Problem 4. Determine the height of an n-element heap and prove its correctness.

Problem 10. Pigeonhole principle applications. Be sure to justify your answer. 10(d) There are 14 3-digit numbers in a list. Can you conclude that there are two distinct subsets of the 14 numbers that have the same sum?

Problem 12-27 (Algorithmic) Andalus Furniture Company has two manufacturing plants, one at Aynor and another at Spartanburg. The cost in dollars of producing a kitchen chair at each of the two plants is given here. Aynor: Cost = 80Q1 +5Q1² + 92 Spartanburg: Cost = 20Q2 +2.502² + 159 Where Q1 = number of chairs produced at Aynor Q2 = number of chairs produced at Spartanburg Andalus needs to manufacture a total of 30 kitchen chairs to meet an order just received. How many chairs should be made at Aynor and how many should be made at Spartanburg in order to minimize total production cost? When required, round your answers to the nearest dollar. The optimal solution is to produce and $ chairs at Aynor for a cost of $ chairs at Spartanburg for a cost of $ The total cost is

Consider the following network, where each number along a link represents the actual distance between the pair of nodes connected by that link. The objective is to find the shortest path from the origin to the destination. (a) For the forward recursion formulation of this problem, how many stages are there, and what arethe states for each stage? Stage 1: n = 1 Stage 2: n = 2 origin) ( Stage 3: n = 3 S A B C (b) Use forward recursion to solve this problem. However, instead of using the usual tables, show your work graphically. In particular, start with the given network, where the answers already are given for f (Sn) for four of the nodes; then solve for and fill in f2 (B) and fi* (0). Draw an arrowhead that shows the optimal link to traverse out of each of the latter two nodes. Finally, identify the optimal path by following the arrows from node O onward to node T. (c) Use forward recursion to solve this problem by manually constructing the usual tables for n = 3, n = 2, and n = 1.…