The restoration lemma by Afek et al. [3] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical shortest path for each node pair, it is no longer guaranteed that one can build replacement paths by concatenating two selected shortest paths. They left as an open problem whether a method of shortest path tiebreaking with this desirable property is generally possible.
We settle this question affirmatively with the first general construction of restorable tiebreaking schemes. We then show applications to various problems in fault-tolerant network design. These include a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact) distance labeling schemes, fault-tolerant subset distance preservers and + 4 additive spanners with improved sparsity, and fast distributed algorithms that construct these objects. For example, an almost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributed construction of sparse fault-tolerant distance preservers resilient to three faults.
Appendices
A Impossibility of Symmetry and Restorability
As observed by Afek et al. [3], one cannot generally have restorability and symmetry at the same time:
Theorem A.1.
There are input graphs that do not admit a tiebreaking scheme that is simultaneously symmetric and 1-restorable.
Proof.
The simplest example is a \(C_4\) :
Assume \(\pi\) is symmetric and consider the selected non-faulty shortest paths \(\pi (s, y)\) and \(\pi (x, t)\) going between the two opposite corners. These paths must intersect on an edge; without loss of generality, suppose this edge is \((s, t)\) . Then \(\pi (s, t)\) is just the single edge \((s, t)\) . Suppose this edge fails, and so the unique replacement \(s \leadsto t\) path is \(q = (s, x, y, t)\) . Since both \(\pi (s, y)\) and \(\pi (x, t)\) use the edge \((s, t)\) , this path does not decompose into two non-faulty shortest paths selected by \(\pi\) . Hence \(\pi\) is not 1-restorable.□
B LOWER BOUND FOR f-FAILURES PRESERVERS WITH A CONSISTENT AND STABLE TIE-BREAKING SCHEME
In this Appendix, we prove Theorem 27 by giving a lower bound constructions for \(S \times V\) distance preservers using a consistent and stable tie-breaking scheme.
We begin by showing the construction for the single source case (i.e., \(\sigma =1\) ) and then extend it to the case of multiple sources. Our construction is based on the graph \(G_f(d)=(V_f,E_f)\) , defined inductively. For \(f=1\) , \(G_1(d)=(V_1, E_1)\) consists of three components:
(1)
a set of vertices \(U=\lbrace u^1_1,\ldots ,u^1_d\rbrace\) connected by a path \(P_1=[u^1_1, \ldots , u^1_d]\) ,
(2)
a set of terminal vertices \(Z=\lbrace z_1,\ldots ,z_d\rbrace\) (viewed by convention as ordered from left to right),
(3)
a collection of \(d\) vertex disjoint paths \(\lbrace Q^1_{i}\rbrace\) , where each path \(Q^1_{i}\) connects \(u^1_i\) and \(z_i\) and has length of \(d-i+1\) edges, for every \(i \in \lbrace 1, \ldots , d\rbrace\) .
The vertex \({\texttt {r}}(G_1(d))=u^1_1\) is fixed as the root of \(G_1(d)\) , hence the edges of the paths \(Q^1_i\) are viewed as directed away from \(u^1_i\) , and the terminal vertices of \(Z\) are viewed as the leaves of the graph, denoted \({\texttt {Leaf}}(G_1(d))=Z\) . Overall, the vertex and edge sets of \(G_1(d)\) are \(V_1=U \cup Z \cup \bigcup _{i=1}^d V(Q^1_i)\) and \(E_1=E(P_1) \cup \bigcup _{i=1}^d E(Q^1_i)\) .
For ease of future analysis, we assign labels to the leaves \(z_i \in {\texttt {Leaf}}(G_1(d))\) . Let \({\texttt {Label}}_f: {\texttt {Leaf}}(G_f(d)) \rightarrow E(G_f(d))^f\) . The label of each leaf corresponds to a set of edge faults under which the path from root to leaf is still maintained (as will be proved later on). Specifically, \({\texttt {Label}}_1(z_i, G_1(d))=(u^1_i,u^1_{i+1})\) for \(i \in [1,d-1]\) . In addition, define \(P(z_i,G_1(d)) = P_1[{\texttt {r}}(G_1(d)),u^1_i] \circ Q^1_i\) to be the path from the root \(u^1_1\) to the leaf \(z_i\) .
