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Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
, 1998
"... . We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than th ..."
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Cited by 49 (11 self)
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. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different tradeoffs have been proposed. The adjacencylist encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted nnode tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...
Lineartime succinct encodings of planar graphs via canonical orderings
 SIAM Journal on Discrete Mathematics
, 1999
"... Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, rough ..."
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Cited by 23 (6 self)
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Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5 + 2 log 3) min{n, f} −7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses ∗
, 2008
"... Let G be a plane graph of n nodes, m edges, f faces, and no selfloop. G need not be connected or simple (i.e., free of multiple edges). We give three sets of coding schemes for G which all take O(m + n) time for encoding and decoding. Our schemes employ new properties of canonical orderings for pla ..."
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Let G be a plane graph of n nodes, m edges, f faces, and no selfloop. G need not be connected or simple (i.e., free of multiple edges). We give three sets of coding schemes for G which all take O(m + n) time for encoding and decoding. Our schemes employ new properties of canonical orderings for planar graphs and new techniques of processing strings of multiple types of parentheses. For applications that need to determine in O(1) time the adjacency of two nodes and the degree of a node, we use 2m + (5 + 1 k)n + o(m + n) bits for any constant k> 0 while the best previous bound by Munro and Raman is 2m + 8n + o(m + n). If G is triconnected or triangulated, our bit count decreases to 2m + 3n + o(m + n) or 2m + 2n + o(m + n), respectively. If G is simple, our bit count is 5 1 3m + (5 + k)n + o(n) for any constant k> 0. Thus, if a simple G is also triconnected or triangulated, then 2m + 2n + o(n) or 2m + n + o(n) bits suffice, respectively. If only adjacency queries are supported, the bit counts for a general G and a simple G become 2m + 14 4 3 n + o(m + n) and 3m + 5n + o(n), respectively. If we only need to reconstruct G from its code, a simple and triconnected G uses
CommonFace Embeddings of Planar Graphs
, 2001
"... Given a planar graph G and a sequence C1,...,Cq, where each Ci is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i ∈ {1,..., q}, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in Ci. This probl ..."
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Given a planar graph G and a sequence C1,...,Cq, where each Ci is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i ∈ {1,..., q}, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in Ci. This problem has applications to the recovery of topological information from geographical data and the design of constrained layouts in VLSI. Let I be the input size, i.e., the total number of vertices and edges in G and the families Ci, counting multiplicity. We show that this problem is NPcomplete in general. We also show that it is solvable in O(I log I) time for the special case where for each input family Ci, each set in Ci induces a connected subgraph of the input graph G. Note that the classical problem of simply finding a planar embedding is a further special case of this case with q = 0. Therefore, the processing of the additional constraints C1,..., Cq only incurs a logarithmic factor of overhead.