Context-Aware Computing

For x and y, we have σ1(x) ≤ K < σ2(x), ${\sigma }_1(x)\le K<{{\sigma }^{\prime }}_2(x),{\sigma }_1(y)\le K<{{\sigma }^{\prime }}_2(y),$ and ${\sigma }_1(y)\le K\le {{\sigma }^{\prime }}_2(y)\cdot \text{If}{\sigma }_2(x)<{\sigma }_2(y),then{{\sigma }^{\prime }}_2(x)\ge {{\sigma }^{\prime }}_2(y).$ We thus have

$C({\sigma }_1(x),{{\sigma }^{\prime }}_2(x))=C({\sigma }_1(x),{\sigma }_2(x))+w_{{\sigma }_2(x)},(11.10)$

$C({\sigma }_1(y),{{\sigma }^{\prime }}_2(y))=C({\sigma }_1(y),{\sigma }_2(y))-w_{{\sigma }_2(x)},(11.11)$

where $W_{{\sigma }_2(x)}$ is the contribution of moving x from position σ2(x) to σ2(x) + 1 to the overall distance. Notice that $w_{{\sigma }_2}(x)$ is also the contribution of moving y from position σ2(y) to σ2(y) – 1.

From formulas (11.10) and (11.11), we get

$C({\sigma }_1(x),{{\sigma }^{\prime }}_2(x))+C({\sigma }_1(y),{{\sigma }^{\prime }}_2(y))=C({\sigma }_1(x),{\sigma }_2(x))+C({\sigma }_1(y),{\sigma }_2(y)).$