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Planning is a critical aspect of multi-step reasoning, yet it remains challenging for large language models (LLMs). In this work, we use pathfinding in graphs as a sandbox for understanding and improving the planning abilities of LLMs. Our results show that while conventional autoregressive training generalizes poorly, an anchoring strategy, whereby a model first predicts a small subset of intermediate nodes along the path, significantly improves the path finding performance. We confirm these gains on two families of graphs with markedly different structures and provide preliminary heuristics for selecting effective anchor nodes, offering guidance for more realistic settings. 

The FuzzyLog is a partially ordered shared log abstraction. Distributed applications can concurrently append to the partial order and play it back. FuzzyLog applications obtain the benefits of an underlying shared log --- extracting strong consistency, durability, and failure atomicity in simple ways --- without suffering from its drawbacks. By exposing a partial order, the FuzzyLog enables three key capabilities for applications: linear scaling for throughput and capacity (without sacrificing atomicity), weaker consistency guarantees, and tolerance to network partitions. We present Dapple, a distributed implementation of the FuzzyLog abstraction that stores the partial order compactly and supports efficient appends / playback via a new ordering protocol. We implement several data structures and applications over the FuzzyLog, including several map variants as well as a ZooKeeper implementation. Our evaluation shows that these applications are compact, fast, and flexible: they retain the simplicity (100s of lines of code) and strong semantics (durability and failure atomicity) of a shared log design while exploiting the partial order of the FuzzyLog for linear scalability, flexible consistency guarantees (e.g., causal+ consistency), and network partition tolerance. On a 6-node Dapple deployment, our FuzzyLogbased ZooKeeper supports 3M/sec single-key writes, and 150K/sec atomic cross-shard renames. 

The FuzzyLog is a partially ordered shared log abstraction. Distributed applications can concurrently append to the partial order and play it back. FuzzyLog applications obtain the benefits of an underlying shared log - extracting strong consistency, durability, and failure atomicity in simple ways - without suffering from its drawbacks. By exposing a partial order, the FuzzyLog enables three key capabilities for applications: linear scaling for throughput and capacity (without sacrificing atomicity), weaker consistency guarantees, and tolerance to network partitions. We present Dapple, a distributed implementation of the FuzzyLog abstraction that stores the partial order compactly and supports efficient appends / playback via a new ordering protocol. We implement several data structures and applications over the FuzzyLog, including several map variants as well as a ZooKeeper implementation. Our evaluation shows that these applications are compact, fast, and flexible: they retain the simplicity (100s of lines of code) and strong semantics (durability and failure atomicity) of a shared log design while exploiting the partial order of the FuzzyLog for linear scalability, flexible consistency guarantees (e.g., causal+ consistency), and network partition tolerance. On a 6-node Dapple deployment, our FuzzyLog-based ZooKeeper supports 3M/sec single-key writes, and 150K/sec atomic cross-shard renames.