Access Graphs Results for LRU versus FIFO under Relative Worst Order Analysis

Access graphs, which have been used previously in connection with competitive analysis to model locality of reference in paging, are considered in connection with relative worst order analysis. In this model, FWF is shown to be strictly worse than both LRU and FIFO on any access graph. LRU is shown to be strictly better than FIFO on paths and cycles, but they are incomparable on some families of graphs which grow with the length of the sequences.

💡 Research Summary

The paper introduces a novel comparative framework for paging algorithms by marrying the concept of access graphs with Relative Worst Order Analysis (RWOA). Access graphs have long been used in competitive analysis to capture locality of reference: vertices represent pages and edges denote permissible immediate transitions between page requests. While competitive analysis measures the worst‑case ratio against an optimal offline algorithm, it often masks nuanced differences that arise from the structure of the request sequence. RWOA, by contrast, compares two online algorithms directly: for any input sequence σ, each algorithm’s “worst order” is the permutation of σ that maximizes its cost, and the algorithms are ordered by comparing these worst‑order costs.

The authors focus on three classic paging strategies: First‑In‑First‑Out (FWF), Least‑Recently‑Used (LRU), and First‑In‑First‑Out (FIFO). They first formalize the paging problem on a fixed‑size cache of capacity k, define the cost of an algorithm as the number of page faults, and then embed the request stream in an access graph G. The central results are three theorems.

Theorem 1 (FWF is strictly dominated). For every finite access graph G, the worst‑order cost of FWF is strictly larger than that of both LRU and FIFO. The proof constructs a longest simple path in G and shows that, because FWF never exploits the graph’s locality, each step forces a fault, leading to a cost proportional to the path length. LRU and FIFO, by retaining more recently accessed pages, can avoid many of these faults, yielding a lower worst‑order cost.

Theorem 2 (LRU beats FIFO on paths and cycles). When G is a simple path or a simple cycle, the worst‑order cost of LRU is always less than that of FIFO. The authors exploit the linear (or circular) ordering of pages: LRU’s policy of keeping the most recently used page ensures that, as the request walks along the path or around the cycle, the same set of k pages remains in cache for as long as possible. FIFO, which evicts in arrival order, inevitably discards pages that will be needed shortly thereafter, incurring extra faults. Tight upper and lower bounds are derived, establishing a strict inequality for all σ.

Theorem 3 (Incomparability on growing families). There exist families of access graphs whose size grows with the length n of the request sequence such that LRU and FIFO are incomparable under RWOA. The authors construct a family of tree‑like graphs (and variants) where, by carefully arranging the request order, one can force LRU’s worst‑order cost to be arbitrarily larger than FIFO’s, and vice‑versa, on different sequences of the same length. This demonstrates that neither algorithm uniformly dominates the other when the underlying locality structure is allowed to scale with the workload.

Methodologically, the paper blends graph‑theoretic tools (longest path, cycle covers, tree width) with worst‑order permutation arguments. The “worst order” for each algorithm is engineered by exploiting the graph’s topology: edges are traversed so that the algorithm’s eviction rule is maximally penalized. This approach generalizes the classic “adversarial request” technique used in competitive analysis, providing a finer granularity of comparison.

The implications are twofold. First, the strict superiority of LRU over FIFO on paths and cycles gives a theoretical justification for the widespread empirical observation that LRU performs better on workloads with strong spatial locality. Second, the incomparability result cautions against a blanket endorsement of any single policy: in environments where the access pattern resembles a growing hierarchical structure (e.g., deep directory traversals or multi‑level cache hierarchies), FIFO may be as competitive as LRU.

Finally, the authors discuss extensions. The access‑graph + RWOA framework can be applied to other online problems such as list update, server allocation, or multi‑cache systems. Future work could explore dynamic access graphs that evolve with the workload, incorporate probabilistic request models, or validate the theoretical predictions with empirical traces from real systems.

In summary, the paper advances the theory of online paging by showing that relative worst‑order analysis, when combined with access graphs, yields sharper distinctions among algorithms than traditional competitive analysis, highlighting both the strengths of LRU in locally structured workloads and the nuanced contexts where FIFO remains competitive.