To complete the inductive construction, let us describe the construction of the graph \(G_{f}(d)=(V_{f}, E_{f})\) , for \(f\ge 2\) , given the graph \(G_{f-1}(\sqrt {d})=(V_{f-1}, E_{f-1})\) . The graph \(G_{f}(d)=(V_{f}, E_{f})\) consists of the following components. First, it contains a path \(P_f=[u^f_1, \ldots , u^f_d]\) , where the vertex \({\texttt {r}}(G_{f}(d))=u^f_1\) is fixed to be the root. In addition, it contains \(d\) disjoint copies of the graph \(G^{\prime }=G_{f-1}(\sqrt {d})\) , denoted by \(G^{\prime }_1, \ldots , G^{\prime }_d\) (viewed by convention as ordered from left to right), where each \(G^{\prime }_i\) is connected to \(u^f_i\) by a collection of \(d\) vertex disjoint paths \(Q^f_i\) , for \(i \in \lbrace 1, \ldots , d\rbrace\) , connecting the vertices \(u^f_i\) with \({\texttt {r}}(G^{\prime }_i)\) . The length of \(Q^f_i\) is \(d-i+1\) , and the leaf set of the graph \(G_{f}(d)\) is the union of the leaf sets of \(G^{\prime }_j\) ’s, \({\texttt {Leaf}}(G_{f}(d))=\bigcup _{j=1}^d {\texttt {Leaf}}(G^{\prime }_j)\) .
Next, define the labels \({\texttt {Label}}_f(z)\) for each \(z \in {\texttt {Leaf}}(G_{f}(d))\) . For every \(j \in \lbrace 1, \ldots , d\rbrace\) and any leaf \(z_{j,i} \in {\texttt {Leaf}}(G^{\prime }_j)\) , let \({\texttt {Label}}_f(z_{j,i}, G_{f}(d))=(u^f_j,u^f_{j+1}) \circ {\texttt {Label}}_{f-1}(z_{j,i}, G^{\prime }_j)\) .
Denote the size (number of vertices) of \(G_f(d)\) by \({\texttt {N}}(f,d)\) , its depth (maximum distance between the root vertex \({\texttt {r}}(G_f(d))\) to a leaf vertex in \({\texttt {Leaf}}(G_f(d))\) ) by \({\texttt {depth}}(f,d)\) , and its number of leaves by \({\texttt {nLeaf}}(f,d) = |{\texttt {Leaf}}(G_f(d))|\) . Note that for \(f=1\) , \({\texttt {N}}(1,d) = 2d+d^2 \le 2d^2\) , \({\texttt {depth}}(1,d)=d\) and \({\texttt {nLeaf}}(1,d)=d\) . We now observe that the following inductive relations hold.
(a) follows by the length of \(Q^f_i\) , which implies that \({\texttt {depth}}(f,d)=d+{\texttt {depth}}(f-1,\sqrt {d})\le 2d\) . (b) follows by the fact that the terminals of the paths starting with \(u_1^f, \ldots , u_d^f\) are the terminals of the graphs \(G^{\prime }_1, \ldots , G^{\prime }_d\) which are disjoint copies of \(G_{f-1}(\sqrt {d})\) , so \({\texttt {nLeaf}}(f,d)=d \cdot {\texttt {nLeaf}}(f-1,\sqrt {d})\) . (c) follows by summing the vertices in the \(d\) copies of \(G^{\prime }_i\) (yielding \(d \cdot {\texttt {N}}(f,d)\) ) and the vertices in \(d\) vertex disjoint paths, namely \(Q^f_1, \ldots , Q^f_d\) of total \(d^2\) vertices, yielding \({\texttt {N}}(f,d)=d \cdot {\texttt {N}}(f-1,\sqrt {d})+d^2\le 2fd^2\) .□
Consider the set of leaves in \(G_f(d)\) , namely, \({\texttt {Leaf}}(G_f(d)) = \bigcup _{i=1}^d {\texttt {Leaf}}(G^{\prime }_i) = \lbrace z_1, \ldots , z_\lambda \rbrace\) , ordered from left to right according to their appearance in \(G_f(d)\) .
