Intelligent behavior in robots is implemented through algorithms. Historically, much of algorithmic robotics research strives to compute outputs that achieve mathematically rigid conditions, such as minimizing path length. But today's robots are increasingly being used to empower the daily lives of people, and experience shows that traditional algorithmic approaches are poorly suited for the unpredictable, idiosyncratic, and adaptive nature of human-robot interaction. This raises a need for entirely new computational, mathematical, and technical approaches for robots to better understand and react to humans. The human-friendly robots of the future will need new algorithms, informed from the ground up by HRI research, to generate interpretable, ethical, socially-acceptable behavior, ensure safety around humans, and execute tasks of value to society.