Programmable photonic meshes can perform general linear transformations, but universal transformations on N modes require O(N²) programmable phase shifters, which cannot simultaneously achieve low drive voltage, gigahertz speed, zero static power, lossless modulation, and a compact footprint. Signal routing does not require universal transformations, which allows considerable circuit simplifications. We present a compact reflective circuit architecture that connects all input-output pairs across N modes in N steps using only O(N) programmable phase shifters. The circuit comprises a single passive N× N interferometer and a single column of programmable reflective phase shifters. We demonstrate the architecture by implementing a 3× 3 router in x-cut thin-film lithium niobate with a passive multimode interferometer and phase shifters based on single-sided photonic-crystal nanobeam cavities. Our programmable devices exhibit a 1.92 GHz/V resonance tuning efficiency via the Pockels effect, a footprint of 100 μm², sub-5-V drive voltage, and a 6.2 GHz electro-optic bandwidth. Through voltage-space characterization and physical modeling, we identify routing states with extinction ratios exceeding 11 dB at each output and demonstrate sequential routing at an equipment-limited speed of 10 MHz. Our results establish cavity-enhanced Pockels switching as a routing layer for classical and quantum computing interconnects across visible and near-infrared wavelengths.