For every leaf vertex \(z \in {\texttt {Leaf}}(G_f(d))\) , we define inductively a path \(P(z, G_f(d))\) connecting the root \({\texttt {r}}(G_{f}(d))=u^f_1\) with the leaf \(z\) . As described above for \(f=1\) , \(P(z_i,G_1(d)) = P_1[{\texttt {r}}(G_1(d)),u^1_i] \circ Q^1_i\) . Consider a leaf \(z \in {\texttt {Leaf}}(G_f(d))\) such that \(z\) is the \(i\text{th}^{}\) leaf in the graph \(G^{\prime }_j\) . We therefore denote \(z\) as \(z_{i,j}\) , and define \(P(z_{j,i},G_f(d)) = P_f[{\texttt {r}}(G_f(d)),u^1_j] \circ Q^f_j \circ P(z_{j,i},G^{\prime }_j)\) . We next claim the following on these paths.
Lemma B.1.
For every leaf \(z_{j,i} \in {\texttt {Leaf}}(G_f(d)),\) it holds that:
(1) The path \(P(z_{j,i}, G_f(d))\) is the only \(u^f_1-z_{j,i}\) path in \(G_f(d)\) .
(3) \(P(z_{j,i}, G_f(d)) \not\subseteq G \setminus {\texttt {Label}}_f(z_{k,\ell }, G_f(d))\) for \(k\lt j\) and every \(\ell \in [1,{\texttt {nLeaf}}(f-1,\sqrt {d})]\) , as well as for \(k= j\) and every \(\ell \in [1, i-1]\) . (4) \(|P(z, G_f(d))| = |P(z^{\prime }, G_f(d))|\) for every \(z,z^{\prime } \in {\texttt {Leaf}}(G_f(d))\) .
Proof.
We prove the claims by induction on \(f\) . For \(f=1\) , the lemma holds by construction. Assume this holds for every \(f^{\prime } \le f-1\) and consider \(G_f(d)\) . Recall that \(P_f=[u^f_1, \ldots , u^f_d]\) , and let \(G^{\prime }_1, \ldots , G^{\prime }_d\) be \(d\) copies of the graph \(G_{f-1}(\sqrt {d})\) , viewed as ordered from left to right, where \(G^{\prime }_j\) is connected to \(u^f_j\) . That is, there are disjoint paths \(Q^f_j\) connecting \(u^f_j\) and \({\texttt {r}}(G^{\prime }_j)\) , for every \(j\in \lbrace 1,\ldots , d\rbrace\) .
Consider a leaf vertex \(z_{j,i}\) , the \(i\text{th}^{}\) leaf vertex in \(G^{\prime }_j\) . By the inductive assumption, there exists a single path \(P(z_{j,i}, G^{\prime }_j)\) between the root \({\texttt {r}}(G^{\prime }_j)\) and the leaf \(z_{j,i}\) , for every \(j \in \lbrace 1,\ldots , d\rbrace\) . We now show that there is a single path between \({\texttt {r}}(G_f(d)) = u^f_1\) and \(z_{j,i}\) for every \(j\in \lbrace 1,\ldots , d\rbrace\) . Since there is a single path \(P^{\prime }\) connecting \({\texttt {r}}(G_f(d))\) and \({\texttt {r}}(G^{\prime }_j)\) given by \(P^{\prime }=P_f[u^f_1, u^f_j]\circ Q^f_j\) , it follows that \(P(z_{j,i}, G_f(d))=P^{\prime } \circ P(z_{j,i}, G^{\prime }_j)\) is a unique path in \(G_f(d)\) .
