Low-Congestion Shortcuts for Graphs Excluding Dense Minors

08/07/2020 ∙ by Mohsen Ghaffari, et al. ∙ ETH Zurich Carnegie Mellon University 0

We prove that any n-node graph G with diameter D admits shortcuts with congestion O(δ D log n) and dilation O(δ D), where δ is the maximum edge-density of any minor of G. Our proof is simple, elementary, and constructive - featuring a Θ̃(δ D)-round distributed construction algorithm. Our results are tight up to Õ(1) factors and generalize, simplify, unify, and strengthen several prior results. For example, for graphs excluding a fixed minor, i.e., graphs with constant δ, only a Õ(D^2) bound was known based on a very technical proof that relies on the Robertson-Seymour Graph Structure Theorem. A direct consequence of our result is that many graph families, including any minor-excluded ones, have near-optimal Θ̃(D)-round distributed algorithms for many fundamental communication primitives and optimization problems including minimum spanning tree, minimum cut, and shortest-path approximations.

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1 Introduction and Related Work

Low-congestion shortcuts are graph-theoretic objects whose quality captures the distributed complexity of a wide range of fundamental graph problems and communication primitives. In this paper, we provide nearly-tight shortcuts for all graphs excluding (dense) minors. Our results significantly strengthen, simplify, generalize and unify several prior results on shortcuts for restricted graph classes. Our results also directly imply simple near-optimal distributed algorithms for a number of well-studied graph problems in excluded minor graphs and many other graph families.

1.1 Background and Definition

Model

We work with the standard synchronous message passing model of computation on networks and distributed systems. The communication network is abstracted as an -node undirected graph with edges. In each communication round, each node can send an -bit message to each of its neighbors in . At the beginning, nodes do not know the topology of . At the end, each node should know its own part of the output, e.g., which of its edges are in the computed minimum spanning tree. This model is sometimes referred to as the model [Pel00].

The Motivation for Shortcuts

The -round lower bound for global distributed graph problems is well-known by now. Concretely, it is known that on worst-case general graphs, for a strikingly wide-range of global graph problems — including minimum spanning tree, minimum cut, maximum flow, single-source shortest path, etc — any distributed algorithm needs a round complexity of  [DSHK11, Elk04, PR99]. This holds even for any (non-trivial) approximation of these problems, and even on graphs with a small, e.g., logarithmic, diameter. On the other hand, the lower bound graphs are carefully crafted pathological topologies which do not occur in practice. Indeed, many real-world networks have small (polylogarithmic) diameters (e.g., the Facebook graph with billions of nodes has average distance below and a diameter of around ) and seem to allow for exponentially faster optimization algorithms with -round complexities. Low-congestion shortcuts [GH16b] were introduced as a graph-theoretic notion to capture and exploit this phenomenon and allow a more fine-grained study of the complexity of global problems and how this complexity relates to the structure of the network topology. In various graph families algorithms based on low-congestion shortcuts have gone well below the barrier often obtaining near-optimal -round algorithms222One can draw parallels between the role that shortcuts have turned out to play for distributed algorithms of global graph problems with the role that separators play as a key algorithmic tool in sequential algorithms for various graph families, such as planar [LT79, Mil84], bounded genus [GHT84], and minor-excluded [AST90]..

The Definition of Shortcuts

Suppose that the set of vertices is partitioned into disjoint subsets , , …, , known as parts, such that the subgraph induced by each part is connected. We call a collection of subgraphs , , …, a shortcut with congestion and dilation if we have the following two properties: (A) , the diameter of the subgraph is , and (B) each edge is in many subgraphs .

Application of Shortcuts

Shortcuts organically lead to fast distributed algorithms. For instance, let us consider the minimum spanning tree problem. Suppose that we are in a graph family for which, we have a distributed algorithm that can compute a shortcut with congestion and dilation in rounds (upon receiving the partition , , …, ). In general, we refer to as the quality of the shortcut. Then, we can use this algorithm to obtain a distributed algorithm for the minimum spanning tree problem, with a round complexity . This follows directly from Boruvka’s 1926 approach [NMN01]. A number of other graph problems can also be solved distributedly using shortcuts, with a similar complexity. This includes min-cut computation [GH16b], single-source shortest-paths approximation [HL18], and many more [GP17, GP17, DG19, LP19].

