In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E \to \mathbb{Z}_{\geq 0}$ and edge lengths $l: E \to \mathbb{Z}_{\geq 0}$ along with a root node $r \in V$ and a length-constraint $h \in \mathbb{Z}_{\geq 0}$. Our goal is to output a spanning tree of minimum weight according to $w$ in which every node is at distance at most $h$ from $r$ according to $l$. We give a polynomial-time algorithm for planar graphs which, for any constant $ε> 0$, outputs an $O\left(\log^{1+ε} n\right)$-approximate solution with every node at distance at most $(1+ε)h$ from $r$ for any constant $ε> 0$. Our algorithm is based on new length-constrained versions of classic planar separators which may be of independent interest. Additionally, our algorithm works for length-constrained Steiner tree. Complementing this, we show that any algorithm on general graphs for length-constrained MST in which nodes are at most $2h$ from $r$ cannot achieve an approximation of $O\left(\log ^{2-ε} n\right)$ for any constant $ε> 0$ under standard complexity assumptions; as such, our results separate the approximability of length-constrained MST in planar and general graphs.

A novel highly-efficient multi-band multi-mode all digital quadrature transmitter is presented. The all digital transmitter uses in-phase (I) codeword and quadrature (Q) codeword to control a switching-mode power amplifier (PA) or digital PA (DPA) consisting of in-phase PA (I-PA) and quadrature PA (Q-PA), where each of the power cells inside I-PA or Q-PA is either on or off. Due to the load interaction between I-PA and Q-PA, a 2-dimensional digital pre-distortion is applied to linearize DPA. The total transmitter is implemented in 40nm CMOS LP process and occupies a die area of 0.7mm2. The digital quadrature transmitter can support 20MHz, 40MHz, and 80MHz WiFi signals, Band 38 and Band 40 LTE signals with class 3 output power, and Bluetooth BDR, EDR2, and EDR3 signals.