## Applications of forbidden 0-1 matrices to search tree and path compression based data structures (2009)

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Citations: | 5 - 4 self |

### BibTeX

@MISC{Pettie09applicationsof,

author = {Seth Pettie},

title = {Applications of forbidden 0-1 matrices to search tree and path compression based data structures },

year = {2009}

}

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### Abstract

In this paper we improve, reprove, and simplify a variety of theorems concerning 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. All of our proofs begin by transcribing the individual operations of a dynamic data structure

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Citation Context ...ccessfully applied to various problems in discrete and computational geometry [13, 4, 14, 27, 8, 28], and, since forbidden submatrix theory largely subsumes the theory of Davenport-Schinzel sequences =-=[1]-=- and Turán-type subgraph avoidance problems, one can often restate proofs based on these concepts in terms of forbidden 0-1 matrices. Despite the pervasive use of the forbidden substructure method in ... |

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Citation Context ... the cost of a sequence of m finds on an n-element universe takes Opm log2 m{n nq time; see [35, 31]. With the additional union-by-rank or union-by-size heuristics the cost becomes Θpn mαpm, nqq; see =-=[35, 36, 31, 21, 11, 18]-=-. Theorem 2.1 The cost of a sequence of n makesets, n 1 unions, and m finds is: Opnq ExptP, P 1 u, m, nq where P and P 1 The proofs of Theorems 2.1, 2.2, and 2.3 reference the compression-node inciden... |

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Citation Context ...hese results we are also able to solve a problem addressed by Sundar [33], namely, to bound the cost of performing a sequence of twists in a binary search tree. In the spirit of the SMAWK appellation =-=[2]-=- we call the operations defined in [3] dilbags, 8 an acronym honoring the authors. A dilbag alters a binary search tree by (1) reshaping the path from the root to the leftmost node via an arbitrary se... |

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Citation Context ... n, ExpP, m, nq ExpP 1 , m, nq p1 op1qqm log1 m{n n. This bound on the length of m path compressions sharpens analyses of [35, 31] and matches, asymptotically, a lower bound of Tarjan and van Leeuwen =-=[37]-=-. 2.2 Postordered Path Compression A sequence of path compressions has the “rising roots” property [23, 5] if, after a compression terminates at a node u, no subsequent compression terminates at a str... |

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Citation Context ...in their entirety. 1 Precursors. It has been known for some time that a specific path compression system could be characterized by forbidden substructure. As an intermediate step in Hart and Sharir’s =-=[17]-=- proof 1 Actually, to get the Opm log1 m{n nq bound on arbitrary path compressions we need a slight generalization of one of Tardos’s [34] results from n n matrices to rectangular m n matrices. The pr... |

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Citation Context ...nalysis of this pattern occupies all of 1/3 of a page and Tardos’s [34] analsysis, tight up to lower order terms, occupies less than a page. In Theorems 2.2 and 2.3 we make use of Marcus and Tardos’s =-=[25]-=- tight analysis of permutation matrices. Their leisurely proof occupies all of 1 page and a terser version would be half that. The proofs of [13, 34, 25] are completely elementary and represent a powe... |

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Citation Context ... the cost of a sequence of m finds on an n-element universe takes Opm log2 m{n nq time; see [35, 31]. With the additional union-by-rank or union-by-size heuristics the cost becomes Θpn mαpm, nqq; see =-=[35, 36, 31, 21, 11, 18]-=-. Theorem 2.1 The cost of a sequence of n makesets, n 1 unions, and m finds is: Opnq ExptP, P 1 u, m, nq where P and P 1 The proofs of Theorems 2.1, 2.2, and 2.3 reference the compression-node inciden... |

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Citation Context ...an algorithm/geometric arrangement directly? This question presupposes that it is possible to perform an ad hoc analysis without mentioning 0-1 matrices. In the geometric applications of 0-1 matrices =-=[13, 4, 27, 8, 17, 39]-=- we are not aware of any alternative analyses without forbidden substructure arguments. With the exception of our bound on deque-ordered path 2 Generalized here means that any subset of the vertices o... |

33 |
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Citation Context ...can be packed into a 0-1 matrix avoiding certain forbidden 0-1 submatrices or patterns. Forbidden submatrices have been successfully applied to various problems in discrete and computational geometry =-=[13, 4, 14, 27, 8, 28]-=-, and, since forbidden submatrix theory largely subsumes the theory of Davenport-Schinzel sequences [1] and Turán-type subgraph avoidance problems, one can often restate proofs based on these concepts... |

30 | The Füredi–Hajnal conjecture implies the Stanley–Wilf conjecture
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Citation Context ...ath compression and other forbidden substructures. The author [29] recently obtained an Opnα pnqq bound on deque-ordered access sequences in splay trees using generalized Davenport-Schinzel sequences =-=[20, 26]-=-, which is a very natural approach for tackling the problem, given Hart and Sharir’s [17] work. It is well known [24] that splaying is an analogue of path compression for binary search trees. Demaine ... |

