Fine-grained reductions have established equivalences between many core problems withÕ(n 3 )-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also haveÕ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn bounds optimal when m n 2 ?Starting from the hypothesis that the minimum weight (2 + 1)-Clique problem in edge weighted graphs requires n 2 +1−o(1) time, we prove that for all sparsities of the form m = Θ(n 1+1/ ), there is no O(n 2 + mn 1−ε ) time algorithm for ε > 0 for any of the below problems• Minimum Weight (2 + 1)-Cycle in a directed weighted graph, • Shortest Cycle in a directed weighted graph, • APSP in a directed or undirected weighted graph, • Radius (or Eccentricities) in a directed or undirected weighted graph, • Wiener index of a directed or undirected weighted graph, • Replacement Paths in a directed weighted graph, • Second Shortest Path in a directed weighted graph, • Betweenness Centrality of a given node in a directed weighted graph. That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem. Our results also lead to new conditional lower bounds from several related hypothesis for unweighted sparse graph problems including k-cycle, shortest cycle, Radius, Wiener index and APSP. * andreali@mit.edu. Supported by the EECS Merrill Lynch Fellowship.†

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Memory efficiency and locality have substantial impact on the performance of programs, particularly when operating on large data sets. Thus, memory-or I/O-efficient algorithms have received significant attention both in theory and practice. The widespread deployment of multicore machines, however, brings new challenges. Specifically, since the memory (RAM) is shared across multiple processes, the effective memory-size allocated to each process fluctuates over time.This paper presents techniques for designing and analyzing algorithms in a cache-adaptive setting, where the RAM available to the algorithm changes over time. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.SPAA '16, July 11 -13, 2016, Pacific Grove, CA, USA We demonstrate the effectiveness of these techniques by deriving several results:• We give a simple recipe for determining whether common divide-and-conquer cache-oblivious algorithms are optimally cache adaptive.• We show how to bound an algorithm's non-optimality.We give a tight analysis showing that a class of cacheoblivious algorithms is a logarithmic factor worse than optimal.• We show the generality of our techniques by analyzing the cache-oblivious FFT algorithm, which is not covered by the above theorems. Nonetheless, the same general techniques can show that it is at most O(log log N ) away from optimal in the cache adaptive setting, and that this bound is tight.These general theorems give concrete results about several algorithms that could not be analyzed using earlier techniques. For example, our results apply to Fast Fourier Transform, matrix multiplication, Jacobi Multipass Filter, and cache-oblivious dynamic-programming algorithms, such as Longest Common Subsequence and Edit Distance.Our results also give algorithm designers clear guidelines for creating optimally cache-adaptive algorithms.