By Al-Jaber Ah.

A number of points concerning the combinatorial houses of heapsort are mentioned during this thesis. A recursion formulation for the variety of lots fulfilling a given situation among any offsprings with a similar father or mother Is given and several other houses of tons are mentioned together with a brand new set of rules to generate the set of all tons of any dimension. additionally during this paintings we outline moment order timber that have a superb value within the learn of the complexity of Williams' algorithms to generate a heap. We speak about this type of timber and we turn out that the producing functionality of the variety of bushes satisfies a nonlinear differential distinction equation. The numerical computation and the asymptotic enlargement for a volume relating to this nonlinear differential distinction equation Is given during this paintings . ultimately, we supply an top certain for the variety of the second one order timber generated from the set of all lots of measurement N the place N has the shape 2-1 for any optimistic integer okay.

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4. Repeat 1-3 M times. 5. Compute the reliability estimate. Develop a pseudocode for the above scheme. 1 Characteristic Property of the Exponential Distribution The notation τ ∼ Exp(λ) means that Fτ (t) = P (τ ≤ t) = 1 − e−λt , t > 0. v. τ is fτ (t) = λe−λt . The characteristic property of the exponential distribution is that the so-called failure rate h(t) is constant: h(t) = fτ (t) = λ = Const. 2) is the following. e. τ > t. Then the conditional probability to fail in the interval [t, t + δt] does not depend on t: P (τ ∈ [t, t + δt]|τ > t) = e−λt − e−λ(t+δt) = 1 − e−λδt e−λt λ · δt.

It may be proved that using Kruskal’s algorithm is more eﬃcient for the case of relatively small number of edges (so-called sparse networks), whereas the Prim’s algorithm is more eﬃcient for the case of relatively large number of edges (so-called dense networks). In this book we will deal with the Kruskal’s algorithm. 2. 1 - Scheme of Kruskal’s Algorithm. 1. Sort all edges by their length in increasing order. Denote the spanning tree by ST . 2. Let index i := 0. Let ST := {}. 3. Choose the next edge (a, b) from E.

U = F node(2) = 2, v = Snode(2) = 3, ucomp = 2, vcomp = 3, r = merge2(2, 3) = 3,Comp = {1, 3, 3, 4, 5, 6, 7}, T comp = {1, 0, 0, 1, 0, 1, 0}, H = {0, 0, 1, 0, 0, 0, 0}, iT comp = 1, ST = {2}. Step 2. u = F node(7) = 1, v = Snode(7) = 7, ucomp = 1, vcomp = 7, r = merge2(1, 7) = 7,Comp = {7, 3, 3, 4, 5, 3, 7}, T comp = {1, 0, 0, 1, 0, 1, 1}, H = {0, 0, 1, 0, 0, 0, 1}, iT comp = 1, ST = {2, 7}. Step 3. u = F node(9) = 2, v = Snode(9) = 6, ucomp = 3, vcomp = 6, r = merge2(3, 6) = 3, Comp = {7, 3, 3, 4, 5, 6, 7}, T comp = {1, 0, 1, 1, 0, 1, 1}, H = {0, 0, 1, 0, 0, 0, 1}, iT comp = 1, ST = {2, 7, 9}.

### Combinatorial properties of heapsort by Al-Jaber Ah.

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