Questions about properties and applications of data structures.

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What are some examples where the Catalan numbers show up in algorithms/data structures?

For some variants of RMQ data structures, the number of Cartesian trees (i.e. the Catalan numbers) is a part of the running-time analysis. What are some other examples where the Cataln numbers show up ...
15
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2answers
1k views

Is the traditional analysis of Bloom filters wrong?

This paper claims that the traditional analysis of the error rate in Bloom filters is incorrect, then provides a lengthy and nontrivial analysis of the actual error rate. The linked paper was ...
4
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1answer
128 views

Heap with $O(1)$ delete-key

Fibonacci heaps have $O(1)$ insertion and $O(\log n)$ delete-min and delete-key (under amortized complexity). Is there a heap data structure with $O(1)$ insertion and delete-key and $O(\log n)$ ...
19
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3answers
932 views

Nontrivial algorithm for computing a sliding window median

I need to calculate the running median: Input: $n$, $k$, vector $(x_1, x_2, \dotsc, x_n)$. Output: vector $(y_1, y_2, \dotsc, y_{n-k+1})$, where $y_i$ is the median of $(x_i, x_{i+1}, \dotsc, ...
4
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0answers
37 views

Has there been any work done on incremental connectivity in path graphs?

This set of lecture notes describes a data structure for decremental connectivity in path graphs that supports queries and removals in amortized O(1) each. Has there been any work done on incremental ...
5
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0answers
123 views

Data structures for Finite Automata

I am a Control Engineer and I have been working on Discrete Event Systems and Supervisory Control, based on Finite Automata Theory. My problem is to represent large automata (about $2 \times 10^6$ ...
-3
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1answer
132 views

Isn't weakly universal hashing even a stronger than truly random? [closed]

So as far as I know the weakly universal hashing is defined as: for any $x, y \subset U, Pr(h(x) = h(y)) \le \frac{1}{m}$ where m is a smaller number than the cardinality of $U$, and h are chosen ...
1
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0answers
73 views

Binary Search Tree DELETE survey

In helping out @bapi-chatterjee on a BST question , when it came to teasing out the combinatorics of BST_DELETE(i) I ran into a wall where even under the conservative assumption that the parent tree ...
2
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0answers
94 views

Height of randomly built binary search tree by insert and delete?

In Introduction to algorithm (CLRS), even in its third edition (published in 2009) it is noted in Sec 12.4 that little is known about height of randomly built binary search tree using insert and ...
3
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1answer
115 views

Algorithm to Bulk Delete nodes from a Treap

I have a Treap, and want to bulk delete nodes in a given key range (i.e. the nodes to be deleted are consecutive nodes in an in-order walk of the tree). If I have $n$ nodes in the Treap, and $k$ nodes ...
1
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0answers
39 views

Atomic snapshot algorithms on tree-structured shared registers

Background: Atomic snapshot memory is a shared memory partitioned into words written (updated) by individual processes, or instantaneously read (*scanned) in its entirety. The Gang of Six algorithm ...
7
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2answers
243 views

Testing boolean vectors orthogonality with fast query-time

Consider the following problems, Problem1: INPUT: a set $S:=\{s_1, \ldots, s_n\}$ of vectors in $d$-dimensional boolean vector space $\{0,1\}^d$ over $\mathbb{F}_2$ TASK: preprocess INPUT in such a ...
4
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1answer
123 views

Deterministic dynamic dictionary on a small universe

We want to maintain a dictionary of $m$ elements with insert/delete and lookup in the word RAM model. Assume $m=O(n)$ at all times, so there can't be too many inserts without deletions. The universe ...
5
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0answers
73 views

Dynamic 2-dimensional orthogonal range reporting in external memory and linear space

Orthogonal 2-dimensional range reporting is the problem of storing a set of values from $U \times V$, where $U$ and $V$ are totally ordered universes, subject to queries of the form "Return all stored ...
4
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0answers
91 views

Concurrent algorithm for strongly connected components (SCCs)

Is anybody aware of a concurrent version of Tarjan's SCCs algorithm, Kosaraju's algorithm or any other fast, O(|V| + |E|) algorithm for finding SCCs? Neither of those algorithms seem to be very hard ...
5
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0answers
102 views

Maintain mex with efficient union

Do you know of any data structure $S[A]$, that maintains a (finite) set $A \subset \mathbb{Z}_{\geq0}$ of non-negative integers, subject to the following operations: Given $S[A],$ calculate minimal ...
2
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1answer
90 views

Why isn't the decrease key operation in a pairing heap $O(1)$

According to the paper (1986) Decrease-key is implemented by first by removing the node from the tree $O(1)$, decreasing the key $O(1)$, then linking it with the root node $O(1)$. The paper admits ...
1
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3answers
288 views

Is there an array structure that allows for O(1) complexity for reverse, zip, slice etc operations?