Of course, the question that remains is this: What is the existential shortcut quality of important graph families, and what is the corresponding construction time ?

Graph Minors and Minor Density

Before proceeding to known results and our contribution, let us briefly recall the definition of minors and their density.

A graph is a minor of graph if can be obtained from by contracting edges and deleting edges and vertices. Equivalently is a minor of graph if there is a mapping from vertices in to disjoint connected subsets of vertices in , each inducing a connected subgraph, such that for every edge there exists a with and .

An important parameter throughout this paper is the minor density of a graph which is defined as:

It is known that the minor density is (up to a small polylogarithmic factor) the same as the size of the largest complete minor in , i.e., its complete-graph minor size .

Lemma 1.1.

[Tho01] , i.e., .

Note if a graph family is closed under taking minors or equivalently excludes a fixed minor of size then very has and therefore also a constant minor density .

1.2 Our Contribution

We first briefly summarize known results on shortcuts (see Section 1.3 for further details). Ghaffari and Haeupler [GH16b, GH16a] provided shortcuts of quality and construction time for planar graphs. This was later extended to graphs with bounded genus, bounded treewidth, and bounded pathwidth [HIZ16b], and with improved construction algorithms [HIZ16a, HHW18]. Haeupler, Li, and Zuzic [HLZ18] gave shortcuts for excluded minor graphs, with quality , using elaborate arguments building on the Graph Structure Theorem of Robertson and Seymour [RS86, RS03]. While excluded minor graphs are vastly more general, encompassing all previous graph classes for constant bounds on the above graph parameters, the question whether near-optimal shortcuts and optimization algorithms for excluded minor topologies are possible remained open.

Existential Results for Graph with Minor Density

In this work, we resolve this question in the positive. We also significantly strengthen, simplify, generalize, and unify all the above results by giving an near-optimal existential shortcut guarantee for any graph , which depends only on its minor density :

Theorem 1.2.

Any -node graph with diameter and minor density admits shortcuts with dilation and congestion .

Besides the important quadratic quantitative improvement from to a near-optimal for excluded minor graphs, i.e., graphs with constant , our proof is significantly simpler than that of [HLZ18]. Instead of the technical proof in [HLZ18] which uses the powerful Graph Structure Theorem [RS86, RS03], we give a short, elegant, and elementary proof. Our proof even provides small concrete constants333No attempt was made to optimize the explicit constants in the and bounds of our main result Theorem 3.1, or any other explicit constants in this paper. Somewhat better constants are likely possible. whereas the Robertson-Seymour Graph Structure Theorem is known to often lead to tower-type dependencies on the size of the excluded minor [LSZ20] and “galactic algorithms” [LR13].

Even more importantly, our result applies to any graph and graph family even if is large or growing. For example it implies that graphs with sub-polynomial minor density or expansion (see [NDM12] for definitions and treatment of such more inclusive graph families) have sub-polynomial shortcuts and fast -round optimization algorithms, which still drastically improve over the -lower bound. Our results are the first that apply to graph classes strictly more general than minor-closed or excluded-minor graph families.

We complement Theorem 1.2 with a matching lower bound which shows that the linear dependency of Theorem 1.2 on the minor-density is necessary and optimal:

Lemma 1.3.

For any there is a graph with diameter and and a collection of parts for which the quality of the best shortcut in is .

Lastly, having the minor density as a parameter in Theorem 1.2 also directly implies often optimal results on shortcuts for other graph parameters in a simple and uniform way. For example, [GH16b] showed that any genus- network admits shortcuts of quality . In [HIZ16b] this dependency of the shortcut quality on the genus was improved to an optimal bound of via sophisticated topological arguments including cutting a genus- graph along fundamental cycles of a shortest-path tree and cleverly combining planar shortcuts for pieces of parts that are cut by these cycles. We obtain the same result as a trivial corollary of Theorem 1.2, given that for any genus- graph.

Corollary 1.4.

Any genus- graph with nodes and diameter admits shortcuts with congestion and dilation .

Bounds for other graph parameters follow similarly as simple corollaries of our main theorem. This includes tight shortcuts for -pathwidth and -treewidth graphs, matching the results in [HIZ16b] (since for such graphs). In contrast to prior works like [HIZ16b], we do not require a completely different proof specific to the graph parameter at hand to obtain these bounds.

