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Table 4 Three-tuples used to determine the edges and labels for the example in Fig. 14

From: Backtracking-based dynamic programming for resolving transmit ambiguities in WSN localization

\(\mathcal {G}_{s}\)

Constraint cliques

Three-tuples of the vertices of \(\mathcal {G}_{i}\)

Action/comment

\(\mathcal {G}_{1}\)

\(\mathcal {C}_{1}\)

\(R_{1}(v_{1}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)

∙ R1(v1)≠R1(v8)⇒ ℓ(v1)≠ℓ(v8)†

  

\(R_{1}(v_{8}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)

 

\(\mathcal {G}_{2}\)

\(\mathcal {C}_{2}\)

\(R_{2}(v_{2}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)

∙ R2(v2)≠R2(v3)⇒ ℓ(v2)≠ℓ(v3)†

  

\(R_{2}(v_{3}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)

 

\(\mathcal {G}_{3}\)

\(\mathcal {C}_{3}\)

\(R_{3}(v_{5}) = (\mathcal {C}_{8}, \mathcal {C}_{6}, 0)\)

∙ R3(v5)≠R3(v9)⇒ ℓ(v5)≠ℓ(v9)†

  

\(R_{3}(v_{9}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)

 

\(\mathcal {G}_{4}\)

\(\mathcal {C}_{4}\)

\(R_{4}(v_{4}) = (\mathcal {C}_{8}, \mathcal {C}_{6}, 0)\)

∙ R4(v4)≠R4(v6)⇒ ℓ(v4)≠ℓ(v6)†

  

\(R_{4}(v_{6}) = (\mathcal {C}_{7}, \mathcal {C}_{9}, 0)\)

 

\(\mathcal {G}_{5}\)

\(\mathcal {C}_{1},\) \(\mathcal {C}_{2}\)

\(R_{5}(v_{1}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)

∙ R5(v i )∼R5(v j ),i,j∈{1,2,3,8} ⇒ complete graph

  

\(R_{5}(v_{2}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)

∙ R5(v1)=R5(v2)⇒ ℓ(v1)=ℓ(v2)

  

\(R_{5}(v_{3}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)

∙ R5(v3)=R5(v8)⇒ ℓ(v3)=ℓ(v8)

  

\(R_{5}(v_{8}) = (\mathcal {C}_{8}, \mathcal {C}_{9}, 0)\)

 

\(\mathcal {G}_{6}\)

\(\mathcal {C}_{3}\), \(\mathcal {C}_{4}\),

\(R_{6}(v_{4}) = (\mathcal {C}_{8}, 0, 0)\)

∙ R6(v4)∼R6(v5)⇒(v4,v5)

  

\(R_{6}(v_{5}) = (\mathcal {C}_{8}, 0, 0)\)

∙ R6(v6)∼R6(v9)⇒(v6,v9)

  

\(R_{6}(v_{6}) = (\mathcal {C}_{7}, \mathcal {C}_{9}, 0)\)

∙ R6(v4)=R6(v5)⇒ ℓ(v4)=ℓ(v5)

  

\(R_{6}(v_{9}) = (\mathcal {C}_{7}, \mathcal {C}_{5}, 0)\)

 

\(\mathcal {G}_{7}\)

\(\mathcal {C}_{1}\), \(\mathcal {C}_{2}\), \(\mathcal {C}_{3}\), \(\mathcal {C}_{4}\), , \(\mathcal {C}_{6}\),,

R7(v1)=(0,0,0)

∙ R7(v1)=R7(v2)=R7(v6)=R7(v9) ⇒ ℓ(v1)=ℓ(v2)=ℓ(v6)=ℓ(v9)

  

R7(v2)=(0,0,0)

∙ R7(v3)=R7(v4)=R7(v5)=R7(v8) ⇒ ℓ(v3)=ℓ(v4)=ℓ(v5)=ℓ(v8)

  

\(R_{7}(v_{3}) = (\mathcal {C}_{8}, 0, 0)\)

∙ R7(v i )∼R7(v j ),(i,j)∈{1,2,6,9}2,i>j ⇒ (v i ,v j )∀(i,j)∈{1,2,6,9}2,i>j

  

\(R_{7}(v_{4}) = (\mathcal {C}_{8}, 0, 0)\)

∙ R7(v i )∼R7(v j ),(i,j)∈{3,4,5,8}2,i>j ⇒ (v i ,v j )∀(i,j)∈{3,4,5,8}2,i>j

  

\(R_{7}(v_{5}) = (\mathcal {C}_{8}, 0, 0)\)

∙ Note that the last two operations introduce edges between the labels\(\left \{ \ell _{1}^{1}, \ell _{2}^{2}, \ell _{2}^{3} \right \}\) and between the labels \(\left \{ \ell _{2}^{1},\ell _{1}^{2} \right \}\)

  

R7(v6)=(0,0,0)

 
  

\(R_{7}(v_{8}) = (\mathcal {C}_{8}, 0, 0)\)

 
  

R7(v9)=(0,0,0)

 
  1. †The edge sets of the leaves are given directly, as the leaves are chosen to be cliques
  2. ∙The red-circled cliques are binding SC cliques, i.e., fully included SC cliques that have not been included before by either of the child nodes