## 3 Main Design
### LACE Measurement  
We use a simple example as follows to illustrate the workflow of LACE. \(S\) periodically transmits data packets to its downstream nodes, \(A\) and \(B\). When receiving the data packets, \(A\) and \(B\) record the packet receptions/losses from \(S\) in bitmaps, with a \"1\" denoting a packet reception and a \"0\" denoting a packet loss. At the same time, \(A\) and \(B\) record the noise and interference, as well as the received signal strength of \(S\)'s transmissions. Furthermore, periodically, receivers \(A\) and \(B\) reply the bitmaps, the noise and interference traces and the signal power traces back to node \(S\), such that \(S\) is able to calculate link correlation \(C_{A,B}^{S}\) as follows:  
\[c_{A,B}^{S}=\alpha\cdot ch_{A,B}^{S}+(1-\alpha)\cdot ci_{A,B}^{S} \tag{1}\]  
where \(ch_{A,B}^{S}\) is the historical correlation value (estimated from the bitmaps), \(ci_{A,B}^{S}\) is the instant correlation value (estimated from the PHY layer information) and \(\alpha\) is a weighting factor. When \(\alpha\) is large, long-term estimation is weighted more, while when \(\alpha\) is small, short-term estimation is weighted more. Long-term estimation: With the bitmaps from \(A\) and \(B\), \(S\) can calculate \(ch_{A,B}^{S}\) as:  
\[ch_{A,B}^{S}=\frac{\sum_{i=1}^{n}(b_{A}[i]\&b_{B}[i])}{\sum_{i=1}^{n}b_{B}[i]} \tag{2}\]  
where & denotes the bitwise AND operation and \(b_{A}[i]\) denotes the \(i\)-th bit in A's bitmap \(b_{A}\). There are different link correlation metrics [13, 16, 17], and the above equation essentially calculates the metric \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Descriptions** \\ \hline \(c_{A,B}^{S}\) & The link correlation between S\(\rightarrow\)A and S\(\rightarrow\)B \\ \(ch_{A,B}^{S}\) & The link correlation value calculated with the data packet trace \\ \(ci_{A,B}^{S}\) & The instant correlation value derived from the PHY information \\ \(ct_{A,B}^{S}(k)\) & The link correlation value with kcommon receptions \\ \(b_{A}[i]\) & The \(i\)-th bit in A’s bitmap \\ \(R(n_{r}=k)\) & The probability with \(k\) common packet receptions \\ \(p_{A}^{S}(i)\) & Packet reception rate of link S\(\rightarrow\)A \\ \(es_{A}^{S}\) & symbol error rate \\ \(ec_{A}^{S}\) & chip error rate \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 2: Lightweight and accurate link correlation estimation (LACE) overview. of the conditional packet reception probability. Different upper layer protocols may require different correlation metrics. It is worth noting that we can support all of the metrics for calculating packet reception/loss correlations. For example, the metric of conditional packet loss probability can be calculated as: \(ch_{A,B}^{S}=\frac{\sum_{i=1}^{n}(-b_{A}[i|k\cdot\
eg b_{B}[i])}{\sum_{i=1}^{n} -b_{B}[i]}\) where \(\
eg\) is the bitwise negation operation. The calculation is similar to the BCM approach. The difference with BCM is that: (1) the bitmaps are recorded for data packet receptions/losses, which are representative of data plane link correlation (as discussed in Section 2, it is the true reflection of the correlations of data packet receptions and losses); (2) the use of data receptions and losses eliminates the need for extra periodic beacon messages, which greatly reduces the transmission overhead; and (3) it is also worth noting that there are also cons to our approach. Our approach can yield accurate link correlation estimation only when there are periodic data packets. When there are few or even no data packet transmissions, our approach cannot work well, since there are no data traces. Short-term estimation: Next, we use RSSI to infer the instant link quality of the two links and further estimate the short-term link correlation.