Worst-case Analysis of Set Union Algorithms
Journal of the ACM (JACM)
Self-adjusting binary search trees
Journal of the ACM (JACM)
Sequential access in splay trees takes linear time
Combinatorica
The cell probe complexity of dynamic data structures
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
The maximum number of unit distances in a convex n-gon
Journal of Combinatorial Theory Series A
An extremal problem on sparse 0-1 matrices
SIAM Journal on Discrete Mathematics
Postorder disjoint set union is linear
SIAM Journal on Computing
Davenport-Schnizel theory of matrices
Discrete Mathematics
SIAM Journal on Computing
Lower bounds for the union-find and the split-find problem on pointer machines
Journal of Computer and System Sciences
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
A near-linear algorithm for the planar segment center problem
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Meldable heaps and boolean union-find
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the Dynamic Finger Conjecture for Splay Trees. Part II: The Proof
SIAM Journal on Computing
On the sequential access theorem and deque conjecture for splay trees
Theoretical Computer Science
Excluded permutation matrices and the Stanley-Wilf conjecture
Journal of Combinatorial Theory Series A
Top-Down Analysis of Path Compression
SIAM Journal on Computing
On 0-1 matrices and small excluded submatrices
Journal of Combinatorial Theory Series A
Splay trees, Davenport-Schinzel sequences, and the deque conjecture
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Faster Algorithms for Incremental Topological Ordering
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Note: On linear forbidden submatrices
Journal of Combinatorial Theory Series A
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
The geometry of binary search trees
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Note: Extremal functions of forbidden double permutation matrices
Journal of Combinatorial Theory Series A
On nonlinear forbidden 0-1 matrices: a refutation of a Füredi-Hajnal conjecture
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Data structures for halfplane proximity queries and incremental voronoi diagrams
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
On nonlinear forbidden 0-1 matrices: a refutation of a Füredi-Hajnal conjecture
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the structure and composition of forbidden sequences, with geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
Origins of Nonlinearity in Davenport-Schinzel Sequences
SIAM Journal on Discrete Mathematics
Tight bounds on the maximum size of a set of permutations with bounded VC-dimension
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Tight bounds on the maximum size of a set of permutations with bounded VC-dimension
Journal of Combinatorial Theory Series A
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In this paper we improve, reprove, and simplify several theorems on the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (non-geometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures. To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object M; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden substructure P, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper bounds on the size of a P-free object M may be reused in different contexts. Among our results, we present the first asymptotically sharp bound on the length of arbitrary path compressions on arbitrary trees, improving analyses of Tarjan [35] and Seidel and Sharir [31]. We reprove the linear bound on postordered path compressions, due to Lucas [23] and Loebel and Nešetřil [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the sequential access theorem for splay trees, originally due to Tarjan [38]. We disprove a conjecture of Aronov et al. [3] related to the efficiency of their data structure for half-plane proximity queries and provide a significantly cleaner analysis of their structure. With the exception of the sequential access theorem, all our proofs are exceptionally simple. Notably absent are calculations of any kind.