Distributed Construction and Applications

In order to be algorithmically useful we also need fast distributed constructions for the new existential shortcut guarantees. We achieve this by proving strong additional structural guarantees for our shortcuts, namely tree-restrictedness and a small block number. Haeupler, Hershkovitz, Izumi, and Zuzic [HIZ16a, HHW18] showed that these structural guarantees are strong enough for a simple uniform shortcut constructions to find a -quality shortcut in only rounds whenever a quality- shortcut with such structure exists.

Theorem 1.5.

There exists a randomized distributed algorithm which, for any -node -edge graph with diameter and minor density , computes a shortcut of quality in

rounds with high probability. There also exists a

-round deterministic algorithm. Both algorithms use only messages.

This, together with all the algorithms that are built on top of the low-congestion shortcut framework, shows that a wide range of fundamental communication primitives and global graph problems — e.g., minimum spanning tree, min-cut, shortest path approximation, etc — can be solved in rounds, in graphs that do not have a minor of density . This vastly widens the range of graph families for which we now know the correct round complexity for basic global graph problems (up to logarithmic factors). We state just two such corollaries as examples.

Corollary 1.6.

There is a distributed algorithms that, for any -node -edge graph with diameter and minor density , computes a minimum spanning tree in rounds with high probability (or rounds deterministically) using messages.

Corollary 1.7.

There is a randomized distributed algorithms that, for any -node -edge graph with diameter and minor density , computes an exact minimum cut in in rounds with high probability using messages.

Similar results can be obtained for for sub-graph connectivity, single-source shortest paths approximations [HL18], as well as several fundamental communication primitives like multiple unicasts or partwise-aggregation (see Section 2).

1.3 Prior Work on Shortcuts

We next overview the known results about the existential quality and construction time of shortcuts.

Let us start with general graphs. It is easy to see that any -node graph , whose diameter is at most , admits shortcuts of quality : Let be a BFS of the graph . Define for any each part with and for any other part. Moreover, this bound is nearly-optimal [GH16b] in general graphs. This bound is implicitly the underlying reason for the seminal round minimum spanning tree algorithm of Kutten and Peleg [KP95, GKP93], although there are more fine-grained aspects there to avoid extra logarithmic factors. Moreover, this round complexity is nearly optimal for solving MST in general graphs, as mentioned before, because of the carefully crafted lower bound graphs of [DSHK11, Elk04, PR99].

Ghaffari and Haeupler [GH16b] showed that any planar graph with diameter at most admits shortcuts of quality , and that this bound is nearly tight, almost matching a lower bound of . The also provided a distributed algorithm for constructing such shortcuts in rounds, assuming a planar embedding of the graph. Combined with the distributed planar embedding algorithm provided by  [GH16a], this led to an round distributed algorithm for MST, and some other graph problems, thus exhibiting a family of graphs in which one can go below the notorious lower bound [DSHK11]. They [GH16b] also showed that graphs embeddable on a surface of genus admit shortcuts of quality .

Haeupler, Izumi, and Zuzic [HIZ16a] showed that one can construct shortcuts with a quality similar to those of [GH16b], even without a planar embedding. Haeupler, Izumi, and Zuzic [HIZ16b] showed that graphs of treewidth or pathwidth admit shortcuts of quality , and they also improved the genus dependency to . Haeupler, Hershkowitz, and Wajc [HHW18] improved and extended the results of  [HIZ16a] by showing that one can obtain algorithms that are near-optimal in both time and message complexity, using shortcuts.

Generalizing the span of shortcuts much further, Haeupler, Li, and Zuzic [HLZ18] gave shortcuts for excluded minor graphs. Concretely, they showed any graph that does not have a clique of constant size as a minor admits a shortcut of quality . This result is fairly involved and it builds on the Graph Structure Theorem of Robertson and Seymour [RS86, RS03]. Using the approach of[HIZ16a, HHW18], all these shortcuts mentioned above[GH16b, HIZ16a, HIZ16b, HLZ18] can be constructed in a round complexity matching their quality, up to logarithmic factors.