Many operations on arrays have $O(n)$ complexity. If we represent arrays as accessors methods, many of them could be done in $O(1)$. For example, the $i$th item in the reverse of an array $A$ of ...
10
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0answers
85 views

Integer priority queue with distribution-sensitive deleteMin

Is there in an integer priority queue that uses $O(n)$ words of space with the following operations, all in worst-case time and without access to randomness: ...
9
votes
2answers
333 views

Select two numbers that sum to $p$, using sub-linear query time

Here is a nearest neighbor problem. Given reals $a_1, \ldots, a_n$ (very large $n$!), plus target real $p$, find $a_i$ and $a_j$ whose SUM is closest to $p$. We allow reasonable ...
2
votes
1answer
93 views

Bloom filter for predecessor queries?

Given a threshold $k$ is it possible to make a succinct data structure $S$ to answer queries of the form, given query $x$ does there exist a value $s$ in $S$ such that $s-k \leq x \leq s+k$? Like a ...
7
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1answer
182 views

Can we perform an n-d range search over an arbitrary box without resorting to simplex methods?

Suppose I have some set of points in d-dimensional space, each with some mass. Our problem size will be the number of points in this set. After some roughly (within polylog factors) linear ...
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votes
1answer
266 views

Find all items which are subsets of an item

I have a problem that I think should have been studied. I am looking for algorithms for it. Each item is a set of key-value pairs. Let $x$ be an item and $F$ be a set of items. Each key and each ...
2
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1answer
375 views

Search for all nearest neighbors within a certain radius of a point in 3D?

I have about 80 million spatial points(3D) and I want to find all the nearest neighbors of a query point which lie under a sphere of a certain radius(can be given as input) with the query point as ...
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0answers
79 views

Linear time algorithm for computing the labels of leaves in a recursively defined tree [closed]

The original copy of the question on MSE. Let $S=(s_0, ..., s_{N-1})$ be a sequence of $N=2^p$ numbers. We consider a labelled binary tree of height $p$ as follows: The root has label $S$, for each ...
3
votes
1answer
52 views

Persistant bag/set with direct access to known elements

I'm looking for a bag or set data structure that will allow for the following operations: Add an element to the set, and get a "pointer" to that element. ...
8
votes
1answer
168 views

Storing a bit vector in uninitialized memory and minimal space

A well-known trick for storing bit vectors using uninitialized memory can allocate a bit vector of size $n$ in which all of the bits are set to $0$ by allocating $(2 n + 1)\lceil \lg n \rceil$ bits of ...
4
votes
1answer
375 views

Multiple-sources dominator trees: compact representation and fast algorithm?

I recently learnt about the concept of dominator trees and was fascinated by it. I was wondering how the problem extends to computing dominators from multiple sources, or even from all vertices in ...
1
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0answers
72 views

Give a simple way to augment Van emde boas tree, to find/delete median in O(log log u) time

I need a simple augmentation to support median/order statistic queries in O(log log n) time,without increasing the time for other operations.
1
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2answers
377 views

How do top researchers keep track new results in datastructures

Is there any twitter or some feed,which constantly sends new results which are being published to your mail.
19
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1answer
380 views

How fast can we compute the set inclusion poset of a set family?

Given a set family $\mathcal{F}$ of subsets of a universe $U$. Let $S_1,S_2 \in \mathcal F$ and we want to answer is $S_1 \subseteq S_2$. I am looking for a data-structure that will allow me to ...
3
votes
1answer
132 views

Outer part of Voronoi diagram in 3D

Given a set of points $V \subset \mathbb{R}^d$, the Voronoi diagram divides $\mathbb{R}^d$ into $|V|$ parts such that for every $v \in V$, the part of $\mathbb{R}^d$ for which $v$ is closer than any ...
3
votes
1answer
129 views

Optimal insertion times in insertion-only data structures beyond Bentley-Saxe

The Bentley-Saxe trick allows us to go from a static decomposable problem to a problem admitting insertions, where the insertion time is off the optimal time by a factor of $\log n$. Is this tight ? ...
9
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1answer
229 views

Heapsort:Heaps =~ Quicksort:BSTs =~ Mergesort:___?

