [U ] , f2 (U) = # of components in G hUi , where G hUi = V , [ u2U δ (u) ! . Then, the potential of U is f (U) = f1 (U) + f2 (U) 1. (b)(a) Figure: U consists of black nodes. In both (a) and (b), f1 (U) = 2, f2 (U) = 1, and f (U) = 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 19 / 49 Potential Function Clearly, f1 (∅) = 0, f2 (∅) = n) f (∅) = n 1; 8∅ 6= U  V , f1 (U)  1, f2 (U)  1) f (U)  1. U is a CDS , f (U) = 1 Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 20 / 49 Potential Function Clearly, f1 (∅) = 0, f2 (∅) = n) f (∅) = n 1; 8∅ 6= U  V , f1 (U)  1, f2 (U)  1) f (U)  1. U is a CDS , f (U) = 1 Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 20 / 49 Gain Denote ∂v f1 (U) = f1 (U) f2 (U [ fvg) , ∂v f2 (U) = f2 (U) f2 (U [ fvg) , Then ∂v f (U) = ∂v f1 (U) + ∂v f2 (U) . If v 2 U then ∂v f1 (U) = ∂v f2 (U) = ∂v f (U) = 0; else ∂v f1 (U) = (# of components of G [U ] adj. to v) 1, ∂v f2 (U) = # of components of G hUi adj. to v . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 22 / 49 Gain Denote ∂v f1 (U) = f1 (U) f2 (U [ fvg) , ∂v f2 (U) = f2 (U) f2 (U [ fvg) , Then ∂v f (U) = ∂v f1 (U) + ∂v f2 (U) . If v 2 U then ∂v f1 (U) = ∂v f2 (U) = ∂v f (U) = 0; else ∂v f1 (U) = (# of components of G [U ] adj. to v) 1, ∂v f2 (U) = # of components of G hUi adj. to v . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 22 / 49 Greedy Algorithm GCDS B ∅; While f (B) > 1 do select v 2 V n B with maximum ∂v f (B) ; B B [ fvg; Output B. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 23 / 49 “Shifted”Supermodularity Lemma Suppose that U and W are two subsets of V satisfying that G [W ] is connected. Then, for any node v 2 V , ∂v f (U [W )  ∂v f (U) + 1. ∂v f1 (U [W )  ∂v f1 (U) + 1, ∂v f2 (U [W )  ∂v f2 (U) . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 26 / 49 “Shifted”Supermodularity Lemma Suppose that U and W are two subsets of V satisfying that G [W ] is connected. Then, for any node v 2 V , ∂v f (U [W )  ∂v f (U) + 1. ∂v f1 (U [W )  ∂v f1 (U) + 1, ∂v f2 (U [W )  ∂v f2 (U) . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 26 / 49 Lower Bound on The Gain Lemma If U is not a CDS, then then at least one node v has gain at least max n 1, f (U )γc 1 o w.r.t. U. W = fvi : 1  i  γcg: a MCDS sorted in the BFS order in G [W ]. Wj = fvi : 1  i  jg with 1  j  γc . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 27 / 49 Lower Bound on The Gain f (U) 1 = f (U) f (U [W ) = ∂v1 f (U) + γc ∑ j=2 ∂vj f (U [Wj1)  ∂v1 f (U) + γc ∑ j=2 ∂vj f (U) + 1
= γc 1+ γc ∑ j=1 ∂vj f (U)  γc 1+ γc
max 1jγc ∂vj f (U) . Hence, max 1jγc ∂vj f (U)  f (U) γc 1. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 28 / 49 Lower Bound on Optimum Lemma γc  n2∆1 . W = fvi : 1  i  γcg: a MCDS sorted in the BFS order in G [W ]. v1 dominates at most ∆+ 1 nodes. Each vi with 2  i  γc dominates ∆ 1 additional nodes. n  (∆+ 1) + (∆ 1) (γc 1) = (∆ 1) γc + 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 29 / 49 Lower Bound on Optimum Lemma γc  n2∆1 . W = fvi : 1  i  γcg: a MCDS sorted in the BFS order in G [W ]. v1 dominates at most ∆+ 1 nodes. Each vi with 2  i  γc dominates ∆ 1 additional nodes. n  (∆+ 1) + (∆ 1) (γc 1) = (∆ 1) γc + 2. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 29 / 49 Upper Bound on Greedy Solution B: output of the greedy algorithm. Bi for 1  i  jB j, : the sequence of the …rst i nodes in B. B0 = ∅. k: be the …rst (smallest) nonnegative integer such that f (Bk ) < 2γc + 2. Claim 1: jB n Bk j  2γc 1. Claim 2: k 1  γc ln (∆ 2) . jB j = k + jB n Bk j  k + 2γc 1  (2+ ln (∆ 2)) γc . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 30 / 49 Upper Bound on Greedy Solution B: output of the greedy algorithm. Bi for 1  i  jB j, : the sequence of the …rst i nodes in B. B0 = ∅. k: be the …rst (smallest) nonnegative integer such that f (Bk ) < 2γc + 2. Claim 1: jB n Bk j  2γc 1. Claim 2: k 1  γc ln (∆ 2) . jB j = k + jB n Bk j  k + 2γc 1  (2+ ln (∆ 2)) γc . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 30 / 49 Upper Bound on Greedy Solution B: output of the greedy algorithm. Bi for 1  i  jB j, : the sequence of the …rst i nodes in B. B0 = ∅. k: be the …rst (smallest) nonnegative integer such that f (Bk ) < 2γc + 2. Claim 1: jB n Bk j  2γc 1. Claim 2: k 1  γc ln (∆ 2) . jB j = k + jB n Bk j  k + 2γc 1  (2+ ln (∆ 2)) γc . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 30 / 49 Proof of Claim 2 `i = f (Bi ) γc for 0  i < k: “shifted”uncoverage n 1 γc = `0 > `1 >


