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If the element at the current index is larger than the search key, the algorithm now knows that the search key, if it is contained in the list at all, is located in the interval formed by the previous search index, 2 j - 1, and the current search index, 2 j. The binary search is then performed with the result of either a failure, if the search ...
First, it can be false in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, rendering it impractical. For example, the problem of deciding whether a graph G contains H as a minor, where H is fixed, can be solved in a running time of O(n 2), [2] where n is the number of vertices in G.
This Lua module is used on approximately 1,360,000 pages, or roughly 2% of all pages. To avoid major disruption and server load, any changes should be tested in the module's /sandbox or /testcases subpages, or in your own module sandbox.
When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n 1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the ...
Another related problem is the bottleneck travelling salesman problem: Find a Hamiltonian cycle in a weighted graph with the minimal weight of the weightiest edge. A real-world example is avoiding narrow streets with big buses. [15] The problem is of considerable practical importance, apart from evident transportation and logistics areas.
Every search problem also has a corresponding decision problem, namely L ( R ) = { x ∣ ∃ y R ( x , y ) } . {\displaystyle L(R)=\{x\mid \exists yR(x,y)\}.\,} This definition may be generalized to n -ary relations using any suitable encoding which allows multiple strings to be compressed into one string (for instance by listing them ...
Euler diagram for P, NP, NP-complete, and NP-hard set of problems. Under the assumption that P ≠ NP, the existence of problems within NP but outside both P and NP-complete was established by Ladner. [1] In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems.
A polynomial-time problem can be very difficult to solve in practice if the polynomial's degree or constants are large enough. In addition, information-theoretic security provides cryptographic methods that cannot be broken even with unlimited computing power. "A large-scale quantum computer would be able to efficiently solve NP-complete problems."