TY - GEN

T1 - Routing in histograms

AU - Chiu, Man Kwun

AU - Cleve, Jonas

AU - Klost, Katharina

AU - Korman, Matias

AU - Mulzer, Wolfgang

AU - van Renssen, André

AU - Roeloffzen, Marcel

AU - Willert, Max

N1 - Funding Information:
M.-K. Chiu, J. Cleve and W. Mulzer partially supported by ERCSTG 757609. J. Cleve partially supported by DFG grant MU 3501/1-2. M. Korman partially supported by MEXT KAKENHI No. 17K12635 and the NSF award CCF-1422311. A. van Renssen partially supported by JSTERATO Grant Number JPMJER1201, Japan.
Funding Information:
M.-K. Chiu, J. Cleve and W. Mulzer partially supported by ERC STG 757609. J. Cleve partially supported by DFG grant MU 3501/1-2. M. Korman partially supported by MEXT KAKENHI No. 17K12635 and the NSF award CCF-1422311. A. van Renssen partially supported by JST ERATO Grant Number JPMJER1201, Japan.
Publisher Copyright:
© Springer Nature Switzerland AG 2020.

PY - 2020

Y1 - 2020

N2 - Let P be an x-monotone orthogonal polygon with n vertices. We call P a simple histogram if its upper boundary is a single edge; and a double histogram if it has a horizontal chord from the left boundary to the right boundary. Two points p and q in P are co-visible if and only if the (axis-parallel) rectangle spanned by p and q completely lies in P. In the r-visibility graph G(P) of P, we connect two vertices of P with an edge if and only if they are co-visible. We consider routing with preprocessing in G(P). We may preprocess P to obtain a label and a routing table for each vertex of P. Then, we must be able to route a packet between any two vertices s and t of P, where each step may use only the label of the target node t, the routing table and the neighborhood of the current node, and the packet header. The routing problem has been studied extensively for general graphs, where truly compact and efficient routing schemes with polylogartihmic routing tables have turned out to be impossible. Thus, special graph classes are of interest. We present a routing scheme for double histograms that sends any data packet along a path of length at most twice the (unweighted) shortest path distance between the endpoints. The labels, routing tables, and headers need O(log n) bits. For the simple histograms, we obtain a routing scheme with optimal routing paths, O(log n)-bit labels, one-bit routing tables, and no headers.

AB - Let P be an x-monotone orthogonal polygon with n vertices. We call P a simple histogram if its upper boundary is a single edge; and a double histogram if it has a horizontal chord from the left boundary to the right boundary. Two points p and q in P are co-visible if and only if the (axis-parallel) rectangle spanned by p and q completely lies in P. In the r-visibility graph G(P) of P, we connect two vertices of P with an edge if and only if they are co-visible. We consider routing with preprocessing in G(P). We may preprocess P to obtain a label and a routing table for each vertex of P. Then, we must be able to route a packet between any two vertices s and t of P, where each step may use only the label of the target node t, the routing table and the neighborhood of the current node, and the packet header. The routing problem has been studied extensively for general graphs, where truly compact and efficient routing schemes with polylogartihmic routing tables have turned out to be impossible. Thus, special graph classes are of interest. We present a routing scheme for double histograms that sends any data packet along a path of length at most twice the (unweighted) shortest path distance between the endpoints. The labels, routing tables, and headers need O(log n) bits. For the simple histograms, we obtain a routing scheme with optimal routing paths, O(log n)-bit labels, one-bit routing tables, and no headers.

UR - http://www.scopus.com/inward/record.url?scp=85080858902&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85080858902&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-39881-1_5

DO - 10.1007/978-3-030-39881-1_5

M3 - Conference contribution

AN - SCOPUS:85080858902

SN - 9783030398804

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 43

EP - 54

BT - WALCOM

A2 - Rahman, M. Sohel

A2 - Sadakane, Kunihiko

A2 - Sung, Wing-Kin

PB - Springer

T2 - 14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020

Y2 - 31 March 2020 through 2 April 2020

ER -