We now show (2). We first show that \(P(z_{j,i}, G_f(d)) \subseteq G \setminus \bigcup _{\ell \ge i ~\mid ~ z_{j,\ell } \in {\texttt {Leaf}}(G^{\prime }_j)} LAB_f(z_{j,\ell }, G_f(d))\) . By the inductive assumption, \(P(z_{j,i}, G^{\prime }_j) \in G \setminus \bigcup _{\ell \ge i} {\texttt {Label}}_{f-1}(z_{j,\ell }, G^{\prime }_j)\) . Since \({\texttt {Label}}_f(z_{j,i}, G_f(d))= (u^f_{j}, u^f_{j+1}) \circ {\texttt {Label}}_{f-1}(z_{j,i}, G^{\prime }_j)\) , it remains to show that \(e_\ell =(u^f_{\ell }, u^f_{\ell +1}) \notin P^{\prime }\) for \(\ell \ge i\) . Since \(P^{\prime }\) diverges from \(P_f\) at the vertex \(u^f_j\) , it holds that \(e_j, \ldots , e_{d-1} \notin P(z_{j,i}, G_f(d))\) . We next complete the proof for every leaf vertex \(z_{k,\ell }\) for \(z_{k,\ell } \in {\texttt {Leaf}}(G^{\prime }_q)\) for \(k \gt j\) and every \(\ell \in {\texttt {nLeaf}}(f-1,\sqrt {d})\) . The claim holds as the edges of \(G^{\prime }_j\) and \(G^{\prime }_k\) are edge-disjoint, and \(e_j, \ldots , e_{d-1} \notin P(z_{j,i}, G_f(d))\) .
Figure B.1.
Figure B.1. Top: Illustration of the graphs \(G_1(d)\) and \(G_f(d)\) . Each graph \(G^{\prime }_i\) is a graph of the form \(G_f(\sqrt {d})\) . Bottom: Extension to \(\sigma\) sources. The collection of leaf nodes \({\texttt {Leaf}}(G^{\prime }_1), \ldots , {\texttt {Leaf}}(G^{\prime }_\sigma)\) are fully connected to a linear size set \(X\) . The size of the resulting bipartite graph \(B^{\prime }\) dominates the size of the construction.
Figure B.2.
Figure B.2. Illustration of the lower bound graph \(G^*_f(V,E,W)\) for \(f=2\) . The edge weights of the bipartite graph are monotone increasing as a function of the leaf index from left to right.
Consider claim (3) and a leaf vertex \(z_{j,i} \in {\texttt {Leaf}}(G^{\prime }_j)\) for some \(j \in \lbrace 1,\ldots , d\rbrace\) and \(i \in {\texttt {nLeaf}}(f-1,\sqrt {d})\) . Let \(Z_1=\lbrace z_{j,\ell } \in {\texttt {Leaf}}(G^{\prime }_j) \mid \ell \lt i\rbrace\) be the set of leaves to the left of \(z_{j,i}\) that belong to \(G^{\prime }_j\) , and let \(Z_2=\lbrace z_{k,\ell } \notin {\texttt {Leaf}}(G^{\prime }_j) \mid j \gt k\rbrace\) be the complementary set of leaves to the left of \(z_{j,i}\) . By the inductive assumption, \(P(z_{j,i}, G^{\prime }_j) \nsubseteq G \setminus {\texttt {Label}}_{f-1}(z_{j,\ell }, G^{\prime }_j)\) for every \(z_{j,\ell } \in Z_1\) . The claim holds for \(Z_1\) as the order of the leaves in \(G^{\prime }_j\) agrees with their order in \(G_f(d)\) , and \({\texttt {Label}}_{f-1}(z_{k,\ell }, G^{\prime }_j) \subset {\texttt {Label}}_{f}(z_{k,\ell }, G_f(d))\) .