Ghaffari, Kuhn, and Su [GKS17] provided shortcut constructions for well-connected graphs. In particular, they showed that any graph where the lazy random walk has mixing time admits shortcuts of quality and they presented distributed algorithms for constructing shortcuts of quality in rounds. These quality and construction time bounds were both improved later to , by Ghaffari and Li [GL18].

Finally, Kitamura et al.[KKOI19] showed that any -chordal graph admits shortcuts of quality , which can be determined in even rounds, and that graph of diameter and admit shortcuts of quality and construction time and respectively. These essentially match the lower bounds for Lotker, Patt-Shamir, and Peleg [LPSP06].

2 Preliminaries: Definitions of Shortcuts

In many distributed algorithms for global graph problems, the problem boils down to the following natural part-wise aggregation task. For instance, this exactly captures the problem of identifying minimum-weight outgoing edges in Boruvka’s classic approach to MST [NMN01].

Definition 2.1.

(The Part-wise Aggregation Problem) Consider a network graph and suppose that the vertices are partitioned into disjoint parts , , …, such that the subgraph induced by each part connected. In the part-wise aggregation problem, the input is a value for each node . The output is that each node should learn an aggregate function of the values held by vertices in its part , e.g., , , or . Alternatively, exactly one node in each part has a message and it should be delivered to all nodes of the part.

Typically, if we can solve the part-wise aggregation problem in a network in time, we can obtain algorithms for various fundamental graph problems with a round complexity of . Thus, we want fast algorithms for part-wise aggregation.

They point to emphasize in the part-wise aggregation problem is that, it is possible that the diameter of the subgraph induced by each part is quite large, much larger than the diameter of the base graph . For instance, the former can be up to while the latter is just (considering a wheel graph with one part being all nodes except the center). Hence, to obtain fast algorithms for part-wise aggregation, we would like to allow some parts to use edges of which are outside the part. Of course, we should limit the number of parts that try to use each single given edge, as that would cause congestion and would slow down the solution. This naturally brings us to the concept of low-congestion shortcuts, as defined in [GH16b].

Definition 2.2.

(Shortcuts) Given a part-wise aggregation problem — i.e., a graph where vertices are partitioned in disjoint parts , , …, , each of which induces a connected subgraph — we call a collection of subgraph , , …, a shortcut with congestion and dilation if we have the following two properties: (I) for each , the diameter of the subgraph is at most , and (II) each edge is in at most many of the subgraphs . We refer to as the quality of the shortcut.

Given a -congestion -dilation shortcut for a part-wise aggregation problem, we can solve the part-wise aggregation problem in rounds, using the random delays technique [LMR94, Gha15, HHW19]. This makes the shortcut quality the dominant parameter which determines the round complexity of shortcut-based algorithms (modulo the shortcut’s construction time).

Tree-Restricted Shortcuts, and their Block-Number

In many graph families [GH16b, HIZ16b, HLZ18], shortcuts can be chosen to come from one low-depth tree of the original graph, which provides a particularly clean and simple structure which can be utilized for efficient shortcut constructions. In particular, one can fix any rooted breadth first search tree — or any other low-depth spanning tree— and restrict each to only include edges from . Note that only edges with descendants in are useful for short-cutting . A particular simple way to construct is to specify a set of edges in and define to be all edges with descendants in the forest . We say such an has block number at most since the graph has at most connected components (of diameter ). More generally we define tree-restricted shortcuts and their block number as follows:

Definition 2.3.

(Tree-Restricted Shortcuts and Block Number) Consider a part-wise aggregation setup in graph with diameter , parts , …, , a rooted tree of with depth . We say the shortcut is tree-restricted or -restricted if all its edges are in , i.e., if . Moreover, for any part , we call the connected components of the graph the blocks of . The block number of a shortcut is the maximum block number of any part.

If in a topology has shortcuts for any collection of parts and any choice of the tree then we say it admits good shortcuts. We also introduce the concept of partial shortcuts which lead to slightly tighter bounds and simpler proofs:

Definition 2.4.

(Admitting Shortcuts) We say a topology with diameter admits tree-restricted -congestion -block shortcuts, if for any tree with depth at most and for any collection of node-disjoint connected parts , there exists a -restricted -congestion -block shortcut.

Definition 2.5.

