We present a novel space-efficient graph coarsening technique for n -vertex planar graphs G , called cloud partition , which partitions the vertices V ( G ) into disjoint sets C of size O (log n ) such that each C induces a connected subgraph of G . Using this partition P we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O ( n /log n ) vertices. The combination of ( F , P ) is referred to as a cloud decomposition . For planar graphs we show that a cloud decomposition can be constructed in O ( n ) time and using O ( n ) bits. Given a cloud decomposition ( F , P ) constructed for a planar graph G we are able to find a balanced separator of G in O ( n /log n ) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to H -minor-free graphs for any fixed graph H . This allows us to construct the succinct encoding scheme for H -minor-free graphs due to Blelloch and Farzan (CPM 2010) in O ( n ) time and O ( n ) bits improving both runtime and space by a factor of Θ(log n ). As an additional application of our cloud decomposition we show that, for H -minor-free graphs, a tree decomposition of width O ( n 1 / 2 + ϵ ) for any ϵ > 0 can be constructed in O ( n ) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O ( k n log n ) for graphs of treewidth k ≤ n in sublinear space and polynomial time.

We call a graph G separable if a balanced separator can be computed for G of size O(n^ε) with ε < 1. Many real-world graphs are separable such as graphs of bounded genus, graphs of constant treewidth, and graphs excluding a fixed minor. In particular, the well-known planar graphs are separable. We present a succinct encoding of separable graphs G such that, after the encoding is computed, any number of depth-first searches (DFS) can be performed from any given start vertex, each in o(n) time and o(n) bits in the word RAM model. After the execution of a DFS, the succinct encoding of G is augmented such that the DFS tree is encoded inside the encoding while maintaining succinctness. Afterward, the encoding provides common DFS-related queries in constant time. These queries include queries such as lowest-common ancestor of two given vertices in the DFS tree or queries that output the lowpoint of a given vertex in the DFS tree. Furthermore, for planar graphs, we show that the succinct encoding can be computed in O(n) bits and expected linear time, and a compact variant can be constructed in O(n) time and bits. For other separable graph classes 𝒢 the runtime and space usage depends on the specific algorithms used to find balanced separators in graphs of 𝒢.

Temporal graphs are graphs where the edge set can change in each time step, and the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been widely studied. We extend the concept of graph automorphisms from static graphs to temporal graphs and show that symmetries enable faster exploration: We prove that a connected temporal graph with $n$ vertices and orbit number $r$ (i.e., $r$ is the number of automorphism orbits) can be explored in $O(r n^{1+ε})$ time steps, for any fixed $ε>0$. For $r=O(n^c)$ for constant $c<1$, this is a significant improvement over the known tight worst-case bound of $Θ(n^2)$ time steps for arbitrary connected temporal graphs. We also give two lower bounds for exploration, showing that $Ω(n \log n)$ time steps are required for some inputs with $r=O(1)$ and that $Ω(rn)$ time steps are required for some inputs for any $r$ with $1\le r\le n$. The techniques we develop for fast exploration are used to derive the following result for rendezvous in connected temporal graphs: Two agents are placed by an adversary at arbitrary vertices and given full information about the temporal graph, except that they do not have consistent vertex labels. The agents can meet at a common vertex after $O(n^{1+ε})$ time steps, for any $ε>0$. For some connected temporal graphs with constant orbit number we present a complementary lower bound of $Ω(n\log n)$ time steps. Finally, we give a randomized algorithm to construct a temporal walk $W$ that visits all vertices of a given orbit with probability at least $1-ε$ for any $0<ε<1$ such that $W$ spans $O((n^{5/3}+rn)\log n)$ time steps. The runtime of this algorithm consists of $O(n^{1/3} \log (n/ε))$ linear-time scans of the snapshots that exist in this time span.

Let $G$ be an unlabeled planar and simple $n$-vertex graph. Unlabeled graphs are graphs where the label-information is either not given or lost during the construction of data-structures. We present a succinct encoding of $G$ that provides induced-minor operations, i.e., edge contractions and vertex deletions. Any sequence of such operations is processed in $O(n)$ time in the word-RAM model. At all times the encoding provides constant time (per element output) neighborhood access and degree queries. Optional hash tables extend the encoding with constant expected time adjacency queries and edge-deletion (thus, all minor operations are supported) such that any number of edge deletions are computed in $O(n)$ expected time. Constructing the encoding requires $O(n)$ bits and $O(n)$ time. The encoding requires $\mathcal{H}(n) + o(n)$ bits of space with $\mathcal{H}(n)$ being the entropy of encoding a planar graph with $n$ vertices. Our data structure is based on the recent result of Holm et al. [ESA 2017] who presented a linear time contraction data structure that allows to maintain parallel edges and works for labeled graphs, but uses $Θ(n \log n)$ bits of space. We combine the techniques used by Holm et al. with novel ideas and the succinct encoding of Blelloch and Farzan [CPM 2010] for arbitrary separable graphs. Our result partially answers the question raised by Blelloch and Farzan whether their encoding can be modified to allow modifications of the graph. As a simple application of our encoding, we present a linear time outerplanarity testing algorithm that uses $O(n)$ bits of space.