> `k1  γc + 2, `i1 `i  f (Ci1) γc 1 = `i1 γc ) `i1 `i `i1  1 γc . Therefore, k 1 γc  k1 ∑ i=1 `i1 `i `i1  ln `0 `k1  ln n 1 γc γc + 2  ln (∆ 1) γc + 2 1 γc γc + 2 = ln  ∆ 2 2∆ 5 γc + 2  < ln (∆ 2) . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 32 / 49 Proof of Claim 2 `i = f (Bi ) γc for 0  i < k: “shifted”uncoverage n 1 γc = `0 > `1 >


> `k1  γc + 2, `i1 `i  f (Ci1) γc 1 = `i1 γc ) `i1 `i `i1  1 γc . Therefore, k 1 γc  k1 ∑ i=1 `i1 `i `i1  ln `0 `k1  ln n 1 γc γc + 2  ln (∆ 1) γc + 2 1 γc γc + 2 = ln  ∆ 2 2∆ 5 γc + 2  < ln (∆ 2) . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 32 / 49 Roadmap Overview MCDS in Arbitrary Networks MCDS in Symmetric Networks MCDS in Plane Geometric Networks Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 33 / 49 Performance of Greedy Algorithms in Plane Geometric Networks 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes n = k + 2+ 2 k ∑ i=1 2i1 = k + 2k+1, ∆ = 2k + k + 1, γc = 2. Greedy solution: vk , vk1,


, v1 Approx. bound:  k2  log ∆ 2 1 2 Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 34 / 49 Greedy Solution 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes Initial degree: vi : 2
2i1 + (k 1) + 2 = 2i + k + 1; u1 and u2: ∑ki=1 2i1 + k + 1 = 2k + k; others: ∑ki=1 2i1 1+ 1+ 1 = 2k . So vk is selected as the …rst dominator. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 35 / 49 Greedy Solution 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes Initial degree: vi : 2
2i1 + (k 1) + 2 = 2i + k + 1; u1 and u2: ∑ki=1 2i1 + k + 1 = 2k + k; others: ∑ki=1 2i1 1+ 1+ 1 = 2k . So vk is selected as the …rst dominator. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 35 / 49 Greedy Solution 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes Initial degree: vi : 2
2i1 + (k 1) + 2 = 2i + k + 1; u1 and u2: ∑ki=1 2i1 + k + 1 = 2k + k; others: ∑ki=1 2i1 1+ 1+ 1 = 2k . So vk is selected as the …rst dominator. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 35 / 49 Greedy Solution 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes After the selection of vk , vk1,


