108,043 research outputs found
Faster space-efficient algorithms for Subset Sum, k -Sum, and related problems
We present randomized algorithms that solve subset sum and knapsack instances with n items in Oβ (20.86n) time, where the Oβ (β ) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve binary integer programming on n variables with few constraints in a similar running time. We also show that for any constant k β₯ 2, random instances of k-Sum can be solved using O(nk -0.5polylog(n)) time and O(log n) space, without the assumption of random access to random bits.Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(log n) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.</p
Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems
We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with items using time and polynomial space, where the notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on variables with few constraints in a similar running time. We also show that for any constant , random instances of -Sum can be solved using time and space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length with integers bounded by a polynomial in share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using space significantly faster than the trivial time algorithm if no value occurs too often in the same list
Equal-Subset-Sum Faster Than the Meet-in-the-Middle
In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey.
Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space
Deterministic Time-Space Tradeoffs for k-SUM
Given a set of numbers, the -SUM problem asks for a subset of numbers
that sums to zero. When the numbers are integers, the time and space complexity
of -SUM is generally studied in the word-RAM model; when the numbers are
reals, the complexity is studied in the real-RAM model, and space is measured
by the number of reals held in memory at any point.
We present a time and space efficient deterministic self-reduction for the
-SUM problem which holds for both models, and has many interesting
consequences. To illustrate:
* -SUM is in deterministic time and space
. In general, any
polylogarithmic-time improvement over quadratic time for -SUM can be
converted into an algorithm with an identical time improvement but low space
complexity as well. * -SUM is in deterministic time and space
, derandomizing an algorithm of Wang.
* A popular conjecture states that 3-SUM requires time on the
word-RAM. We show that the 3-SUM Conjecture is in fact equivalent to the
(seemingly weaker) conjecture that every -space algorithm for
-SUM requires at least time on the word-RAM.
* For , -SUM is in deterministic time and
space
The Flexible Group Spatial Keyword Query
We present a new class of service for location based social networks, called
the Flexible Group Spatial Keyword Query, which enables a group of users to
collectively find a point of interest (POI) that optimizes an aggregate cost
function combining both spatial distances and keyword similarities. In
addition, our query service allows users to consider the tradeoffs between
obtaining a sub-optimal solution for the entire group and obtaining an
optimimized solution but only for a subgroup.
We propose algorithms to process three variants of the query: (i) the group
nearest neighbor with keywords query, which finds a POI that optimizes the
aggregate cost function for the whole group of size n, (ii) the subgroup
nearest neighbor with keywords query, which finds the optimal subgroup and a
POI that optimizes the aggregate cost function for a given subgroup size m (m
<= n), and (iii) the multiple subgroup nearest neighbor with keywords query,
which finds optimal subgroups and corresponding POIs for each of the subgroup
sizes in the range [m, n]. We design query processing algorithms based on
branch-and-bound and best-first paradigms. Finally, we provide theoretical
bounds and conduct extensive experiments with two real datasets which verify
the effectiveness and efficiency of the proposed algorithms.Comment: 12 page
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