Solving kk-SUM using few linear queries

The $k$-SUM problem is given $n$ input real numbers to determine whether any $k$ of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within $P$, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil \frac{k}{2} \rceil$. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth $O(n^3\log^3 n)$ solving $k$-SUM. Furthermore, we show that there exists a randomized algorithm that runs in $\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8})$ time, and performs $O(n^3\log^3 n)$ linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the $+8$) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of $k$. The $O(n^3\log^3 n)$ bound on the number of linear queries is also a tighter bound than any known algorithm solving $k$-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-$P$. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist $o(n)$-linear decision trees of depth $o(n^4)$

Keyword-aware Optimal Route Search

Identifying a preferable route is an important problem that finds applications in map services. When a user plans a trip within a city, the user may want to find "a most popular route such that it passes by shopping mall, restaurant, and pub, and the travel time to and from his hotel is within 4 hours." However, none of the algorithms in the existing work on route planning can be used to answer such queries. Motivated by this, we define the problem of keyword-aware optimal route query, denoted by KOR, which is to find an optimal route such that it covers a set of user-specified keywords, a specified budget constraint is satisfied, and an objective score of the route is optimal. The problem of answering KOR queries is NP-hard. We devise an approximation algorithm OSScaling with provable approximation bounds. Based on this algorithm, another more efficient approximation algorithm BucketBound is proposed. We also design a greedy approximation algorithm. Results of empirical studies show that all the proposed algorithms are capable of answering KOR queries efficiently, while the BucketBound and Greedy algorithms run faster. The empirical studies also offer insight into the accuracy of the proposed algorithms.Comment: VLDB201

Optimal web-scale tiering as a flow problem

We present a fast online solver for large scale parametric max-flow problems as they occur in portfolio optimization, inventory management, computer vision, and logistics. Our algorithm solves an integer linear program in an online fashion. It exploits total unimodularity of the constraint matrix and a Lagrangian relaxation to solve the problem as a convex online game. The algorithm generates approximate solutions of max-flow problems by performing stochastic gradient descent on a set of flows. We apply the algorithm to optimize tier arrangement of over 84 million web pages on a layered set of caches to serve an incoming query stream optimally

Partial-sum queries in OLAP data cubes using covering codes

A partial-sum query obtains the summation over a set of specified cells of a data cube. We establish a connection between the covering problem in the theory of error-correcting codes and the partial-sum problem and use this connection to devise algorithms for the partial-sum problem with efficient space-time trade-offs. For example, using our algorithms, with 44 percent additional storage, the query response time can be improved by about 12 percent; by roughly doubling the storage requirement, the query response time can be improved by about 34 percent

Improved Low-qubit Hidden Shift Algorithms

Hidden shift problems are relevant to assess the quantum security of various cryptographic constructs. Multiple quantum subexponential time algorithms have been proposed. In this paper, we propose some improvements on a polynomial quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use subset-sum algorithms to significantly reduce its complexity. We also propose new tradeoffs between quantum queries, classical time and classical memory to solve this problem

Data Structure for a Time-Based Bandwidth Reservations Problem

We discuss a problem of handling resource reservations. The resource can be reserved for some time, it can be freed or it can be queried what is the largest amount of reserved resource during a time interval. We show that the problem has a lower bound of $\Omega(\log n)$ per operation on average and we give a matching upper bound algorithm. Our solution also solves a dynamic version of the related problems of a prefix sum and a partial sum