By Hans-Otto Günther, Paul van Beek

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**Sample text**

The size of an input for a computer program is usually deﬁned by the length of a binary encoding of the input. For example, in a binary encoding an integer number a is represented as binary number using log2 a bits, an array with m numbers needs m log2 a bits when a is the largest number in the array. On the other hand, in a so-called unary encoding a number a is represented by a bits. Since in most cases it is very diﬃcult to determine the average running time of an algorithm for an input with size n, often the worst-case running time of an algorithm is studied.

In this case we also set pred(j) := i to indicate that i is the predecessor 30 Chapter 2 Algorithms and Complexity from j on a shortest s-j-path. e. from j to s by the sequence j, pred[j], pred[pred[j]], pred[pred[pred[j]]], etc. until pred[k] = s for some node k ∈ V holds). 2 gives a formal description of Dijkstra’s algorithm. While S denotes the set of permanently labeled nodes, S = V \ S is the set of temporarily labeled nodes. Algorithm Dijkstra 1. S := ∅; S := V ; 2. d(s) := 0; pred(s) := 0; 3.

Cn , 0, . . , 0), ⎛ ··· a1n .. am1 · · · amn a11 ⎜ A = ⎝ ... ⎞ ⎛ ⎞ x1 b1 ⎜ ⎟ ⎜ ⎟ b = ⎝ ... ⎠ and x = ⎝ ... ⎠ . bm xn+m ⎞ 0 ⎟ .. 34) c j xj j ∈B / i∈B 50 Chapter 2 Algorithms and Complexity in matrix form we set x = (xB , xN ), c = (cB , cN ), A = (AB , AN ) where the components of x have been reordered in such a way that the basic variables are followed by the non-basic variables. The components in c and the columns in A are reordered accordingly. t. AB xB + AN xN = b xB ≥ 0, xN ≥ 0. 36) has a unique solution, which can be seen as follows.