Next, consider the complementary leaf set \(Z_2\) to the left of \(z_{j,i}\) . Since for every \(z_{k,\ell } \in Z_2\) , the divergence point of \(P(z_{k,\ell }, G_f(d))\) and \(P_f\) is at \(u^f_k\) for \(k \lt j\) , it follows that \(e_k=(u^f_k, u^f_{k+1}) \in P(z_{j,i}, G_f(d))\) , and thus \(P(z_{j,i}, G_f(d)) \nsubseteq G \setminus {\texttt {Label}}_f(z_{k,\ell }, G_f(d))\) for every \(z_{k,\ell } \in Z_2\) . Finally, consider (4). By setting the length of the paths \(Q^f_j\) to \(d-j+1\) for every \(j \in \lbrace 1,\ldots , d\rbrace\) , we have that \(\operatorname{dist}(u^f_1, {\texttt {r}}(G^{\prime }_j))=d\) for every \(j \in [1,d]\) . The proof then follows by induction as well, since \(|P(z_{j,i}, G^{\prime }_j)|=|P(z_{k,\ell }, G^{\prime }_k)|\) for every \(k, j \in [1,d]\) and \(i,\ell \in [1, {\texttt {nLeaf}}(f-1,\sqrt {d}]\) .□
Finally, we turn to describe the graph \(G^*_f(V, E,W)\) which establishes our lower bound, where \(W\) is a particular bad edge weight function that determines the consistent tie-breaking scheme which provides the lower bound, See Figure B.2 for a picture of the following construction. The graph \(G^*_f(V, E,W)\) consists of three components. The first is the graph \(G_{f}(d)\) for \(d=\lfloor \sqrt {n/(4f)} \rfloor\) . By Observation 1, \({\texttt {N}}(f,d)=|V(G_{f}(d))|\le n/2\) . The second component of \(G^*_f(V, E,W)\) is a set of vertices \(X=\lbrace x_1, \ldots , x_\chi \rbrace\) , where the last vertex of \(P_f\) , namely, \(u^f_d\) is connected to all the vertices of \(X\) . The cardinality of \(X\) is \(\chi =n-{\texttt {N}}(f,d)-1\) . The third component of \(G^*_f(V, E,W)\) is a complete bipartite graph \(B\) connecting the vertices of \(X\) with the leaf set \({\texttt {Leaf}}(G_f(d))\) , i.e., the disjoint leaf sets \({\texttt {Leaf}}(G^{\prime }_1), \ldots , {\texttt {Leaf}}(G^{\prime }_d)\) . We finally define the weight function \(W: E \rightarrow (1,1+1/n^2)\) . Let \(W(e)=1\) for every \(e \in E \setminus E(B)\) . The weights of the bipartite graph edges \(B\) are defined as follows. Consider all leaf vertices \({\texttt {Leaf}}(G_f(d))\) from left to right given by \(\lbrace z_1, \ldots , z_\lambda \rbrace\) . Then, \(W(z_j,x_i)=(\lambda -j)/n^4\) for every \(z_j\) and every \(x_i \in X\) . The vertex set of the resulting graph is thus \(V=V(G_{f}(d))\cup \lbrace v^{*}\rbrace \cup X\) and hence \(|V|=n\) . By Prop. (b) of Observation 1, \({\texttt {nLeaf}}(G_f(d))=d^{2-1/2^{f-1}}=\Theta ((n/f)^{1-1/2^f}),\) hence \(|E(B)|=\Theta ((n/f)^{2-1/2^f})\) .
We now complete the proof of Theorem 27 for the single source case.
Theorem 27 for |S|=1.
Theorem 27 for \(|S|=1\) . Let \(s=u^f_1\) be the chosen source in the graph \(G^*_f(V, E,W)\) . We first claim that under the weights \(W\) , there is a unique shortest path, denoted by \(\pi (s,x_i ~\mid ~ F)\) for every \(x_i \in X\) and every fault set \(F \in \lbrace {\texttt {Label}}_f(z_1, G_f(d)), \ldots , {\texttt {Label}}_f(z_\ell , G_f(d))\rbrace\) . By Lemma B.1(1), there is a unique shortest path from each \(s\) to each \(z_j \in {\texttt {Leaf}}(G_f(d))\) denoted by \(P(z_j, G_f(d))\) .
In addition, by Lemma B.1(4), the unweighted length of all the \(s\) - \(z_j\) paths are the same for every \(z_j\) . Since each \(x_i\) is connected to each \(z_j\) with a distinct edge weight in \((1,1+1/n^2)\) , we get that each \(x_i\) has a unique shortest path from \(s\) in each subgraph \(G \setminus {\texttt {Label}}_f(z_j, G_f(d))\) . Note that since the uniqueness of \(\pi\) is provided by the edge weights it is both consistent and stable. Also note that the weights of \(W\) are sufficiently small so that they only use to break the ties between equally length paths.