(Admitting Partial Shortcuts) We say a topology with diameter admits tree-restricted -congestion -block partial shortcuts, if for any tree with depth at most and for any collection of node-disjoint connected parts , there are at least of the parts with a -restricted -congestion -block shortcut.

It is easy to see that a small block number directly implies a small dilation and that admitting partial shortcuts is essentially the same as admitting shortcuts – up to a factor in the congestion.

Observation 2.6.

Any -block -restricted shortcut in a graph with diameter has dilation at most

Proof.

For each part , the graph is a forest with at most connected components, each of them has a diameter of at most the twice the depth of which is . These components are connected via edges of , because induces a connected subgraph. Hence, the diameter of , i.e., the dilation for this part, is at most . ∎

Observation 2.7.

Any -node graph that admits tree-restricted -congestion -block partial shortcuts also admits tree-restricted -congestion -block shortcuts.

Proof.

For a collection of parts and a tree consider iterations in which one takes a -restricted -congestion -block partial shortcut for any still remaining parts. Given that such a partial shortcut defines sets of shortcut edges for at least half of the remaining parts, all parts will have a set of shortcut edges from in the end. Taking this as a full -restricted shortcut might lead to a -congestion but leaves the block number of for every part unaffected. ∎

Efficient Shortcut Constructions for Tree-restricted Shortcuts

The main reason for Haeupler, Izumi and Zuzic [HIZ16a] to introduce the concept of -block tree-restricted shortcuts is that this additional structure can be used to obtain a very simple and efficient shortcut construction. This construction was further improved by Haeupler, Hershkowitz, and Wajc [HHW18] to give a slightly faster, deterministic, and message optimal construction:

Lemma 2.8.

[[HHW18, HIZ16a]] There exists a simple distributed algorithm which, for any -node -edge -diameter graph which admits -congestion -block partial shortcuts with quality , computes a quality- shortcut for any given collection of parts in rounds with high probability using message (or deterministically in rounds).

By proving that the shortcuts from Theorem 1.2 can be chosen to be tree-restricted with a small block number we get an efficient construction algorithm, and therefore Theorem 1.5, “for free”. We remark that proving this additional tree-restriction structure and thus having a fast construction algorithm is crucial for the algorithmic usability of shortcuts and generally quite hard. Indeed, efficiently constructing good general shortcuts for all graph families that admit them remains a major open problem. Even in the special case of well-connected graphs (with a small mixing time, e.g., expander or random graphs), for which shortcuts generally cannot be chosen to be tree-restricted, there is currently a gap between the construction time and the shortcut quality[GL18, GKS17].

3 Shortcuts for Graphs with Minor Density

3.1 Main Result

We prove the following main result, which directly implies Theorem 1.2 and Theorem 1.5.

Figure 3.1: A schematic illustration of a tree (indicated in blue), the one part that we are focusing on (indicated as a gray area with white nodes in it) along with a few others (indicated just as gray areas), and the overcongested edges (indicated in red) that have descendants in this part, which we call representatives (indicates with red cross marks). When the blue tree goes through a part (a gray area), we indicate that this section of the tree has some vertex of that part.
Theorem 3.1.

Every with diameter and minor density admits tree-restricted -congestion -block partial shortcuts.

Indeed, using creftypecap 2.7 and creftypecap 2.6 the existence of a -congestion -block partial shortcut directly implies the existence of an -congestion -dilation shortcut and therefore Theorem 1.2. The constructive main theorem Theorem 1.5 directly follows from using Lemma 2.8 on the tree-restricted shortcuts of Theorem 3.1.

The general idea to prove Theorem 3.1 is to ”run” the shortcut construction algorithm from [HIZ16a] and prove that if it fails to find a sufficiently good tree-restricted shortcut, then contains a minor with density exceeding .

Proof of Theorem 3.1.

Let be any rooted spanning tree in of depth at most . Let be a collection of connected node-disjoint parts. We set our desired congestion to be . For a tree edge , let be the endpoint of that is further away from the root.

Defining overcongested edges

Initially, let . We process tree edges in order of decreasing depths, level by level. For any edge , let be the parts that have a non-empty intersection with the descendants of in . If we say is overcongested and we add to .

