Algorithm to Element as Product of Strong Generators Input : a group G acting on f~ = { 1,2 ..... n }; an element g of G; a base and strong generating set for G; Schreier vectors v (i), 1 < i < k, for the stabiliser chain; Output : a symbolic product, word, expressing g in terms of the strong generators;

Algorithm for Testing MembershipInput : a group G acting on f~ = { 1,2 ..... n };a permutation g of f~ = { 1,2 ..... n };a base and strong ~enerating set for G;Schreier vectors v (i) , 1 < i < k, for the stabiliser chain;Output: a boolean value answer, indicating whether g ~ G;function is_in_group(p : permutation; i : 1..k+l ) : boolean;(* return true if the permutation p is in the group G (i) *)

Consider a recursively defined set S such that  1 is an element of S (x-y) is an element of S if both x and y are elements of S. S is equal to the ----------------------------------.

Algorithm Element as a Strong Generator ProductInput: a group G operating on f = 1,2,.....n; an element g of G; a base and strong generating set for G; Schreier vectors v I 1 I k, for the stabiliser chainOutput: a symbolic product, a word that expresses g in terms of the strong generators.

Given the definition in Haskell: fmap :: (a -> b) -> [a] -> [b] fmap g xs = reverse (map g xs)   With Functor law: (fmap g . fmap h) xs = fmap (g . h) xs   Using list xs :: [Int], prove the definition does not satisfy the law:

3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k.   Show that the Strongly Independent Set Problem is NP-Complete.

True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n) <= |h(n)-h(m)|. (b) If an A* heuristics is admissible then it satisfies the monotone restriction. (c) Best-first search guarantees optimality in its returned solution. (d) Least-cost-first search guarantees optimality in its returned solution. (e) If all edges are with unit cost, then Breadth-first search guarantees optimality in its returned solution.

Refer to image: (Computation and Automata) Provide new and correct solution for positive feedback!

Liskov Substitution Principle (LSP)a. Propose a way to fix the problem in there.

Show that the vertices reached in each invocation of the recursive method from the constructor are in a strong component in a DFS of a digraph G where marked vertices are taken into account in reverse postorder provided by a DFS of the digraph's reverse GR (Kosaraju's algorithm)