, vj , the “residue” degree: vi with i < j : 2
2i1 = 2i ; u1 and u2: ∑j1i=1 2 i1 = 2j1 1; others: ∑j1i=1 2 i1 1 = 2j1 2. So vj1 is selected as a dominator. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 36 / 49 Greedy Solution 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes After the selection of vk , vk1,


, vj , the “residue” degree: vi with i < j : 2
2i1 = 2i ; u1 and u2: ∑j1i=1 2 i1 = 2j1 1; others: ∑j1i=1 2 i1 1 = 2j1 2. So vj1 is selected as a dominator. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 36 / 49 Greedy Solution 2 u u2 v2 vkv1 1 k-120 21 20 2k-121 # of nodes # of nodes After the selection of vk , vk1,


, vj , the “residue” degree: vi with i < j : 2
2i1 = 2i ; u1 and u2: ∑j1i=1 2 i1 = 2j1 1; others: ∑j1i=1 2 i1 1 = 2j1 2. So vj1 is selected as a dominator. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 36 / 49 Independence Number vs. Connected Domination Number α: independence number of G γc : connected domination number of G α  8<: 5 if γc = 1; 8 if γc = 2; min  3.4306γc + 4.8185, 323γc + 1 if γc  3. (a) (b) (c) Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 37 / 49 A Conjecture Conjecture: if γc  3, then α  3γc + 3. (a) (b) 4 y1 p1 q1 x1 o0 p2 o5 x2 y2 q2 v6 o6 u2 v 1 u3 u4 u5 w5w4w3w1 v2 v1 v3 v4 v5 o1 o2 o3 o4 y1 p1 q1 x1 p2 x2 y2 q2 o0 o5 w1 w2 w3 w4 u1 u2 u3 u4 u 2 v1 w2 v3 v4 v5 o1 o2 o3 o Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 38 / 49 Algorithm for Phase 1 1 Construct be an arbitrary rooted spanning tree T 2 Select an MIS I in the …rst-…t manner in the BFS ordering in T . hv1, v2,


, vni: BFS ordering of V in T . Initialization: I fv1g. First-…t selection: For i = 2 up to n, add vi to I if vi is not adjacent to any node in I . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 40 / 49 Algorithm for Phase 1 1 Construct be an arbitrary rooted spanning tree T 2 Select an MIS I in the …rst-…t manner in the BFS ordering in T . hv1, v2,


, vni: BFS ordering of V in T . Initialization: I fv1g. First-…t selection: For i = 2 up to n, add vi to I if vi is not adjacent to any node in I . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 40 / 49 Algorithm for Phase 1 1 Construct be an arbitrary rooted spanning tree T 2 Select an MIS I in the …rst-…t manner in the BFS ordering in T . hv1, v2,


, vni: BFS ordering of V in T . Initialization: I fv1g. First-…t selection: For i = 2 up to n, add vi to I if vi is not adjacent to any node in I . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 40 / 49 Algorithm for Phase 2 GC C ∅; While f (C ) > 1 do select v 2 V n (I [ C ) with maximum ∂v f (C ) ; C C [ fvg; Output C . For any subset U  V n I , f (U) = # of components in G [I [U ]. Gain of a node v w.r.t. U: ∂w f (U) = f (U) f (U [ fxg) . Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 42 / 49 Lower Bound on Gain Lemma If f (U) > 1, then at least one node w in V n (I [U) has gain at least max f1, df (U) /γc e 1g . Since the set I has 2-hop separation property, at least one node w 2 V n (I [U) is adjacent to at least two connected components of G [I [U ]. Since each component of G [I [U ] must be adjacent to some node in OPT n (I [U), at lease some node w 2 OPT n (I [U) is adjacent to f (U) jOPT n (I [U)j    f (U) γc  components of G [I [U ]. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 43 / 49 Lower Bound on Gain Lemma If f (U) > 1, then at least one node w in V n (I [U) has gain at least max f1, df (U) /γc e 1g . Since the set I has 2-hop separation property, at least one node w 2 V n (I [U) is adjacent to at least two connected components of G [I [U ]. Since each component of G [I [U ] must be adjacent to some node in OPT n (I [U), at lease some node w 2 OPT n (I [U) is adjacent to f (U) jOPT n (I [U)j    f (U) γc  components of G [I [U ]. Peng-Jun Wan () Minimum Connected Dominating Set in Multihop Wireless Networks 43 / 49