Title: Fast Binary Functional Search on Graph. Abstract: The large-scale search is an essential task in modern information systems. Numerous learning based models are proposed to capture semantic level similarity measures for searching or ranking. However, these measures are usually complicated and beyond metric distances. As Approximate Nearest Neighbor Search (ANNS) techniques have specifications on metric distances, efficient searching by advanced measures is still an open question. In this paper, we formulate large-scale search as a general task, Optimal Binary Functional Search (OBFS), which contains ANNS as special cases. We analyze existing OBFS methods' limitations and explain they are not applicable for complicated searching measures. We propose a flexible graph-based solution for OBFS, Search on L2 Graph (SL2G). SL2G approximates gradient decent in Euclidean space, with accessible conditions. Experiments demonstrate SL2G's efficiency in searching by advanced matching measures (i.e., Neural Network based measures).