Read e-book online An Introduction to the Analysis of Algorithms (2nd Edition) PDF

By Michael Soltys

ISBN-10: 9814401153

ISBN-13: 9789814401159

A successor to the 1st version, this up-to-date and revised e-book is a smart spouse consultant for college kids and engineers alike, in particular software program engineers who layout trustworthy code. whereas succinct, this version is mathematically rigorous, overlaying the principles of either laptop scientists and mathematicians with curiosity in algorithms.
along with masking the normal algorithms of computing device technology corresponding to grasping, Dynamic Programming and Divide & overcome, this variation is going additional through exploring sessions of algorithms which are frequently missed: Randomised and on-line algorithms -- with emphasis put on the set of rules itself.
The insurance of either fields are well timed because the ubiquity of Randomised algorithms are expressed during the emergence of cryptography whereas on-line algorithms are crucial in different fields as different as working platforms and inventory industry predictions.
whereas being really brief to make sure the essentiality of content material, a powerful concentration has been put on self-containment, introducing the assumption of pre/post-conditions and loop invariants to readers of all backgrounds. Containing programming routines in Python, options can be put on the book's web site.
Readership: scholars of undergraduate classes in algorithms and programming.

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Additional resources for An Introduction to the Analysis of Algorithms (2nd Edition)

Example text

Finally, show how to use this loop invariant to prove that T is connected. 8. Suppose that G = (V, E) is not connected. Show that in this case, when G is given to Kruskal’s algorithm as input, the algorithm computes a spanning forest of G. Define first precisely what is a spanning forest (first define the notion of a connected component). 7. To show that the spanning tree resulting from the algorithm is in fact a MCST, we reason that after each iteration of the loop, the set T of edges can be extended to a MCST using edges that have not yet been considered.

Therefore c(ei ) ≤ c(ej ), so c(T3 ) ≤ c(T1 ), so T3 must also be a MCST. Since T ∪ {ei } ⊆ T3 , it follows that T ∪ {ei } is promising. This finishes the proof of the induction step. 10 (Exchange Lemma). Let G be a connected graph, and let T1 and T2 be any two spanning trees for G. For every edge e in T2 − T1 there is an edge e in T1 − T2 such that T1 ∪ {e} − {e } is a spanning tree for G. ) T1 T2 e’ e Fig. 1 ◦ e1 ◦ Exchange lemma. ◦ e1 ◦ ◦ e2 e4 ◦ e3 ◦ ◦ e3 ◦ e1 ◦ e4 e2 ◦ ◦ Fig. 2 Example of the exchange lemma: the left-most and the middle graphs are two different spanning trees of the same graph.

12. Suppose that edge e1 has a smaller cost than any of the other edges; that is, c(e1 ) < c(ei ), for all i > 1. Show that there is at least one MCST for G that includes edge e1 . 13. , sorts edges of the same cost—arbitrarily. Show that for every MCST T of a graph G, there exists a particular way of breaking the ties so that the algorithm returns T . 14. Write a Python program that takes as input the description of a grid, and outputs its minimum cost spanning tree. An n-grid is a graph consisting of n2 nodes, organized as a square array of n × n points.

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