27 | Davenport-Schinzel theory of matrices
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Citation Context ...terization for any one of the categories. Nonetheless, large swaths of forbidden submatrices have successfully been categorized. Marcus and Tardos [25], answering a problem posed by Füredi and Hajnal =-=[14]-=-, showed that every permutation matrix A has ExpA, nq Opnq. Recently Geneson [15] showed that double permutation matrices are also linear. From the quasilinear bounds on generalized Davenport-Schinzel... |

22 |
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Citation Context ...e proximity queries and provide a significantly cleaner analysis of their structure. Finally, we give tight bounds on the number of “right twists” in a binary search tree, closing a problem of Sundar =-=[33]-=-. With the exception of the sequential access theorem, all our proofs are exceptionally simple.1 Introduction In this paper we demonstrate a simple and effective method for analyzing dynamic data str... |

20 | On 0-1 matrices and small excluded submatrices - Tardos |

18 |
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Citation Context ... the cost of a sequence of m finds on an n-element universe takes Opm log2 m{n nq time; see [35, 31]. With the additional union-by-rank or union-by-size heuristics the cost becomes Θpn mαpm, nqq; see =-=[35, 36, 31, 21, 11, 18]-=-. Theorem 2.1 The cost of a sequence of n makesets, n 1 unions, and m finds is: Opnq ExptP, P 1 u, m, nq where P and P 1 The proofs of Theorems 2.1, 2.2, and 2.3 reference the compression-node inciden... |

16 |
An extremal Problem on sparse 0-1 matrices
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Citation Context ...can be packed into a 0-1 matrix avoiding certain forbidden 0-1 submatrices or patterns. Forbidden submatrices have been successfully applied to various problems in discrete and computational geometry =-=[13, 4, 14, 27, 8, 28]-=-, and, since forbidden submatrix theory largely subsumes the theory of Davenport-Schinzel sequences [1] and Turán-type subgraph avoidance problems, one can often restate proofs based on these concepts... |

16 | Improved bounds and new techniques for Davenport–Schinzel sequences and their generalizations
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Citation Context ...ath compression and other forbidden substructures. The author [29] recently obtained an Opnα pnqq bound on deque-ordered access sequences in splay trees using generalized Davenport-Schinzel sequences =-=[20, 26]-=-, which is a very natural approach for tackling the problem, given Hart and Sharir’s [17] work. It is well known [24] that splaying is an analogue of path compression for binary search trees. Demaine ... |

15 | Data-structural bootstrapping, linear path compression, and catenable heap-ordered double-ended queues
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Citation Context ...e the linear upper bounds on postordered path compressions, due to Lucas [23] and Loebel and Ne˘set˘ril [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan =-=[5]-=-, and the linearity of sequential access in 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 p... |

15 | A near-linear algorithm for the planar segment center problem
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Citation Context ...can be packed into a 0-1 matrix avoiding certain forbidden 0-1 submatrices or patterns. Forbidden submatrices have been successfully applied to various problems in discrete and computational geometry =-=[13, 4, 14, 27, 8, 28]-=-, and, since forbidden submatrix theory largely subsumes the theory of Davenport-Schinzel sequences [1] and Turán-type subgraph avoidance problems, one can often restate proofs based on these concepts... |

15 | Splay trees, Davenport-Schinzel sequences, and the deque conjecture
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Citation Context ...nection 3 (nor has anyone since) and, rather unexpectedly, there was no followup work exploring the relationship between other brands of path compression and other forbidden substructures. The author =-=[29]-=- recently obtained an Opnα pnqq bound on deque-ordered access sequences in splay trees using generalized Davenport-Schinzel sequences [20, 26], which is a very natural approach for tackling the proble... |

12 |
On the sequential access theorem and deque conjecture for splay trees
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Citation Context ...is section we reprove the sequential access theorem, which states that the time to splay each node in order is Opnq. Our proof is fairly complex relative other proofs of the sequential access theorem =-=[33, 9]-=-. The purpose of this section is to illustrate that forbidden 0-1 matrices can be used in situations where there is no apparent forbidden substructure. We assume, for notational simplicity, that nodes... |

12 |
Forbidden paths and cycles in ordered graphs and matrices
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Citation Context |

12 |
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Citation Context ...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 Ne˘set˘ril [22], the linear bound on deque-ordered path compressions, due to... |

11 |
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10 | Data structures for halfplane proximity queries and incremental voronoi diagrams
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Citation Context ...ue-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the linearity of sequential access in 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. Finally, we give tight bounds on the number of “rig... |

10 |
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Citation Context ...ions on arbitrary trees, improving analyses of Tarjan [35] and Seidel and Sharir [31]. We reprove the linear upper bounds on postordered path compressions, due to Lucas [23] and Loebel and Ne˘set˘ril =-=[22]-=-, the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the linearity of sequential access in splay trees, originally due to Tarjan [38]. We disprove a con... |

9 |
Postorder disjoint set union is linear
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Citation Context ...ngth of arbitrary path compressions on arbitrary trees, improving analyses of Tarjan [35] and Seidel and Sharir [31]. We reprove the linear upper bounds on postordered path compressions, due to Lucas =-=[23]-=- and Loebel and Ne˘set˘ril [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the linearity of sequential access in splay trees, originally due to... |

