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 O∗ (20.86n) time, where the O∗ (∙ ) 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 ≥ 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

Role of Sthanic Chikitsa in Stree Roga: An Ayurveda Review

Ayurveda put health of women on prime focus and gives detailed description on Stree Roga and their management through Ayurveda approaches. The common gynecological problems are Yonidaha, Yonikandu, Yoni paicchilya, Yoni strava, Yoni karkashata, Vamini and Upapluta yonivyapad, etc. The disturbed pattern of menstruation also causes many gynecological problems associated with female health. Ayurveda explains many ways for curing gynecological disorders especially Yoni Roga requiring local therapeutic measures, Sthanik Chikitsa is one of them which offers therapeutic relieves in various Stree Roga. Yoni-Pichu, Yoni Dhoopan, Yonidhawan, Yoni-Lepana, Uttarbasti, Yoni-Varti and Agnikarma, etc. are common approaches of Sthanik Chikitsa which are useful in many gynecological problems. These all approaches of Ayurveda help to maintain good health status of women and relives symptoms of white discharge, itching, burning micturation, foul smell and discharge, etc. These non-surgical and less invasive techniques offers health benefits to retain reproductive health of female and does not imparts severe side effects. Present article explains role of Sthanic Chikitsa in various Stree Roga

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 $n$ items using $O^*(2^{0.86n})$ time and polynomial space, where the $O^*(\cdot)$ 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 $n$ variables with few constraints in a similar running time. We also show that for any constant $k\geq 2$, random instances of $k$-Sum can be solved using $O(n^{k-0.5}polylog(n))$ 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 $O(n^2)$ time algorithm if no value occurs too often in the same list

Faster space-efficient algorithms for subset sum, k -sum, and related problems

\u3cp\u3eWe present randomized algorithms that solve subset sum and knapsack instances with n items in O\u3csup\u3e∗\u3c/sup\u3e (2\u3csup\u3e0.86n\u3c/sup\u3e) time, where the O\u3csup\u3e∗\u3c/sup\u3e (∙ ) 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 ≥ 2, random instances of k-sum can be solved using O(n\u3csup\u3ek\u3c/sup\u3e -\u3csup\u3e0.5\u3c/sup\u3epolylog(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(n\u3csup\u3e2\u3c/sup\u3e) time algorithm if no value occurs too often in the same list.\u3c/p\u3