O(1) QUERY TIME ALGORITHM FOR ALL PAIRS SHORTEST DISTANCES ON INTERVAL GRAPHS

1999 ◽  
Vol 10 (04) ◽  
pp. 465-472 ◽  
Author(s):  
ALAN P. SPRAGUE ◽  
TADAO TAKAOKA
Keyword(s):  
Time Algorithm ◽  
Interval Graph ◽  
Query Time ◽  
Circular Arc ◽  
Interval Model ◽  
Erew Pram ◽  
Pram Model ◽  
Shortest Distance ◽  
Distance Problem ◽  
Circular Arc Graphs

We present an algorithm for the All Pairs Shortest Distance problem on an interval graph on n vertices: after O(n) preprocessing time, the algorithm can deliver a response to a distance query in O(1) time. The method used here is simpler than the method of Chen et al. [4], which has the same preprocessing and query time. It is assumed that an interval model for the graph is given, and ends of intervals are already sorted by coordinate. The preprocessing algorithm can be executed in the EREW PRAM model in O( log n) time, using n/ log n processors. These algorithms (sequential and parallel) may be extended to circular arc graphs, with the same time and processor bounds.

DYNAMIC PROGRAMMING ON INTERVALS

1993 ◽  
Vol 03 (03) ◽  
pp. 323-330 ◽  
Author(s):  
TAKAO ASANO
Keyword(s):  
Dynamic Programming ◽  
Dominating Set ◽  
Minimum Weight ◽  
Time Algorithm ◽  
Interval Graph ◽  
Efficient Implementation ◽  
Maximum Weight ◽  
Circular Arc ◽  
Maximum Weight Clique

We consider problems on intervals which can be solved by dynamic programming. Specifically, we give an efficient implementation of dynamic programming on intervals. As an application, an optimal sequential partition of a graph G=(V, E) can be obtained in O(m log n) time, where n=|V| and m=|E|. We also present an O(n log n) time algorithm for finding a minimum weight dominating set of an interval graph G=(V, E), and an O(m log n) time algorithm for finding a maximum weight clique of a circular-arc graph G=(V, E), provided their intersection models of n intervals (arcs) are given.

PARALLEL VERTEX COLOURING OF INTERVAL GRAPHS

1999 ◽  
Vol 10 (01) ◽  
pp. 19-31 ◽  
Author(s):  
G. SAJITH ◽  
SANJEEV SAXENA
Keyword(s):  
Chromatic Number ◽  
Interval Graph ◽  
Interval Graphs ◽  
Chromatic Numbers ◽  
Erew Pram ◽  
Pram Model ◽  
Interval Representation ◽  
Left And Right ◽  
Crcw Pram ◽  
Vertex Colouring

Evidence is given to suggest that minimally vertex colouring an interval graph may not be in NC 1. This is done by showing that 3-colouring a linked list is NC 1-reducible to minimally colouring an interval graph. However, it is shown that an interval graph with a known interval representation and an O(1) chromatic number can be minimally coloured in NC 1. For the CRCW PRAM model, an o( log n) time, polynomial processors algorithm is obtained for minimally colouring an interval graph with o( log n) chromatic number and a known interval representation. In particular, when the chromatic number is O(( log n)1-ε), 0<ε<1, the algorithm runs in O( log n/ log log n) time. Also, an O( log n) time, O(n) cost, EREW PRAM algorithm is found for interval graphs of arbitrary chromatic numbers. The following lower bound result is also obtained: even when the left and right endpoints of the interval are separately sorted, minimally colouring an interval graph needs Ω( log n/ log log n) time, on a CRCW PRAM, with a polynomial number of processors.

L(h,k)-Labeling of Intersection Graphs

2020 ◽  
pp. 135-170
Author(s):  
Sk. Amanathulla ◽  
Madhumangal Pal
Keyword(s):  
Coding Theory ◽  
Chordal Graph ◽  
Interval Graph ◽  
Graph Labeling ◽  
Frequency Assignment ◽  
Permutation Graph ◽  
Intersection Graphs ◽  
Circular Arc ◽  
Trapezoid Graph ◽  
Circular Arc Graphs

One important problem in graph theory is graph coloring or graph labeling. Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in (mobile) communication system, coding theory, ray crystallography, radar, circuit design, etc. For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers such that if and if, where it represents the distance between the nodes. Intersection graph is a very important subclass of graph. Unit disc graph, chordal graph, interval graph, circular-arc graph, permutation graph, trapezoid graph, etc. are the important subclasses of intersection graphs. In this chapter, the authors discuss labeling for intersection graphs, specially for interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, etc., and have presented a lot of results for this problem.

