An Improved Enumeration Scheme for Subset Sum Problem

نویسنده

  • Changlin Wan
چکیده

The subset sum problem (SSP) can be briefly stated as: given a target integer E and a set A containing n positive integer aj , find a subset of A summing to E. The density d of an SSP instance is defined by the ratio of n to m, where m is the logarithm of the largest integer within A. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected O(n log n) time when density d ≥ c · √n/ log n, while the previously best density scope is d ≥ c · n/(log n). In addition, the overall expected time and space requirement in the average case are proven to be O(n log n) and O(n) respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. Specifically, the worst-case time complexity of our algorithm is proved to be O((n − 6)2 + n), while the previously best result is O(n2).

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Solving Medium-Density Subset Sum Problems in Expected Polynomial Time

The subset sum problem (SSP) can be briefly stated as: given a target integer E and a set A containing n positive integer aj , find a subset of A summing to E. The density d of an SSP instance is defined by the ratio of n to m, where m is the logarithm of the largest integer within A. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme fo...

متن کامل

Enumeration of Dominant Solutions: An Application in Transport Network Design

A One-Dimensional Binary Integer Programming Problem (1DB-IPP) is concerned with selecting a subset from a set of k items in budget constraint to optimize an objective function. In this problem a dominant solution is defined as a feasible selection to which no further item could be added in budget constraint. This paper presents a simple algorithm for Enumeration of Dominant Solutions (EDS) and...

متن کامل

Design and Evaluation of Alternate Enumeration Techniques for Subset Sum Problem

The subset sum problem, also referred as SSP, is a NP-Hard computational problem. SSP has its applications in broad domains like cryptography, number theory, operation research and complexity theory. The most famous algorithm for solving SSP is Backtracking Algorithm which has exponential time complexity. Therefore, our goal is to design and develop better alternate enumeration techniques for f...

متن کامل

A fast integer-based batch full-homomorphic encryption scheme over finite field

In view of the problems that the plaintext space is too small in the existing schemes. In this paper, a new improved scheme is presented by improving the DGHV scheme. The plaintext space of the improved scheme is extended from finite prime field F2 in the original scheme to finite prime field Fp. Combine and apply the method of encryption in the batch encryption scheme was proposed in 2013, and...

متن کامل

Attacking the Chor-Rivest Cryptosystem by Improved Lattice Reduction

We introduce algorithms for lattice basis reduction that are improvements of the famous L 3-algorithm. If a random L 3 {reduced lattice basis b1; : : : ; bn is given such that the vector of reduced Gram{ Schmidt coeecients (fi;jg 1 j < i n) is uniformly distributed in 0; 1) (n 2) , then the pruned enumeration nds with positive probability a shortest lattice vector. We demonstrate the power of t...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • CoRR

دوره abs/0712.3203  شماره 

صفحات  -

تاریخ انتشار 2007