Call traces, which record the sequence of function calls and returns, have wide applications to program analysis activities such as bug reproduction, fault diagnosis, performance analysis, program comprehension and anomaly detection. However, the conventional call trace collection by instrumenting additional code at each function call/return site induces high space and time overhead. This paper aims at reducing the recording overheads by instrumenting only a small amount of call sites while keeping the capability to recover the trace with its full fidelity. We propose a call trace model and a logged call trace model based on an LL(1) grammar, which enables us to define the criteria of a feasible solution to call trace collection. Based on the two models, we prove that to collect call traces with minimal instrumentation is an NP-hard problem. We then propose an efficient approach to obtaining a suboptimal solution. We implemented our approach as a tool {\sf Casper} and evaluated it using the DaCapo benchmark suite. The experiment results show that our approach causes significantly lower runtime (and space) overhead than two state-of-the-arts approaches.