We present a novel space-efficient graph coarsening technique for $n$-vertex planar graphs $G$, called cloud partition, which partitions the vertices $V(G)$ into disjoint sets $C$ of size $O(\log n)$ such that each $C$ induces a connected subgraph of $G$. Using this partition $P$ we construct a so-called structure-maintaining minor $F$ of $G$ via specific contractions within the disjoint sets such that $F$ has $O(n/\log n)$ vertices. The combination of $(F, P)$ is referred to as a cloud decomposition. For planar graphs we show that a cloud decomposition can be constructed in $O(n)$ time and using $O(n)$ bits. Given a cloud decomposition $(F, P)$ constructed for a planar graph $G$ we are able to find a balanced separator of $G$ in $O(n/\log n)$ time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to $H$-minor-free graphs for any fixed graph $H$. This allows us to construct the succinct encoding scheme for $H$-minor-free graphs due to Blelloch and Farzan (CPM 2010) in $O(n)$ time and $O(n)$ bits improving both runtime and space by a factor of $Θ(\log n)$. As an additional application of our cloud decomposition we show that, for $H$-minor-free graphs, a tree decomposition of width $O(n^{1/2 + ε})$ for any $ε> 0$ can be constructed in $O(n)$ bits and a time linear in the size of the tree decomposition. Finally, we implemented our cloud decomposition algorithm and experimentally verified its practical effectiveness on both randomly generated graphs and real-world graphs such as road networks. The obtained data shows that a simplified version of our algorithms suffices in a practical setting, as many of the theoretical worst-case scenarios are not present in the graphs we encountered.

Abstract For n-vertex graphs with treewidth $$k = O(n^{1/2-\epsilon })$$ k = O ( n 1 / 2 - ϵ ) and an arbitrary $$\epsilon >0$$ ϵ > 0 , we present a word-RAM algorithm to compute vertex separators using only O(n) bits of working memory. As an application of our algorithm, we give an O(1)-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in $$c^k n (\log \log n) \log ^* n$$ c k n ( log log n ) log ∗ n time using O(n) bits for some constant $$c > 0$$ c > 0 . Together with the result of Banerjee et al. (Proceedings of 21st international conference on computing and combinatorics (COCOON 2015). LNCS, vol 9198, Springer, pp 349–360, 2015. https://doi.org/10.1007/978-3-319-21398-9_28 ) we are able to compute a solution for all monadic-second-order problems (MSO) with $$O(n + \tau (k) \cdot p (\log _{p} n) \log n)$$ O ( n + τ ( k ) · p ( log p n ) log n ) bits in $$O(\tau (k) \cdot n^{2 + (2/\log p)})$$ O ( τ ( k ) · n 2 + ( 2 / log p ) ) time where k is the treewidth of the given graph, p is some arbitrary parameter with $$2 \le p \le n$$ 2 ≤ p ≤ n and $$\tau $$ τ is some function depending on the MSO formula. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover, Independent Set, Dominating Set, MaxCut and q-Coloring by using polynomial time and O(n) bits as long as the treewidth of the graph is smaller than $$c' \log n$$ c ′ log n for some problem dependent constant $$0< c' < 1$$ 0 < c ′ < 1 .

A temporal graph is a graph whose edge set can change over time. We only require that the edge set in each time step forms a connected graph. The temporal exploration problem asks for a temporal walk that starts at a given vertex, moves over at most one edge in each time step, visits all vertices, and reaches the last unvisited vertex as early as possible. We show in this paper that every temporal graph with n vertices can be explored in O(n^{1.75}) time steps provided that either the degree of the graph is bounded in each step or the temporal walk is allowed to make two moves per step. This result is interesting because it breaks the lower bound of Omega(n^2) steps that holds for the worst-case exploration time if only one move per time step is allowed and the graph in each step can have arbitrary degree. We complement this main result by a logarithmic inapproximability result and a proof that for sparse temporal graphs (i.e., temporal graphs with O(n) edges in the underlying graph) making O(1) moves per time step can improve the worst-case exploration time at most by a constant factor.

We present space-efficient algorithms for computing cut vertices in a given graph with n vertices and m edges in linear time using $$O(n+\min \{m,n\log \log n\})$$ bits. With the same time and using $$O(n+m)$$ bits, we can compute the biconnected components of a graph. We use this result to show an algorithm for the recognition of (maximal) outerplanar graphs in $$O(n\log \log n)$$ time using O(n) bits.

A c-color choice dictionary of size n in N is a fundamental data structure in the development of space-efficient algorithms that stores the colors of n elements and that supports operations to get and change the color of an element as well as an operation choice that returns an arbitrary element of that color. For an integer f>0 and a constant c=2^f, we present a word-RAM algorithm for a c-color choice dictionary of size n that supports all operations above in constant time and uses only nf+1 bits, which is optimal if all operations have to run in o(n/w) time where w is the word size. In addition, we extend our choice dictionary by an operation union without using more space.