We now claim that a collection of \(\lbrace s\rbrace \times X\) replacement paths (chosen based on the weights of \(W\) ) contains all edges of the bipartite graph \(B\) . Formally, letting
we will show that \(E(B) \subseteq \bigcup _{P \in \mathcal {P}} P\) which will complete the proof. To see this we show that \(\pi (s,x_i ~\mid ~ {\texttt {Label}}_f(z_j, G_f(d)))=P(z_j, G_f(d)) \circ (z_j,x_i)\) . Indeed, by Lemma B.1(2), we have that \(P(z_j, G_f(d)) \subseteq G \setminus {\texttt {Label}}_f(z_j, G_f(d))\) . It remains to show that the shortest \(s\) - \(x_i\) path (based on edge weights) in \(G \setminus {\texttt {Label}}_f(z_j, G_f(d))\) goes through \(z_j\) . By Lemma B.1(2,3), the only \(z_k\) vertices in \({\texttt {Leaf}}(G_f(d))\) that are connected to \(s\) in \(G \setminus {\texttt {Label}}_f(z_j, G_f(d))\) are \(\lbrace z_1,\ldots , z_j\rbrace\) . Since \(W(z_1,x_i) \gt W(z_2,x_i)\gt \ldots \gt W(z_j,x_i)\) , we have that \((z_j, x_i)\) is the last edge of \(\pi (s,x_i ~\mid ~ {\texttt {Label}}_f(z_j, G_f(d)))\) . As this holds for every \(x_i \in X\) and every \(z_j \in {\texttt {Leaf}}(G_f(d))\) , the claim follows.□
Extension to Multiple Sources. Given a parameter \(\sigma\) representing the number of sources, the lower bound graph \(G\) includes \(\sigma\) copies, \(G^{\prime }_1, \ldots , G^{\prime }_\sigma\) , of \(G_f(d)\) , where \(d=O(\sqrt {(n / 4f\sigma)})\) . By Observation 1, each copy consists of at most \(n/2\sigma\) vertices. We now add to \(G\) a collection \(X\) of \(\Theta (n)\) vertices connected to the \(\sigma\) leaf sets \({\texttt {Leaf}}(G^{\prime }_1), \ldots , {\texttt {Leaf}}(G^{\prime }_\sigma)\) by a complete bipartite graph \(B^{\prime }\) . See Figure B.1 for an illustration. We adjust the size of the set \(X\) in the construction so that \(|V(G)|=n\) . Since \({\texttt {nLeaf}}(G^{\prime }_i)=\Omega ((n / (f\sigma))^{1-1/2^f})\) (see Observation 1), overall \(|E(G)| = \Omega (n \cdot \sigma \cdot {\texttt {nLeaf}}(G_f(d))) = \Omega (\sigma ^{1/2^f}\cdot (n/f)^{2-1/2^f})\) . The weights of all graph edges not in \(B^{\prime }\) are set to 1. For every \(i \in \lbrace 1,\ldots , \sigma \rbrace\) , the edge weights of the bipartite graph \(B_j=({\texttt {Leaf}}(G^{\prime }_1),X)\) are set in the same manner as for the single source case. Since the path from each source \(s_i\) to \(X\) cannot aid the vertices of \(G^{\prime }_j\) for \(j \ne i\) , the analysis of the single-source case can be applied to show that each of the bipartite graph edges in necessary upon a certain sequence of at most \(f\) -edge faults. This completes the proof of Theorem 27.
References
[1]
Ittai Abraham, Shiri Chechik, and Cyril Gavoille. 2012. Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels. In 44th Annual ACM Symposium on Theory of Computing. 1199–1218.
Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. 2018. Efficient oracles and routing schemes for replacement paths. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France. 13:1–13:15.
Greg Bodwin, Keerti Choudhary, Merav Parter, and Noa Shahar. 2020. New fault tolerant subset preservers. In 47th International Colloquium on Automata, Languages, and Programming (ICALP’20) (Leibniz International Proceedings in Informatics (LIPIcs)), Artur Czumaj, Anuj Dawar, and Emanuela Merelli (Eds.), Vol. 168. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 15:1–15:19.
