Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 21.4, Problem 6E
Program Plan Intro
To show that the worst case running time of m MAKE-SET, UNION and FIND-SET operations, n of which are MAKE-SET operations on a disjoint set forest is
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for G such that1. cq,0c2 ..... cx r e A, and2. G c~l,0h ..... ct, is the pointwise stabilizer of A in G.Applying the base change algorithm if necessary, we may assume a strong generating set S ofU relative to B is known.Let us return to the above example where G is the symmetries of the square acting on pairs ofpoints and A is A 1 , the set of edges of the square. The points (x I =1 and cz2=3 form a base forG, so G1,3 = G1,3,4,6 = < identity > (and s=0). Hence, ~1 =1 and ~2=2 form a base for imrThe stabiliser G 1 is generated by b• so {a,b,bxa} is a strong generating set of Grelative to the chosen base. Furthermore, the stabiliser of 1 in imr is generated by bxa=(2,3).Hence, the set of images { -d, b, bxa } = { (1,3,4,2), (1,2)(3,4), (2,3) } is a strong generatingset of im(p relative to the base [1,2]. The kernel of the homomorphism is the trivial subgroup,< identity >
perform each of the basic tasks :
Prove the following problem is NPC: Given n sets S1,...,Sn, can we find a set A with size at most k, so that ∀i, Si ∩ A ̸= ∅ ?
Give an approximation factor preserving reduction from the set cover problem to the following problem, thereby showing that it is unlikely to have a better approximation guarantee than O(log n).
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- You are given an adjacency list representation of a weighted directed graph G = (V, E) with n vertices, m edges, and no negative cycles. Furthermore, you are given two designated vertices s, t ∈ V and a subset P ⊆ E of the edges are labeled as “premium” edges. Describe a O(mn)-time algorithm to find the shortest path from s to t that uses at most one premium edge (along with any number of non-premium edges).arrow_forwardLet R=ABCDEGHK and F= {ABK→C, A→DG, B→K, K→ADH, H→GE} . Is it in BCNF? Prove your answer.arrow_forwardProve f(n) = n3 - (n2 log2 n) + 2n f(n) ∈ O(n3) by giving the constants (c, n0) and arguing that your constants hold as n goes to infinity by either "chaining up", "chaining down", carefully treating the inequality as an equality and doing some algebra, or even weak induction. Also, what is the minimum val4ue for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.arrow_forward
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