The bipartite graph

We define the bipartite graph whose node set consists of edge-nodes corresponding to overcongested edges on the one side and part-nodes corresponding to the parts from on the other side. The edges of indicate which part contributed to which edge being overcongested. We associate every edge with some representative node , that is a descendant of and can be reached from via . A schematic illustration of this setup is given in Figure 3.1. Let be the set of representative nodes in . Note that is equal to the degree of node . The degree of any edge-node is , as we only have overcongested edges represented in .

Next, we argue that that one of the following two cases applies: either (I) there exists a good partial shortcut, or (II) graph has a minor of density exceeding – contradicting the assumption that .

(I) Either there exists a good partial shortcut

If at least half of all parts have a degree of at most in , then defining for any such part to be all ancestor edges of in the forest identifies a -congestion -block partial shortcut.

(II) Or there is a dense minor in

If we are not in case (I), then at least half of all parts in have degree at least in . In this case, the average degree among part-nodes in is at least , because at least half of the parts have degree at least . Moreover, the average degree among edge-nodes (and in fact even their minimum degree) in is at least .

Let be a random subset of in which each part is included independently at random with probability . We define a subgraph of the bipartite graph which is also a minor of , as follows. The part-nodes in are exactly the and in the minor-mapping such a node is mapped to the vertex set of . All edges with are edge-nodes in . The vertex set in of such an edge-node is exactly the vertices in the connected component containing in the forest .

To define which edges are in , we say is potentially present if the tree path between and the representative , including the deeper endpoint but excluding the representative node , does not contain any node from . We say edge is actually present and add to if it is both potentially present and . Note that is indeed a minor of under the mapping function since the vertex sets corresponding to nodes in are disjoint and connected in and edges in are a subset of the edges produced when contracting these vertex sets in .

Density of the graph

Let be the number of edges in . Every edge has a probability of at least to be potentially present. This probability is independent from the probability for to be in . Hence, .

The nodes in on the other hand consist of (A) at most some edge-nodes, given that they have degree at least in , each of which is included in with probability and (B) at most some part-nodes, with average degree more than in , each of which is included in with probability . Hence,

Therefore, by linearity of expectation, we can conclude that

which implies that . That is, with a non-zero probability444With a slightly more careful argument, we can show that there is probability to find a minor of density exceeding , but for our existence proof, just a positive probability suffices., the minor in has density exceeding , giving the desired contradiction. ∎

We remark that the above proof of Theorem 3.1 can easily be made constructive directly. A trivial implementation would lead to an deterministic -round algorithm. Using the sampling idea from [HIZ16a, HHW18] to identify overcongested-edges one can speed this up to . Overall, this would safe a factor in the quality of the computed (partial) shortcut. One could also make the algorithm certifying, i.e, output a dense minor if a (partial) shortcut of desired quality cannot be found. For example, one can obtain an algorithm which when run on a graph with tree of depth and a collection of parts terminates in rounds for some and outputs both an -block -congestion partial shortcut and a -dense bipartite minor, which explains/certifies why no better shortcut was found.

3.2 Optimality of the Main Result

Next we prove that our main result Theorem 3.1 is existentially optimal up to only small explicitly given constant factors in the congestion and block number of partial shortcuts. This directly implies the slightly weaker Lemma 1.3 for full shortcuts presented in Section 1.2.

Lemma 3.2.

For every with there exists a topology (with nodes) and a set of node-disjoint paths such that:

  • has diameter and every minor of has density less than and

  • the best partial shortcut quality for the set of paths in is at least , i.e., any partial shortcut has either congestion at least or dilation at least .

Figure 3.2: The lower bound graph for Lemma 3.2 with , and . The special path of length at most is at the top, and below are the paths/parts of length . Every steps on these parts there is a column connecting all paths and (green) connections every steps on this column to the special path on top.
Proof.

Let , , . Note that , , . The topology is made of one special path of length at the top, along with many paths of length at the bottom, known as rows. In every column, every row is connected to a node in the top path, such that all these nodes of the same column are connected to one node in the top node. See Figure 3.2 for an illustration. More formally, the graph is defined as follows:

  • The edges are such that the -nodes form a path of length . Moreover, for any , the vertices -nodes form a path of length . Also, for any , the nodes form a path of length of which every node connects to , that is, for every .