Finding a Maximum Clique in a Set of Proper Circular Arcs in Time O(n) with Applications

1997 ◽  
Vol 08 (04) ◽  
pp. 443-467 ◽  
Author(s):  
Glenn K. Manacher ◽  
Terrance A. Mankus
Keyword(s):  
Time Complexity ◽  
Maximum Clique ◽  
Special Property ◽  
Maximum Weight ◽  
Time And Space ◽  
Circular Arc ◽  
Interval Model ◽  
Circular Arcs ◽  
The One ◽  
Circular Arc Graphs

A maximum clique is sought in a set of n proper circular arcs (PCAS). By means of several passes, each O(n) in time and space, a PCAS is transformed initially into a set of circle chords and finally into a set of intervals. This interval model inherits a special property from the PCAS which ensures the discovery of a maximum overlap clique in time O(n). The one-to-one arc/interval correspondence guarantees the identification of the maximum clique in the PCAS in O(n) time and space. The present paper gives new, simpler proofs for the lemmas first outlined by us in Ref. [9], extending the methods outlined in that paper so that the time bound is improved from O(n log n) to O(n). The method depends only on certain interconnections between constructions related to the computation of longest increasing subsequences. Independently, Hell, Huang and Bhattacharya5 recently discovered a completely different approach that also achieves the same complexity, and can moreover be applied to the weighted case and to the coloring problem on proper circular arcs. The previous best result, due to Apostolico and Hambrusch2 applies to general circular arc models and has time complexity O(n2 log log n) and space complexity O(n). As applications of the method, we show that maximum weight clique of a set of weighted proper circular arcs can be found in time O(n2) and space O(n). The previous best result was O(n2 log log n) for dense general circular arc graphs.13 We also show that, for n chords with randomly placed endpoints (1) the average cardinality of a maximum clique is cn1/2 ± o(n1/2), where 21/2< c < e21/2, and (2) a maximum clique may be found in average time O(n3/2) and space θ(n). The previous best average time complexity, derived from Ref. [1], was O(n3/2 log n).

scholarly journals Isomorphism of graph classes related to the circular-ones property

2013 ◽  
Vol Vol. 15 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Andrew R. Curtis ◽  
Min Chih Lin ◽  
Ross M. Mcconnell ◽  
Yahav Nussbaum ◽  
Francisco Juan Soulignac ◽  
...  
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Circular Arc ◽  
Graph Classes ◽  
Discrete Algorithms ◽  
International Audience ◽  
Related Graph ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Discrete Algorithms International audience We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.

L(2,1,1)-Labeling of Circular-Arc Graphs

2021 ◽  
Vol 13 (2) ◽  
pp. 537-544
Author(s):  
S. Amanathulla ◽  
B. Bera ◽  
M. Pal
Keyword(s):  
Circuit Design ◽  
Coding Theory ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Graph Labeling ◽  
Frequency Assignment ◽  
Circular Arc ◽  
X Ray Crystallography ◽  
The Difference ◽  
Circular Arc Graphs

Graph labeling problem has been broadly studied in recent past for its wide applications, in mobile communication system for frequency assignment, radar, circuit design, X-ray crystallography, coding theory, etc. An L211-labeling  (L211L) of a graph G = (V, E) is a function γ : V → Z∗ such that |γ(u) − γ(v)| ≥ 2, if d(u, v) = 1 and |γ(u) − γ(v)| ≥ 1, if  d(u, v) = 1 or 2, where  Z∗  be the set of non-negative integers and d(u, v) represents the distance between the nodes u and v. The L211L numbers of a graph G, are denoted by λ2,1,1(G) which is the difference between largest and smallest labels used in L211L. In this article, for circular-arc graph (CAG) G we have proved that λ2,1,1(G) ≤ 6∆ − 4, where ∆ represents the degree of the graph. Beside this we have designed a polynomial time algorithm to label a CAG satisfying the conditions of L211L. The time complexity of the algorithm is O(n∆2), where n is the number of nodes of the graph G.

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