Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels

Abstract

International audienceThis paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbidden-set labeling schemes for pla- nar graphs. For a given n-vertex planar graph G with edge weights drawn from [1,M] and parameter ε > 0, our forbidden-set labeling scheme uses labels of length λ = O(ε−1 log2 n log (nM ) * (ε−1 + log n)). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1 + ε), in O(|F|2λ) time. We then present a general method to transform (1 + ε) forbidden-set labeling schemas into a fully dynamic (1 + ε) distance oracle. Our fully dynamic (1 + ε) distance oracle is of size O(n log n * (ε−1 + log n)) and has O ̃(n1/2) query and update time, both the query and the update time are worst case. This improves on the best previously known (1 + ε) dynamic distance oracle for planar graphs, which has worst case query time O ̃(n2/3) and amortized update time of O ̃(n2/3). Our (1 + ε) forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch (1 + ε)

Near-Optimal Compression for the Planar Graph Metric

“…The root of K is also marked. Then, for every marked node v ∈ K, the slice of v is the subgraph of G enclosed by C v and not strictly enclosed 4 by C u for any marked descendent u of v. The embedding of slices is inherited from the embedding of G. Thus, the boundary of the infinite face of the slice s of v is C v . The cycle C v is also called the boundary of the slice s. Each cycle C u corresponding to a marked descendant u of v such that there are no other marked nodes on the v-to-u path becomes a face in the slice s. Such a face is called a hole of s, and C u is called the boundary of the hole.…”

“…Adding these edges can be done consistently with the embedding of s because the path in T F V can be embedded on the same plane as s such that s and T F V only intersect at vertices of s. See Figure 2. We introduce an artificial vertex v s embedded in the infinite face of s and triangulate the infinite face of s by adding edges between v s and every vertex of the 4 A vertex or an edge x is enclosed by a cycle C if x is incident to a face enclosed by C. If x is enclosed by C but x ∈ C then x is said to be strictly enclosed by C. Proof. By construction of s , for every grandparent to grandchild path in T F V there is a corresponding edge in s .…”

“…Finding so-called distance labeling schemes in which the labels are as short as possible is a classical subject of study [38,45,50,67]. Such labels are used for efficient algorithms both in theory [4,72] and practice [30]. A famous open question is to close the rare polynomial gap in the bounds for planar graphs that has been embarrassingly open since the work of Gavoile et al [38]: the upper bound is O(n 1/2 ) bits per label (due to [42] who shaved a log factor over [38]), and the lower bound is Ω(n 1/3 ).…”

Decrementai Transitive Closure and Shortest Paths for Planar Digraphs and Beyond

“…Even though the decremental setting may seem very restricted, decremental algorithms are important both theoretically and from the point of view of their applications. Many of the state-of-the-art fully-dynamic algorithms use their decremental counterparts in a black-box manner (e.g., [1,3,30]). Decremental graph algorithms are often used in the role of a data structure to design efficient static algorithms (e.g., [10,15,26]).…”

“…This bound was achieved with so-called Even-Shiloach trees [8], and it stood for over 30 years. Only recently, Henzinger et al [12] presented a randomized decremental single-source reachability algorithm with total update time O(mn 0.984+o (1) ), which they later improved to O(mn 0.9+o (1) ) [13]. Very recently, Chechik et al [5] improved the total update time to O(m √ n ) 1 .…”