9 |
Top-down analysis of path compression
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(Show Context)
Citation Context ... structure. 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 upper bounds on postordered path compressions, due to Lucas [23] and Loebel and Ne˘set˘ril [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, ... |

8 | The geometry of binary search tree
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(Show Context)
Citation Context ... is a very natural approach for tackling the problem, given Hart and Sharir’s [17] work. It is well known [24] that splaying is an analogue of path compression for binary search trees. Demaine et al. =-=[7]-=- showed an interesting equivalence between the intrinsic cost of an access sequence in a binary search tree and a problem on 0-1 matrices that does not involve forbidden substructure. 4 Pros and Cons ... |

8 | Faster Algorithms for Incremental Topological Ordering
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Citation Context ...structure type analysis, of which we name just a few. A very appealing data structure of Haeupler 11 (the proof of the sequential access theorem being the exception that proves the rule.) � 10et al. =-=[16]-=- maintains an explicit topological order over the strongly connected components of a graph as edges are inserted one by one. It seems likely that the ways in which this permutation evolves could be ca... |

7 | On linear forbidden submatrices
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Citation Context ...symptotic bounds on all forbidden submatrices with weight 4, as well as some tight bounds when there are multiple forbidden submatrices. At present we have a decent understanding of weight 5 matrices =-=[34, 19, 12, 28, 15]-=-. and an inadequate understanding of weight 6 and larger matrices. All the known upper and lower bounds on ExpA, nq, for various A, are either linear, quasilinear (Θpn2αOp1qpnqq, where α is the invers... |

7 |
Lower bounds for the union-find and the split-find problem on pointer machines
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Citation Context |

6 |
Linear bound on extremal functions of some forbidden patterns
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(Show Context)
Citation Context ...s on all forbidden submatrices with weight 4, as well as some tight bounds when there are multiple forbidden submatrices. At present we have an incomplete understanding of weight-5 forbidden matrices =-=[34, 19, 12, 28, 15, 30]-=-. All the known upper and lower bounds on ExpP, nq, for various P , are either linear, quasilinear (of the form n2αOp1qpnq , where α is the inverse-Ackermann function), Θpnpolylogpnqq, or polynomial, ... |

5 |
Extremal functions of forbidden double permutation matrices
- Geneson
- 2009
(Show Context)
Citation Context ...symptotic bounds on all forbidden submatrices with weight 4, as well as some tight bounds when there are multiple forbidden submatrices. At present we have a decent understanding of weight 5 matrices =-=[34, 19, 12, 28, 15]-=-. and an inadequate understanding of weight 6 and larger matrices. All the known upper and lower bounds on ExpA, nq, for various A, are either linear, quasilinear (Θpn2αOp1qpnqq, where α is the invers... |

5 | On nonlinear forbidden 0–1 matrices: A refutation of a Füredi–Hajnal conjecture
- Pettie
(Show Context)
Citation Context ...too big. It consistent with current knowledge that if ExpA, nq is npolylogpnq then it is also Opn log c nq for some fixed c. Füredi and Hajnal [14] conjectured that c 1. A recent result of the author =-=[30]-=- disproves this conjecture. 7 This is a weaker version of Füredi and Hajnal’s earlier conjecture [14] that ExpA, nq Opn log nq for acyclic A. This conjecture happens to be false [30]. 3Results of Kes... |

4 |
On the competitiveness of splay trees; Relations to the Union-Find Problem
- Lucas
- 1992
(Show Context)
Citation Context ...access sequences in splay trees using generalized Davenport-Schinzel sequences [20, 26], which is a very natural approach for tackling the problem, given Hart and Sharir’s [17] work. It is well known =-=[24]-=- that splaying is an analogue of path compression for binary search trees. Demaine et al. [7] showed an interesting equivalence between the intrinsic cost of an access sequence in a binary search tree... |

3 |
Sequential access in play trees takes linear time
- Tarjan
- 1985
(Show Context)
Citation Context ...bel and Ne˘set˘ril [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the linearity of sequential access in 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. Fina... |

2 |
Efficiency of equivalence algorithms. In Complexity of computer computations (Proc
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- 1972
(Show Context)
Citation Context ...new proof that m path compressions on an n-node tree have total length Opm log1 m{n nq, which matches asymptotic upper bounds of Tarjan [35] and Seidel and Sharir [31] and matches a known lower bound =-=[10]-=- up to lower order terms. We present considerably simpler proofs that postordered path compressions with “rising roots” take Opm nq time (matching Loebl and Ne˘set˘ril [22] and Lucas [23]) and that de... |

1 |
On the dynamic finger conjecture for splay trees II: The proof
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(Show Context)
Citation Context ...ing some forbidden substructure argument? Given the existing applications of the method to splay trees (Theorem 2.6 and [29]), one would hope that Cole’s notorious proof of the dynamic finger theorem =-=[6]-=- could be simplified. A couple corollaries of the dynamic optimality conjecture for splay trees, namely the split [24] and traversal [32] conjectures, strongly resemble the deque-ordered path compress... |