Greg Bodwin, Fabrizio Grandoni, Merav Parter, and Virginia Vassilevska Williams. 2017. Preserving distances in very faulty graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP’17). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Gilad Braunschvig, Shiri Chechik, and David Peleg. 2012. Fault tolerant additive spanners. In International Workshop on Graph-Theoretic Concepts in Computer Science. Springer, 206–214.
Shiri Chechik and Sarel Cohen. 2019. Near optimal algorithms for the single source replacement paths problem. In 30th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2090–2109.
Bruno Courcelle, Cyril Gavoille, M. Kanté, and Andrew Twigg. 2007. Forbidden-set labeling on graphs. In 2nd Workshop on Locality Preserving Distributed Computing Methods (LOCALITY) (Co-located with PODC).
Bruno Courcelle and Andrew Twigg. 2007. Compact forbidden-set routing. In Annual Symposium on Theoretical Aspects of Computer Science. Springer, 37–48.
Michael Dinitz and Robert Krauthgamer. 2011. Fault-tolerant spanners: Better and simpler. In 30th Annual ACM Symposium on Principles of Distributed Computing (PODC 2011) (San Jose, CA, June 6–8). 169–178.
Michael Dinitz and Caleb Robelle. 2020. Efficient and simple algorithms for fault-tolerant spanners. In ACM Symposium on Principles of Distributed Computing (PODC’20).
Joan Feigenbaum, David R. Karger, Vahab S. Mirrokni, and Rahul Sami. 2005. Subjective-cost policy routing. In International Workshop on Internet and Network Economics. Springer, 174–183.
Mohsen Ghaffari and Merav Parter. 2016. Near-optimal distributed algorithms for fault-tolerant tree structures. In 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2016) (Asilomar State Beach/Pacific Grove, CA, July 11–13). 387–396.
Manoj Gupta and Shahbaz Khan. 2017. Multiple source dual fault tolerant BFS trees. In 44th International Colloquium on Automata, Languages, and Programming (ICALP’17). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
John Hershberger and Subhash Suri. 2001. Vickrey prices and shortest paths: What is an edge worth?. In 42nd IEEE Symposium on Foundations of Computer Science. IEEE, 252–259.
Frank Thomson Leighton, Bruce M. Maggs, and Satish B. Rao. 1994. Packet routing and job-shop scheduling ino (congestion+ dilation) steps. Combinatorica 14, 2 (1994), 167–186.
Kavindra Malik, Ashok K. Mittal, and Santosh K. Gupta. 1989. The k most vital arcs in the shortest path problem. Operations Research Letters 8, 4 (1989), 223–227.
Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. 1987. Matching is as easy as matrix inversion. In 19th Annual ACM Symposium on Theory of Computing. 345–354.
Raimund Seidel. 1995. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences 51, 3 (1995), 400–403.
Avi Shoshan and Uri Zwick. 1999. All pairs shortest paths in undirected graphs with integer weights. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039). IEEE, 605–614.
Jan van den Brand and Thatchaphol Saranurak. 2019. Sensitive distance and reachability oracles for large batch updates. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS’19). IEEE, 424–435.
Oren Weimann and Raphael Yuster. 2010. Replacement paths via fast matrix multiplication. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science. IEEE, 655–662.
Oren Weimann and Raphael Yuster. 2013. Replacement paths and distance sensitivity oracles via fast matrix multiplication. ACM Transactions on Algorithms (TALG) 9, 2 (2013), 14.
Virginia Vassilevska Williams and Ryan Williams. 2010. Subcubic equivalences between path, matrix and triangle problems. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science. IEEE, 645–654.
PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing
The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original ...
For a graph G and a collection of vertex pairs {(s"1,t"1),...,(s"k,t"k)}, the k disjoint paths problem is to find k vertex-disjoint paths P"1,...,P"k, where P"i is a path from s"i to t"i for each i=1,...,k. In the corresponding optimization problem, the ...
ISAAC '09: Proceedings of the 20th International Symposium on Algorithms and Computation
For a graph and a set of vertex pairs {(s1, t1), ..., (sk , tk )}, the k disjoint paths problem is to find k vertex-disjoint paths P1, ..., Pk, where Pi is a path from si to ti for each i = 1, ..., k. In the corresponding ...
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Greg Bodwin, 
