Einführung in die Informatik (0434 L 350)

Suppose you have a program (a “ “Oracle”) for the decision variant of the round trip problem (TSP): Is there a round trip with costs ≤ K? How can you solve the associated optimization problem using the oracle call polynomial often with different inputs? You shouldn't just have the value the shortest round trip, but also the round trip itself. ¨ Assume that the input graph G is a complete graph with n vertices and positive integer costs cij. Estimate the number of calls to the oracle and the running time of the reduction algorithm itself.

We insert items at the end of a list stored as a field. Every time, ¨ if the length exceeds a power of two, the list will be doubled long field resaved: This means, for example: If the list L has a length of 16 and a new element x is inserted, then a new list ¨ L ′ of length 32, and the previous ones 16 elements are added to the new list L ′ copied (“restored”). Then you can there the new element x is appended. The old list L is thrown away. (a) How many restore operations of individual elements have in total after n insert operations, if we are dealing with a list of length 1 begin? Provide an exact formula. (b) If n varies between 10 and 100, for what values ​​of ¨ n in this interval is the ratio of memory operations to n the largest? the smallest?

We insert items at the end of a list stored as a field. Every time, ¨ if the length exceeds a power of two, the list will be doubled long field resaved: This means, for example: If the list L has a length of 16 and a new element x is inserted, then a new list ¨ L ′ of length 32, and the previous ones 16 elements are added to the new list L ′ copied (“restored”). Then you can there the new element x is appended. The old list L is thrown away. (a) How many restore operations of individual elements have in total after n insert operations, if we are dealing with a list of length 1 begin? Provide an exact formula. (b) If n varies between 10 and 100, for what values ​​of ¨ n in this interval is the ratio of memory operations to n the largest? the smallest?

An undirected graph G is given. We are looking for a subset that is as large as possible of nodes in which no two nodes are adjacent. Show that this problem is NP-hard by reducing 3-colorability to it. Instructions: Take three disjoint copies (one ” red”, one” green”, and a ¨ ” blue”) of the graphene whose 3-colorability is to be tested, and connect each three corresponding nodes in the different copies through three edges that a Form a triangle. (What do you have to show to prove that this reduction works?)

Extend the following dictionary data structures (if that makes sense) so that: it is efficiently possible to find the k-smallest key in the stored key when k is entered in the saved  key quantity to find. Describe and analyze the necessary ones too Modifications for the other dictionary operations: (c) Hash table (d) 2-3 tree

Extend the following dictionary data structures (if that makes sense) so that: it is efficiently possible to find the k-smallest key in the stored key when k is entered in the saved  key quantity to find. Describe and analyze the necessary ones too Modifications for the other dictionary operations:  (c) Hash table (d) 2-3 tree