We first argue that the graph has diameter at most . From every node one can reach a -node by going at most steps to the closest node in its -path which is in the same column as of a -node, then going at most steps up or down to a node that has a -node neighbor, and then doing one more step to that -node. From any -node one can reach in at most steps. Overall the diameter of is therefore at most .

Now we argue that has no minor with density . Notice that is planar after deleting the edges between the nodes and any -path except . Any minor of with nodes therefore has, according to Euler’s formula, at most edges and thus an edge density of at most .

Finally, we argue about the shortcut quality. We have paths , each of length . For each path , the only way to shorten the distance between the two endpoints is to use the edges of the top-path. In fact, unless a part has at least of the edges of the top path in its shortcut , the dilation of its part, i.e., the diameter of would be at least . Therefore, if the shortcut quality is smaller than , each part needs to have at least half of the edges of the top path in its shortcut. But then, overall, the edges of the top path appear at least times, in total, in shortcuts. Hence, at least one edge has congestion at least . Lastly . ∎

We note that there is a factor gap in the congestion (but not the dilation) between the upper bound for full shortcuts in Theorem 1.2 and Lemma 1.3. This gap stems from the loss of creftypecap 2.7, which goes from partial shortcuts to (full) shortcuts. A planar lower bound topology with shortcuts given in [GH16b] shows that the congestion-gap between partial shortcuts and full shortcuts can be at least . Whether this is the maximal gap between partial and full shortcuts and whether Theorem 1.2 can be improved to shortcuts of quality independent of , e.g., , are interesting questions – albeit not ones of particular importance to the algorithmic applications of shortcuts.

3.3 Shortcuts for other Graph Parameters and Algorithmic Applications

We finish by giving the few remaining technical details for the direct implications of our existential and algorithmic shortcut guarantees from Theorem 3.1 and Theorem 1.5, which are stated in Section 1.2.

As discussed in Section 1.2, our shortcut guarantees in terms of the minor density directly imply bounds for other graph parameters of interest. Consider, for example, the following easy and existentially tight bounds on the minor density in terms of graph parameters considered in [HIZ16b]:

Lemma 3.3.

The following bounds on the minor density of a graph hold:

  • If has genus, non-orientable genus, or Euler genus of , then .

  • If has treewidth or pathwidth at most , then .

Proof.

Being embeddable into an orientable or non-orientable surface is a graph property that is closed under minor-operations. Moreover, An -node graph having a genus, non-orientable genus, or Euler genus of implies that this graph has at most edges. Since any graph with density has at least nodes and thus at least edges, the (minor) density of any such graph is at most . Similarly, having treewidth or pathwidth at most are graph properties closed under taking minors. Moreover, a graph of treewidth (or pathwidth) at most and nodes has less than edges and therefore its (minor) density is at most . ∎

Combining Theorem 3.1 with Lemma 3.3 now directly implies Corollary 1.4 and the following analogous corollary for treewidth- graphs:

Corollary 3.4.

Any graph with nodes, diameter , and treewidth at most admits shortcuts with congestion and dilation .

Note that while completely different and highly nontrivial proofs specific to planar graphs, bounded genus graphs, and bounded treewidth graphs were given in [GH16b, HIZ16b], we obtain the same existentially optimal results by simply plugging in bounds on the minor density in terms of the desired graph parameter into Theorem 3.1. Constructive results of these shortcuts follow similarly from Theorem 1.5.

Algorithmic applications, such as, the fast distributed MST and minimum-cut algorithms claimed in Corollary 1.6 and Corollary 1.7 follow immediately and in a completely modular fashion from our new constructive shortcuts given in Theorem 1.5 and shortcut-based algorithms like the ones given in [GH16b, HHW18].

Proof of Corollary 1.6 and Corollary 1.7.

The min-cut algorithm follows from a randomized algorithm given in [GH16b] that computes a approximation in time, with high probability, given a shortcut of quality which is can be constructed according to Theorem 1.5. To convert this into an exact algorithm we observe that the minimum degree of and therefore also its min-cut is of size at most given that the density of can be at most . Setting therefore implies an exact algorithm. The MST statement follows directly from Theorem 1.5 and Boruvka’s distributed MST algorithm, as described in [GH16b, HHW18]. ∎

Acknowledgements

The second author thanks Anupam Gupta for suggesting the question that led to this paper.

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