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Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and $o(\log n)$ rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on planar graphs. Such problems include, for example, counting connected components, bipartition, minimum spanning tree problem, (approximate) shortest paths, and (approximate) diameter/radius. In this paper, we show a way to get around this limitation. Specifically, we show that if we have a ``nice'' (for example, straight-line) embedding of the input graph, all the mentioned problems can be solved with $O(n^{2/3+\epsilon})$ space per machine in $O(1)$ rounds. In conjunction with existing algorithms for computing the Delaunay triangulation, our results imply an MPC algorithm for exact Euclidean minimum spanning thee (EMST) that uses $O(n^{2/3 + \epsilon})$ space per machine and finishes in $O(1)$ rounds. This is the first improvement over a straightforward use of the standard Bor\r{u}vka's algorithm with the Dauleanay triangulation algorithm of Goodrich [SODA 1997] which results in $\Theta(\log n)$ rounds. This also partially negatively answers a question of Andoni, Nikolov, Onak, and Yaroslavtsev [STOC 2014], asking for lower bounds for exact EMST. We extend our algorithms to work with embeddings consisting of curves that are not ``too squiggly" (as formalized by the total absolute curvature). We do this via a new lemma which we believe is of independent interest and could be used to parameterize other geometric problems by the total absolute curvature. We also state several open problems regarding massively parallel computation on planar graphs., Comment: To appear at SODA 2023

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs., LIPIcs, Vol. 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), pages 42:1-42:18

We show that every labelled planar graph $G$ can be assigned a canonical embedding $\phi(G)$, such that for any planar $G'$ that differs from $G$ by the insertion or deletion of one edge, the number of local changes to the combinatorial embedding needed to get from $\phi(G)$ to $\phi(G')$ is $O(\log n)$. In contrast, there exist embedded graphs where $\Omega(n)$ changes are necessary to accommodate one inserted edge. We provide a matching lower bound of $\Omega(\log n)$ local changes, and although our upper bound is worst-case, our lower bound hold in the amortized case as well. Our proof is based on BC trees and SPQR trees, and we develop \emph{pre-split} variants of these for general graphs, based on a novel biased heavy-path decomposition, where the structural changes corresponding to edge insertions and deletions in the underlying graph consist of at most $O(\log n)$ basic operations of a particularly simple form. As a secondary result, we show how to maintain the pre-split trees under edge insertions in the underlying graph deterministically in worst case $O(\log^3 n)$ time. Using this, we obtain deterministic data structures for incremental planarity testing, incremental planar embedding, and incremental triconnectivity, that each have worst case $O(\log^3 n)$ update and query time, answering an open question by La Poutr\'e and Westbrook from 1998., Comment: Accepted for publication at SODA'20

Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is $2$-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph is partitioned into trails. Can we orient the trails such that the resulting directed graph is strongly connected? We show that $2$-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation, which is both optimal and the first polynomial time algorithm for deciding this problem. The generalised Robbins' theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs states that the undirected edges of a mixed multigraph can be oriented making the resulting directed graph strongly connected exactly when the mixed graph is connected and the underlying graph is bridgeless. We show that as long as all cuts have at least $2$ undirected edges or directed edges both ways, then there exists an orientation making the resulting directed graph strongly connected. This provides the first polynomial time algorithm for this problem and a very simple polynomial time algorithm to the previous problem., Comment: Earlier version submitted to SODA'17

We present a data structure that, given a graph $G$ of $n$ vertices and $m$ edges, and a suitable pair of nested $r$-divisions of $G$, preprocesses $G$ in $O(m+n)$ time and handles any series of edge-deletions in $O(m)$ total time while answering queries to pairwise biconnectivity in worst-case $O(1)$ time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case $O(1)$ time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in Õ((log n)2) amortized time, and can find a bridge in the component of any given vertex, or a bridge separating any two given vertices, in O(log n/ log log n) worst case time. Our bounds match the current best for bounds for deterministic fully-dynamic connectivity up to log log n factors.The previous best dynamic bridge finding was an Õ((log n)3) amortized time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on the O((log n)4) amortized time algorithm by Holm et al.[STOC98, JACM2001].Our approach is based on a different and purely combinatorial improvement of the algorithim of Holm et al., which by itself gives a new combinatorial Õ((log n)3) amortized time algorithm. Combining it with Thorup's bittrick, we get down to the claimed Õ((log n)2) amortized time.Essentially the same new trick can be applied to the biconnectivity data structure from [STOC98, JACM2001], improving the amortized update time to Õ((log n)3).We also offer improvements in space. We describe a general trick which applies to both of our new algorithms, and to the old ones, to get down to linear space, where the previous best use O(m + n log n log log n).Our result yields an improved running time for deciding whether a unique perfect matching exists in a static graph.