Please excuse the terseness of the title, I may have sacrificed clarity on the altar of conciseness. One can see that inserting elements of an array into a binary search tree and reading them back ...
2
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1answer
117 views

Verifying consistency of strict and non-strict partial orders constraints

I am building a set of constraints of the kind $x < y$ and $x \leq y$, where $<$ is a strict order and $\leq$ is a non-strict order on the same set, and $x$ and $y$ are abstract variables ...
3
votes
1answer
171 views

What is the fastest deterministic algorithm for incremental dynamic tree reachability?

As the title. The dynamic algorithm maintains the transitive closure of a tree when the tree undergoes a series of edge insertions (but no deletions)? And the algorithm supports constant query time. ...
12
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0answers
223 views

Data structure for updates on intervals and querying number of zeros

I am looking for a data structure that would maintain an integer table t of size n, and allowing the following operations in time $O(\log n)$. increase(a,b), which increases t[a],t[a+1],..,t[b]. ...
7
votes
2answers
180 views

Almost universal string hashing in $Z_{2^n}$ and sublinear space

Here are two families of hash functions on strings $\vec{x} = \langle x_0 x_1 x_2 \dots x_m \rangle$: For $p$ prime and $x_i \in \mathbb{Z_p}$, $h^1_{a}(\vec{x}) = \sum a^i x_i \bmod p$ for $a \in ...
3
votes
1answer
112 views

How to find the first $k$ points of high enough level using a priority search tree?

In reading Chan's paper, Closest Point Problems Simplified on a RAM, the following came up as a sub-problem: Given a set $P$ of points in the plane, and a query point $q$, find the first $k$ points ...
2
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0answers
204 views

Dynamic shortest path data structure for DAG

Let $G$ be a dynamic DAG (directed acyclic graph) where new vertices and new edges can be inserted. I am looking for an efficient data structure/algorithm to maintain the shortest path from a fixed ...
12
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0answers
204 views

How much independence is required for separate chaining?

If $n$ balls are placed into $n$ bins uniformly at random, the heaviest loaded bin has $O(\lg n/\lg \lg n)$ balls in it with high probability. In "The Power of Simple Tabulation Hashing", Pătraşcu and ...
3
votes
1answer
225 views

Breadth first search and Eppstein K shortest paths algorithm

I'm trying to understand the algorithm for finding K shortest paths in a graph described by Eppstein in this paper: http://www.ics.uci.edu/~eppstein/pubs/Epp-SJC-98.pdf I have trouble particularly ...
3
votes
2answers
613 views

Efficient algorithm to find overlapping circles of various sizes

I have a collection of N circles in the plane with various position and radius. Circles move around according to one force and become bound to each other once they overlap. I need a fast way to ...
4
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1answer
251 views

Dynamic Data structure for All nearest smaller values

I need a data structure that stores a sequence of numbers and supports the following operations. The input to each operation includes the position of an item in the current sequence (not the value or ...
8
votes
1answer
294 views

logic in the presence of doubt, uncertainty, lies

I was reading Harry Frankfurt's On Bulls*t, a 1986 philosophical essay about this blurry notion between truth and falsity. This is not a gratuitous exercise. This may have applications to computer ...
11
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4answers
464 views

Shortest number of editing move between two words

I am looking for a data structure and an algorithm to compute the minimum number of changes required to transform one word into another, given the two words as inputs, where the only allowed changes ...
4
votes
1answer
258 views

Multidimensional B+ tree

I've got an idea for indexing multidimensional data. I haven't been able to find anything equivalent and am wondering if it is indeed a novel approach. The idea is a 'stacked' B+ tree implementation ...
15
votes
2answers
502 views

Exponential Speedup in External Memory

Background The external memory, or DAM model, defines the cost of an algorithm by the number of I/Os it performs (essentially, the number of cache misses). These running times are generally given in ...
5
votes
1answer
191 views

Revision Tracking Graph

Define the Revision Tracking Graph (RTG), which is an oriented graph (without circles) where each node x has a set C(x) associated with it. C(x) contains all edges on all paths from a node 0 ( C(0) = ...
12
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1answer
227 views

Minimal elements of a monotonic predicate over the powerset $2^{|n|}$

Consider a monotonic predicate $P$ over the powerset $2^{|n|}$ (ordered by inclusion). By "monotonic" I mean: $\forall x, y \in 2^{|n|}$ such that $x \subset y$, if $P(x)$ then $P(y)$. I am looking ...