Data Structures and Algorithms | Set 27 Last Updated : 13 Dec, 2022 Comments Improve Suggest changes Like Article Like Report Following questions have been asked in GATE CS 2011 exam. 1) An undirected graph G(V, E) contains n ( n > 2 ) nodes named v1 , v2 ,....vn. Two nodes vi , vj are connected if and only if 0 < |i - j| <= 2. Each edge (vi, vj ) is assigned a weight i + j. A sample graph with n = 4 is shown below. What will be the cost of the minimum spanning tree (MST) of such a graph with n nodes? (A) 1/12(11n^2 - 5n) (B) n^2 - n + 1 (C) 6n - 11 (D) 2n + 1 Answer: (B) Minimum spanning tree for 2 nodes would be (v1) _ (v2) Total weight 3 Minimum spanning tree for 3 nodes would be (v1) _ (v2) | (v3) Total weight= 3 + 4 = 7 Minimum spanning tree for 4 nodes would be (v1) _ (v2) _ (v4) | (v3) Total weight= 3 + 4 + 6 = 13 Minimum spanning tree for 5 nodes would be (v1) _ (v2) _ (v4) | (v3) | (v5) Total weight= 3 + 4 + 6 + 8 = 21 Minimum spanning tree for 6 nodes would be (v1) _ (v2) _ (v4) _ (v6) | (v3) | (v5) Total weight= 3 + 4 + 6 + 8 + 10 = 31 We can observe from above examples that when we add kth node, the weight of spanning tree increases by 2k-2. Let T(n) be the weight of minimum spanning tree. T(n) can be written as T(n) = T(n-1) + (2n-2) for n > 2 T(1) = 0 and T(2) = 3 The recurrence can be written as sum of series (2n - 2) + (2n-4) + (2n-6) + (2n-8) + .... 3 and solution of this recurrence is n^2 - n + 1. 2) The length of the path from v5 to v6 in the MST of previous question with n = 10 is (A) 11 (B) 25 (C) 31 (D) 41 Answer: (C) Any MST which has more than 5 nodes will have the same distance between v5 and v6 as the basic structure of all MSTs (with more than 5 nodes) would be following. (v1) _ (v2) _ (v4) _ (v6) _ . . (more even numbered nodes) | (v3) | (v5) | . . (more odd numbered nodes) Distance between v5 and v6 = 3 + 4 + 6 + 8 + 10 = 31 3) Consider two binary operators '↑ ' and '↓' with the precedence of operator ↓ being lower than that of the ↑ operator. Operator ↑ is right associative while operator ↓ is left associative. Which one of the following represents the parse tree for expression (7 ↓ 3 ↑ 4 ↑ 3 ↓ 2)? Answer: (B) Let us consider the given expression (7 ↓ 3 ↑ 4 ↑ 3 ↓ 2). Since the precedence of ↑ is higher, the sub-expression ([3 ↑ 4 ↑ 3) will be evaluated first. In this sub-expression, 4 ↑ 3 would be evaluated first because ↑ is right to left associative. So the expression is evaluated as ((7 ↓ (3 ↑ (4 ↑ 3))) ↓ 2). Also, note that among the two ↓ operators, first one is evaluated before the second one because the associativity of ↓ is left to right. Please see GATE Corner for all previous year paper/solutions/explanations, syllabus, important dates, notes, etc. Please write comments if you find any of the answers/explanations incorrect, or you want to share more information about the topics discussed above. 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