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in Õ((log n)2) amortized time, and can find a bridge in the component of any given vertex, or a bridge separating any two given vertices, in O(log n= log log n) worst case time. Our bounds match the current best for bounds for deterministic fully-dynamic connectivity up to log log n factors. The previous best dynamic bridge finding was an Õ((log n)3) amortized time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on the O((log n)4) amortized time algorithm by Holm et al.[STOC98, JACM2001]. Our approach is based on a different and purely combinatorial improvement of the algorithm of Holm et al., which by itself gives a new combinatorial Õ((log n)3) amortized time algorithm. Combining it with Thorup's bittrick, we get down to the claimed Õ((log n)2) amortized time. Essentially the same new trick can be applied to the biconnectivity data structure from [STOC98, JACM2001], improving the amortized update time to Õ((log n)3). We also offer improvements in space. We describe a general trick which applies to both of our new algorithms, and to the old ones, to get down to linear space, where the previous best use O(m + n log n log log n). Our result yields an improved running time for deciding whether a unique perfect matching exists in a static graph.

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized $O(\log^2 n)$ replacements per insertion, where $n$ is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for \emph{any} replacement strategy, almost matching the $\Omega(\log n)$ lower bound. The previous best strategy known achieved amortized $O(\sqrt{n})$ replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial $O(n)$ bound was known except in special cases. Our analysis immediately implies the same upper bound of $O(\log^2 n)$ reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load $L$, then the SAP protocol makes amortized $O( \min\{L \log^2 n , \sqrt{n}\log n\})$ reassignments. We also show that this is close to tight because $\Omega(\min\{L, \sqrt{n}\})$ reassignments can be necessary.

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in $\tilde{O}((\log n)^2)$ amortized time, and can find a bridge in the component of any given vertex, or a bridge separating any two given vertices, in $O(\log n / \log \log n)$ worst case time. Our bounds match the current best for bounds for deterministic fully-dynamic connectivity up to $\log\log n$ factors. The previous best dynamic bridge finding was an $\tilde{O}((\log n)^3)$ amortized time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on the $O((\log n)^4)$ amortized time algorithm by Holm et al.[STOC98, JACM2001]. Our approach is based on a different and purely combinatorial improvement of the algorithm of Holm et al., which by itself gives a new combinatorial $\tilde{O}((\log n)^3)$ amortized time algorithm. Combining it with Thorup's bittrick, we get down to the claimed $\tilde{O}((\log n)^2)$ amortized time. Essentially the same new trick can be applied to the biconnectivity data structure from [STOC98, JACM2001], improving the amortized update time to $\tilde{O}((\log n)^3)$. We also offer improvements in space. We describe a general trick which applies to both of our new algorithms, and to the old ones, to get down to linear space, where the previous best use $O(m + n\log n\log\log n)$. Finally, we show how to obtain $O(\log n/\log \log n)$ query time, matching the optimal trade-off between update and query time. Our result yields an improved running time for deciding whether a unique perfect matching exists in a static graph., Comment: v1 Submitted to SODA'18 v2 Added note about improvement to biconnectivity v3 Small changes. Drop unproven claim about finding first k bridges in O(k) additional time

We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fully-dynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of each tree in the forest. The forest can be updated by insertion and deletion of edges and by changes to vertex and edge weights. Each update is supported in O(log n) time, where n is the size of the tree(s) involved in the update. Also, we show how to support nearest common ancestor queries and level ancestor queries with respect to arbitrary roots in O(log n) time. Finally, with marked and unmarked vertices, we show how to compute distances to a nearest marked vertex. The later has applications to approximate nearest marked vertex in general graphs, and thereby to static optimization problems over shortest path metrics. Technically speaking, top trees are easily implemented either with Frederickson's topology trees [Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, SIAM J. Comput. 26 (2) pp. 484-538, 1997] or with Sleator and Tarjan's dynamic trees [A Data Structure for Dynamic Trees. J. Comput. Syst. Sc. 26 (3) pp. 362-391, 1983]. However, we claim that the interface is simpler for many applications, and indeed our new bounds are quadratic improvements over previous bounds where they exist., Comment: Preliminary versions of this work presented at ICALP'97 and SWAT'00. The new version takes layered top trees into account