Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation such as the Congested Clique model and the Massively Parallel Computation (MPC) model, as well as in related big data models such as streaming and semi-streaming. We provide algorithms for constructing the well-known Thorup-Zwick distance oracle (or distance sketches) in these different computational models and discuss the tradeoffs between various complexity measures such as space, time, stretch, and message complexity. One key component we use to construct distance sketches are the hopsets of Elkin and Neiman 2016. In particular, we show that these hopsets can be constructed efficiently in the MPC model, even in the very low memory setting. This result has additional applications such as the first subpolynomial algorithm for approximate single-source shortest paths for weighted graphs in the low memory MPC setting.
We note that there are no bounds for distace computation for weighted graphs in the MPC model that we are aware of. It is known that many PRAM algorithms can be simulated in the MPC model ( @cite_18 , @cite_7 ), and shortest path computation is well studied in the PRAM model (e.g. @cite_4 , @cite_5 ). However, most of these algorithms use @math number of processors, in which case the simulations of @cite_18 and @cite_7 do not directly apply in MPC model with total memory of @math . Hence, these PRAM algorithm do not apply to the MPC setting directly.
@cite_5 @cite_18 @cite_4 @cite_7
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The power of shared-memory in models of parallel computation is studied, and a novel distributed data structure that eliminates the need for shared memory without significantly increasing the run time of the parallel computation is described. More specifically, it is shown how a complete network of processors can deterministically simulate one PRAM step in O (log n (log log n ) 2 ) time when both models use n processors and the size of the PRAM's shared memory is polynomial in n . (The best previously known upper bound was the trivial O ( n )). It is established that this upper bound is nearly optimal, and it is proved that an on-line simulation of T PRAM steps by a complete network of processors requires O( T (log n log log n )) time. A simple consequence of the upper bound is that an Ultracomputer (the currently feasible general-purpose parallel machine) can simulate one step of a PRAM (the most convenient parallel model to program) in O ((log n ) 2 log log n ) steps. The problem of constructing in parallel a maximal independent set of a given graph is considered. A new deterministic NC-algorithm implemented in the EREW PRAM model is presented. On graphs with n vertices and m edges, it uses @math processors and runs in @math time. This reduces by a factor of @math both the running time and the processor count of the previously fastest deterministic algorithm that solves the problem using a linear number of processors. We present oblivious implementations of several data structures for secure multiparty computation (MPC) such as arrays, dictionaries, and priority queues. The resulting oblivious data structures have only polylogarithmic overhead compared with their classical counterparts. To achieve this, we give secure multiparty protocols for the ORAM of (Asiacrypt ‘11) and the Path ORAM scheme of (CCS ‘13), and we compare the resulting implementations. We subsequently use our oblivious priority queue for secure computation of Dijkstra’s shortest path algorithm on general graphs, where the graph structure is secret. To the best of our knowledge, this is the first implementation of a non-trivial graph algorithm in multiparty computation with polylogarithmic overhead. We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of Ω (n + D) rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where n is the number of nodes in the network and D is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs O(n^2 3 D^1 3 + n^5 6) rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing O(√n D) rounds and the second performing O(√n D^1 4 + n^3 5 + D) rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a (1+e)-approximation O((√n D^1 4+D) e)-round algorithm for directed SSSP and a new work depth trade-off for exact SSSP on directed graphs in the PRAM model.