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) |