There are 100000 documents in the database. The statistic data for one query is shown in the following table. The precision is 20%.

In the master method, we define $T(N)T(N)$ in the recursive form $T(N)=aT(N\mathrm{/}b)+f(N)T(N)\; =\; aT(N/b)\; +\; f(N)$. It means that the algorithm divides the problem into $aa$ parts, with each part being $\frac{1}{b}\backslash frac\{1\}\{b\}$ the size of the original. It costs $f(N)f(N)$ to gather results from subproblems and to derive its own result.

A Las Vegas algorithm is a randomized algorithm that always gives the correct result, however the runtime of a Las Vegas algorithm differs depending on the input.
A Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability. The running time for the algorithm is fixed however.
Then if a Monte Carlo algorithm runs in $O(n^2)O(n^2)$ time, with the probability 50% of producing a correct solution, then there must be a Las Vegas algorithm that can get a solution in $O(n^2)O(n^2)$ time in expectation.

To evaluate the prefix-sums of a sequence of 16 numbers by the parallel algorithm with Balanced Binary Trees, $C(1,3)C(1,3)$ is found before $C(2,1)C(2,1)$.

$C(h,i)={\sum }_{k=1}^{a}A(k)C(h,i)\; =\; \backslash sum\_\{k=1\}^\{a\}\; A(k)$ where $(0,a)(0,\; a)$ is the rightmost descendant leaf of node $(h,i)(h,\; i)$.

In a Turnpike Reconstruction problem, given the distance set $D=\{1,2,2,3,3,3,5,5,6,8\}D\; =\; \backslash \{\; 1,\; 2,\; 2,\; 3,\; 3,\; 3,\; 5,\; 5,\; 6,\; 8\; \backslash \}$, there must be a point placed at 6.

Suppose we have a potential function $\mathrm{\Phi }\backslash Phi$ such that for all $\mathrm{\Phi }(D_i)\ge \mathrm{\Phi }(D_0)\backslash Phi(D\_i)\; \backslash geq\; \backslash Phi(D\_0)$ for all $ii$, but $\mathrm{\Phi }(D_0)\mathrm{\ne }0\backslash Phi(D\_0)\; \backslash neq\; 0$. Then there exists a potential ${\mathrm{\Phi }}^{\mathrm{\prime }}\backslash Phi^\backslash prime$ such that ${\mathrm{\Phi }}^{\mathrm{\prime }}(D_0)=0\backslash Phi\; ^\backslash prime(D\_0)\; =0$, ${\mathrm{\Phi }}^{\mathrm{\prime }}(D_i)\ge 0\backslash Phi\; ^\backslash prime(D\_i)\; \backslash geq\; 0$ for all $i\ge 1i\; \backslash geq\; 1$, and the amortized costs using ${\mathrm{\Phi }}^{\mathrm{\prime }}\backslash Phi^\backslash prime$ are the same as the amortized costs using $\mathrm{\Phi }\backslash Phi$.

Suppose that $TT$ is an minimum spanning tree of the graph $GG$, and $SS$ is a connected sub-graph of $GG$. Then $T\cap ST\backslash cap\; S$ must belong to an minimum spanning tree of $SS$.

Considering leftist heaps of $nn$ nodes, the Merge operation has a higher time complexity than the DeleteMin operation.

The height of an AVL tree of 30 nodes can be 5. (The height of an empty tree is defined to be -1)

NP-complete problems are the hardest problems in NP-hard problems in terms of computational complexity.

For an approximation algorithm for a minimization problem, given that the algorithm does not guarantee to find the optimal solution, the best approximation ratio possible to achieve is a constant $\alpha >1\backslash alpha\; >\; 1$.

Suppose that the replacement selection is applied to generate longer runs with a priority queue of size 3. Given the sequence of numbers { 81, 94, 11, 96, 12, 99, 17，35，28， 58， 41，75，15}. Then 99 will belong to the 1st run.

Without any assumptions on the distances, if P $\mathrm{\ne }\backslash neq$ NP, there is no $\rho \rho$-approximation for TSP (Travelling Salesman Problem) for any $\rho \ge 1\rho \; \backslash ge\; 1$.