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Comput."],"published-print":{"date-parts":[[2025,6]]},"abstract":"Abstract<\/jats:title>\n We present fast algorithms for approximate shortest paths in the massively parallel computation (MPC) model. We provide randomized algorithms that take \n \n $$\\textrm{poly}(\\log {\\log {n}})$$<\/jats:tex-math>\n \n \n poly<\/mml:mtext>\n (<\/mml:mo>\n log<\/mml:mo>\n \n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> rounds in the near-linear memory MPC model. Our results are for unweighted undirected graphs with n<\/jats:italic> vertices and m<\/jats:italic> edges. Our first contribution is a \n \n $$(1+\\epsilon )$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \u03f5<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-approximation algorithm for Single-Source Shortest Paths (SSSP) that takes \n \n $$\\textrm{poly}(\\log {\\log {n}})$$<\/jats:tex-math>\n \n \n poly<\/mml:mtext>\n (<\/mml:mo>\n log<\/mml:mo>\n \n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> rounds in the near-linear MPC model, where the memory per machine is \n \n $$\\tilde{O}(n)$$<\/jats:tex-math>\n \n \n \n O<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the total memory is \n \n $$\\tilde{O}(mn^{\\rho })$$<\/jats:tex-math>\n \n \n \n O<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n m<\/mml:mi>\n \n n<\/mml:mi>\n \u03c1<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \n \n $$\\rho $$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a small constant. Our second contribution is a distance oracle that allows to approximate the distance between any pair of vertices. The distance oracle is constructed in \n \n $$\\textrm{poly}(\\log {\\log {n}})$$<\/jats:tex-math>\n \n \n poly<\/mml:mtext>\n (<\/mml:mo>\n log<\/mml:mo>\n \n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> rounds and allows to query a \n \n $$(1+\\epsilon )(2k-1)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \u03f5<\/mml:mi>\n )<\/mml:mo>\n (<\/mml:mo>\n 2<\/mml:mn>\n k<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-approximate distance between any pair of vertices u<\/jats:italic> and v<\/jats:italic> in O<\/jats:italic>(1) additional rounds. The algorithm is for the near-linear memory MPC model with total memory of size \n \n $$\\tilde{O}((m+n^{1+\\rho })n^{1\/k})$$<\/jats:tex-math>\n \n \n \n O<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n \n (<\/mml:mo>\n m<\/mml:mi>\n +<\/mml:mo>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n \u03c1<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \n \n $$\\rho $$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a small constant. While our algorithms are for the near-linear MPC model, in fact they only use one machine with \n \n $$\\tilde{O}(n)$$<\/jats:tex-math>\n \n \n \n O<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> memory, where the rest of machines can have sublinear memory of size \n \n $$O(n^{\\gamma })$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n n<\/mml:mi>\n \u03b3<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for a small constant \n \n $$\\gamma < 1$$<\/jats:tex-math>\n \n \n \u03b3<\/mml:mi>\n <<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. All previous algorithms for approximate shortest paths in the near-linear MPC model either required \n \n $$\\Omega (\\log {n})$$<\/jats:tex-math>\n \n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> rounds or had an \n \n $$\\Omega (\\log {n})$$<\/jats:tex-math>\n \n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> approximation. Our approach is based on fast construction of near-additive emulators, limited-scale hopsets and limited-scale distance sketches that are tailored for the MPC model. While our end-results are for the near-linear MPC model, many of the tools we construct such as hopsets and emulators are constructed in the more restricted sublinear MPC model.<\/jats:p>","DOI":"10.1007\/s00446-025-00482-y","type":"journal-article","created":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T03:38:02Z","timestamp":1745379482000},"page":"131-162","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Massively parallel algorithms for approximate shortest paths"],"prefix":"10.1007","volume":"38","author":[{"given":"Michal","family":"Dory","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaked","family":"Matar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,23]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Assadi, S., Bateni, M. 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