Many succinct data structures on the word RAM require precomputed tables to start operating. Usually, the tables can be constructed in sublinear time. In this time, most of a data structure is not initialized, i.e., there is plenty of unused space allocated for the data structure. We present a general framework to store temporarily extra buffers between the real data so that the data can be processed immediately, stored first in the buffers, and then moved into the real data structure after finishing the tables. As an application, we apply our framework to Dodis, Patrascu, and Thorup's data structure (STOC 2010) that emulates c-ary memory and to Farzan and Munro's succinct encoding of arbitrary graphs (TCS 2013). We also use our framework to present an in-place dynamical initializable array.

The choice dictionary is introduced as a data structure that can be initialized with a parameter $n\in\mathbb{N}=\{1,2,\ldots\}$ and subsequently maintains an initially empty subset $S$ of $\{1,\ldots,n\}$ under insertion, deletion, membership queries and an operation choice that returns an arbitrary element of $S$. The choice dictionary appears to be fundamental in space-efficient computing. We show that there is a choice dictionary that can be initialized with $n$ and an additional parameter $t\in\mathbb{N}$ and subsequently occupies $n+O(n(t/w)^t+\log n)$ bits of memory and executes each of the four operations insert, delete, contains (i.e., a membership query) and choice in $O(t)$ time on a word RAM with a word length of $w=Ω(\log n)$ bits. In particular, with $w=Θ(\log n)$, we can support insert, delete, contains and choice in constant time using $n+O(n/(\log n)^t)$ bits for arbitrary fixed $t$. We extend our results to maintaining several pairwise disjoint subsets of $\{1,\ldots,n\}$. We study additional space-efficient data structures for subsets $S$ of $\{1,\ldots,n\}$, including one that supports only insertion and an operation extract-choice that returns and deletes an arbitrary element of $S$. All our main data structures can be initialized in constant time and support efficient iteration over the set $S$, and we can allow changes to $S$ while an iteration over $S$ is in progress. We use these abilities crucially in designing the most space-efficient algorithms known for solving a number of graph and other combinatorial problems in linear time. In particular, given an undirected graph $G$ with $n$ vertices and $m$ edges, we can output a spanning forest of $G$ in $O(n+m)$ time with at most $(1+ε)n$ bits of working memory for arbitrary fixed $ε>0$.

We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of n items and that the available workspace is Theta(s) bits, where lg n <= s <= n * lg n. Three techniques that can be used as general tools in different space-efficient algorithms are introduced and employed within our algorithms. In particular, we give an almost-optimal algorithm for finding the closest pair among a set of n points that runs in O(n^2 /s + n * lg s) time. We also give a simple algorithm to enumerate the intersections of n line segments that runs in O((n^2 /s^{2/3}) * lg s + k) time, where k is the number of intersections. The counting version can be solved in O((n^2/s^{2/3}) * lg s) time. When the segments are axis-parallel, we give an O((n^2/s) * lg^{4/3} s + n^{4/3} * lg^{1/3} n)-time algorithm that counts the intersections and an O((n^2/s) * lg s * lg lg s + n * lg s + k)-time algorithm that enumerates the intersections, where k is the number of intersections. We finally present an algorithm that runs in O((n^2 /s + n * lg s) * sqrt{(n/s) * lg n}) time to calculate Klee's measure of axis-parallel rectangles.

We reconsider basic algorithmic graph problems in a setting where an n-vertex input graph is read-only and the computation must take place in a working memory of O(n) bits or little more than that. For computing connected components and performing breadth-first search, we match the running times of standard algorithms that have no memory restrictions, for depth-first search and related problems we come within a factor of \Theta(\log\log n), and for computing minimum spanning forests and single-source shortest-paths trees we come close for sparse input graphs.

Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time. We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk3 log k). The previous best algorithm with a running time subexponential in k was the algorithm of Gu and Tamaki [12] with a running time of O(n1+ε log n) and an approximation ratio 1.5 + 1/∊ for any ∊ > 0. The running time of our algorithm is also better than the running time of O(f(k) · n log n) of Reed's algorithm [18] for general graphs, where f is a function exponential in k.

We study the problem of maximising the lifetime of a sensor network for fault-tolerant target coverage in a setting with composite events. Here, a composite event is the simultaneous occurrence of a combination of atomic events, such as the detection of smoke and high temperature. We are given sensor nodes that have an initial battery level and can monitor certain event types, and a set of points at which composite events need to be detected. The points and sensor nodes are located in the Euclidean plane, and all nodes have the same sensing radius. The goal is to compute a longest activity schedule with the property that at any point in time, each event point is monitored by at least two active sensor nodes. We present a (6+ε)-approximation algorithm for this problem by devising an approximation algorithm with the same ratio for the dual problem of minimising the weight of a fault-tolerant sensor cover and applying the Garg-Könemann algorithm. Our algorithm for the minimum-weight fault-tolerant sensor cover problem generalises previous approximation algorithms for geometric set cover with weighted unit disks and is obtained by enumerating properties of the optimal solution that guide a dynamic programming approach.