6 research outputs found
New Paths from Splay to Dynamic Optimality
Consider the task of performing a sequence of searches in a binary search
tree. After each search, an algorithm is allowed to arbitrarily restructure the
tree, at a cost proportional to the amount of restructuring performed. The cost
of an execution is the sum of the time spent searching and the time spent
optimizing those searches with restructuring operations. This notion was
introduced by Sleator and Tarjan in (JACM, 1985), along with an algorithm and a
conjecture. The algorithm, Splay, is an elegant procedure for performing
adjustments while moving searched items to the top of the tree. The conjecture,
called "dynamic optimality," is that the cost of splaying is always within a
constant factor of the optimal algorithm for performing searches. The
conjecture stands to this day. In this work, we attempt to lay the foundations
for a proof of the dynamic optimality conjecture.Comment: An earlier version of this work appeared in the Proceedings of the
Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. arXiv admin note:
text overlap with arXiv:1907.0630
Sorting Pattern-Avoiding Permutations via 0-1 Matrices Forbidding Product Patterns
We consider the problem of comparison-sorting an -permutation that
avoids some -permutation . Chalermsook, Goswami, Kozma, Mehlhorn, and
Saranurak prove that when is sorted by inserting the elements into the
GreedyFuture binary search tree, the running time is linear in the extremal
function . This is the maximum number
of 1s in an 0-1 matrix avoiding , where
is the permutation matrix of , the Kronecker
product, and . The
same time bound can be achieved by sorting with Kozma and Saranurak's
SmoothHeap.
In this paper we give nearly tight upper and lower bounds on the density of
-free matrices in terms of the inverse-Ackermann
function . \mathrm{Ex}(P_\pi\otimes \text{hat},n) =
\left\{\begin{array}{ll} \Omega(n\cdot 2^{\alpha(n)}), & \mbox{for most
$\pi$,}\\ O(n\cdot 2^{O(k^2)+(1+o(1))\alpha(n)}), & \mbox{for all $\pi$.}
\end{array}\right. As a consequence, sorting -free sequences can be
performed in time. For many corollaries of the
dynamic optimality conjecture, the best analysis uses forbidden 0-1 matrix
theory. Our analysis may be useful in analyzing other classes of access
sequences on binary search trees
Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition
Greedy BST (or simply Greedy) is an online self-adjusting binary search tree
defined in the geometric view ([Lucas, 1988; Munro, 2000; Demaine, Harmon,
Iacono, Kane, Patrascu, SODA 2009). Along with Splay trees (Sleator, Tarjan
1985), Greedy is considered the most promising candidate for being dynamically
optimal, i.e., starting with any initial tree, their access costs on any
sequence is conjectured to be within factor of the offline optimal.
However, in the past four decades, the question has remained elusive even for
highly restricted input.
In this paper, we prove new bounds on the cost of Greedy in the ''pattern
avoidance'' regime. Our new results include:
The (preorder) traversal conjecture for Greedy holds up to a factor of
, improving upon the bound of in
(Chalermsook et al., FOCS 2015). This is the best known bound obtained by any
online BSTs.
We settle the postorder traversal conjecture for Greedy.
The deque conjecture for Greedy holds up to a factor of ,
improving upon the bound in (Chalermsook, et al., WADS
2015).
The split conjecture holds for Greedy up to a factor of .
Key to all these results is to partition (based on the input structures) the
execution log of Greedy into several simpler-to-analyze subsets for which
classical forbidden submatrix bounds can be leveraged. Finally, we show the
applicability of this technique to handle a class of increasingly complex
pattern-avoiding input sequences, called -increasing sequences.
As a bonus, we discover a new class of permutation matrices whose extremal
bounds are polynomially bounded. This gives a partial progress on an open
question by Jacob Fox (2013).Comment: Accepted to SODA 202
Optimization with pattern-avoiding input
Permutation pattern-avoidance is a central concept of both enumerative and
extremal combinatorics. In this paper we study the effect of permutation
pattern-avoidance on the complexity of optimization problems.
In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC
1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015)
conjectured that the amortized access cost of an optimal binary search tree
(BST) is whenever the access sequence avoids some fixed pattern. They
showed a bound of , which was recently improved to
by Chalermsook, Pettie, and Yingchareonthawornchai
(2023); here is the BST size and the inverse-Ackermann
function. In this paper we resolve the conjecture, showing a tight
bound. This indicates a barrier to dynamic optimality: any candidate online BST
(e.g., splay trees or greedy trees) must match this optimum, but current
analysis techniques only give superconstant bounds.
More broadly, we argue that the easiness of pattern-avoiding input is a
general phenomenon, not limited to BSTs or even to data structures. To
illustrate this, we show that when the input avoids an arbitrary, fixed, a
priori unknown pattern, one can efficiently compute a -server solution of
requests from a unit interval, with total cost , in
contrast to the worst-case bound; and a traveling salesman tour
of points from a unit box, of length , in contrast to the
worst-case bound; similar results hold for the euclidean
minimum spanning tree, Steiner tree, and nearest-neighbor graphs.
We show both results to be tight. Our techniques build on the Marcus-Tardos
proof of the Stanley-Wilf conjecture, and on the recently emerging concept of
twin-width; we believe our techniques to be more generally applicable